In [181]:

    

from __future__ import unicode_literals, print_function
import json
import pathlib
import random

import spacy
from spacy.pipeline import EntityRecognizer
from spacy.gold import GoldParse
from spacy.tagger import Tagger
from spacy.matcher import Matcher
from spacy.attrs import IS_PUNCT, LOWER, TAG, ORTH

import os
import re
 
try:
    unicode
except:
    unicode = str


    

In [182]:

    

nlp = spacy.load('en')


    

In [183]:

    

#initialize the triggering rule 'let $' for the rule-based annotation
matcher = Matcher(nlp.vocab)
matcher.add_pattern("let", [{LOWER: "let"}, {ORTH: "$"}])


    

In [184]:

    

def tex2doc(tex_file):  #read the whole tex file in the spaCy doc object
    with open(tex_file, 'r') as tex:
        data=tex.read()
    doc = nlp(data)
    return doc


    

In [185]:

    

def rule_based_annotation(doc):
    annotation = []
    for m in matcher(doc):
        let_position = m[2]
        dollar_position = m[3]
        annotation += add_letDollarBe_entity(doc, let_position, dollar_position)
    return (doc.text, annotation)


def add_letDollarBe_entity(doc, let_position, dollar_position):
    new_annotation=[]
    sentence = doc[let_position : dollar_position].sent   #stay in the current sentence, so to respect sentence boundaries
    print('===== sent', sentence, '========\n')
    
    part = doc[let_position : sentence.end]  #forget the already-processed first part of the sentence
    
    for regex_match in re.finditer('^let (\$+[^\$]+\$+) be (an?|the) \S+', part.text, re.IGNORECASE): #there is at most one match in this for loop because of ^
        #compute left and right char offset for candidate VAR
        left_offset = part.start_char + regex_match.span(1)[0]
        right_offset = part.start_char + regex_match.span(1)[1]
        
        print('regex_match', regex_match.group(1))
        for nn in part.noun_chunks:
            print('nn candidate', nn)
            if nn.start_char >= right_offset + 3: #we only consider noun chunks after the '$ be '
                
                
                
                new_annotation += ((left_offset, right_offset, 'VAR'),)
                
                #left and right char offset for TYPE are already among the nn attributes
                new_annotation += ((nn.start_char, nn.end_char, 'TYPE'),)
                print('nn', nn.text)
                break  #we only consider the 1st noun chunk after the 'be' 
                
    
    print('new_annotation', new_annotation)
                    
    return new_annotation


    

In [244]:

    

annotated_data=[]

directory = os.fsencode('tex_files/')
list_of_texs = [os.fsdecode(file) for file in os.listdir(directory)[0:10]]  #remove [0:10] to get the full data
for filename in list_of_texs:
    print("file: ", filename)
    doc = tex2doc(os.path.join(os.fsdecode(directory), filename))
    annotated_data.append(rule_based_annotation(doc))


    

    




file:  pione.tex
===== sent Before we state the result we introduce the category of $G$-sets for a
topological group $G$.

\begin{definition}
\label{definition-G-set-continuous}
Let $G$ be a topological group.
 ========

regex_match $G$
nn candidate a topological group
nn a topological group
new_annotation [(794, 797, 'VAR'), (801, 820, 'TYPE')]
===== sent \medskip\noindent
Recall that if $L/K$ is an infinite Galois extension then the
Galois group $G = \text{Gal}(L/K)$ comes endowed with a canonical
topology, see Fields, Section \ref{fields-section-infinite-galois}.

\begin{lemma}
\label{lemma-sheaves-point}
Let $K$ be a field. ========

regex_match $K$
nn candidate a field
nn a field
new_annotation [(1863, 1866, 'VAR'), (1870, 1877, 'TYPE')]
===== sent Let $K^{sep}$ a separable closure of $K$.
Consider the profinite group $G = \text{Gal}(K^{sep}/K)$.
 ========

new_annotation []
===== sent In this section we discuss some of the material the reader can
find in \cite[Expos\'e V, Sections 4, 5, and 6]{SGA1}.

\medskip\noindent
Let $F : \mathcal{C} \to \textit{Sets}$ be a functor.
 ========

regex_match $F : \mathcal{C} \to \textit{Sets}$
nn candidate $F
nn candidate a functor
nn a functor
new_annotation [(3573, 3608, 'VAR'), (3612, 3621, 'TYPE')]
===== sent \begin{lemma}
\label{lemma-aut-inverse-limit}
Let $\mathcal{C}$ be a category and let $F : \mathcal{C} \to \textit{Sets}$
be a functor. ========

regex_match $\mathcal{C}$
nn candidate a category
nn a category
new_annotation [(5201, 5214, 'VAR'), (5218, 5228, 'TYPE')]
===== sent \begin{lemma}
\label{lemma-aut-inverse-limit}
Let $\mathcal{C}$ be a category and let $F : \mathcal{C} \to \textit{Sets}$
be a functor. ========

new_annotation []
===== sent \end{lemma}

\begin{proof}
Let $\xi = (\gamma_X) \in ========

new_annotation []
===== sent \end{proof}

\begin{example}
\label{example-galois-category-G-sets}
Let $G$ be a topological group. ========

regex_match $G$
nn candidate a topological group
nn a topological group
new_annotation [(6342, 6345, 'VAR'), (6349, 6368, 'TYPE')]
===== sent The category $G\textit{-Sets}$ of $G$-sets is defined in
Definition \ref{definition-G-set-continuous}.
\end{example}

\noindent
Let $G$ be a topological group. ========

regex_match $G$
nn candidate a topological group
nn a topological group
new_annotation [(6798, 6801, 'VAR'), (6805, 6824, 'TYPE')]
===== sent The universal property of the profinite completion is
that any continuous map $G \to H$ to a profinite group $H$ factors
canonically as $G \to G^\wedge \to H$.

\begin{lemma}
\label{lemma-single-out-profinite}
Let $G$ be a topological group. ========

regex_match $G$
nn candidate a topological group
nn a topological group
new_annotation [(7325, 7328, 'VAR'), (7332, 7351, 'TYPE')]
===== sent Let $\gamma \in \text{Aut}(F_G)$ and let $X \in \Ob(\mathcal{C})$.
We will show there is a $g \in G$ such that $\gamma$ and $g$
induce the same action on $F_G(X)$. ========

new_annotation []
===== sent Let $\gamma \in \text{Aut}(F_G)$ and let $X \in \Ob(\mathcal{C})$.
We will show there is a $g \in G$ such that $\gamma$ and $g$
induce the same action on $F_G(X)$. ========

new_annotation []
===== sent F_G(G/U) = G/U$. Let $e \in G$ be the neutral element and
say that $\gamma(eU) = g_0U$ for some $g_0 \in G$. For any
$g_1 \in G$ the morphism
$$
R_{g_1} : G/U \longrightarrow G/U,\quad gU \longmapsto gg_1U
$$
of $\textit{Finite-}G\textit{-Sets}$ commutes with the action of
$\gamma$. Hence
$$
\gamma(g_1U) = \gamma(R_{g_1}(eU)) = R_{g_1}(\gamma(eU)) =
R_{g_1}(g_0U) = g_0g_1U
$$
Thus we see that $g = g_0$ works.
 ========

regex_match $e \in G$
nn candidate the neutral element
nn the neutral element
new_annotation [(9216, 9225, 'VAR'), (9229, 9248, 'TYPE')]
===== sent \begin{lemma}
\label{lemma-second-fundamental-functor}
Let $G$ be a topological group. ========

regex_match $G$
nn candidate a topological group
nn a topological group
new_annotation [(9887, 9890, 'VAR'), (9894, 9913, 'TYPE')]
===== sent \end{lemma}

\begin{proof}
Let $X$ be a nonempty object of $\textit{Finite-}G\textit{-Sets}$.
The diagram
$$
\xymatrix{
X \ar[r] \ar[d] & \{*\} \ar[d] \\
\{*\} \ar[r] & \{*\}
}
$$
is cocartesian. ========

regex_match $X$
nn candidate a nonempty object
nn a nonempty object
new_annotation [(10124, 10127, 'VAR'), (10131, 10148, 'TYPE')]
===== sent Let $U \subset G$ be an open, normal subgroup with finite index.
 ========

regex_match $U \subset G$
nn candidate $U
nn candidate an open, normal subgroup
nn an open, normal subgroup
new_annotation [(10336, 10349, 'VAR'), (10353, 10377, 'TYPE')]
===== sent \end{proof}

\begin{example}
\label{example-from-C-F-to-G-sets}
Let $\mathcal{C}$ be a category and let $F : \mathcal{C} \to \textit{Sets}$
be a functor such that $F(X)$ is finite for all $X \in \Ob(\mathcal{C})$.
 ========

regex_match $\mathcal{C}$
nn candidate a category
nn a category
new_annotation [(12244, 12257, 'VAR'), (12261, 12271, 'TYPE')]
===== sent \end{proof}

\begin{example}
\label{example-from-C-F-to-G-sets}
Let $\mathcal{C}$ be a category and let $F : \mathcal{C} \to \textit{Sets}$
be a functor such that $F(X)$ is finite for all $X \in \Ob(\mathcal{C})$.
 ========

new_annotation []
===== sent \begin{definition}
\label{definition-galois-category}
\begin{reference}
Different from the definition in \cite[Expos\'e V, Definition 5.1]{SGA1}.
Compare with \cite[Definition 7.2.1]{BS}.
\end{reference}
Let $\mathcal{C}$ be a category and let $F : \mathcal{C} \to \textit{Sets}$
be a functor. ========

regex_match $\mathcal{C}$
nn candidate a category
nn a category
new_annotation [(13293, 13306, 'VAR'), (13310, 13320, 'TYPE')]
===== sent \begin{definition}
\label{definition-galois-category}
\begin{reference}
Different from the definition in \cite[Expos\'e V, Definition 5.1]{SGA1}.
Compare with \cite[Definition 7.2.1]{BS}.
\end{reference}
Let $\mathcal{C}$ be a category and let $F : \mathcal{C} \to \textit{Sets}$
be a functor. ========

new_annotation []
===== sent \begin{lemma}
\label{lemma-epi-mono}
Let $(\mathcal{C}, F)$ be a Galois category. ========

regex_match $(\mathcal{C}, F)$
nn candidate F)$
nn candidate a Galois category
nn a Galois category
new_annotation [(14677, 14695, 'VAR'), (14699, 14716, 'TYPE')]
===== sent Let $E$ be the equalizer of $a$ and $b$. ========

regex_match $E$
nn candidate the equalizer
nn the equalizer
new_annotation [(15748, 15751, 'VAR'), (15755, 15768, 'TYPE')]
===== sent Equivalently, $X$ is Galois if it is connected and
$\text{Aut}(X)$ acts transitively on $F(X)$.

\begin{lemma}
\label{lemma-galois}
Let $(\mathcal{C}, F)$ be a Galois category. ========

regex_match $(\mathcal{C}, F)$
nn candidate F)$
nn candidate a Galois category
nn a Galois category
new_annotation [(18042, 18060, 'VAR'), (18064, 18081, 'TYPE')]
===== sent Let $n = |F(X)|$. Consider $X^n$ endowed with its natural action of
$S_n$. Let
$$
X^n = \coprod\nolimits_{t \in T} Z_t
$$
be the decomposition into connected objects. ========

new_annotation []
===== sent Namely, if not, say
$s'_i = s'_j$, then $Z_t$ is the image of an connected component of
$X^{n - 1}$ under the diagonal morphism
$$
\Delta_{ij} : X^{n - 1} \longrightarrow X^n
$$
Since morphisms of connected objects are epimorphisms and induce
surjections after applying $F$ it would follow that $s_i = s_j$
which is not the case.

\medskip\noindent
Let $G \subset S_n$ be the subgroup of elements with $g(Z_t) = ========

regex_match $G \subset S_n$
nn candidate $G \subset S_n$
nn candidate the subgroup
nn the subgroup
new_annotation [(19013, 19028, 'VAR'), (19032, 19044, 'TYPE')]
===== sent \begin{lemma}
\label{lemma-tame}
\begin{reference}
Compare with \cite[Definition 7.2.4]{BS}.
\end{reference}
Let $(\mathcal{C}, F)$ be a Galois category. ========

regex_match $(\mathcal{C}, F)$
nn candidate F)$
nn candidate a Galois category
nn a Galois category
new_annotation [(19685, 19703, 'VAR'), (19707, 19724, 'TYPE')]
===== sent Let $G = \text{Aut}(F)$
be as in Example \ref{example-from-C-F-to-G-sets}. For any connected
$X$ in $\mathcal{C}$ the action of $G$ on $F(X)$ is transitive.
 ========

new_annotation []
===== sent Let $I$ be the set of isomorphism classes of Galois objects in $\mathcal{C}$.
For each $i \in I$ let $X_i$ be a representative of the isomorphism class.
 ========

regex_match $I$
nn candidate the set
nn the set
new_annotation [(19993, 19996, 'VAR'), (20000, 20007, 'TYPE')]
===== sent Let $I$ be the set of isomorphism classes of Galois objects in $\mathcal{C}$.
For each $i \in I$ let $X_i$ be a representative of the isomorphism class.
 ========

regex_match $X_i$
nn candidate a representative
nn a representative
new_annotation [(20090, 20095, 'VAR'), (20099, 20115, 'TYPE')]
===== sent To show surjectivity, let $\gamma \in F(X)$. ========

new_annotation []
===== sent Namely, just let $h_{ii'}(a) = a' : X_{i'} \to X_{i'}$
be the unique automorphism such that
$F(a')(\gamma_{i'}) = F(f_{ii'} \circ a)(\gamma_i)$.
As before this makes the diagram commute and moreover the choice
is unique.
 ========

new_annotation []
===== sent \end{proof}

\begin{proposition}
\label{proposition-galois}
\begin{reference}
This is a weak version of \cite[Expos\'e V]{SGA1}.
The proof is borrowed from \cite[Theorem 7.2.5]{BS}.
\end{reference}
Let $(\mathcal{C}, F)$ be a Galois category. ========

regex_match $(\mathcal{C}, F)$
nn candidate F)$
nn candidate a Galois category
nn a Galois category
new_annotation [(23150, 23168, 'VAR'), (23172, 23189, 'TYPE')]
===== sent Let $G = \text{Aut}(F)$
be as in Example \ref{example-from-C-F-to-G-sets}. ========

new_annotation []
===== sent Let $X$ and $Y$ be objects and let
$s : F(X) \to F(Y)$ be a map. ========

new_annotation []
===== sent Since
$F(Y)$ surjects onto $F(X_i)$ for each $i$ we see that
$U \subset H$. Let $M \subset \text{Aut}(Y)$ be the finite subgroup
corresponding to
$$
(H/U)^{opp} \subset (G/U)^{opp} = \text{Aut}_{G\textit{-Sets}}(G/U)
= \text{Aut}(Y).
 ========

regex_match $M \subset \text{Aut}(Y)$
nn candidate the finite subgroup
nn the finite subgroup
new_annotation [(25115, 25140, 'VAR'), (25144, 25163, 'TYPE')]
===== sent \end{proof}

\begin{lemma}
\label{lemma-functoriality-galois}
Let $(\mathcal{C}, F)$ and $(\mathcal{C}', F')$ be Galois categories.
 ========

new_annotation []
===== sent Let $H : \mathcal{C} \to \mathcal{C}'$ be an exact functor.
 ========

regex_match $H : \mathcal{C} \to \mathcal{C}'$
nn candidate $H
nn candidate an exact functor
nn an exact functor
new_annotation [(25563, 25597, 'VAR'), (25601, 25617, 'TYPE')]
===== sent \end{proof}





\section{Functors and homomorphisms}
\label{section-translation}

\noindent
Let $(\mathcal{C}, F)$, $(\mathcal{C}', F')$, $(\mathcal{C}'', F'')$
be Galois categories. ========

new_annotation []
===== sent Set $G = \text{Aut}(F)$, $G' = \text{Aut}(F')$, and
$G'' = \text{Aut}(F'')$. Let $H : \mathcal{C} \to \mathcal{C}'$
and $H' : \mathcal{C}' \to \mathcal{C}''$ be exact functors.
 ========

new_annotation []
===== sent Let $h : G' \to G$ and $h' : G'' \to G'$ be the corresponding
continuous homomorphism as in Lemma \ref{lemma-functoriality-galois}.
In this section we consider the corresponding $2$-commutative diagram
\begin{equation}
\label{equation-translation}
\vcenter{
\xymatrix{
\mathcal{C} \ar[r]_H \ar[d] &
\mathcal{C}' \ar[r]_{H'} \ar[d] &
\mathcal{C}'' \ar[d] \\
 ========

new_annotation []
===== sent Let $X$ be a connected object of $\mathcal{C}$ and let
$*' \to H(X)$ be a morphism. ========

regex_match $X$
nn candidate a connected object
nn a connected object
new_annotation [(28594, 28597, 'VAR'), (28601, 28619, 'TYPE')]
===== sent Let $N \subset G'$ be the smallest closed normal
subgroup containing the image of $h'$. ========

regex_match $N \subset G'$
nn candidate $
nn candidate N
nn candidate the smallest closed normal
subgroup
nn the smallest closed normal
subgroup
new_annotation [(31687, 31701, 'VAR'), (31705, 31740, 'TYPE')]
===== sent This means that $N = \Ker(h)$. Let $X'$ be a
finite $G'$-set such that $G''$ acts trivially. ========

new_annotation []
===== sent Let $H'' \subset G''$
be an open subgroup. ========

new_annotation []
===== sent Then the desired diagram is
$$
G''/H'' \leftarrow G''/(h')^{-1}(H') \rightarrow G'/H'
$$
Conversely, assume (2) holds for the functor
$\textit{Finite-}G'\textit{-Sets} \to \textit{Finite-}G''\textit{-Sets}$.
Let $g'' \in \Ker(h')$. Pick any open subgroup $H'' \subset G''$.
 ========

new_annotation []
===== sent \medskip\noindent
Let $X$ be a scheme. ========

regex_match $X$
nn candidate a scheme
nn a scheme
new_annotation [(35654, 35657, 'VAR'), (35661, 35669, 'TYPE')]
===== sent It turns out that there is a stack $p : \textit{F\'Et} \to \Sch$
over the category of schemes whose fibre over $X$ is the category
$\textit{F\'Et}_X$ just defined. See Examples of Stacks, Section
\ref{examples-stacks-section-finite-etale}.

\begin{example}
\label{example-finite-etale-geometric-point}
Let $k$ be an algebraically closed field and $X = \Spec(k)$. In this case
$\textit{F\'Et}_X$ is equivalent to the category of finite sets. ========

regex_match $k$
nn candidate an algebraically closed field
nn an algebraically closed field
new_annotation [(36513, 36516, 'VAR'), (36520, 36549, 'TYPE')]
===== sent The reason is
that a scheme \'etale over $k$ is the disjoint union of spectra of
fields finite separable over $k$, see
Morphisms, Lemma \ref{morphisms-lemma-etale-over-field}.
\end{example}

\begin{lemma}
\label{lemma-finite-etale-covers-limits-colimits}
Let $X$ be a scheme. The category $\textit{F\'Et}_X$ has finite limits and
finite colimits and for any morphism $X' \to X$ the base change functor
$\textit{F\'Et}_X \to \textit{F\'Et}_{X'}$ is exact.
 ========

regex_match $X$
nn candidate a scheme
nn a scheme
new_annotation [(36977, 36980, 'VAR'), (36984, 36992, 'TYPE')]
===== sent Let $a, b : Z \to Y$ be two morphisms of $\textit{F\'Et}_X$.
Since $Z \to X$ and $Y \to X$ are finite \'etale we can write
$Z = \underline{\Spec}(\mathcal{C})$ and $Y = \underline{\Spec}(\mathcal{B})$
for some finite locally free $\mathcal{O}_X$-algebras $\mathcal{C}$
and $\mathcal{B}$. The morphisms $a, b$ induce two maps
$a^\sharp, b^\sharp : \mathcal{B} \to \mathcal{C}$.
Let $\mathcal{A} = \text{Eq}(a^\sharp, b^\sharp)$ be their
equalizer. ========

new_annotation []
===== sent Let $a, b : Z \to Y$ be two morphisms of $\textit{F\'Et}_X$.
Since $Z \to X$ and $Y \to X$ are finite \'etale we can write
$Z = \underline{\Spec}(\mathcal{C})$ and $Y = \underline{\Spec}(\mathcal{B})$
for some finite locally free $\mathcal{O}_X$-algebras $\mathcal{C}$
and $\mathcal{B}$. The morphisms $a, b$ induce two maps
$a^\sharp, b^\sharp : \mathcal{B} \to \mathcal{C}$.
Let $\mathcal{A} = \text{Eq}(a^\sharp, b^\sharp)$ be their
equalizer. ========

new_annotation []
===== sent $
After a further replacement by the members of an open covering
we may assume that $a, b$ correspond to
maps $a_s, b_s : \{1, \ldots, m\} \to \{1, \ldots, n\}$, i.e.,
the summand $X$ of $Z$ corresponding to the index $j$ maps into
the summand $X$ of $Y$ corresponding to the index $a_s(j)$, resp.\ $b_s(j)$
under the morphism $a$, resp.\ $b$.
Let $\{1, \ldots, n\} \to T$ be the coequalizer of $a_s, b_s$.
 ========

regex_match $\{1, \ldots, n\} \to T$
nn candidate the coequalizer
nn the coequalizer
new_annotation [(39596, 39620, 'VAR'), (39624, 39639, 'TYPE')]
===== sent \end{proof}

\begin{remark}
\label{remark-colimits-commute-forgetful}
Let $X$ be a scheme. ========

regex_match $X$
nn candidate a scheme
nn a scheme
new_annotation [(39968, 39971, 'VAR'), (39975, 39983, 'TYPE')]
===== sent Hence the statement follows from the fact that being a coequalizer
is fpqc local as formulate precisely in
Descent, Lemma \ref{descent-lemma-coequalizer-fpqc-local}.
\end{remark}

\begin{lemma}
\label{lemma-internal-hom-finite-etale}
Let $X$ be a scheme. ========

regex_match $X$
nn candidate a scheme
nn a scheme
new_annotation [(41340, 41343, 'VAR'), (41347, 41355, 'TYPE')]
===== sent \end{proof}

\noindent
Let $X$ be a scheme. ========

regex_match $X$
nn candidate a scheme
nn a scheme
new_annotation [(42563, 42566, 'VAR'), (42570, 42578, 'TYPE')]
===== sent We often use $\overline{x}$ to denote the scheme $\Spec(k)$ as well as
the morphism, and we use $\kappa(\overline{x})$
to denote $k$. We say $\overline{x}$ {\it lies over} $x$
to indicate that $x \in X$ is the image of $\overline{x}$.
We will discuss this further in
\'Etale Cohomology, Section \ref{etale-cohomology-section-stalks}.
Given $\overline{x}$ and an \'etale morphism $U \to X$ we can
consider
$$
|U_{\overline{x}}| : \text{the underlying set of points of the
scheme }U_{\overline{x}} = U \times_X \overline{x}
$$
Since $U_{\overline{x}}$ as a scheme over $\overline{x}$
is a disjoint union of copies of $\overline{x}$
(Morphisms, Lemma \ref{morphisms-lemma-etale-over-field})
we can also describe this set as
$$
|U_{\overline{x}}| =
\left\{
\begin{matrix}
\text{commutative} \\
\text{diagrams}
\end{matrix}
\vcenter{
\xymatrix{
\overline{x} \ar[rd]_{\overline{x}} \ar[r]_{\overline{u}} & U \ar[d] \\
& X
}
}
\right\}
$$
The assignment $U \mapsto |U_{\overline{x}}|$ is a functor
which is often denoted $F_{\overline{x}}$.

\begin{lemma}
\label{lemma-finite-etale-connected-galois-category}
Let $X$ be a connected scheme. ========

regex_match $X$
nn candidate a connected scheme
nn a connected scheme
new_annotation [(43872, 43875, 'VAR'), (43879, 43897, 'TYPE')]
===== sent Let $\overline{x}$ be a geometric point.
 ========

regex_match $\overline{x}$
nn candidate a geometric point
nn a geometric point
new_annotation [(43903, 43917, 'VAR'), (43921, 43938, 'TYPE')]
===== sent Then it follows from
Topology, Lemma \ref{topology-lemma-finite-fibre-connected-components}
that $Y$ is a finite coproduct of its connected components
both as a scheme and in the sense of
Definition \ref{definition-galois-category}.

\medskip\noindent
Let $Y \to Z$ be a morphism in $\textit{F\'Et}_X$ which induces a
bijection $F_{\overline{x}}(Y) \to F_{\overline{x}}(Z)$. We have to
show that $Y \to Z$ is an isomorphism. ========

regex_match $Y \to Z$
nn candidate $Y
nn candidate a morphism
nn a morphism
new_annotation [(45799, 45808, 'VAR'), (45812, 45822, 'TYPE')]
===== sent The following definition makes sense thanks to
Lemma \ref{lemma-finite-etale-connected-galois-category}.

\begin{definition}
\label{definition-fundamental-group}
Let $X$ be a connected scheme. ========

regex_match $X$
nn candidate a connected scheme
nn a connected scheme
new_annotation [(46629, 46632, 'VAR'), (46636, 46654, 'TYPE')]
===== sent Let $\overline{x}$ be a geometric point
of $X$. The {\it fundamental group} of $X$ with
{\it base point} $\overline{x}$ is the group
$$
\pi_1(X, \overline{x}) = \text{Aut}(F_{\overline{x}})
$$
of automorphisms of the fibre functor
$F_{\overline{x}} : \textit{F\'Et}_X \to \textit{Sets}$
endowed with its canonical profinite topology from
Lemma \ref{lemma-aut-inverse-limit}.
\end{definition}

\noindent
 ========

regex_match $\overline{x}$
nn candidate a geometric point
nn a geometric point
new_annotation [(46660, 46674, 'VAR'), (46678, 46695, 'TYPE')]
===== sent \begin{theorem}
\label{theorem-fundamental-group}
Let $X$ be a connected scheme. ========

regex_match $X$
nn candidate a connected scheme
nn a connected scheme
new_annotation [(47219, 47222, 'VAR'), (47226, 47244, 'TYPE')]
===== sent Let $\overline{x}$ be a geometric point
of $X$.
\begin{enumerate}
\item The fibre functor $F_{\overline{x}}$ defines an equivalence of
categories
$$
\textit{F\'Et}_X \longrightarrow
\textit{Finite-}\pi_1(X, \overline{x})\textit{-Sets}
$$
\item ========

regex_match $\overline{x}$
nn candidate a geometric point
nn a geometric point
new_annotation [(47250, 47264, 'VAR'), (47268, 47285, 'TYPE')]
===== sent \end{proof}

\begin{remark}
\label{remark-finite-etale-under-galois}
Let $X$ be a connected scheme with geometric point $\overline{x}$.
 ========

regex_match $X$
nn candidate a connected scheme
nn a connected scheme
new_annotation [(49040, 49043, 'VAR'), (49047, 49065, 'TYPE')]
===== sent \end{remark}

\begin{lemma}
\label{lemma-fundamental-group-Galois-group}
Let $K$ be a field and set $X = \Spec(K)$. Let $\overline{K}$ be an
algebraic closure and denote $\overline{x} : \Spec(\overline{K}) \to X$
the corresponding geometric point. ========

regex_match $K$
nn candidate a field
nn a field
new_annotation [(50342, 50345, 'VAR'), (50349, 50356, 'TYPE')]
===== sent \end{remark}

\begin{lemma}
\label{lemma-fundamental-group-Galois-group}
Let $K$ be a field and set $X = \Spec(K)$. Let $\overline{K}$ be an
algebraic closure and denote $\overline{x} : \Spec(\overline{K}) \to X$
the corresponding geometric point. ========

new_annotation []
===== sent Let $K^{sep} \subset \overline{K}$
be the separable algebraic closure.
 ========

new_annotation []
===== sent \begin{lemma}
\label{lemma-what-equivalence-gives}
Let $f : X \to Y$ be a morphism of quasi-compact and quasi-separated schemes
such that the base change functor $\textit{F\'Et}_Y \to \textit{F\'Et}_X$
is an equivalence of categories. ========

regex_match $f : X \to Y$
nn candidate X \to
nn candidate a morphism
nn a morphism
new_annotation [(53399, 53412, 'VAR'), (53416, 53426, 'TYPE')]
===== sent In this case
\begin{enumerate}
\item $f$ induces a homeomorphism $\pi_0(X) \to \pi_0(Y)$,
\item if $X$ or equivalently $Y$ is connected, then
$\pi_1(X, \overline{x}) = \pi_1(Y, \overline{y})$.
\end{enumerate}
\end{lemma}

\begin{proof}
Let $Y = Y_0 \amalg Y_1$ be a decomposition into nonempty open and closed
subschemes. ========

regex_match $Y = Y_0 \amalg Y_1$
nn candidate Y
nn candidate a decomposition
nn a decomposition
new_annotation [(53819, 53839, 'VAR'), (53843, 53858, 'TYPE')]
===== sent \begin{lemma}
\label{lemma-gabber}
Let $(A, I)$ be a henselian pair. ========

regex_match $(A, I)$
nn candidate $(A
nn candidate a henselian pair
nn a henselian pair
new_annotation [(55686, 55694, 'VAR'), (55698, 55714, 'TYPE')]
===== sent \begin{lemma}
\label{lemma-thickening}
Let $X \subset X'$ be a thickening of schemes. ========

regex_match $X \subset X'$
nn candidate $X
nn candidate a thickening
nn a thickening
new_annotation [(56309, 56323, 'VAR'), (56327, 56339, 'TYPE')]
===== sent \end{lemma}

\begin{proof}
For a discussion of thickenings see
More on Morphisms, Section \ref{more-morphisms-section-thickenings}.
Let $U' \to X'$ be an \'etale morphism such that $U = U' \times_{X'} X \to X$
is finite \'etale. ========

regex_match $U' \to X'$
nn candidate U
nn candidate an \'etale morphism
nn an \'etale morphism
new_annotation [(56630, 56641, 'VAR'), (56645, 56664, 'TYPE')]
===== sent \medskip\noindent
Let $X' = \bigcup X_i'$ be an affine open covering. ========

regex_match $X' = \bigcup X_i'$
nn candidate $X
nn candidate affine
nn affine
new_annotation [(57181, 57200, 'VAR'), (57207, 57213, 'TYPE')]
===== sent \end{proof}

\noindent
The ``correct'' way to prove the following proposition would be to
deduce it from the invariance of the \'etale site, see
\'Etale Cohomology, Theorem
\ref{etale-cohomology-theorem-topological-invariance}.

\begin{proposition}
\label{proposition-universal-homeomorphism}
Let $f : X \to Y$ be a universal homeomorphism of schemes. ========

regex_match $f : X \to Y$
nn candidate X \to
nn candidate a universal homeomorphism
nn a universal homeomorphism
new_annotation [(58549, 58562, 'VAR'), (58566, 58591, 'TYPE')]
===== sent Instead of stating and proving the results in the connected case
we prove the results in general and we leave it to the reader to deduce
the result for fundamental groups using
Lemma \ref{lemma-what-equivalence-gives}.

\begin{lemma}
\label{lemma-finite-etale-on-proper-over-henselian}
Let $A$ be a henselian local ring. ========

regex_match $A$
nn candidate a henselian local ring
nn a henselian local ring
new_annotation [(61146, 61149, 'VAR'), (61153, 61175, 'TYPE')]
===== sent Let $X$ be a proper scheme over $A$
with closed fibre $X_0$. ========

regex_match $X$
nn candidate a proper scheme
nn a proper scheme
new_annotation [(61181, 61184, 'VAR'), (61188, 61203, 'TYPE')]
===== sent Let $X_n = X \times_{\Spec(A)} \Spec(A/\mathfrak m^{n + 1})$.
 ========

new_annotation []
===== sent Let $U \to X$ and $V \to X$ be finite \'etale morphisms and
let $\varphi_0 : U_0 \to V_0$ be a morphism over $X_0$. Look at
the morphism
$$
\Gamma_{\varphi_0} : U_0 \longrightarrow U_0 \times_{X_0} V_0
$$
This morphism is both finite \'etale and a closed immersion.
 ========

new_annotation []
===== sent Let $U \to X$ and $V \to X$ be finite \'etale morphisms and
let $\varphi_0 : U_0 \to V_0$ be a morphism over $X_0$. Look at
the morphism
$$
\Gamma_{\varphi_0} : U_0 \longrightarrow U_0 \times_{X_0} V_0
$$
This morphism is both finite \'etale and a closed immersion.
 ========

regex_match $\varphi_0 : U_0 \to V_0$
nn candidate a morphism
nn a morphism
new_annotation [(63013, 63038, 'VAR'), (63042, 63052, 'TYPE')]
===== sent Let $U_0 \to X_0$ be a finite \'etale morphism.
 ========

regex_match $U_0 \to X_0$
nn candidate a finite \'etale morphism
nn a finite \'etale morphism
new_annotation [(63825, 63838, 'VAR'), (63842, 63867, 'TYPE')]
===== sent Let $A^\wedge$ be the completion of $A$ with respect to the maximal ideal.
 ========

regex_match $A^\wedge$
nn candidate the completion
nn the completion
new_annotation [(63873, 63883, 'VAR'), (63887, 63901, 'TYPE')]
===== sent Let $X^\wedge$ be the base change of $X$ to $A^\wedge$.
By the result above there exists a finite \'etale morphism
$V \to X^\wedge$ whose special fibre is $U_0$.
Write $A^\wedge = \colim A_i$ with $A \to A_i$ of finite type.
 ========

regex_match $X^\wedge$
nn candidate the base change
nn the base change
new_annotation [(63948, 63958, 'VAR'), (63962, 63977, 'TYPE')]
===== sent Let $(A, \mathfrak m)$ be a henselian local ring.
 ========

regex_match $(A, \mathfrak m)$
nn candidate \mathfrak
nn candidate a henselian local ring
nn a henselian local ring
new_annotation [(65467, 65485, 'VAR'), (65489, 65511, 'TYPE')]
===== sent By Limits, Lemma
\ref{limits-lemma-proper-limit-of-proper-finite-presentation-noetherian}
we can write $X \to \Spec(A)$ as a cofiltered limit of
proper morphisms $X_i \to S_i$ with $S_i$ of finite type over $\mathbf{Z}$.
For each $i$ let $s_i \in S_i$ be the image of $s$.
Since $S = \lim S_i$ and $A = \mathcal{O}_{S, s}$ we have
$A = \colim \mathcal{O}_{S_i, s_i}$. ========

regex_match $s_i \in S_i$
nn candidate the image
nn the image
new_annotation [(65809, 65822, 'VAR'), (65826, 65835, 'TYPE')]
===== sent \end{proof}

\begin{lemma}
\label{lemma-finite-etale-invariant-over-proper}
Let $k \subset k'$ be an extension of algebraically closed fields.
 ========

regex_match $k \subset k'$
nn candidate an extension
nn an extension
new_annotation [(67020, 67034, 'VAR'), (67038, 67050, 'TYPE')]
===== sent Let $X$ be a proper scheme over $k$. Then the functor
 ========

regex_match $X$
nn candidate a proper scheme
nn a proper scheme
new_annotation [(67087, 67090, 'VAR'), (67094, 67109, 'TYPE')]
===== sent Let $U' \to X_{k'}$ be a finite \'etale morphism.
 ========

regex_match $U' \to X_{k'}$
nn candidate U
nn candidate a finite \'etale morphism
nn a finite \'etale morphism
new_annotation [(67361, 67376, 'VAR'), (67380, 67405, 'TYPE')]
===== sent Since $k$ is algebraically closed we can find a
$k$-valued point $t$ in $\Spec(A_i)$. Let $U = (U_i)_t$ be the
fibre of $U_i$ over $t$. Let $A_i^h$ be the
henselization of $(A_i)_{\mathfrak m}$ where $\mathfrak m$ is
the maximal ideal corresponding to the point $t$. ========

new_annotation []
===== sent Since $k$ is algebraically closed we can find a
$k$-valued point $t$ in $\Spec(A_i)$. Let $U = (U_i)_t$ be the
fibre of $U_i$ over $t$. Let $A_i^h$ be the
henselization of $(A_i)_{\mathfrak m}$ where $\mathfrak m$ is
the maximal ideal corresponding to the point $t$. ========

new_annotation []
===== sent In this section we ask when $\pi_1(U) \to \pi_1(X)$ is surjective
for $U$ a dense open of a scheme $X$. We will see that this is the
case (roughly) when $U \cap B$ is connected for any small
``ball'' $B$ around a point $x \in X \setminus U$.

\begin{lemma}
\label{lemma-dense-faithful}
Let $f : X \to Y$ be a morphism of schemes. If $f(X)$ is dense in $Y$
then the base change functor $\textit{F\'Et}_Y ========

regex_match $f : X \to Y$
nn candidate X \to
nn candidate a morphism
nn a morphism
new_annotation [(69023, 69036, 'VAR'), (69040, 69050, 'TYPE')]
===== sent The condition in the following lemma that the punctured spectrum
of the strict henselization is connected follows for example from
the assumption that the local ring is geometrically unibranch, see
More on Algebra, Lemma \ref{more-algebra-lemma-geometrically-unibranch}.
There is a partial converse in
Properties, Lemma \ref{properties-lemma-geometrically-unibranch}.

\begin{lemma}
\label{lemma-same-etale-extensions}
Let $(A, \mathfrak m)$ be a local ring. ========

regex_match $(A, \mathfrak m)$
nn candidate \mathfrak
nn candidate a local ring
nn a local ring
new_annotation [(70447, 70465, 'VAR'), (70469, 70481, 'TYPE')]
===== sent Set $X = \Spec(A)$
and let $U = X \setminus ========

new_annotation []
===== sent Since the category of finite \'etale coverings has an
internal hom (Lemma \ref{lemma-internal-hom-finite-etale})
it suffices to prove the following: Given $Y$ finite \'etale over $X$
any morphism $s : U \to Y$ over $X$ extends to a morphism $t : X \to Y$
over $Y$. Let $A^{sh}$ be the strict henselization of $A$ and denote
$X^{sh} = \Spec(A^{sh})$, $U^{sh} = U \times_X X^{sh}$,
$Y^{sh} = Y \times_X X^{sh}$. By the first paragraph and our assumption
on $A$, we can extend the base change $s^{sh} : U^{sh} \to Y^{sh}$ of $s$ to
$t^{sh} : X^{sh} \to Y^{sh}$. Set $A' = A^{sh} \otimes_A A^{sh}$.
 ========

regex_match $A^{sh}$
nn candidate the strict henselization
nn the strict henselization
new_annotation [(71444, 71452, 'VAR'), (71456, 71480, 'TYPE')]
===== sent \begin{lemma}
\label{lemma-depth-2-connected-punctured-spectrum}
\begin{reference}
\cite[Proposition 2.1]{Hartshorne-connectedness}
\end{reference}
\begin{slogan}
Hartshorne's connectedness
\end{slogan}
Let $A$ be a Noetherian local ring of depth $\geq 2$.
Then the punctured spectra of $A$, $A^h$, and $A^{sh}$ are connected.
 ========

regex_match $A$
nn candidate a Noetherian local ring
nn a Noetherian local ring
new_annotation [(72730, 72733, 'VAR'), (72737, 72760, 'TYPE')]
===== sent \end{lemma}

\begin{proof}
Let $U$ be the punctured spectrum of $A$.
 ========

regex_match $U$
nn candidate the punctured spectrum
nn the punctured spectrum
new_annotation [(72881, 72884, 'VAR'), (72888, 72910, 'TYPE')]
===== sent Thus $\text{depth}(A) \leq 1$ by
Dualizing Complexes, Lemma \ref{dualizing-lemma-depth}.
To see the result for $A^h$ and $A^{sh}$ use
More on Algebra, Lemma \ref{more-algebra-lemma-henselization-depth}.
\end{proof}

\begin{lemma}
\label{lemma-quasi-compact-dense-open-connected-at-infinity-Noetherian}
Let $X$ be a scheme. ========

regex_match $X$
nn candidate a scheme
nn a scheme
new_annotation [(73618, 73621, 'VAR'), (73625, 73633, 'TYPE')]
===== sent Let $U \subset X$ be a dense open. ========

regex_match $U \subset X$
nn candidate $
nn candidate U
new_annotation []
===== sent \end{lemma}

\begin{proof}
Let $Y_1, Y_2$ be finite \'etale over $X$ and let
$\varphi : (Y_1)_U \to (Y_2)_U$ be a morphism over $U$. We have to show that
$\varphi$ lifts uniquely to a morphism $Y_1 \to Y_2$ over $X$.
Uniqueness follows from Lemma \ref{lemma-dense-faithful}.

\medskip\noindent
Let $x \in X \setminus U$ be a generic point of an irreducible component
of $X \setminus U$. Set $V = U \times_X \Spec(\mathcal{O}_{X, x})$.
By our choice of $x$ this is the punctured spectrum of
$\Spec(\mathcal{O}_{X, x})$. By
Lemma \ref{lemma-same-etale-extensions}
we can extend the morphism $\varphi_V : (Y_1)_V \to (Y_2)_V$
uniquely to a morphism
$(Y_1)_{\Spec(\mathcal{O}_{X, x})} \to (Y_2)_{\Spec(\mathcal{O}_{X, x})}$.
By Limits, Lemma \ref{limits-lemma-glueing-near-point}
we find an open $U \subset U'$ containing $x$ and an extension
$\varphi' : (Y_1)_{U'} \to (Y_2)_{U'}$ of $\varphi$.
 ========

new_annotation []
===== sent \end{lemma}

\begin{proof}
Let $Y_1, Y_2$ be finite \'etale over $X$ and let
$\varphi : (Y_1)_U \to (Y_2)_U$ be a morphism over $U$. We have to show that
$\varphi$ lifts uniquely to a morphism $Y_1 \to Y_2$ over $X$.
Uniqueness follows from Lemma \ref{lemma-dense-faithful}.

\medskip\noindent
Let $x \in X \setminus U$ be a generic point of an irreducible component
of $X \setminus U$. Set $V = U \times_X \Spec(\mathcal{O}_{X, x})$.
By our choice of $x$ this is the punctured spectrum of
$\Spec(\mathcal{O}_{X, x})$. By
Lemma \ref{lemma-same-etale-extensions}
we can extend the morphism $\varphi_V : (Y_1)_V \to (Y_2)_V$
uniquely to a morphism
$(Y_1)_{\Spec(\mathcal{O}_{X, x})} \to (Y_2)_{\Spec(\mathcal{O}_{X, x})}$.
By Limits, Lemma \ref{limits-lemma-glueing-near-point}
we find an open $U \subset U'$ containing $x$ and an extension
$\varphi' : (Y_1)_{U'} \to (Y_2)_{U'}$ of $\varphi$.
 ========

regex_match $x \in X \setminus U$
nn candidate X \setminus
nn candidate a generic point
nn a generic point
new_annotation [(74265, 74286, 'VAR'), (74290, 74305, 'TYPE')]
===== sent \end{proof}

\begin{lemma}
\label{lemma-retrocompact-dense-open-connected-at-infinity-closed}
Let $X$ be a scheme. ========

regex_match $X$
nn candidate a scheme
nn a scheme
new_annotation [(75128, 75131, 'VAR'), (75135, 75143, 'TYPE')]
===== sent Let $U \subset X$ be a dense open. ========

regex_match $U \subset X$
nn candidate $
nn candidate U
new_annotation []
===== sent \end{lemma}

\begin{proof}
Let $Y_1, Y_2$ be finite \'etale over $X$ and let
$\varphi : (Y_1)_U \to (Y_2)_U$ be a morphism over $U$. We have to show that
$\varphi$ lifts uniquely to a morphism $Y_1 \to Y_2$ over $X$.
Uniqueness follows from Lemma \ref{lemma-dense-faithful}.

\medskip\noindent
Let $x \in X \setminus U$. Set $V = U \times_X \Spec(\mathcal{O}_{X, x})$.
 ========

new_annotation []
===== sent \end{lemma}

\begin{proof}
Let $Y_1, Y_2$ be finite \'etale over $X$ and let
$\varphi : (Y_1)_U \to (Y_2)_U$ be a morphism over $U$. We have to show that
$\varphi$ lifts uniquely to a morphism $Y_1 \to Y_2$ over $X$.
Uniqueness follows from Lemma \ref{lemma-dense-faithful}.

\medskip\noindent
Let $x \in X \setminus U$. Set $V = U \times_X \Spec(\mathcal{O}_{X, x})$.
 ========

new_annotation []
===== sent \end{proof}

\begin{lemma}
\label{lemma-quasi-compact-dense-open-connected-at-infinity}
Let $X$ be a scheme. ========

regex_match $X$
nn candidate a scheme
nn a scheme
new_annotation [(76752, 76755, 'VAR'), (76759, 76767, 'TYPE')]
===== sent Let $U \subset X$ be a dense open. ========

regex_match $U \subset X$
nn candidate $
nn candidate U
new_annotation []
===== sent \end{lemma}

\begin{proof}
Let $Y_1, Y_2$ be finite \'etale over $X$ and let
$\varphi : (Y_1)_U \to (Y_2)_U$ be a morphism over $U$. We have to show that
$\varphi$ lifts uniquely to a morphism $Y_1 \to Y_2$ over $X$.
Uniqueness follows from Lemma \ref{lemma-dense-faithful}.
We will prove existence by showing that we can enlarge $U$
if $U \not = X$ and using Zorn's lemma to finish the proof.

\medskip\noindent
Let $x \in X \setminus U$ be a generic point of an irreducible component
of $X \setminus U$. Set $V = U \times_X \Spec(\mathcal{O}_{X, x})$.
By our choice of $x$ this is the punctured spectrum of
$\Spec(\mathcal{O}_{X, x})$. By
Lemma \ref{lemma-same-etale-extensions}
we can extend the morphism $\varphi_V : (Y_1)_V \to (Y_2)_V$
(uniquely) to a morphism
$(Y_1)_{\Spec(\mathcal{O}_{X, x})} \to (Y_2)_{\Spec(\mathcal{O}_{X, x})}$.
Choose an affine neighbourhood $W \subset X$ of $x$.
 ========

new_annotation []
===== sent \end{lemma}

\begin{proof}
Let $Y_1, Y_2$ be finite \'etale over $X$ and let
$\varphi : (Y_1)_U \to (Y_2)_U$ be a morphism over $U$. We have to show that
$\varphi$ lifts uniquely to a morphism $Y_1 \to Y_2$ over $X$.
Uniqueness follows from Lemma \ref{lemma-dense-faithful}.
We will prove existence by showing that we can enlarge $U$
if $U \not = X$ and using Zorn's lemma to finish the proof.

\medskip\noindent
Let $x \in X \setminus U$ be a generic point of an irreducible component
of $X \setminus U$. Set $V = U \times_X \Spec(\mathcal{O}_{X, x})$.
By our choice of $x$ this is the punctured spectrum of
$\Spec(\mathcal{O}_{X, x})$. By
Lemma \ref{lemma-same-etale-extensions}
we can extend the morphism $\varphi_V : (Y_1)_V \to (Y_2)_V$
(uniquely) to a morphism
$(Y_1)_{\Spec(\mathcal{O}_{X, x})} \to (Y_2)_{\Spec(\mathcal{O}_{X, x})}$.
Choose an affine neighbourhood $W \subset X$ of $x$.
 ========

regex_match $x \in X \setminus U$
nn candidate X \setminus
nn candidate a generic point
nn a generic point
new_annotation [(77533, 77554, 'VAR'), (77558, 77573, 'TYPE')]
===== sent By the argument above
we see that this maximal element is an extension of $\varphi$
over all of $X$.
\end{proof}

\begin{lemma}
\label{lemma-local-exact-sequence}
Let $(A, \mathfrak m)$ be a local ring. ========

regex_match $(A, \mathfrak m)$
nn candidate \mathfrak
nn candidate a local ring
nn a local ring
new_annotation [(79487, 79505, 'VAR'), (79509, 79521, 'TYPE')]
===== sent Let $U^{sh}$ be the punctured spectrum
of the strict henselization $A^{sh}$ of $A$.
 ========

regex_match $U^{sh}$
nn candidate the punctured spectrum
nn the punctured spectrum
new_annotation [(79585, 79593, 'VAR'), (79597, 79619, 'TYPE')]
===== sent Write $X^{sh} = \Spec(A^{sh})$. Let $Y \to X$ be a finite \'etale morphism.
 ========

regex_match $Y \to X$
nn candidate $Y
nn candidate a finite \'etale morphism
nn a finite \'etale morphism
new_annotation [(80131, 80140, 'VAR'), (80144, 80169, 'TYPE')]
===== sent Let $\xi \in X^{sh} \times_X X^{sh}$ be a point not
in $U^{sh} ========

regex_match $\xi \in X^{sh} \times_X X^{sh}$
nn candidate \in X^{sh
nn candidate a point
nn a point
new_annotation [(81518, 81550, 'VAR'), (81554, 81561, 'TYPE')]
===== sent Finally, $Y \to X$ is finite \'etale as $Y^{sh} \to X^{sh}$ is, see
Descent, Lemmas \ref{descent-lemma-descending-property-etale} and
\ref{descent-lemma-descending-property-finite}.
\end{proof}

\noindent
Let $X$ be an irreducible scheme. ========

regex_match $X$
nn candidate an irreducible scheme
nn an irreducible scheme
new_annotation [(83458, 83461, 'VAR'), (83465, 83486, 'TYPE')]
===== sent Let $\eta \in X$ be the geometric
point. ========

regex_match $\eta \in X$
nn candidate $
nn candidate the geometric
point
nn the geometric
point
new_annotation [(83492, 83504, 'VAR'), (83508, 83527, 'TYPE')]
===== sent The canonical morphism $\eta \to X$ induces a canonical map
\begin{equation}
\label{equation-inclusion-generic-point}
\text{Gal}(\kappa(\eta)^{sep}/\kappa(\eta)) = \pi_1(\eta, \overline{\eta})
\longrightarrow \pi_1(X, \overline{\eta})
\end{equation}
The identification on the left hand side is
Lemma \ref{lemma-fundamental-group-Galois-group}.

\begin{lemma}
\label{lemma-irreducible-geometrically-unibranch}
Let $X$ be an irreducible, geometrically unibranch scheme.
 ========

regex_match $X$
new_annotation []
===== sent Hence $\textit{F\'Et}_\eta$ is the colimit of the
categories $\textit{F\'Et}_U$ over such $U$, see
Limits, Lemmas \ref{limits-lemma-descend-finite-presentation},
\ref{limits-lemma-descend-finite-finite-presentation}, and
\ref{limits-lemma-descend-etale}.
A formal argument then shows that fully faithfulness for
$\textit{F\'Et}_X \to \textit{F\'Et}_\eta$ follows from the
fully faithfulness of the functors $\textit{F\'Et}_X \to \textit{F\'Et}_U$.
\end{proof}

\begin{lemma}
\label{lemma-exact-sequence-finite-nr-closed-pts}
Let $X$ be a scheme. ========

regex_match $X$
nn candidate a scheme
nn a scheme
new_annotation [(85709, 85712, 'VAR'), (85716, 85724, 'TYPE')]
===== sent Let $x_1, \ldots, x_n \in X$ be a finite
number of closed points such that
\begin{enumerate}
\item $U = X \setminus \{x_1, \ldots, x_n\}$ is connected and is
a retrocompact open of $X$, and
\item for each $i$ the punctured spectrum $U_i^{sh}$ of the
strict henselization of $\mathcal{O}_{X, x_i}$ is connected.
 ========

regex_match $x_1, \ldots, x_n \in X$
nn candidate x_n \in
nn candidate a finite
number
nn a finite
number
new_annotation [(85730, 85754, 'VAR'), (85758, 85773, 'TYPE')]
===== sent Write
$x = x_1$, $U^{sh} = U_1^{sh}$,
set $A = \mathcal{O}_{X, x}$, and let $A^{sh}$ be the strict henselization.
 ========

regex_match $A^{sh}$
nn candidate the strict henselization
nn the strict henselization
new_annotation [(86768, 86776, 'VAR'), (86780, 86804, 'TYPE')]
===== sent \end{proof}









\section{Fundamental groups of normal schemes}
\label{section-normal}

\noindent
Let $X$ be an integral, geometrically unibranch scheme. ========

regex_match $X$
nn candidate an integral, geometrically unibranch scheme
nn an integral, geometrically unibranch scheme
new_annotation [(87659, 87662, 'VAR'), (87666, 87709, 'TYPE')]
===== sent \medskip\noindent
Let $X$ be an integral normal scheme with function field $K$.
Let $K \subset L$ be a finite extension. ========

regex_match $X$
nn candidate an integral normal scheme
nn an integral normal scheme
new_annotation [(88176, 88179, 'VAR'), (88183, 88208, 'TYPE')]
===== sent \medskip\noindent
Let $X$ be an integral normal scheme with function field $K$.
Let $K \subset L$ be a finite extension. ========

regex_match $K \subset L$
nn candidate a finite extension
nn a finite extension
new_annotation [(88238, 88251, 'VAR'), (88255, 88273, 'TYPE')]
===== sent Let $x \in X$.
We can choose an \'etale neighbourhood $(U, u) \to (X, x)$ such that
$$
Y \times_X U = \coprod V_j \longrightarrow U
$$
is a disjoint union of closed immersions, see
\'Etale Morphisms, Lemma \ref{etale-lemma-finite-unramified-etale-local}.
Shrinking we may assume $U$ is quasi-compact.
 ========

new_annotation []
===== sent \end{proof}

\begin{lemma}
\label{lemma-finite-etale-covering-normal-unramified}
Let $X$ be a normal integral scheme with function field $K$.
Let $Y \to X$ be a finite \'etale morphism. ========

regex_match $X$
nn candidate a normal integral scheme
nn a normal integral scheme
new_annotation [(90977, 90980, 'VAR'), (90984, 91008, 'TYPE')]
===== sent \end{proof}

\begin{lemma}
\label{lemma-finite-etale-covering-normal-unramified}
Let $X$ be a normal integral scheme with function field $K$.
Let $Y \to X$ be a finite \'etale morphism. ========

regex_match $Y \to X$
nn candidate a finite \'etale morphism
nn a finite \'etale morphism
new_annotation [(91038, 91047, 'VAR'), (91051, 91076, 'TYPE')]
===== sent \medskip\noindent
Let $L$ be the function field of $Y$ and let $Y' \to X$ be the normalization
of $X$ in $L$. By
Morphisms, Lemma \ref{morphisms-lemma-characterize-normalization}
we obtain a factorization $Y' \to Y \to X$ and $Y' \to Y$ is
the normalization of $Y$ in $L$. Since $Y$ is normal it is clear
that $Y' = Y$ (this can also be deduced from
Morphisms, Lemma \ref{morphisms-lemma-finite-birational-over-normal}).
 ========

regex_match $L$
nn candidate the function field
nn the function field
new_annotation [(91804, 91807, 'VAR'), (91811, 91829, 'TYPE')]
===== sent \medskip\noindent
Let $L$ be the function field of $Y$ and let $Y' \to X$ be the normalization
of $X$ in $L$. By
Morphisms, Lemma \ref{morphisms-lemma-characterize-normalization}
we obtain a factorization $Y' \to Y \to X$ and $Y' \to Y$ is
the normalization of $Y$ in $L$. Since $Y$ is normal it is clear
that $Y' = Y$ (this can also be deduced from
Morphisms, Lemma \ref{morphisms-lemma-finite-birational-over-normal}).
 ========

regex_match $Y' \to X$
nn candidate Y
nn candidate the normalization
nn the normalization
new_annotation [(91845, 91855, 'VAR'), (91859, 91876, 'TYPE')]
===== sent \end{proof}

\begin{proposition}
\label{proposition-normal}
Let $X$ be a normal integral scheme with function field $K$.
Then the canonical map (\ref{equation-inclusion-generic-point})
$$
\text{Gal}(K^{sep}/K) = \pi_1(\eta, \overline{\eta})
\longrightarrow \pi_1(X, \overline{\eta})
$$
is identified with the quotient map
$\text{Gal}(K^{sep}/K) \to \text{Gal}(M/K)$ where $M \subset K^{sep}$
is the union of the finite subextensions $L$
such that $X$ is unramified in $L$.
\end{proposition}

\begin{proof}
The normal scheme $X$ is geometrically unibranch
(Properties, Lemma \ref{properties-lemma-normal-geometrically-unibranch}).
 ========

regex_match $X$
nn candidate a normal integral scheme
nn a normal integral scheme
new_annotation [(92267, 92270, 'VAR'), (92274, 92298, 'TYPE')]
===== sent \end{proof}

\begin{lemma}
\label{lemma-local-exact-sequence-normal}
Let $(A, \mathfrak m)$ be a normal local ring.
Set $X = \Spec(A)$. Let $A^{sh}$ be the strict henselization of $A$.
 ========

regex_match $(A, \mathfrak m)$
nn candidate \mathfrak
nn candidate a normal local ring
nn a normal local ring
new_annotation [(94816, 94834, 'VAR'), (94838, 94857, 'TYPE')]
===== sent \end{proof}

\begin{lemma}
\label{lemma-local-exact-sequence-normal}
Let $(A, \mathfrak m)$ be a normal local ring.
Set $X = \Spec(A)$. Let $A^{sh}$ be the strict henselization of $A$.
 ========

regex_match $A^{sh}$
nn candidate the strict henselization
nn the strict henselization
new_annotation [(94883, 94891, 'VAR'), (94895, 94919, 'TYPE')]
===== sent Let $K$ and $K^{sh}$ be the fraction fields of $A$ and $A^{sh}$.
 ========

new_annotation []
===== sent Write $X^{sh} = \Spec(A^{sh})$. Let $Y \to X$ be a finite \'etale morphism.
 ========

regex_match $Y \to X$
nn candidate $Y
nn candidate a finite \'etale morphism
nn a finite \'etale morphism
new_annotation [(95444, 95453, 'VAR'), (95457, 95482, 'TYPE')]
===== sent To finish the proof, it suffices according to
Lemma \ref{lemma-functoriality-galois-ses}
to show the following: Given a finite \'etale morphism
$V \to \Spec(K)$ such that $V \times_{\Spec(K)} \Spec(K^{sh})$
is a disjoint union of copies of $\Spec(K^{sh})$, we can find a
finite \'etale morphism
$Y \to X$ with $V \cong Y \times_X \Spec(K)$ over $\Spec(K)$.
Write $V = \Spec(L)$, so $L$ is a finite product of
finite separable extensions of $K$.
Let $B \subset L$ be the integral closure of $A$ in $L$.
If $A \to B$ is \'etale, then we can take $Y = \Spec(B)$
and the proof is complete. ========

regex_match $B \subset L$
nn candidate $B \subset
nn candidate the integral closure
nn the integral closure
new_annotation [(96319, 96332, 'VAR'), (96336, 96356, 'TYPE')]
===== sent \begin{lemma}
\label{lemma-get-algebraic-closure}
Let $A$ be a normal domain whose fraction field is separably algebraically
closed. ========

regex_match $A$
nn candidate a normal domain
nn a normal domain
new_annotation [(97319, 97322, 'VAR'), (97326, 97341, 'TYPE')]
===== sent Let $\mathfrak p \subset A$ be a nonzero prime ideal.
 ========

regex_match $\mathfrak p \subset A$
nn candidate p \subset
nn candidate a nonzero prime ideal
nn a nonzero prime ideal
new_annotation [(97402, 97425, 'VAR'), (97429, 97450, 'TYPE')]
===== sent \end{lemma}

\begin{proof}
Let $(A, \mathfrak m, \kappa)$ be normal local with separably
closed fraction field $K$. If $ ========

new_annotation []
===== sent Let $f \in A[T]$ be monic and let $a_0 \in \kappa$ be a root
of multiplicity $1$ of the reduction $\overline{f} \in \kappa[T]$.
 ========

new_annotation []
===== sent Let $f \in A[T]$ be monic and let $a_0 \in \kappa$ be a root
of multiplicity $1$ of the reduction $\overline{f} \in \kappa[T]$.
 ========

regex_match $a_0 \in \kappa$
nn candidate a root
nn a root
new_annotation [(98994, 99010, 'VAR'), (99014, 99020, 'TYPE')]
===== sent Let $f = \prod f_i$ be the factorization in $K[T]$.
 ========

regex_match $f = \prod f_i$
nn candidate the factorization
nn the factorization
new_annotation [(99092, 99107, 'VAR'), (99111, 99128, 'TYPE')]
===== sent Thus $A$ is henselian, see
Algebra, Definition \ref{algebra-definition-henselian}.
\end{proof}

\begin{lemma}
\label{lemma-inertia-base-change}
Let $G$ be a finite group acting on a ring $R$. Let $R^G \to A$ be a ring
map. ========

regex_match $G$
nn candidate a finite group
nn a finite group
new_annotation [(99651, 99654, 'VAR'), (99658, 99672, 'TYPE')]
===== sent Thus $A$ is henselian, see
Algebra, Definition \ref{algebra-definition-henselian}.
\end{proof}

\begin{lemma}
\label{lemma-inertia-base-change}
Let $G$ be a finite group acting on a ring $R$. Let $R^G \to A$ be a ring
map. ========

regex_match $R^G \to A$
nn candidate a ring
map
nn a ring
map
new_annotation [(99699, 99710, 'VAR'), (99714, 99724, 'TYPE')]
===== sent Let $\mathfrak q' \subset A \otimes_{R^G} R$ be a prime lying
over the prime $\mathfrak q \subset R$. Then
$$
I_\mathfrak q = \{\sigma \in G \mid
\sigma(\mathfrak q) = \mathfrak q\text{ and }
\sigma \bmod \mathfrak q = \text{id}_{\kappa(\mathfrak q)}\}
$$
is equal to
$$
I_{\mathfrak q'} = \{\sigma \in G \mid
\sigma(\mathfrak q') = \mathfrak q'\text{ and }
\sigma \bmod \mathfrak q' = \text{id}_{\kappa(\mathfrak q')}\}
$$
\end{lemma}

\begin{proof}
 ========

regex_match $\mathfrak q' \subset A \otimes_{R^G} R$
nn candidate A \otimes_{R^G
nn candidate a prime lying
nn a prime lying
new_annotation [(99730, 99770, 'VAR'), (99774, 99787, 'TYPE')]
===== sent \end{proof}

\begin{lemma}
\label{lemma-inertia-invariants-etale}
Let $G$ be a finite group acting on a ring $R$. Let $\mathfrak q \subset ========

regex_match $G$
nn candidate a finite group
nn a finite group
new_annotation [(100655, 100658, 'VAR'), (100662, 100676, 'TYPE')]
===== sent \end{proof}

\begin{lemma}
\label{lemma-inertia-invariants-etale}
Let $G$ be a finite group acting on a ring $R$. Let $\mathfrak q \subset ========

new_annotation []
===== sent Let $R^G \to A$ be an \'etale ring map. ========

regex_match $R^G \to A$
nn candidate an \'etale ring map
nn an \'etale ring map
new_annotation [(101106, 101117, 'VAR'), (101121, 101140, 'TYPE')]
===== sent Let $\mathfrak p = R^G \cap \mathfrak q$.
By More on Algebra, Lemma \ref{more-algebra-lemma-one-orbit}
the fibre of $\Spec(R) \to \Spec(R^G)$ over $\mathfrak p$ is
finite. ========

new_annotation []
===== sent \end{proof}

\noindent
The following lemma generalizes
More on Algebra, Lemma \ref{more-algebra-lemma-inertial-invariants-unramified}.

\begin{lemma}
\label{lemma-inertial-invariants-unramified}
Let $A$ be a normal domain with fraction field $K$.
 ========

regex_match $A$
nn candidate a normal domain
nn a normal domain
new_annotation [(105791, 105794, 'VAR'), (105798, 105813, 'TYPE')]
===== sent Let $L/K$ be a (possibly infinite) Galois extension.
 ========

regex_match $L/K$
nn candidate a (possibly infinite) Galois extension
nn a (possibly infinite) Galois extension
new_annotation [(105843, 105848, 'VAR'), (105852, 105890, 'TYPE')]
===== sent Let $G = \text{Gal}(L/K)$ and let
$B$ be the integral closure of $A$ in $L$.
Let $\mathfrak q \subset B$. Set
 ========

new_annotation []
===== sent Let $G = \text{Gal}(L/K)$ and let
$B$ be the integral closure of $A$ in $L$.
Let $\mathfrak q \subset B$. Set
 ========

new_annotation []
===== sent \medskip\noindent
Let $(A, \mathfrak m, \kappa)$ be a normal local ring with
fraction field $K$. Choose a separable algebraic closure $K^{sep}$. Let
$A^{sep}$ be the integral closure of $A$ in $K^{sep}$.
Choose maximal ideal $\mathfrak m^{sep} \subset A^{sep}$.
 ========

regex_match $(A, \mathfrak m, \kappa)$
nn candidate A
nn candidate a normal local ring
nn a normal local ring
new_annotation [(106892, 106918, 'VAR'), (106922, 106941, 'TYPE')]
===== sent Let $A \subset A^h \subset A^{sh}$ be the henselization and strict
henselization. ========

regex_match $A \subset A^h \subset A^{sh}$
nn candidate $A \subset A^h \subset
nn candidate the henselization and strict
henselization
nn the henselization and strict
henselization
new_annotation [(107136, 107166, 'VAR'), (107170, 107212, 'TYPE')]
===== sent Thus
$$
\text{Aut}(A^{sh}/A) = \text{Aut}(A^{sh}/A^h) =
\text{Gal}(\kappa(\mathfrak m^{sh})/\kappa)
$$
Let $\kappa \subset \kappa'$ be a finite Galois extension with
Galois group $G$. Let $A^h \subset A'$ be the finite \'etale extension
corresponding to $\kappa \subset \kappa'$ by
Algebra, Lemma \ref{algebra-lemma-henselian-cat-finite-etale}.
 ========

regex_match $\kappa \subset \kappa'$
nn candidate a finite Galois extension
nn a finite Galois extension
new_annotation [(111665, 111689, 'VAR'), (111693, 111718, 'TYPE')]
===== sent Thus
$$
\text{Aut}(A^{sh}/A) = \text{Aut}(A^{sh}/A^h) =
\text{Gal}(\kappa(\mathfrak m^{sh})/\kappa)
$$
Let $\kappa \subset \kappa'$ be a finite Galois extension with
Galois group $G$. Let $A^h \subset A'$ be the finite \'etale extension
corresponding to $\kappa \subset \kappa'$ by
Algebra, Lemma \ref{algebra-lemma-henselian-cat-finite-etale}.
 ========

regex_match $A^h \subset A'$
nn candidate the finite \'etale extension
nn the finite \'etale extension
new_annotation [(111746, 111762, 'VAR'), (111766, 111794, 'TYPE')]
===== sent \end{proof}

\noindent
Let $X$ be a normal integral scheme with function field $K$.
Let $K^{sep}$ be a separable algebraic closure of $K$.
Let $X^{sep} \to X$ be the normalization of $X$ in $K^{sep}$.
Since $G = \text{Gal}(K^{sep}/K)$ acts on $K^{sep}$
we obtain a right action of $G$ on $X^{sep}$.
For $y \in X^{sep}$ define
$$
D_y = \{\sigma \in G \mid \sigma(y) = y\} \supset
I_y = \{\sigma \in D \mid \sigma \bmod \mathfrak m_y =
\text{id}_{\kappa(y)} \}
$$
similarly to the above. ========

regex_match $X$
nn candidate a normal integral scheme
nn a normal integral scheme
new_annotation [(112923, 112926, 'VAR'), (112930, 112954, 'TYPE')]
===== sent \end{proof}

\noindent
Let $X$ be a normal integral scheme with function field $K$.
Let $K^{sep}$ be a separable algebraic closure of $K$.
Let $X^{sep} \to X$ be the normalization of $X$ in $K^{sep}$.
Since $G = \text{Gal}(K^{sep}/K)$ acts on $K^{sep}$
we obtain a right action of $G$ on $X^{sep}$.
For $y \in X^{sep}$ define
$$
D_y = \{\sigma \in G \mid \sigma(y) = y\} \supset
I_y = \{\sigma \in D \mid \sigma \bmod \mathfrak m_y =
\text{id}_{\kappa(y)} \}
$$
similarly to the above. ========

regex_match $K^{sep}$
nn candidate a separable algebraic closure
nn a separable algebraic closure
new_annotation [(112984, 112993, 'VAR'), (112997, 113026, 'TYPE')]
===== sent \end{proof}

\noindent
Let $X$ be a normal integral scheme with function field $K$.
Let $K^{sep}$ be a separable algebraic closure of $K$.
Let $X^{sep} \to X$ be the normalization of $X$ in $K^{sep}$.
Since $G = \text{Gal}(K^{sep}/K)$ acts on $K^{sep}$
we obtain a right action of $G$ on $X^{sep}$.
For $y \in X^{sep}$ define
$$
D_y = \{\sigma \in G \mid \sigma(y) = y\} \supset
I_y = \{\sigma \in D \mid \sigma \bmod \mathfrak m_y =
\text{id}_{\kappa(y)} \}
$$
similarly to the above. ========

regex_match $X^{sep} \to X$
nn candidate $X^{sep
nn candidate the normalization
nn the normalization
new_annotation [(113039, 113054, 'VAR'), (113058, 113075, 'TYPE')]
===== sent On the other hand, for $x \in X$
let $\mathcal{O}_{X, x}^{sh}$ be a strict henselization,
let $K_x^{sh}$ be the fraction field of $\mathcal{O}_{X, x}^{sh}$
and choose a $K$-embedding $K_x^{sh} \to K^{sep}$.

\begin{lemma}
\label{lemma-normal-pione-quotient-inertia}
Let $X$ be a normal integral scheme with function field $K$.
 ========

regex_match $\mathcal{O}_{X, x}^{sh}$
nn candidate $\mathcal{O}_{X
nn candidate a strict henselization
nn a strict henselization
new_annotation [(113419, 113444, 'VAR'), (113448, 113470, 'TYPE')]
===== sent On the other hand, for $x \in X$
let $\mathcal{O}_{X, x}^{sh}$ be a strict henselization,
let $K_x^{sh}$ be the fraction field of $\mathcal{O}_{X, x}^{sh}$
and choose a $K$-embedding $K_x^{sh} \to K^{sep}$.

\begin{lemma}
\label{lemma-normal-pione-quotient-inertia}
Let $X$ be a normal integral scheme with function field $K$.
 ========

regex_match $K_x^{sh}$
nn candidate the fraction field
nn the fraction field
new_annotation [(113476, 113486, 'VAR'), (113490, 113508, 'TYPE')]
===== sent On the other hand, for $x \in X$
let $\mathcal{O}_{X, x}^{sh}$ be a strict henselization,
let $K_x^{sh}$ be the fraction field of $\mathcal{O}_{X, x}^{sh}$
and choose a $K$-embedding $K_x^{sh} \to K^{sep}$.

\begin{lemma}
\label{lemma-normal-pione-quotient-inertia}
Let $X$ be a normal integral scheme with function field $K$.
 ========

regex_match $X$
nn candidate a normal integral scheme
nn a normal integral scheme
new_annotation [(113652, 113655, 'VAR'), (113659, 113683, 'TYPE')]
===== sent Let $Y \to X$ be the normalization of $X$ in $L$.
The inclusion $L \subset K^{sep}$ induces a morphism
$\pi : X^{sep} \to Y$. For $y ========

regex_match $Y \to X$
nn candidate $Y
nn candidate the normalization
nn the normalization
new_annotation [(115139, 115148, 'VAR'), (115152, 115169, 'TYPE')]
===== sent \end{proof}

\begin{example}
\label{example-bigger-codim}
Let $X$ be a normal integral Noetherian scheme with function field $K$.
Purity of branch locus (see below) tells us that if $X$ is regular, then
it suffices in Lemma \ref{lemma-normal-pione-quotient-inertia}
to consider the inertia groups $I = \pi_1(\Spec(K_x^{sh}))$
for points $x$ of codimension $1$ in $X$.
In general this is not enough however. ========

regex_match $X$
nn candidate a normal integral Noetherian scheme
nn a normal integral Noetherian scheme
new_annotation [(115908, 115911, 'VAR'), (115915, 115950, 'TYPE')]
===== sent Namely, let
$Y = \mathbf{A}_k^n = \Spec(k[t_1, \ldots, t_n])$
where $k$ is a field not of characteristic $2$.
Let $G = \{\pm 1\}$ be the group of order $2$ acting on $Y$
by multiplication on the coordinates. ========

regex_match $G = \{\pm 1\}$
nn candidate $G
nn candidate the group
nn the group
new_annotation [(116367, 116382, 'VAR'), (116386, 116395, 'TYPE')]
===== sent Set
$$
X = \Spec(k[t_it_j, i, j \in \{1, \ldots, n\}])
$$
The embedding $k[t_it_j] \subset k[t_1, \ldots, t_n]$
defines a degree $2$ morphism $Y \to X$ which is unramified everywhere
except over the maximal ideal $\mathfrak m = (t_it_j)$
which is a point of codimension $n$ in $X$.
\end{example}

\begin{lemma}
\label{lemma-unramified}
Let $X$ be an integral normal scheme with function field $K$.
 ========

regex_match $X$
nn candidate an integral normal scheme
nn an integral normal scheme
new_annotation [(116801, 116804, 'VAR'), (116808, 116833, 'TYPE')]
===== sent Let $L/K$ be a finite extension. ========

regex_match $L/K$
nn candidate a finite extension
nn a finite extension
new_annotation [(116863, 116868, 'VAR'), (116872, 116890, 'TYPE')]
===== sent Let $Y \to X$ be the normalization
of $X$ in $L$. The following are equivalent
\begin{enumerate}
\item $X$ is unramified in $L$ as defined in Section \ref{section-normal},
\item $Y \to X$ is an unramified morphism of schemes,
\item $Y \to X$ is an \'etale morphism of schemes,
\item $Y \to X$ is a finite \'etale morphism of schemes,
\item for $x \in X$ the projection
$Y \times_X \Spec(\mathcal{O}_{X, x}) \to \Spec(\mathcal{O}_{X, x})$
is unramified,
\item same as in (5) but with $\mathcal{O}_{X, x}^h$,
\item same as in (5) but with $\mathcal{O}_{X, x}^{sh}$,
\item for $x \in X$ the scheme theoretic fibre $Y_x$
is \'etale over $x$ of degree $\geq [L : K]$.
\end{enumerate}
If $L/K$ is Galois with Galois group $G$, then these are also
equivalent to
\begin{enumerate}
\item[(9)] for $y \in Y$ the group
$I_y = \{g \in G \mid g(y) = y\text{ and }
g \bmod \mathfrak m_y = \text{id}_{\kappa(y)}\}$ is trivial.
 ========

regex_match $Y \to X$
nn candidate $Y
nn candidate the normalization
nn the normalization
new_annotation [(116896, 116905, 'VAR'), (116909, 116926, 'TYPE')]
===== sent Let $L^{sh} = L \otimes_K K_x^{sh}$ where $K_x^{sh}$ is the fraction field
of $\mathcal{O}_{X, x}^{sh}$. Then $L^{sh} = \prod_{i = 1, \ldots, n} L_i$
with $L_i/K_x^{sh}$ finite separable. ========

new_annotation []
===== sent \begin{lemma}
\label{lemma-structure-decomposition}
Let $A$ be a discrete valuation ring with fraction field $K$.
 ========

regex_match $A$
nn candidate a discrete valuation ring
nn a discrete valuation ring
new_annotation [(122395, 122398, 'VAR'), (122402, 122427, 'TYPE')]
===== sent Let $L/K$ be a (possibly infinite) Galois extension.
 ========

regex_match $L/K$
nn candidate a (possibly infinite) Galois extension
nn a (possibly infinite) Galois extension
new_annotation [(122457, 122462, 'VAR'), (122466, 122504, 'TYPE')]
===== sent Let $B$ be the integral closure of $A$ in $L$.
Let $\mathfrak m$ be a maximal ideal of $B$.
Let $G = \text{Gal}(L/K)$,
$D = \{\sigma \in G \mid \sigma(\mathfrak m) = \mathfrak m\}$, and
$I = \{\sigma \in D \mid \sigma \bmod \mathfrak m =
\text{id}_{\kappa(\mathfrak ========

regex_match $B$
nn candidate the integral closure
nn the integral closure
new_annotation [(122510, 122513, 'VAR'), (122517, 122537, 'TYPE')]
===== sent Let $B$ be the integral closure of $A$ in $L$.
Let $\mathfrak m$ be a maximal ideal of $B$.
Let $G = \text{Gal}(L/K)$,
$D = \{\sigma \in G \mid \sigma(\mathfrak m) = \mathfrak m\}$, and
$I = \{\sigma \in D \mid \sigma \bmod \mathfrak m =
\text{id}_{\kappa(\mathfrak ========

regex_match $\mathfrak m$
nn candidate a maximal ideal
nn a maximal ideal
new_annotation [(122557, 122570, 'VAR'), (122574, 122589, 'TYPE')]
===== sent Let $B$ be the integral closure of $A$ in $L$.
Let $\mathfrak m$ be a maximal ideal of $B$.
Let $G = \text{Gal}(L/K)$,
$D = \{\sigma \in G \mid \sigma(\mathfrak m) = \mathfrak m\}$, and
$I = \{\sigma \in D \mid \sigma \bmod \mathfrak m =
\text{id}_{\kappa(\mathfrak ========

new_annotation []
===== sent To construct the second short exact sequence let $\Lambda$ be the set
of finite Galois subextensions, i.e., $\lambda \in \Lambda$ corresponds
to $L/L_\lambda/K$. Set $G_\lambda = \text{Gal}(L_\lambda/K)$.
Recall that $G_\lambda$ is an inverse system of finite groups with surjective
transition maps and that $G = \lim_{\lambda \in \Lambda} G_\lambda$, see
Fields, Lemma \ref{fields-lemma-infinite-galois-limit}.
We let $B_\lambda$ be the integral closure of $A$ in $L_\lambda$.
 ========

regex_match $\Lambda$
nn candidate the set
nn the set
new_annotation [(124196, 124205, 'VAR'), (124209, 124216, 'TYPE')]
===== sent To construct the second short exact sequence let $\Lambda$ be the set
of finite Galois subextensions, i.e., $\lambda \in \Lambda$ corresponds
to $L/L_\lambda/K$. Set $G_\lambda = \text{Gal}(L_\lambda/K)$.
Recall that $G_\lambda$ is an inverse system of finite groups with surjective
transition maps and that $G = \lim_{\lambda \in \Lambda} G_\lambda$, see
Fields, Lemma \ref{fields-lemma-infinite-galois-limit}.
We let $B_\lambda$ be the integral closure of $A$ in $L_\lambda$.
 ========

regex_match $B_\lambda$
nn candidate the integral closure
nn the integral closure
new_annotation [(124566, 124577, 'VAR'), (124581, 124601, 'TYPE')]
===== sent I_{\lambda'} \ar[r]^-{\theta_{\lambda', can}} &
\mu_{n_{\lambda'}}(\kappa(\mathfrak m_{\lambda'}))
}
$$
see
More on Algebra, Remark \ref{more-algebra-remark-canonical-inertia-character}.

\medskip\noindent
Let $S \subset \mathbf{N}$ be the collection of integers $n_\lambda$.
Since $\Lambda$ is directed, we see that $S$ is multiplicatively directed.
 ========

regex_match $S \subset \mathbf{N}$
nn candidate $S
nn candidate the collection
nn the collection
new_annotation [(126728, 126750, 'VAR'), (126754, 126768, 'TYPE')]
===== sent \end{proof}

\begin{lemma}
\label{lemma-structure-decomposition-separable-closure}
Let $A$ be a discrete valuation ring with fraction field $K$.
Let $K^{sep}$ be a separable closure of $K$.
Let $A^{sep}$ be the integral closure of $A$ in $K^{sep}$.
Let $\mathfrak m^{sep}$ be a maximal ideal of $A^{sep}$.
Let $\mathfrak m = \mathfrak m^{sep} \cap A$, let
$\kappa = A/\mathfrak m$, and let
$\overline{\kappa} = A^{sep}/\mathfrak m^{sep}$.
Then $\overline{\kappa}$ is an algebraic closure of $\kappa$.
Let $G = \text{Gal}(K^{sep}/K)$,
$D = \{\sigma \in G \mid \sigma(\mathfrak m^{sep}) = \mathfrak m^{sep}\}$, and
$I = \{\sigma \in D \mid \sigma \bmod \mathfrak m^{sep} =
\text{id}_{\kappa(\mathfrak ========

regex_match $A$
nn candidate a discrete valuation ring
nn a discrete valuation ring
new_annotation [(129123, 129126, 'VAR'), (129130, 129155, 'TYPE')]
===== sent \end{proof}

\begin{lemma}
\label{lemma-structure-decomposition-separable-closure}
Let $A$ be a discrete valuation ring with fraction field $K$.
Let $K^{sep}$ be a separable closure of $K$.
Let $A^{sep}$ be the integral closure of $A$ in $K^{sep}$.
Let $\mathfrak m^{sep}$ be a maximal ideal of $A^{sep}$.
Let $\mathfrak m = \mathfrak m^{sep} \cap A$, let
$\kappa = A/\mathfrak m$, and let
$\overline{\kappa} = A^{sep}/\mathfrak m^{sep}$.
Then $\overline{\kappa}$ is an algebraic closure of $\kappa$.
Let $G = \text{Gal}(K^{sep}/K)$,
$D = \{\sigma \in G \mid \sigma(\mathfrak m^{sep}) = \mathfrak m^{sep}\}$, and
$I = \{\sigma \in D \mid \sigma \bmod \mathfrak m^{sep} =
\text{id}_{\kappa(\mathfrak ========

regex_match $K^{sep}$
nn candidate a separable closure
nn a separable closure
new_annotation [(129185, 129194, 'VAR'), (129198, 129217, 'TYPE')]
===== sent \end{proof}

\begin{lemma}
\label{lemma-structure-decomposition-separable-closure}
Let $A$ be a discrete valuation ring with fraction field $K$.
Let $K^{sep}$ be a separable closure of $K$.
Let $A^{sep}$ be the integral closure of $A$ in $K^{sep}$.
Let $\mathfrak m^{sep}$ be a maximal ideal of $A^{sep}$.
Let $\mathfrak m = \mathfrak m^{sep} \cap A$, let
$\kappa = A/\mathfrak m$, and let
$\overline{\kappa} = A^{sep}/\mathfrak m^{sep}$.
Then $\overline{\kappa}$ is an algebraic closure of $\kappa$.
Let $G = \text{Gal}(K^{sep}/K)$,
$D = \{\sigma \in G \mid \sigma(\mathfrak m^{sep}) = \mathfrak m^{sep}\}$, and
$I = \{\sigma \in D \mid \sigma \bmod \mathfrak m^{sep} =
\text{id}_{\kappa(\mathfrak ========

regex_match $A^{sep}$
nn candidate the integral closure
nn the integral closure
new_annotation [(129230, 129239, 'VAR'), (129243, 129263, 'TYPE')]
===== sent \end{proof}

\begin{lemma}
\label{lemma-structure-decomposition-separable-closure}
Let $A$ be a discrete valuation ring with fraction field $K$.
Let $K^{sep}$ be a separable closure of $K$.
Let $A^{sep}$ be the integral closure of $A$ in $K^{sep}$.
Let $\mathfrak m^{sep}$ be a maximal ideal of $A^{sep}$.
Let $\mathfrak m = \mathfrak m^{sep} \cap A$, let
$\kappa = A/\mathfrak m$, and let
$\overline{\kappa} = A^{sep}/\mathfrak m^{sep}$.
Then $\overline{\kappa}$ is an algebraic closure of $\kappa$.
Let $G = \text{Gal}(K^{sep}/K)$,
$D = \{\sigma \in G \mid \sigma(\mathfrak m^{sep}) = \mathfrak m^{sep}\}$, and
$I = \{\sigma \in D \mid \sigma \bmod \mathfrak m^{sep} =
\text{id}_{\kappa(\mathfrak ========

regex_match $\mathfrak m^{sep}$
nn candidate a maximal ideal
nn a maximal ideal
new_annotation [(129289, 129308, 'VAR'), (129312, 129327, 'TYPE')]
===== sent \end{proof}

\begin{lemma}
\label{lemma-structure-decomposition-separable-closure}
Let $A$ be a discrete valuation ring with fraction field $K$.
Let $K^{sep}$ be a separable closure of $K$.
Let $A^{sep}$ be the integral closure of $A$ in $K^{sep}$.
Let $\mathfrak m^{sep}$ be a maximal ideal of $A^{sep}$.
Let $\mathfrak m = \mathfrak m^{sep} \cap A$, let
$\kappa = A/\mathfrak m$, and let
$\overline{\kappa} = A^{sep}/\mathfrak m^{sep}$.
Then $\overline{\kappa}$ is an algebraic closure of $\kappa$.
Let $G = \text{Gal}(K^{sep}/K)$,
$D = \{\sigma \in G \mid \sigma(\mathfrak m^{sep}) = \mathfrak m^{sep}\}$, and
$I = \{\sigma \in D \mid \sigma \bmod \mathfrak m^{sep} =
\text{id}_{\kappa(\mathfrak ========

new_annotation []
===== sent \end{proof}

\begin{lemma}
\label{lemma-structure-decomposition-separable-closure}
Let $A$ be a discrete valuation ring with fraction field $K$.
Let $K^{sep}$ be a separable closure of $K$.
Let $A^{sep}$ be the integral closure of $A$ in $K^{sep}$.
Let $\mathfrak m^{sep}$ be a maximal ideal of $A^{sep}$.
Let $\mathfrak m = \mathfrak m^{sep} \cap A$, let
$\kappa = A/\mathfrak m$, and let
$\overline{\kappa} = A^{sep}/\mathfrak m^{sep}$.
Then $\overline{\kappa}$ is an algebraic closure of $\kappa$.
Let $G = \text{Gal}(K^{sep}/K)$,
$D = \{\sigma \in G \mid \sigma(\mathfrak m^{sep}) = \mathfrak m^{sep}\}$, and
$I = \{\sigma \in D \mid \sigma \bmod \mathfrak m^{sep} =
\text{id}_{\kappa(\mathfrak ========

new_annotation []
===== sent Then the only other assertion that needs a proof is the fact that
with $S$ as in Lemma \ref{lemma-structure-decomposition} the
limit $\lim_{n \in S} \mu_n(\overline{\kappa})$ is equal to
$\lim_{n\text{ prime to }p} \mu_n(\kappa^{sep})$. To see this
it suffices to show that every integer $n$ prime to $p$
divides an element of $S$.
Let $\pi \in A$ be a uniformizer and consider the splitting
field $L$ of the polynomial $X^n - \pi$. ========

regex_match $\pi \in A$
nn candidate a uniformizer
nn a uniformizer
new_annotation [(131335, 131346, 'VAR'), (131350, 131363, 'TYPE')]
===== sent In this section we work out what happens when comparing the
fundamental group of a scheme $X$ over a field $k$ with the
fundamental group of $X_{\overline{k}}$ where $\overline{k}$
is the algebraic closure of $k$.

\begin{lemma}
\label{lemma-limit}
Let $I$ be a directed set. ========

regex_match $I$
nn candidate a directed set
nn a directed set
new_annotation [(132323, 132326, 'VAR'), (132330, 132344, 'TYPE')]
===== sent Let $X_i$ be an
inverse system of quasi-compact and quasi-separated schemes
over $I$ with affine transition morphisms.
 ========

new_annotation []
===== sent Let ========

new_annotation []
===== sent \end{proof}

\begin{lemma}
\label{lemma-perfection}
Let $k$ be a field with perfection $k^{perf}$. Let $X$ be a connected scheme
over $k$. Then $X_{k^{perf}}$ is connected and
$\pi_1(X_{k^{perf}}) \to \pi_1(X)$ is an isomorphism.
 ========

regex_match $k$
nn candidate a field
nn a field
new_annotation [(133146, 133149, 'VAR'), (133153, 133160, 'TYPE')]
===== sent \end{proof}

\begin{lemma}
\label{lemma-perfection}
Let $k$ be a field with perfection $k^{perf}$. Let $X$ be a connected scheme
over $k$. Then $X_{k^{perf}}$ is connected and
$\pi_1(X_{k^{perf}}) \to \pi_1(X)$ is an isomorphism.
 ========

regex_match $X$
nn candidate a connected scheme
nn a connected scheme
new_annotation [(133193, 133196, 'VAR'), (133200, 133218, 'TYPE')]
===== sent See Proposition \ref{proposition-universal-homeomorphism}.
To see that $\Spec(k^{perf}) \to \Spec(k)$ is a universal
homeomorphism you can use
Algebra, Lemma \ref{algebra-lemma-radicial-integral-bijective}.
\end{proof}

\begin{lemma}
\label{lemma-ses-field}
Let $k$ be a field with algebraic closure $\overline{k}$.
Let $X$ be a quasi-compact and quasi-separated scheme over $k$.
If the base change $X_{\overline{k}}$ is connected, then
there is a short exact sequence
$$
1 \to \pi_1(X_{\overline{k}}) \to \pi_1(X) \to \pi_1(\Spec(k)) \to 1
$$
of profinite topological groups.
 ========

regex_match $k$
nn candidate a field
nn a field
new_annotation [(133674, 133677, 'VAR'), (133681, 133688, 'TYPE')]
===== sent See Proposition \ref{proposition-universal-homeomorphism}.
To see that $\Spec(k^{perf}) \to \Spec(k)$ is a universal
homeomorphism you can use
Algebra, Lemma \ref{algebra-lemma-radicial-integral-bijective}.
\end{proof}

\begin{lemma}
\label{lemma-ses-field}
Let $k$ be a field with algebraic closure $\overline{k}$.
Let $X$ be a quasi-compact and quasi-separated scheme over $k$.
If the base change $X_{\overline{k}}$ is connected, then
there is a short exact sequence
$$
1 \to \pi_1(X_{\overline{k}}) \to \pi_1(X) \to \pi_1(\Spec(k)) \to 1
$$
of profinite topological groups.
 ========

regex_match $X$
nn candidate a quasi-compact and quasi-separated scheme
nn a quasi-compact and quasi-separated scheme
new_annotation [(133732, 133735, 'VAR'), (133739, 133781, 'TYPE')]
===== sent Therefore the composition
$\pi_1(X_{\overline{k}}) \to \pi_1(X) \to \pi_1(\Spec(k))$
is the trivial homomorphism by Lemma \ref{lemma-composition-trivial}.

\medskip\noindent
Let $U \to X$ be a finite \'etale morphism with $U$ connected.
 ========

regex_match $U \to X$
nn candidate $U
nn candidate a finite \'etale morphism
nn a finite \'etale morphism
new_annotation [(135029, 135038, 'VAR'), (135042, 135067, 'TYPE')]
===== sent \medskip\noindent
Let $V \to X_{\overline{k}}$ be a finite \'etale cover. ========

regex_match $V \to X_{\overline{k}}$
nn candidate $V
nn candidate a finite \'etale cover
nn a finite \'etale cover
new_annotation [(136309, 136333, 'VAR'), (136337, 136359, 'TYPE')]
===== sent Let $f : X \to S$ be a flat proper morphism of
finite presentation whose
geometric fibres are connected and reduced.
 ========

regex_match $f : X \to S$
nn candidate X \to
nn candidate a flat proper morphism
nn a flat proper morphism
new_annotation [(137878, 137891, 'VAR'), (137895, 137917, 'TYPE')]
===== sent Assume $S$ is connected and let $\overline{s}$
be a geometric point of $S$. Then there is an exact
sequence
$$
\pi_1(X_{\overline{s}}) \to \pi_1(X) \to \pi_1(S) \to 1
$$
of fundamental groups. ========

new_annotation []
===== sent \end{reference}
Let $f : X \to S$ be a proper morphism of schemes.
Let $X \to S' \to S$ be the Stein factorization of $f$, see
More on Morphisms, Theorem
\ref{more-morphisms-theorem-stein-factorization-general}.
If $f$ is of finite presentation, flat, with geometrically
reduced fibres, then $S' \to S$ is finite \'etale.
 ========

regex_match $f : X \to S$
nn candidate X \to
nn candidate a proper morphism
nn a proper morphism
new_annotation [(138384, 138397, 'VAR'), (138401, 138418, 'TYPE')]
===== sent \end{reference}
Let $f : X \to S$ be a proper morphism of schemes.
Let $X \to S' \to S$ be the Stein factorization of $f$, see
More on Morphisms, Theorem
\ref{more-morphisms-theorem-stein-factorization-general}.
If $f$ is of finite presentation, flat, with geometrically
reduced fibres, then $S' \to S$ is finite \'etale.
 ========

regex_match $X \to S' \to S$
nn candidate $X
nn candidate the Stein factorization
nn the Stein factorization
new_annotation [(138435, 138451, 'VAR'), (138455, 138478, 'TYPE')]
===== sent \end{lemma}

\begin{proof}
Let $s \in S$. Set $n$ be the number of connected components of
the geometric fibre $X_{\overline{s}}$. Note that $n < \infty$ as the geometric
fibre of $X \to S$ at $s$ is a proper scheme over a field, hence Noetherian,
hence has a finite number of connected components.
 ========

new_annotation []
===== sent \end{proof}

\begin{proposition}
\label{proposition-first-homotopy-sequence}
Let $f : X \to S$ be a flat proper morphism of finite presentation whose
geometric fibres are connected and reduced. ========

regex_match $f : X \to S$
nn candidate X \to
nn candidate a flat proper morphism
nn a flat proper morphism
new_annotation [(140409, 140422, 'VAR'), (140426, 140448, 'TYPE')]
===== sent Assume $S$ is connected and
let $\overline{s}$ be a geometric point of $S$. Then there is an exact
sequence
$$
\pi_1(X_{\overline{s}}) \to \pi_1(X) \to \pi_1(S) \to 1
$$
of fundamental groups.
 ========

regex_match $\overline{s}$
nn candidate a geometric point
nn a geometric point
new_annotation [(140554, 140568, 'VAR'), (140572, 140589, 'TYPE')]
===== sent \end{proposition}

\begin{proof}
Let $Y \to X$ be a finite \'etale morphism. ========

regex_match $Y \to X$
nn candidate $Y
nn candidate a finite \'etale morphism
nn a finite \'etale morphism
new_annotation [(140752, 140761, 'VAR'), (140765, 140790, 'TYPE')]
===== sent \medskip\noindent
It is immediate that the composition
$\textit{F\'Et}_S \to \textit{F\'Et}_X \to \textit{F\'Et}_{X_{\overline{s}}}$
sends any $U$ to a disjoint union of copies of $X_{\overline{s}}$.
Hence $\pi_1(X_{\overline{s}}) \to \pi_1(X) \to \pi_1(S)$ is trivial
by Lemma \ref{lemma-composition-trivial}.

\medskip\noindent
Let $Y \to X$ be a finite \'etale morphism with $Y$ connected such that
$Y \times_X X_{\overline{s}}$ contains a connected component $Z$
isomorphic to $X_{\overline{s}}$. Consider the Stein factorization $T$
as above. ========

regex_match $Y \to X$
nn candidate $Y
nn candidate a finite \'etale morphism
nn a finite \'etale morphism
new_annotation [(142446, 142455, 'VAR'), (142459, 142484, 'TYPE')]
===== sent Let $\overline{t} \in T_{\overline{s}}$ be the point corresponding
to the fibre $Z$. Observe that $T$ is connected (as the image of a connected
scheme) and by the surjectivity above $T \times_S X$ is connected.
 ========

regex_match $\overline{t} \in T_{\overline{s}}$
nn candidate $\overline{t
nn candidate the point
nn the point
new_annotation [(142664, 142699, 'VAR'), (142703, 142712, 'TYPE')]
===== sent Now consider the factorization
$$
\pi : Y \longrightarrow T \times_S X
$$
Let $\overline{x} \in X_{\overline{s}}$ be any closed point. ========

new_annotation []
===== sent Let $f : X \to S$ be a proper morphism of schemes
with geometrically connected fibres.
 ========

regex_match $f : X \to S$
nn candidate X \to
nn candidate a proper morphism
nn a proper morphism
new_annotation [(143939, 143952, 'VAR'), (143956, 143973, 'TYPE')]
===== sent Let $s' \leadsto s$ be a specialization of points in $S$.
Let $\overline{s}$ and $\overline{s}'$ be geometric points
lying over $s$ and $s'$. ========

regex_match $s' \leadsto s$
nn candidate s
nn candidate a specialization
nn a specialization
new_annotation [(144026, 144041, 'VAR'), (144045, 144061, 'TYPE')]
===== sent Let $s' \leadsto s$ be a specialization of points in $S$.
Let $\overline{s}$ and $\overline{s}'$ be geometric points
lying over $s$ and $s'$. ========

new_annotation []
===== sent Let $A$ be the
strict henselization of $\mathcal{O}_{S, s}$ with respect to
$\kappa(s) \subset \kappa(s)^{sep} \subset \kappa(\overline{s})$, see
Algebra, Definition \ref{algebra-definition-henselization}.
 ========

new_annotation []
===== sent Let $t' \leadsto t$ be a specialization of points in $T$
and consider a specialization map
$sp : \pi_1(Y_{\overline{t}'}) \to \pi_1(Y_{\overline{t}})$ as above.
 ========

regex_match $t' \leadsto t$
nn candidate $t
nn candidate a specialization
nn a specialization
new_annotation [(146005, 146020, 'VAR'), (146024, 146040, 'TYPE')]
===== sent Then there is a commutative diagram
$$
\xymatrix{
\pi_1(Y_{\overline{t}'}) \ar[r]_{sp} \ar[d] & \pi_1(Y_{\overline{t}}) \ar[d] \\
\pi_1(X_{\overline{s}'}) \ar[r]^{sp} & \pi_1(X_{\overline{s}})
}
$$
of specialization maps where $\overline{s}$ and $\overline{s}'$
are the images of $\overline{t}$ and $\overline{t}'$.
\end{lemma}

\begin{proof}
Let $B$ be the strict henselization of $\mathcal{O}_{T, t}$ with respect to
$\kappa(t) \subset \kappa(t)^{sep} \subset \kappa(\overline{t})$.
Pick $\psi : \overline{t}' \to \Spec(B)$ lifting $\overline{t}' \to T$
as in the construction of the specialization map.
 ========

regex_match $B$
nn candidate the strict henselization
nn the strict henselization
new_annotation [(146509, 146512, 'VAR'), (146516, 146540, 'TYPE')]
===== sent Let $s$ and $s'$ denote the images of $t$ and $t'$ in $S$.
Let $A$ be the strict henselization of $\mathcal{O}_{S, s}$
with respect to
$\kappa(s) \subset \kappa(s)^{sep} \subset \kappa(\overline{s})$.
Since $\kappa(\overline{s}) = \kappa(\overline{t})$,
by the functoriality of strict henselization
(Algebra, Lemma \ref{algebra-lemma-strictly-henselian-functorial})
we obtain a ring map $A \to B$ fitting into the commutative diagram
$$
\xymatrix{
\overline{t}' \ar[r]_-\psi \ar[d] & \Spec(B) \ar[d] \ar[r] & T \ar[d] \\
\overline{s}' \ar[r]^-\varphi & \Spec(A) \ar[r] & S
}
 ========

new_annotation []
===== sent Let $s$ and $s'$ denote the images of $t$ and $t'$ in $S$.
Let $A$ be the strict henselization of $\mathcal{O}_{S, s}$
with respect to
$\kappa(s) \subset \kappa(s)^{sep} \subset \kappa(\overline{s})$.
Since $\kappa(\overline{s}) = \kappa(\overline{t})$,
by the functoriality of strict henselization
(Algebra, Lemma \ref{algebra-lemma-strictly-henselian-functorial})
we obtain a ring map $A \to B$ fitting into the commutative diagram
$$
\xymatrix{
\overline{t}' \ar[r]_-\psi \ar[d] & \Spec(B) \ar[d] \ar[r] & T \ar[d] \\
\overline{s}' \ar[r]^-\varphi & \Spec(A) \ar[r] & S
}
 ========

regex_match $A$
nn candidate the strict henselization
nn the strict henselization
new_annotation [(146831, 146834, 'VAR'), (146838, 146862, 'TYPE')]
===== sent \end{proof}

\begin{lemma}
\label{lemma-specialization-map-composition}
Let $f : X \to S$ be a proper morphism with geometrically connected fibres.
 ========

regex_match $f : X \to S$
nn candidate X \to
nn candidate a proper morphism
nn a proper morphism
new_annotation [(147852, 147865, 'VAR'), (147869, 147886, 'TYPE')]
===== sent Let $s'' \leadsto s' \leadsto s$ be specializations of points of $S$.
A composition of specialization maps
$\pi_1(X_{\overline{s}''}) \to \pi_1(X_{\overline{s}'}) \to
\pi_1(X_{\overline{s}})$ is a specialization map
$\pi_1(X_{\overline{s}''}) \to \pi_1(X_{\overline{s}})$.
\end{lemma}

\begin{proof}
Let $\mathcal{O}_{S, s} \to A$ be the strict henselization
constructed using $\kappa(s) \to \kappa(\overline{s})$.
Let $A \to \kappa(\overline{s}')$ be the map used to construct
the first specialization map. ========

new_annotation []
===== sent Let $s'' \leadsto s' \leadsto s$ be specializations of points of $S$.
A composition of specialization maps
$\pi_1(X_{\overline{s}''}) \to \pi_1(X_{\overline{s}'}) \to
\pi_1(X_{\overline{s}})$ is a specialization map
$\pi_1(X_{\overline{s}''}) \to \pi_1(X_{\overline{s}})$.
\end{lemma}

\begin{proof}
Let $\mathcal{O}_{S, s} \to A$ be the strict henselization
constructed using $\kappa(s) \to \kappa(\overline{s})$.
Let $A \to \kappa(\overline{s}')$ be the map used to construct
the first specialization map. ========

regex_match $\mathcal{O}_{S, s} \to A$
nn candidate $\mathcal{O}_{S
nn candidate the strict henselization
nn the strict henselization
new_annotation [(148228, 148254, 'VAR'), (148258, 148282, 'TYPE')]
===== sent Let $s'' \leadsto s' \leadsto s$ be specializations of points of $S$.
A composition of specialization maps
$\pi_1(X_{\overline{s}''}) \to \pi_1(X_{\overline{s}'}) \to
\pi_1(X_{\overline{s}})$ is a specialization map
$\pi_1(X_{\overline{s}''}) \to \pi_1(X_{\overline{s}})$.
\end{lemma}

\begin{proof}
Let $\mathcal{O}_{S, s} \to A$ be the strict henselization
constructed using $\kappa(s) \to \kappa(\overline{s})$.
Let $A \to \kappa(\overline{s}')$ be the map used to construct
the first specialization map. ========

regex_match $A \to \kappa(\overline{s}')$
nn candidate the map
nn the map
new_annotation [(148343, 148372, 'VAR'), (148376, 148383, 'TYPE')]
===== sent Let $\mathcal{O}_{S, s'} \to A'$
be the strict henselization constructed using
$\kappa(s') \subset \kappa(\overline{s}')$.
By functoriality of strict henselization, there is a map
$A \to A'$ such that the composition with $A' \to \kappa(\overline{s}')$
is the given map
(Algebra, Lemma \ref{algebra-lemma-map-into-henselian-colimit}).
 ========

new_annotation []
===== sent Next, let $A' \to \kappa(\overline{s}'')$ be the map used to
construct the second specialization map. ========

regex_match $A' \to \kappa(\overline{s}'')$
nn candidate the map
nn the map
new_annotation [(148777, 148808, 'VAR'), (148812, 148819, 'TYPE')]
===== sent Then it is clear that
the composition of the first and second specialization maps
is the specialization map
$\pi_1(X_{\overline{s}''}) \to \pi_1(X_{\overline{s}})$
constructed using $A \to A' \to \kappa(\overline{s}'')$.
\end{proof}

\noindent
Let $X \to S$ be a proper morphism with geometrically connected fibres.
 ========

regex_match $X \to S$
nn candidate $X
nn candidate a proper morphism
nn a proper morphism
new_annotation [(149117, 149126, 'VAR'), (149130, 149147, 'TYPE')]
===== sent Let $R$ be a strictly henselian valuation ring with algebraically
closed fraction field and let $\Spec(R) \to ========

regex_match $R$
nn candidate a strictly henselian valuation ring
nn a strictly henselian valuation ring
new_annotation [(149189, 149192, 'VAR'), (149196, 149231, 'TYPE')]
===== sent Let $R$ be a strictly henselian valuation ring with algebraically
closed fraction field and let $\Spec(R) \to ========

new_annotation []
===== sent Let $\eta, s \in \Spec(R)$ be the generic and closed point.
 ========

regex_match $\eta, s \in \Spec(R)$
nn candidate $\eta
nn candidate the generic and closed point
nn the generic and closed point
new_annotation [(149317, 149339, 'VAR'), (149343, 149371, 'TYPE')]
===== sent \begin{lemma}
\label{lemma-specialization-map-valuation-ring}
Let $f : X \to S$ be a proper morphism with geometrically connected fibres.
 ========

regex_match $f : X \to S$
nn candidate X \to
nn candidate a proper morphism
nn a proper morphism
new_annotation [(149654, 149667, 'VAR'), (149671, 149688, 'TYPE')]
===== sent Let $s' \leadsto s$ be a specialization of points of $S$ and let
$sp : \pi_1(X_{\overline{s}'}) \to \pi_1(X_{\overline{s}})$
be a specialization map. ========

regex_match $s' \leadsto s$
nn candidate s
nn candidate a specialization
nn a specialization
new_annotation [(149730, 149745, 'VAR'), (149749, 149765, 'TYPE')]
===== sent \end{lemma}

\begin{proof}
Let $\mathcal{O}_{S, s} \to A$ be the strict henselization
constructed using $\kappa(s) \to \kappa(\overline{s})$.
Let $A \to \kappa(\overline{s}')$ be the map used to construct $sp$.
Let $R \subset \kappa(\overline{s}')$ be a valuation ring with
fraction field $\kappa(\overline{s}')$ dominating the image of $A$.
 ========

regex_match $\mathcal{O}_{S, s} \to A$
nn candidate $\mathcal{O}_{S
nn candidate the strict henselization
nn the strict henselization
new_annotation [(150069, 150095, 'VAR'), (150099, 150123, 'TYPE')]
===== sent \end{lemma}

\begin{proof}
Let $\mathcal{O}_{S, s} \to A$ be the strict henselization
constructed using $\kappa(s) \to \kappa(\overline{s})$.
Let $A \to \kappa(\overline{s}')$ be the map used to construct $sp$.
Let $R \subset \kappa(\overline{s}')$ be a valuation ring with
fraction field $\kappa(\overline{s}')$ dominating the image of $A$.
 ========

regex_match $A \to \kappa(\overline{s}')$
nn candidate the map
nn the map
new_annotation [(150184, 150213, 'VAR'), (150217, 150224, 'TYPE')]
===== sent \end{lemma}

\begin{proof}
Let $\mathcal{O}_{S, s} \to A$ be the strict henselization
constructed using $\kappa(s) \to \kappa(\overline{s})$.
Let $A \to \kappa(\overline{s}')$ be the map used to construct $sp$.
Let $R \subset \kappa(\overline{s}')$ be a valuation ring with
fraction field $\kappa(\overline{s}')$ dominating the image of $A$.
 ========

regex_match $R \subset \kappa(\overline{s}')$
nn candidate $R
nn candidate a valuation ring
nn a valuation ring
new_annotation [(150253, 150286, 'VAR'), (150290, 150306, 'TYPE')]
===== sent \end{proof}

\noindent
Let $X \to S$ be a proper morphism with geometrically connected fibres.
 ========

regex_match $X \to S$
nn candidate $X
nn candidate a proper morphism
nn a proper morphism
new_annotation [(150668, 150677, 'VAR'), (150681, 150698, 'TYPE')]
===== sent Let $R$ be a strictly henselian discrete valuation ring and let
$\Spec(R) \to S$ be a morphism. ========

regex_match $R$
nn candidate a strictly henselian discrete valuation ring
nn a strictly henselian discrete valuation ring
new_annotation [(150740, 150743, 'VAR'), (150747, 150791, 'TYPE')]
===== sent Let $\eta, s \in \Spec(R)$ be the
generic and closed point. ========

new_annotation []
===== sent \begin{lemma}
\label{lemma-specialization-map-discrete-valuation-ring}
Let $f : X \to S$ be a proper morphism with geometrically connected fibres.
 ========

regex_match $f : X \to S$
nn candidate X \to
nn candidate a proper morphism
nn a proper morphism
new_annotation [(151177, 151190, 'VAR'), (151194, 151211, 'TYPE')]
===== sent Let $s' \leadsto s$ be a specialization of points of $S$ and let
$sp : \pi_1(X_{\overline{s}'}) \to \pi_1(X_{\overline{s}})$
be a specialization map. ========

regex_match $s' \leadsto s$
nn candidate s
nn candidate a specialization
nn a specialization
new_annotation [(151253, 151268, 'VAR'), (151272, 151288, 'TYPE')]
===== sent \end{lemma}

\begin{proof}
Let $\mathcal{O}_{S, s} \to A$ be the strict henselization
constructed using $\kappa(s) \to \kappa(\overline{s})$.
Let $A \to \kappa(\overline{s}')$ be the map used to construct $sp$.
Let $R \subset \kappa(\overline{s}')$ be a discrete valuation ring
dominating the image of $A$, see Algebra, Lemma \ref{algebra-lemma-exists-dvr}.
Choose a diagram of fields
$$
\xymatrix{
\kappa(\overline{s}) \ar[r] & k \\
 ========

regex_match $\mathcal{O}_{S, s} \to A$
nn candidate $\mathcal{O}_{S
nn candidate the strict henselization
nn the strict henselization
new_annotation [(151582, 151608, 'VAR'), (151612, 151636, 'TYPE')]
===== sent \end{lemma}

\begin{proof}
Let $\mathcal{O}_{S, s} \to A$ be the strict henselization
constructed using $\kappa(s) \to \kappa(\overline{s})$.
Let $A \to \kappa(\overline{s}')$ be the map used to construct $sp$.
Let $R \subset \kappa(\overline{s}')$ be a discrete valuation ring
dominating the image of $A$, see Algebra, Lemma \ref{algebra-lemma-exists-dvr}.
Choose a diagram of fields
$$
\xymatrix{
\kappa(\overline{s}) \ar[r] & k \\
 ========

regex_match $A \to \kappa(\overline{s}')$
nn candidate the map
nn the map
new_annotation [(151697, 151726, 'VAR'), (151730, 151737, 'TYPE')]
===== sent \end{lemma}

\begin{proof}
Let $\mathcal{O}_{S, s} \to A$ be the strict henselization
constructed using $\kappa(s) \to \kappa(\overline{s})$.
Let $A \to \kappa(\overline{s}')$ be the map used to construct $sp$.
Let $R \subset \kappa(\overline{s}')$ be a discrete valuation ring
dominating the image of $A$, see Algebra, Lemma \ref{algebra-lemma-exists-dvr}.
Choose a diagram of fields
$$
\xymatrix{
\kappa(\overline{s}) \ar[r] & k \\
 ========

regex_match $R \subset \kappa(\overline{s}')$
nn candidate $R
nn candidate a discrete valuation ring
nn a discrete valuation ring
new_annotation [(151766, 151799, 'VAR'), (151803, 151828, 'TYPE')]
===== sent Let $R^{sh}$ be the strict
henselization of $R$ constructed using $R \to k$. Then
$R^{sh}$ is a discrete valuation ring by
More on Algebra, Lemma \ref{more-algebra-lemma-henselization-dvr}.
Denote $\eta, o$ the generic and closed point of $\Spec(R^{sh})$.
 ========

regex_match $R^{sh}$
nn candidate the strict
henselization
nn the strict
henselization
new_annotation [(152080, 152088, 'VAR'), (152092, 152116, 'TYPE')]
===== sent \begin{situation}
\label{situation-local-lefschetz}
Let $(A, \mathfrak m)$ be a Noetherian local ring and $f \in \mathfrak m$.
We set $X = \Spec(A)$ and $X_0 = \Spec(A/fA)$ and we
let $U = X \setminus \{\mathfrak m\}$ and
$U_0 = X_0 \setminus \{\mathfrak m\}$ be the punctured spectrum of
$A$ and $A/fA$.
\end{situation}

\noindent
Recall that for a scheme $X$ the category of schemes finite
\'etale over $X$ is denoted $\textit{F\'Et}_X$, see
Section \ref{section-finite-etale}.
In Situation \ref{situation-local-lefschetz}
we will study the base change functors
$$
\xymatrix{
\textit{F\'Et}_X \ar[d] \ar[r] & \textit{F\'Et}_U \ar[d] \\
 ========

regex_match $(A, \mathfrak m)$
nn candidate \mathfrak
nn candidate a Noetherian local ring
nn a Noetherian local ring
new_annotation [(153116, 153134, 'VAR'), (153138, 153161, 'TYPE')]
===== sent \begin{situation}
\label{situation-local-lefschetz}
Let $(A, \mathfrak m)$ be a Noetherian local ring and $f \in \mathfrak m$.
We set $X = \Spec(A)$ and $X_0 = \Spec(A/fA)$ and we
let $U = X \setminus \{\mathfrak m\}$ and
$U_0 = X_0 \setminus \{\mathfrak m\}$ be the punctured spectrum of
$A$ and $A/fA$.
\end{situation}

\noindent
Recall that for a scheme $X$ the category of schemes finite
\'etale over $X$ is denoted $\textit{F\'Et}_X$, see
Section \ref{section-finite-etale}.
In Situation \ref{situation-local-lefschetz}
we will study the base change functors
$$
\xymatrix{
\textit{F\'Et}_X \ar[d] \ar[r] & \textit{F\'Et}_U \ar[d] \\
 ========

new_annotation []
===== sent \end{lemma}

\begin{proof}
Let $a, b : V \to W$ be two morphisms of schemes finite \'etale over $U$
whose restriction to $U_0$ are the same. ========

new_annotation []
===== sent In Situation \ref{situation-local-lefschetz}. Let $V \to U$ be a finite
morphism.   ========

regex_match $V \to U$
nn candidate $V
nn candidate a finite
morphism
nn a finite
morphism
new_annotation [(155088, 155097, 'VAR'), (155101, 155118, 'TYPE')]
===== sent Let $A^\wedge$ be the $\mathfrak m$-adic completion of $A$,
let $X' = \Spec(A^\wedge)$ and let $U'$ and $V'$ be the base changes of
$U$ and $V$ to $X'$. ========

regex_match $A^\wedge$
nn candidate the $\mathfrak m$-adic completion
nn the $\mathfrak m$-adic completion
new_annotation [(155125, 155135, 'VAR'), (155139, 155172, 'TYPE')]
===== sent Let $A^\wedge$ be the $\mathfrak m$-adic completion of $A$,
let $X' = \Spec(A^\wedge)$ and let $U'$ and $V'$ be the base changes of
$U$ and $V$ to $X'$. ========

new_annotation []
===== sent Let $A^\wedge$ be the $\mathfrak m$-adic completion of $A$,
let $X' = \Spec(A^\wedge)$ and let $U'$ and $V'$ be the base changes of
$U$ and $V$ to $X'$. ========

new_annotation []
===== sent Let $A^\wedge$ be the $\mathfrak m$-adic completion of $A$,
let $X' = \Spec(A^\wedge)$ and let $U'$ and $U'_0$ be the base changes of
$U$ and $U_0$ to $X'$. ========

regex_match $A^\wedge$
nn candidate the $\mathfrak m$-adic completion
nn the $\mathfrak m$-adic completion
new_annotation [(156862, 156872, 'VAR'), (156876, 156909, 'TYPE')]
===== sent Let $A^\wedge$ be the $\mathfrak m$-adic completion of $A$,
let $X' = \Spec(A^\wedge)$ and let $U'$ and $U'_0$ be the base changes of
$U$ and $U_0$ to $X'$. ========

new_annotation []
===== sent Let $A^\wedge$ be the $\mathfrak m$-adic completion of $A$,
let $X' = \Spec(A^\wedge)$ and let $U'$ and $U'_0$ be the base changes of
$U$ and $U_0$ to $X'$. ========

new_annotation []
===== sent Let $Y_1 \subset Y$ be the open and closed subscheme corresponding to $e$.
Then $Y_1 \times_X X' = s'(X')$ which implies that $Y_1 \to X$ is
an isomorphism (by faithfully flat descent) and gives the desired section.

\medskip\noindent
 ========

regex_match $Y_1 \subset Y$
nn candidate the open and closed subscheme
nn the open and closed subscheme
new_annotation [(159139, 159154, 'VAR'), (159158, 159187, 'TYPE')]
===== sent Here we use that $s'$ is
a lift of $s'_0$. Namely, let $Y_{0, 1} \subset Y_0 = Y \times_X X_0$
be the closure of $s_0(U_0) \subset V_0 = Y_0 \times_{X_0} U_0$.
As $X' \to X$ is flat, the base change $Y'_{0, 1} \subset Y'_0$
is the closure of $s'_0(U'_0)$ which is equal to $X'_0 \subset Y'_0$
(see Morphisms, Lemma
\ref{morphisms-lemma-flat-base-change-scheme-theoretic-image}).
 ========

new_annotation []
===== sent Since $Y'_0 \to Y_0$ is submersive
(Morphisms, Lemma \ref{morphisms-lemma-fpqc-quotient-topology})
we conclude that $Y_{0, 1}$ is open and closed in $Y_0$.
Let $e_0 \in B/fB$ be the corresponding idempotent.
 ========

regex_match $e_0 \in B/fB$
nn candidate $e_0 \in B
nn candidate the corresponding idempotent
nn the corresponding idempotent
new_annotation [(159956, 159970, 'VAR'), (159974, 160002, 'TYPE')]
===== sent For $n \geq 1$ let $U_n$ be the punctured spectrum of $A/f^{n + 1}A$
and let $V_n \to U_n$ be the base change of $V \to U$. By
\'Etale Morphisms, Theorem \ref{etale-theorem-remarkable-equivalence}
we conclude that there is a unique decomposition
$V_n = U_n \amalg V'_n$
into open and closed subschemes whose base change to $U_0$ recovers
the given decomposition.

 ========

regex_match $U_n$
nn candidate the punctured spectrum
nn the punctured spectrum
new_annotation [(161431, 161436, 'VAR'), (161440, 161462, 'TYPE')]
===== sent For $n \geq 1$ let $U_n$ be the punctured spectrum of $A/f^{n + 1}A$
and let $V_n \to U_n$ be the base change of $V \to U$. By
\'Etale Morphisms, Theorem \ref{etale-theorem-remarkable-equivalence}
we conclude that there is a unique decomposition
$V_n = U_n \amalg V'_n$
into open and closed subschemes whose base change to $U_0$ recovers
the given decomposition.

 ========

regex_match $V_n \to U_n$
nn candidate the base change
nn the base change
new_annotation [(161489, 161502, 'VAR'), (161506, 161521, 'TYPE')]
===== sent \end{proof}



























\section{Purity in local case, I}
\label{section-local-purity}

\noindent
Let $(A, \mathfrak m)$ be a Noetherian local ring. ========

regex_match $(A, \mathfrak m)$
nn candidate \mathfrak
nn candidate a Noetherian local ring
nn a Noetherian local ring
new_annotation [(166930, 166948, 'VAR'), (166952, 166975, 'TYPE')]
===== sent Set $X = \Spec(A)$
and let $U = X \setminus \{\mathfrak m\}$ be the punctured spectrum.
 ========

regex_match $U = X \setminus \{\mathfrak m\}$
nn candidate $U = X \setminus
nn candidate the punctured spectrum
nn the punctured spectrum
new_annotation [(167004, 167037, 'VAR'), (167041, 167063, 'TYPE')]
===== sent These results will be useful to proceed by induction on dimension
in the proofs of our main results on local purity, namely,
Lemma \ref{lemma-local-purity} and
Proposition \ref{proposition-purity-complete-intersection}.

\begin{lemma}
\label{lemma-sections-over-punctured-spec}
Let $(A, \mathfrak m)$ be a Noetherian local ring. ========

regex_match $(A, \mathfrak m)$
nn candidate \mathfrak
nn candidate a Noetherian local ring
nn a Noetherian local ring
new_annotation [(167734, 167752, 'VAR'), (167756, 167779, 'TYPE')]
===== sent Set $X = \Spec(A)$
and let $U = X \setminus \{\mathfrak m\}$.
 ========

new_annotation []
===== sent Let $\pi : Y \to X$ be a finite morphism such that
$\text{depth}(\mathcal{O}_{Y, y}) \geq 2$ for all closed points
$y \in Y$.
Then $Y$ is the spectrum of $B = \mathcal{O}_Y(\pi^{-1}(U))$.
\end{lemma}

\begin{proof}
Set $V = \pi^{-1}(U)$ and denote $\pi' : V \to U$ the restriction of $\pi$.
Consider the $\mathcal{O}_X$-module map
$$
\pi_*\mathcal{O}_Y \longrightarrow j_*\pi'_*\mathcal{O}_V
$$
where $j : U \to X$ is the inclusion morphism. ========

regex_match $\pi : Y \to X$
nn candidate $\pi
nn candidate a finite morphism
nn a finite morphism
new_annotation [(167847, 167862, 'VAR'), (167866, 167883, 'TYPE')]
===== sent Let $x \in X$.
If $x \in U$, then the map is an
isomorphism on stalks as $V = Y \times_X U$.
 ========

new_annotation []
===== sent Let $y_1, \ldots, y_n \in Y$ be the points mapping to $x$.
 ========

regex_match $y_1, \ldots, y_n \in Y$
nn candidate y_n
nn candidate the points mapping
nn the points mapping
new_annotation [(168823, 168847, 'VAR'), (168851, 168869, 'TYPE')]
===== sent \end{proof}

\begin{lemma}
\label{lemma-reformulate-purity}
Let $(A, \mathfrak m)$ be a Noetherian local ring. ========

regex_match $(A, \mathfrak m)$
nn candidate \mathfrak
nn candidate a Noetherian local ring
nn a Noetherian local ring
new_annotation [(169161, 169179, 'VAR'), (169183, 169206, 'TYPE')]
===== sent Set $X = \Spec(A)$
and let $U = X \setminus \{\mathfrak m\}$.
 ========

new_annotation []
===== sent Let $V$ be finite \'etale
over $U$. Assume $A$ has depth $\geq 2$. The following are equivalent
\begin{enumerate}
\item $V = Y \times_X U$ for some $Y \to X$ finite \'etale,
\item $B = \Gamma(V, \mathcal{O}_V)$ is finite \'etale over $A$.
 ========

new_annotation []
===== sent Let $x \in X$ be the point
corresponding to $\mathfrak m$.
Let $y \in Y$ be a point mapping to $x$. We claim that
$\text{depth}(\mathcal{O}_{Y, y}) \geq 2$.
This is true because $Y \to X$ is \'etale and hence
$A = \mathcal{O}_{X, x}$ and $\mathcal{O}_{Y, y}$ have
the same depth (Algebra, Lemma \ref{algebra-lemma-apply-grothendieck}).
 ========

regex_match $x \in X$
nn candidate the point
nn the point
new_annotation [(169643, 169652, 'VAR'), (169656, 169665, 'TYPE')]
===== sent Let $x \in X$ be the point
corresponding to $\mathfrak m$.
Let $y \in Y$ be a point mapping to $x$. We claim that
$\text{depth}(\mathcal{O}_{Y, y}) \geq 2$.
This is true because $Y \to X$ is \'etale and hence
$A = \mathcal{O}_{X, x}$ and $\mathcal{O}_{Y, y}$ have
the same depth (Algebra, Lemma \ref{algebra-lemma-apply-grothendieck}).
 ========

regex_match $y \in Y$
nn candidate $y
nn candidate a point mapping
nn a point mapping
new_annotation [(169702, 169711, 'VAR'), (169715, 169730, 'TYPE')]
===== sent \end{proof}

\begin{lemma}
\label{lemma-reformulate-purity-normal}
Let $(A, \mathfrak m)$ be a Noetherian local ring. ========

regex_match $(A, \mathfrak m)$
nn candidate \mathfrak
nn candidate a Noetherian local ring
nn a Noetherian local ring
new_annotation [(170223, 170241, 'VAR'), (170245, 170268, 'TYPE')]
===== sent Set $X = \Spec(A)$
and let $U = X \setminus \{\mathfrak m\}$. Assume $A$ is normal
of dimension $\geq 2$. The functor
$$
\textit{F\'Et}_U \longrightarrow
\left\{
\begin{matrix}
\text{finite normal }A\text{-algebras }B\text{ such} \\
 ========

new_annotation []
===== sent Let $f \in \mathfrak m$. Then $B_f = \Gamma(V \times_U D(f), \mathcal{O}_V)$
(Properties, Lemma \ref{properties-lemma-invert-f-sections}).
 ========

new_annotation []
===== sent \end{proof}

\begin{lemma}
\label{lemma-purity-and-completion}
Let $(A, \mathfrak m)$ be a Noetherian local ring. ========

regex_match $(A, \mathfrak m)$
nn candidate \mathfrak
nn candidate a Noetherian local ring
nn a Noetherian local ring
new_annotation [(173012, 173030, 'VAR'), (173034, 173057, 'TYPE')]
===== sent Set $X = \Spec(A)$
and let $U = X \setminus \{\mathfrak m\}$.
 ========

new_annotation []
===== sent Let $V$ be finite \'etale over $U$.
Let $A^\wedge$ be the $\mathfrak m$-adic completion of $A$,
let $X' = \Spec(A^\wedge)$ and let $U'$ and $V'$ be the base changes of
$U$ and $V$ to $X'$. The following are equivalent
\begin{enumerate}
\item $V = Y \times_X U$ for some $Y \to X$ finite \'etale, and
\item $V' = Y' \times_{X'} U'$ for some $Y' \to X'$ finite \'etale.
 ========

new_annotation []
===== sent Let $V$ be finite \'etale over $U$.
Let $A^\wedge$ be the $\mathfrak m$-adic completion of $A$,
let $X' = \Spec(A^\wedge)$ and let $U'$ and $V'$ be the base changes of
$U$ and $V$ to $X'$. The following are equivalent
\begin{enumerate}
\item $V = Y \times_X U$ for some $Y \to X$ finite \'etale, and
\item $V' = Y' \times_{X'} U'$ for some $Y' \to X'$ finite \'etale.
 ========

regex_match $A^\wedge$
nn candidate the $\mathfrak m$-adic completion
nn the $\mathfrak m$-adic completion
new_annotation [(173161, 173171, 'VAR'), (173175, 173208, 'TYPE')]
===== sent Let $V$ be finite \'etale over $U$.
Let $A^\wedge$ be the $\mathfrak m$-adic completion of $A$,
let $X' = \Spec(A^\wedge)$ and let $U'$ and $V'$ be the base changes of
$U$ and $V$ to $X'$. The following are equivalent
\begin{enumerate}
\item $V = Y \times_X U$ for some $Y \to X$ finite \'etale, and
\item $V' = Y' \times_{X'} U'$ for some $Y' \to X'$ finite \'etale.
 ========

new_annotation []
===== sent Let $V$ be finite \'etale over $U$.
Let $A^\wedge$ be the $\mathfrak m$-adic completion of $A$,
let $X' = \Spec(A^\wedge)$ and let $U'$ and $V'$ be the base changes of
$U$ and $V$ to $X'$. The following are equivalent
\begin{enumerate}
\item $V = Y \times_X U$ for some $Y \to X$ finite \'etale, and
\item $V' = Y' \times_{X'} U'$ for some $Y' \to X'$ finite \'etale.
 ========

new_annotation []
===== sent \end{enumerate}
\end{lemma}

\begin{proof}
The implication (1) $\Rightarrow$ (2) follows from taking the base change
of a solution $Y \to X$. Let $Y' \to X'$ be as in (2).
 ========

new_annotation []
===== sent In Situation \ref{situation-local-lefschetz}.
Let $V$ be finite \'etale over $U$. Assume
\begin{enumerate}
\item $f$ is a nonzerodivisor,
\item $A$ has depth $\geq 3$,
\item $V_0 = V \times_U U_0$ is equal to $Y_0 \times_{X_0} U_0$
for some $Y_0 \to X_0$ finite \'etale.
 ========

new_annotation []
===== sent \begin{lemma}
\label{lemma-lift-purity-general}
In Situation \ref{situation-local-lefschetz}.
Let $V$ be finite \'etale over $U$. Assume
\begin{enumerate}
\item $f$ is a nonzerodivisor,
\item $H^1_\mathfrak m(A)$ is a finite $A$-module,
\item a power of $f$ annihilates $H^2_\mathfrak m(A)$,
 ========

new_annotation []
===== sent \end{proof}










\section{Purity of branch locus}
\label{section-purity}

\noindent
We will use the discriminant of a finite locally free morphism. See
Discriminants, Section \ref{discriminant-section-discriminant}.

\begin{lemma}
\label{lemma-find-point-codim-1}
Let $(A, \mathfrak m)$ be a Noetherian local ring with $\dim(A) \geq 1$.
Let $f \in \mathfrak m$. Then there exist a $\mathfrak p \in V(f)$ with
$\dim(A_\mathfrak p) = 1$.
\end{lemma}

\begin{proof}
 ========

regex_match $(A, \mathfrak m)$
nn candidate \mathfrak
nn candidate a Noetherian local ring
nn a Noetherian local ring
new_annotation [(176897, 176915, 'VAR'), (176919, 176942, 'TYPE')]
===== sent \end{proof}










\section{Purity of branch locus}
\label{section-purity}

\noindent
We will use the discriminant of a finite locally free morphism. See
Discriminants, Section \ref{discriminant-section-discriminant}.

\begin{lemma}
\label{lemma-find-point-codim-1}
Let $(A, \mathfrak m)$ be a Noetherian local ring with $\dim(A) \geq 1$.
Let $f \in \mathfrak m$. Then there exist a $\mathfrak p \in V(f)$ with
$\dim(A_\mathfrak p) = 1$.
\end{lemma}

\begin{proof}
 ========

new_annotation []
===== sent If $\dim(A) > 1$, then let $Z \subset \Spec(A)$ be an irreducible
component of dimension $> 1$. ========

regex_match $Z \subset \Spec(A)$
nn candidate $Z \subset
nn candidate an irreducible
component
nn an irreducible
component
new_annotation [(177204, 177224, 'VAR'), (177228, 177252, 'TYPE')]
===== sent \end{proof}

\begin{lemma}
\label{lemma-ramification-quasi-finite-flat}
Let $f : X \to Y$ be a morphism of locally Noetherian schemes.
 ========

regex_match $f : X \to Y$
nn candidate X \to
nn candidate a morphism
nn a morphism
new_annotation [(177973, 177986, 'VAR'), (177990, 178000, 'TYPE')]
===== sent Let $x \in X$. Assume
\begin{enumerate}
\item $f$ is flat,
\item $f$ is quasi-finite at $x$,
\item $x$ is not a generic point of an irreducible component of $X$,
\item for specializations $x' \leadsto x$ with
$\dim(\mathcal{O}_{X, x'}) = 1$ our $f$ is unramified at $x'$.
\end{enumerate}
Then $f$ is \'etale at $x$.
\end{lemma}

\begin{proof}
Observe that the set of points where $f$ is unramified is the same as
the set of points where $f$ is \'etale and that this set is open.
 ========

new_annotation []
===== sent \end{proof}

\begin{lemma}
\label{lemma-local-purity}
Let $(A, \mathfrak m)$ be a regular local ring of dimension $d \geq 2$.
Set $X = \Spec(A)$ and $U = X \setminus \{\mathfrak m\}$. Then
\begin{enumerate}
\item the functor $\textit{F\'Et}_X \to \textit{F\'Et}_U$
is essentially surjective,
\item any finite $A \to B$ with $B$ normal which
induces a finite \'etale morphism on punctured spectra is \'etale.
 ========

regex_match $(A, \mathfrak m)$
nn candidate \mathfrak
nn candidate a regular local ring
nn a regular local ring
new_annotation [(180258, 180276, 'VAR'), (180280, 180300, 'TYPE')]
===== sent The case $d \geq 3$. Let $V \to U$ be finite \'etale.
 ========

new_annotation []
===== sent Let $f \in \mathfrak m_A$, $f \not ========

new_annotation []
===== sent \in \mathfrak m_A^2$.
Then $A/fA$ is a regular local ring of dimension $d - 1 \geq 2$, see
Algebra, Lemma \ref{algebra-lemma-regular-ring-CM}.
Let $U_0$ be the punctured spectrum of $A/fA$ and let
$V_0 = V \times_U U_0$.
 ========

regex_match $U_0$
nn candidate the punctured spectrum
nn the punctured spectrum
new_annotation [(181996, 182001, 'VAR'), (182005, 182027, 'TYPE')]
===== sent 3.1]{SGA1}
\end{reference}
\begin{history}
This result was first stated and proved by Zariski in
geometric form in \cite{Zariski-Purity}.
The generalization to nonperfect ground fields by Nagata
was published as the next article in the same volume of the
Proceedings of the National Academy of Sciences of the United States of America
in \cite{Nagata-Remarks-Purity}. In the following year Nagata
proved the result for Noetherian local rings in \cite{Nagata-Purity}.
His proof uses a result of Chow which is a Bertini theorem for
complete local domains, see \cite{Chow-Bertini};
the history of Bertini's theorems is discussed in
Kleiman's historical article \cite{Kleiman-Bertini}.
A few years later a completely different proof was found by
Auslander, see \cite{Auslander-Purity}.
\end{history}
Let $f : X \to Y$ be a morphism of locally Noetherian schemes.
 ========

regex_match $f : X \to Y$
nn candidate X \to
nn candidate a morphism
nn a morphism
new_annotation [(183258, 183271, 'VAR'), (183275, 183285, 'TYPE')]
===== sent Let $x \in X$ and set $y = f(x)$. Assume
\begin{enumerate}
\item $\mathcal{O}_{X, x}$ is normal,
\item $\mathcal{O}_{Y, y}$ is regular,
\item $f$ is quasi-finite at $x$,
\item $\dim(\mathcal{O}_{X, x}) = \dim(\mathcal{O}_{Y, y}) \geq 1$
\item for specializations $x' \leadsto x$ with
$\dim(\mathcal{O}_{X, x'}) = 1$ our $f$ is unramified at $x'$.
\end{enumerate}
Then $f$ is \'etale at $x$.
\end{lemma}

\begin{proof}
We will prove the lemma by induction on
$d = \dim(\mathcal{O}_{X, x}) = \dim(\mathcal{O}_{Y, y})$.

\medskip\noindent
 ========

new_annotation []
===== sent \end{proof}

















\section{Finite \'etale covers of punctured spectra, II}
\label{section-pi1-punctured-spec-II}

\noindent
In this section we prove some variants of the material discussed
in Section \ref{section-pi1-punctured-spec}. Suppose
we have a Noetherian local ring $(A, \mathfrak m)$ and $f \in \mathfrak m$.
We set $X = \Spec(A)$ and $X_0 = \Spec(A/fA)$ and we
let $U = X \setminus \{\mathfrak m\}$ and
$U_0 = X_0 \setminus \{\mathfrak m\}$ be the punctured spectrum of
$A$ and $A/fA$. All of this is exactly as in
Situation \ref{situation-local-lefschetz}.
The difference is that we will consider the functor
$$
\colim_{U_0 \subset U' \subset U} \textit{F\'Et}_{U'}
\longrightarrow \textit{F\'Et}_{U_0},\quad
V' \longmapsto V_0 = V' \times_{U'} U_0
$$
 ========

new_annotation []
===== sent In other words, we will not try to lift finite \'etale coverings
of $U_0$ to all of $U$, but just to some open neighbourhood
$U'$ of $U_0$ in $U$.

\begin{lemma}
\label{lemma-faithful-general}
In Situation \ref{situation-local-lefschetz}.
Let $U' \subset U$ be open and contain $U_0$.
Assume $\dim(A/\mathfrak p) \geq 2$ for every minimal prime
$\mathfrak p \subset A$ corresponding to a point of $U'$. ========

new_annotation []
===== sent \end{lemma}

\begin{proof}
Let $a, b : V' \to W'$ be two morphisms of schemes finite \'etale
over $U'$ whose restriction to $U_0$ are the same. ========

new_annotation []
===== sent 2$.
The assumption therefore implies that every
irreducible component of $U'$ meets $U_0$.
The image of any irreducible component of $V'$ is an
irreducible component of $U'$ and hence meets $U_0$.
Hence $V_0$ meets every connected component of $V'$ and
we conclude that $a = b$ by \'Etale Morphisms, Proposition
\ref{etale-proposition-equality}.
To see the existence of such a $U'$ note that if
$\mathfrak p \subset A$ is a prime with $\dim(A/\mathfrak p) = 1$
then $\mathfrak p$ corresponds to a closed point of $U$.
\end{proof}

\begin{lemma}
\label{lemma-fully-faithful-general-better}
In Situation \ref{situation-local-lefschetz} assume
\begin{enumerate}
\item $A$ has a dualizing complex and is $f$-adically complete,
\item $f$ is a nonzerodivisor,
\item for $x \in X \setminus X_0$ whose closure $\overline{\{x\}}$
in $X$ meets $U_0$ we have $\text{depth}(\mathcal{O}_{X, x}) \geq 1$
or $\text{depth}(\mathcal{O}_{X, x}) + \dim(\overline{\{x\}}) > 2$.
\end{enumerate}
Let $V'$, $W'$ be finite \'etale over an open $U' \subset U$
which contains $U_0$. Let
$\varphi_0 : V' \times_{U'} U_0 \to W' \times_{U'} U_0$
be a morphism over $U_0$.
Then there exists an open $U'' \subset U'$
containing $U_0$ and a morphism
$\varphi : V' \times_{U'} U'' \to W' \times_{U'} U''$
lifting $\varphi_0$.
\end{lemma}

\begin{proof}
Since the category of finite \'etale coverings has an internal hom
(Lemma \ref{lemma-internal-hom-finite-etale})
it suffices to prove the following: Given $V'$ finite \'etale over $U'$
any section $U_0 \to V' \times_{U'} U_0$ extends to a section of $V'$
over some open $U'' \subset U'$ containing $U_0$.
Given our section we obtain a decomposition
$V' \times_{U'} U_0 = U_0 \amalg R_0$ into open and closed subschemes.
 ========

new_annotation []
===== sent For $n \geq 1$ let $U_n$ be the punctured spectrum of $A/f^{n + 1}A$. By
\'Etale Morphisms, Theorem \ref{etale-theorem-remarkable-equivalence}
we conclude that there is a unique decomposition
$V' \times_{U'} U_n = U_n \amalg R_n$
into open and closed subschemes whose base change to $U_0$ recovers
the given decomposition.

 ========

regex_match $U_n$
nn candidate the punctured spectrum
nn the punctured spectrum
new_annotation [(190273, 190278, 'VAR'), (190282, 190304, 'TYPE')]
===== sent \end{enumerate}
Let $V'$, $W'$ be finite \'etale over an open $U' \subset U$
which contains $U_0$. Let
$\varphi_0 : V' \times_{U'} U_0 \to W' \times_{U'} U_0$
be a morphism over $U_0$.
Then there exists an open $U'' \subset U'$
containing $U_0$ and a morphism
$\varphi : V' \times_{U'} U'' \to W' \times_{U'} U''$
lifting $\varphi_0$.
\end{lemma}

\begin{proof}
This lemma is a variant of
Lemma \ref{lemma-fully-faithful-general-better}
and if $A$ is a complete local ring, then it follows from that lemma.
 ========

new_annotation []
===== sent For $n \geq 1$ let $U_n$ be the punctured spectrum of $A/f^{n + 1}A$. By
\'Etale Morphisms, Theorem \ref{etale-theorem-remarkable-equivalence}
we conclude that there is a unique decomposition
$V' \times_{U'} U_n = U_n \amalg R_n$
into open and closed subschemes whose base change to $U_0$ recovers
the given decomposition.

 ========

regex_match $U_n$
nn candidate the punctured spectrum
nn the punctured spectrum
new_annotation [(192534, 192539, 'VAR'), (192543, 192565, 'TYPE')]
===== sent Let $\mathfrak q \in U_0$ be a prime. ========

regex_match $\mathfrak q \in U_0$
new_annotation []
===== sent For $n \geq 1$ let $U_n$ be the punctured spectrum of $A/f^{n + 1}A$.
By \'Etale Morphisms, Theorem \ref{etale-theorem-remarkable-equivalence}
we conclude that there is a unique finite \'etale morphism
$\pi_n : V_n \to U_n$ whose base change to $U_0$ recovers $V_0 \to U_0$.
 ========

regex_match $U_n$
nn candidate the punctured spectrum
nn the punctured spectrum
new_annotation [(195362, 195367, 'VAR'), (195371, 195393, 'TYPE')]
===== sent For $n \geq 1$ let $U_n$ be the punctured spectrum of $A/f^{n + 1}A$.
By \'Etale Morphisms, Theorem \ref{etale-theorem-remarkable-equivalence}
we conclude that there is a unique finite \'etale morphism
$\pi_n : V_n \to U_n$ whose base change to $U_0$ recovers $V_0 \to U_0$.
 ========

regex_match $U_n$
nn candidate the punctured spectrum
nn the punctured spectrum
new_annotation [(198393, 198398, 'VAR'), (198402, 198424, 'TYPE')]
===== sent Set
$$
B_n = \Gamma(V_n, \mathcal{O}_{V_n}) = \Gamma(U, \mathcal{F}_n)
$$
We conclude that the system $(B_n)$ satisfies the Mittag-Leffler condition,
that $B = \lim B_n$ is a finite $A$-algebra, that $f$ is a nonzerodivisor
on $B$ and that $B/fB \subset B_0$. To finish the proof,
we will show that the finite morphism
$\Spec(B) \to \Spec(A)$ (a) becomes isomorphic to $V_0 \to U_0$
after base change to $U_0$ and (b) is \'etale at all points lying
over $U_0$.

\medskip\noindent
Let $\mathfrak q \in U_0$ be a prime. ========

regex_match $\mathfrak q \in U_0$
new_annotation []
===== sent \end{proof}

\begin{remark}
\label{remark-combine}
Let $(A, \mathfrak m)$ be a complete local ring and $f \in \mathfrak m$
a nonzerodivisor. ========

regex_match $(A, \mathfrak m)$
nn candidate \mathfrak
nn candidate a complete local ring
nn a complete local ring
new_annotation [(201108, 201126, 'VAR'), (201130, 201151, 'TYPE')]
===== sent Let $U$, resp.\ $U_0$ be the punctured spectrum of
$A$, resp.\ $A/fA$. Assume
\begin{enumerate}
\item if $\mathfrak p \in V(f) \setminus \{\mathfrak m\}$, then
$\text{depth}((A/f)_\mathfrak p) + \dim(A/\mathfrak p) > 1$, and
\item if $\mathfrak p \not \in V(f)$ and
$V(\mathfrak p) \cap V(f) \not = \{\mathfrak m\}$, then
$\text{depth}(A_\mathfrak p) + \dim(A/\mathfrak p) > 3$.
\end{enumerate}
 ========

new_annotation []
===== sent \begin{lemma}
\label{lemma-purity-inherited-by-hypersurface-better}
Let $(A, \mathfrak m)$ be a Noetherian local ring.
 ========

regex_match $(A, \mathfrak m)$
nn candidate \mathfrak
nn candidate a Noetherian local ring
nn a Noetherian local ring
new_annotation [(202628, 202646, 'VAR'), (202650, 202673, 'TYPE')]
===== sent Let $f \in \mathfrak m$. Assume
\begin{enumerate}
\item $A$ has a dualizing complex and is $f$-adically complete,
\item $f$ is a nonzerodivisor,
\item if $\mathfrak p \in V(f) \setminus \{\mathfrak m\}$, then
$\text{depth}((A/f)_\mathfrak p) + \dim(A/\mathfrak p) > 1$, and
\item if $\mathfrak p \not \in V(f)$ and
$V(\mathfrak p) \cap V(f) \not = \{\mathfrak m\}$, then
$\text{depth}(A_\mathfrak p) + \dim(A/\mathfrak p) > 3$,
\item for every maximal ideal $\mathfrak p \subset A_f$
purity holds for $(A_f)_\mathfrak p$, and
\item purity holds for $A$.
 ========

new_annotation []
===== sent Similarly we have $X_0 = \Spec(A/fA)$
and $U_0 = X_0 \setminus \{\mathfrak m\}$.
Let $V_0 \to U_0$ be a finite \'etale morphism. ========

regex_match $V_0 \to U_0$
nn candidate a finite \'etale morphism
nn a finite \'etale morphism
new_annotation [(203471, 203484, 'VAR'), (203488, 203513, 'TYPE')]
===== sent Then the restriction of
$Y$ to $X_0$ is the desired extension of $V_0 \to U_0$.
\end{proof}

\begin{lemma}
\label{lemma-purity-inherited-by-hypersurface}
Let $(A, \mathfrak m)$ be a Noetherian local ring.
 ========

regex_match $(A, \mathfrak m)$
nn candidate \mathfrak
nn candidate a Noetherian local ring
nn a Noetherian local ring
new_annotation [(204435, 204453, 'VAR'), (204457, 204480, 'TYPE')]
===== sent Let $f \in \mathfrak m$. Assume
\begin{enumerate}
\item $A$ is $f$-adically complete,
\item $f$ is a nonzerodivisor,
\item $H^1_\mathfrak m(A/fA)$ and $H^2_\mathfrak m(A/fA)$ are finite
$A$-modules,
\item for every maximal ideal $\mathfrak p \subset A_f$
purity holds for $(A_f)_\mathfrak p$,
\item purity holds for $A$.
 ========

new_annotation []
===== sent See the discussion in Divided Power Algebra, Section \ref{dpa-section-lci}.

\begin{proposition}
\label{proposition-purity-complete-intersection}
Let $(A, \mathfrak m)$ be a Noetherian local ring. ========

regex_match $(A, \mathfrak m)$
nn candidate \mathfrak
nn candidate a Noetherian local ring
nn a Noetherian local ring
new_annotation [(205562, 205580, 'VAR'), (205584, 205607, 'TYPE')]
===== sent Let $f : X \to S$ be a proper smooth morphism of schemes.
 ========

regex_match $f : X \to S$
nn candidate X \to
nn candidate a proper smooth morphism
nn a proper smooth morphism
new_annotation [(207188, 207201, 'VAR'), (207205, 207229, 'TYPE')]
===== sent Let $s \leadsto s'$ be a specialization of points in $S$.
Then the specialization map
$$
sp : \pi_1(X_{\overline{s}}) \longrightarrow \pi_1(X_{\overline{s}'})
$$
of Section \ref{section-specialization-map}
is surjective and
\begin{enumerate}
\item if the characteristic of $\kappa(s')$ is zero, then it is
an isomorphism, or
\item if the characteristic of $\kappa(s')$ is $p > 0$, then it
induces an isomorphism on maximal prime-to-$p$ quotients.
 ========

regex_match $s \leadsto s'$
nn candidate $s
nn candidate a specialization
nn a specialization
new_annotation [(207246, 207261, 'VAR'), (207265, 207281, 'TYPE')]
===== sent \end{enumerate}

\begin{lemma}
\label{lemma-specialization-map-surjective}
Let $f : X \to S$ be a flat proper morphism with geometrically
connected fibres. ========

regex_match $f : X \to S$
nn candidate X \to
nn candidate a flat proper morphism
nn a flat proper morphism
new_annotation [(207768, 207781, 'VAR'), (207785, 207807, 'TYPE')]
===== sent Let $s' \leadsto s$ be a specialization.
 ========

regex_match $s' \leadsto s$
nn candidate s
nn candidate a specialization
nn a specialization
new_annotation [(207849, 207864, 'VAR'), (207868, 207884, 'TYPE')]
===== sent More on Morphisms, Lemma \ref{more-morphisms-lemma-geometrically-reduced-open}.
Let $\mathcal{O}_{S, s} \to A \to \kappa(\overline{s}')$ be as
in the construction of the specialization map, see
Section \ref{section-specialization-map}.
 ========

new_annotation []
===== sent This follows from
Proposition \ref{proposition-first-homotopy-sequence}
and $\pi_1(\Spec(A)) = \{1\}$.
\end{proof}

\begin{proposition}
\label{proposition-specialization-map-isomorphism}
Let $f : X \to S$ be a smooth proper morphism with geometrically
connected fibres. ========

regex_match $f : X \to S$
nn candidate X \to
nn candidate a smooth proper morphism
nn a smooth proper morphism
new_annotation [(208692, 208705, 'VAR'), (208709, 208733, 'TYPE')]
===== sent Let $s' \leadsto s$ be a specialization.
 ========

regex_match $s' \leadsto s$
nn candidate s
nn candidate a specialization
nn a specialization
new_annotation [(208775, 208790, 'VAR'), (208794, 208810, 'TYPE')]
===== sent Let $K$ be the fraction field of $A$.
 ========

regex_match $K$
nn candidate the fraction field
nn the fraction field
new_annotation [(209921, 209924, 'VAR'), (209928, 209946, 'TYPE')]
===== sent Choose an algebraic closure $\overline{K}$ which
corresponds to a geometric generic point $\overline{\eta}$ of $\Spec(A)$.
For $\overline{K}/L/K$ finite separable, let $B \subset L$ be the
integral closure of $A$ in $L$. This is a discrete
valuation ring by
More on Algebra, Remark \ref{more-algebra-remark-finite-separable-extension}.

\medskip\noindent
Let $X \to \Spec(A)$ be as in the previous paragraph.
 ========

new_annotation []
===== sent Choose an algebraic closure $\overline{K}$ which
corresponds to a geometric generic point $\overline{\eta}$ of $\Spec(A)$.
For $\overline{K}/L/K$ finite separable, let $B \subset L$ be the
integral closure of $A$ in $L$. This is a discrete
valuation ring by
More on Algebra, Remark \ref{more-algebra-remark-finite-separable-extension}.

\medskip\noindent
Let $X \to \Spec(A)$ be as in the previous paragraph.
 ========

new_annotation []
===== sent Given $V$ we can first descend $V$ to $V' \to X_{K^{sep}}$ by
Lemma \ref{lemma-perfection} and then to
$V'' \to X_L$ by Lemma \ref{lemma-limit}.
Let $Z \to X_B$ be the normalization of $X_B$ in $V''$.
Observe that $Z$ is normal and that $Z_L = V''$ as schemes
over $X_L$. Hence $Z \to X_B$ is finite \'etale over
the generic fibre. ========

regex_match $Z \to X_B$
nn candidate the normalization
nn the normalization
new_annotation [(210850, 210861, 'VAR'), (210865, 210882, 'TYPE')]
===== sent Hence the special fibre of $X_B \to \Spec(B)$ is irreducible;
let $\xi_B$ be its generic point. ========

new_annotation []
===== sent Let $e_i$ be
the ramification index of this extension. ========

new_annotation []
===== sent \medskip\noindent
To get rid of the ramification we are going to choose a further finite
separable extension $K^{sep}/L'/L/K$ such that the ramification
index $e$ of the induced extensions $B'/B$ is divisible by $e_i$.
Consider the normalized base change $Z'$ of $Z$ with respect to
$\Spec(B') \to \Spec(B)$, see discussion in
More on Morphisms, Section \ref{more-morphisms-section-reduced-fibre-theorem}.
Let $\xi_{i, j}$ be the points of $Z'$ mapping to $\xi_{B'}$
and to $\xi_i$ in $Z$. ========

regex_match $\xi_{i, j}$
nn candidate the points
nn the points
new_annotation [(212135, 212147, 'VAR'), (212151, 212161, 'TYPE')]
===== sent \end{proof}

\noindent
Let $G$ be a profinite group. ========

regex_match $G$
nn candidate a profinite group
nn a profinite group
new_annotation [(213259, 213262, 'VAR'), (213266, 213283, 'TYPE')]
===== sent Let $p$ be a prime number.
 ========

regex_match $p$
nn candidate a prime number
nn a prime number
new_annotation [(213289, 213292, 'VAR'), (213296, 213310, 'TYPE')]
===== sent The {\it maximal prime-to-$p$ quotient} is by definition
$$
G' = \lim_{U \subset G\text{ open, normal, index prime to }p} G/U
$$
If $X$ is a connected scheme and $p$ is given, then the maximal
prime-to-$p$ quotient of $\pi_1(X)$ is denoted $\pi'_1(X)$.

\begin{theorem}
\label{theorem-specialization-map-isomorphism-prime-to-p}
Let $f : X \to S$ be a smooth proper morphism with geometrically
connected fibres. ========

regex_match $f : X \to S$
nn candidate X \to
nn candidate a smooth proper morphism
nn a smooth proper morphism
new_annotation [(213644, 213657, 'VAR'), (213661, 213685, 'TYPE')]
===== sent Let $s' \leadsto s$ be a specialization.
 ========

regex_match $s' \leadsto s$
nn candidate s
nn candidate a specialization
nn a specialization
new_annotation [(213727, 213742, 'VAR'), (213746, 213762, 'TYPE')]
===== sent Hence the morphism $X_{\kappa_B} \to X_{\kappa_A}$
is a universal homeomorphism and induces an isomorphism
of fundamental groups by Proposition \ref{proposition-universal-homeomorphism}.
\end{enumerate}
\end{proof}







\section{Tame ramification}
\label{section-tame}

\noindent
Let $X \to Y$ be a finite \'etale morphism of schemes of finite type
over $\mathbf{Z}$. There are many ways to define what it means for $f$
to be tamely ramified at $\infty$. The article \cite{Kerz-Schmidt}
discusses to what extent these notions agree.

 ========

regex_match $X \to Y$
nn candidate $X
nn candidate a finite \'etale morphism
nn a finite \'etale morphism
new_annotation [(215265, 215274, 'VAR'), (215278, 215303, 'TYPE')]
file:  intersection.tex
===== sent \section{Cycles}
\label{section-cycles}

\noindent
Let $X$ be a variety. ========

regex_match $X$
nn candidate a variety
nn a variety
new_annotation [(5391, 5394, 'VAR'), (5398, 5407, 'TYPE')]
===== sent Let $Z_i$ be the irreducible components of $Z$ of
dimension $k$ and let $n_i$ be the {\it multiplicity of $Z_i$ in $Z$}
defined as
$$
n_i = \text{length}_{\mathcal{O}_{X, Z_i}} \mathcal{O}_{Z, Z_i}
$$
where $\mathcal{O}_{X, Z_i}$, resp.\ $\mathcal{O}_{Z, Z_i}$ is the
local ring of $X$, resp.\ $Z$ at the generic point of $Z_i$.
We define the $k$-cycle associated to $Z$ to be the $k$-cycle
$$
[Z]_k = \sum n_i [Z_i].
 ========

regex_match $Z_i$
nn candidate the irreducible components
nn the irreducible components
new_annotation [(6201, 6206, 'VAR'), (6210, 6236, 'TYPE')]
===== sent Let $Z_i$ be the irreducible components of $Z$ of
dimension $k$ and let $n_i$ be the {\it multiplicity of $Z_i$ in $Z$}
defined as
$$
n_i = \text{length}_{\mathcal{O}_{X, Z_i}} \mathcal{O}_{Z, Z_i}
$$
where $\mathcal{O}_{X, Z_i}$, resp.\ $\mathcal{O}_{Z, Z_i}$ is the
local ring of $X$, resp.\ $Z$ at the generic point of $Z_i$.
We define the $k$-cycle associated to $Z$ to be the $k$-cycle
$$
[Z]_k = \sum n_i [Z_i].
 ========

regex_match $n_i$
nn candidate the {\it multiplicity
nn the {\it multiplicity
new_annotation [(6269, 6274, 'VAR'), (6278, 6299, 'TYPE')]
===== sent \noindent
Suppose that $X$ is a variety and that
$\mathcal{F}$ is a coherent $\mathcal{O}_X$-module with
$\dim(\text{Supp}(\mathcal{F})) \leq k$.
Let $Z_i$ be the irreducible components of $\text{Supp}(\mathcal{F})$
of dimension $k$ and let $n_i$ be the
{\it multiplicity of $Z_i$ in $\mathcal{F}$} defined as
$$
n_i = \text{length}_{\mathcal{O}_{X, Z_i}} \mathcal{F}_{\xi_i}
$$
where $\mathcal{O}_{X, Z_i}$ is the
local ring of $X$ at the generic point $\xi_i$ of $Z_i$
and $\mathcal{F}_{\xi_i}$ is the stalk of $\mathcal{F}$ at this point.
 ========

regex_match $Z_i$
nn candidate the irreducible components
nn the irreducible components
new_annotation [(6931, 6936, 'VAR'), (6940, 6966, 'TYPE')]
===== sent \noindent
Suppose that $X$ is a variety and that
$\mathcal{F}$ is a coherent $\mathcal{O}_X$-module with
$\dim(\text{Supp}(\mathcal{F})) \leq k$.
Let $Z_i$ be the irreducible components of $\text{Supp}(\mathcal{F})$
of dimension $k$ and let $n_i$ be the
{\it multiplicity of $Z_i$ in $\mathcal{F}$} defined as
$$
n_i = \text{length}_{\mathcal{O}_{X, Z_i}} \mathcal{F}_{\xi_i}
$$
where $\mathcal{O}_{X, Z_i}$ is the
local ring of $X$ at the generic point $\xi_i$ of $Z_i$
and $\mathcal{F}_{\xi_i}$ is the stalk of $\mathcal{F}$ at this point.
 ========

new_annotation []
===== sent Let $Z \subset X$ be a $k$-dimensional
closed subvariety. ========

regex_match $Z \subset X$
nn candidate $Z \subset
nn candidate a $k$-dimensional
closed subvariety
nn a $k$-dimensional
closed subvariety
new_annotation [(7763, 7776, 'VAR'), (7780, 7815, 'TYPE')]
===== sent Morphisms, Definition \ref{morphisms-definition-degree}.
Let $\alpha = \sum n_i [Z_i]$ be a $k$-cycle on $Y$. The
{\it pushforward} of $\alpha$ is the sum $f_* \alpha = \sum n_i f_*[Z_i]$
where each $f_*[Z_i]$ is defined as above. ========

regex_match $\alpha = \sum n_i [Z_i]$
nn candidate $\alpha
nn candidate a $k$-cycle
nn a $k$-cycle
new_annotation [(8089, 8114, 'VAR'), (8118, 8129, 'TYPE')]
===== sent Let $\mathcal{F}$ be a coherent sheaf with
$\dim(\text{Supp}(\mathcal{F})) \leq k$, then
$f_*[\mathcal{F}]_k = [f_*\mathcal{F}]_k$. In particular, if
$Z \subset X$ is a closed subscheme of dimension $\leq k$, then
$f_*[Z] = [f_*\mathcal{O}_Z]_k$.
\end{lemma}

\begin{proof}
See Chow Homology, Lemma \ref{chow-lemma-cycle-push-sheaf}.
\end{proof}

\begin{lemma}
\label{lemma-compose-pushforward}
Let $f : X \to Y$ and $g : Y \to Z$ be proper morphisms of
varieties. ========

regex_match $\mathcal{F}$
nn candidate a coherent sheaf
nn a coherent sheaf
new_annotation [(8582, 8595, 'VAR'), (8599, 8615, 'TYPE')]
===== sent Let $\mathcal{F}$ be a coherent sheaf with
$\dim(\text{Supp}(\mathcal{F})) \leq k$, then
$f_*[\mathcal{F}]_k = [f_*\mathcal{F}]_k$. In particular, if
$Z \subset X$ is a closed subscheme of dimension $\leq k$, then
$f_*[Z] = [f_*\mathcal{O}_Z]_k$.
\end{lemma}

\begin{proof}
See Chow Homology, Lemma \ref{chow-lemma-cycle-push-sheaf}.
\end{proof}

\begin{lemma}
\label{lemma-compose-pushforward}
Let $f : X \to Y$ and $g : Y \to Z$ be proper morphisms of
varieties. ========

new_annotation []
===== sent This follows from
Algebra, Lemma \ref{algebra-lemma-CM-over-regular-flat} and
Varieties, Lemma \ref{varieties-lemma-dimension-fibres-locally-algebraic}
showing $\dim(X) = \dim(Y) + r$.}.
Let $Z \subset X$ be a $k$-dimensional closed subvariety. ========

regex_match $Z \subset X$
nn candidate $Z \subset
nn candidate a $k$-dimensional closed subvariety
nn a $k$-dimensional closed subvariety
new_annotation [(9858, 9871, 'VAR'), (9875, 9910, 'TYPE')]
===== sent This defines a homomorphism
$$
f^* : Z_k(Y) \longrightarrow Z_{k + r}(X)
$$
See Chow Homology, Section \ref{chow-section-flat-pullback}.

\begin{lemma}
\label{lemma-pullback}
Let $f : X \to Y$ be a flat morphism of varieties. Set $r = \dim(X) - \dim(Y)$.
Then $f^*[\mathcal{F}]_k = [f^*\mathcal{F}]_{k + r}$
if $\mathcal{F}$ is a coherent sheaf on $Y$ and the dimension of the
support of $\mathcal{F}$ is at most $k$.
\end{lemma}

\begin{proof}
See Chow Homology, Lemma \ref{chow-lemma-pullback-coherent}.
\end{proof}

\begin{lemma}
\label{lemma-compose-flat-pullback}
Let $f : X \to Y$ and $g : Y \to Z$ be flat morphisms of
varieties. ========

regex_match $f : X \to Y$
nn candidate X \to
nn candidate a flat morphism
nn a flat morphism
new_annotation [(10389, 10402, 'VAR'), (10406, 10421, 'TYPE')]
===== sent This defines a homomorphism
$$
f^* : Z_k(Y) \longrightarrow Z_{k + r}(X)
$$
See Chow Homology, Section \ref{chow-section-flat-pullback}.

\begin{lemma}
\label{lemma-pullback}
Let $f : X \to Y$ be a flat morphism of varieties. Set $r = \dim(X) - \dim(Y)$.
Then $f^*[\mathcal{F}]_k = [f^*\mathcal{F}]_{k + r}$
if $\mathcal{F}$ is a coherent sheaf on $Y$ and the dimension of the
support of $\mathcal{F}$ is at most $k$.
\end{lemma}

\begin{proof}
See Chow Homology, Lemma \ref{chow-lemma-pullback-coherent}.
\end{proof}

\begin{lemma}
\label{lemma-compose-flat-pullback}
Let $f : X \to Y$ and $g : Y \to Z$ be flat morphisms of
varieties. ========

new_annotation []
===== sent \medskip\noindent
Let $X$ be a variety. ========

regex_match $X$
nn candidate a variety
nn a variety
new_annotation [(11493, 11496, 'VAR'), (11500, 11509, 'TYPE')]
===== sent Let $W \subset X \times \mathbf{P}^1$
be a closed subvariety of dimension $k + 1$. Let $a, b$ be distinct closed
points of $\mathbf{P}^1$. ========

new_annotation []
===== sent Let $W \subset X \times \mathbf{P}^1$
be a closed subvariety of dimension $k + 1$. Let $a, b$ be distinct closed
points of $\mathbf{P}^1$. ========

new_annotation []
===== sent A traditional
choice is to choose $a = 0$ and $b = \infty$.

\medskip\noindent
More generally, let $\alpha = \sum n_i [W_i]$ be a $(k + 1)$-cycle on
$X \times \mathbf{P}^1$.  Let $a_i, b_i$ be pairs of distinct closed points of
$\mathbf{P}^1$. ========

regex_match $\alpha = \sum n_i [W_i]$
nn candidate a $(k + 1)$-cycle
nn a $(k + 1)$-cycle
new_annotation [(13061, 13086, 'VAR'), (13090, 13107, 'TYPE')]
===== sent A traditional
choice is to choose $a = 0$ and $b = \infty$.

\medskip\noindent
More generally, let $\alpha = \sum n_i [W_i]$ be a $(k + 1)$-cycle on
$X \times \mathbf{P}^1$.  Let $a_i, b_i$ be pairs of distinct closed points of
$\mathbf{P}^1$. ========

new_annotation []
===== sent \medskip\noindent
We define
$$
A_k(X) = Z_k(X)/ \sim_{rat}
$$
to be the {\it Chow group of $k$-cycles on $X$}. We will see in
Lemma \ref{lemma-rational-equivalence}
that this agrees with the Chow group as defined in
Chow Homology, Definition \ref{chow-definition-rational-equivalence}.


\section{Rational equivalence and rational functions}
\label{section-alternative}

\noindent
Let $X$ be a variety. ========

regex_match $X$
nn candidate a variety
nn a variety
new_annotation [(14157, 14160, 'VAR'), (14164, 14173, 'TYPE')]
===== sent Let $W \subset X$ be a subvariety
of dimension $k + 1$. Let $f \in \mathbf{C}(W)^*$ be a nonzero rational
function on $W$. For every subvariety $Z \subset W$ of dimension $k$
one can define the order of vanishing $\text{ord}_{W, Z}(f)$ of $f$ at
at $Z$. If $f$ is an element of the local ring $\mathcal{O}_{W, Z}$,
then one has
$$
\text{ord}_{W, Z}(f) =
\text{length}_{\mathcal{O}_{X, z}} \mathcal{O}_{W, Z}/f\mathcal{O}_{W, Z}
$$
where $\mathcal{O}_{X, Z}$, resp.\ $\mathcal{O}_{W, Z}$ is the
local ring of $X$, resp.\ $W$ at the generic point of $Z$. ========

regex_match $W \subset X$
nn candidate $
nn candidate a subvariety
nn a subvariety
new_annotation [(14179, 14192, 'VAR'), (14196, 14208, 'TYPE')]
===== sent Let $W \subset X$ be a subvariety
of dimension $k + 1$. Let $f \in \mathbf{C}(W)^*$ be a nonzero rational
function on $W$. For every subvariety $Z \subset W$ of dimension $k$
one can define the order of vanishing $\text{ord}_{W, Z}(f)$ of $f$ at
at $Z$. If $f$ is an element of the local ring $\mathcal{O}_{W, Z}$,
then one has
$$
\text{ord}_{W, Z}(f) =
\text{length}_{\mathcal{O}_{X, z}} \mathcal{O}_{W, Z}/f\mathcal{O}_{W, Z}
$$
where $\mathcal{O}_{X, Z}$, resp.\ $\mathcal{O}_{W, Z}$ is the
local ring of $X$, resp.\ $W$ at the generic point of $Z$. ========

regex_match $f \in \mathbf{C}(W)^*$
nn candidate a nonzero rational
function
nn a nonzero rational
function
new_annotation [(14235, 14258, 'VAR'), (14262, 14289, 'TYPE')]
===== sent The {\it principal divisor
associated to $f$} is
$$
\text{div}_W(f) = \sum \text{ord}_{W, Z}(f)[Z]
$$
in $Z_k(W)$. Since $W \subset X$ is a closed subvariety we may think
of $\text{div}_W(f)$ as a cycle on $X$.
See Chow Homology, Section \ref{chow-section-principal-divisors}.

\begin{lemma}
\label{lemma-rational-equivalence}
Let $X$ be a variety. ========

regex_match $X$
nn candidate a variety
nn a variety
new_annotation [(15118, 15121, 'VAR'), (15125, 15134, 'TYPE')]
===== sent Let $W \subset X$ be a subvariety
of dimension $k + 1$. Let $f \in \mathbf{C}(W)^*$ be a nonzero rational
function on $W$. Then $\text{div}_W(f)$ is rationally equivalent to zero on
$X$. ========

regex_match $W \subset X$
nn candidate $
nn candidate a subvariety
nn a subvariety
new_annotation [(15140, 15153, 'VAR'), (15157, 15169, 'TYPE')]
===== sent Let $W \subset X$ be a subvariety
of dimension $k + 1$. Let $f \in \mathbf{C}(W)^*$ be a nonzero rational
function on $W$. Then $\text{div}_W(f)$ is rationally equivalent to zero on
$X$. ========

regex_match $f \in \mathbf{C}(W)^*$
nn candidate a nonzero rational
function
nn a nonzero rational
function
new_annotation [(15196, 15219, 'VAR'), (15223, 15250, 'TYPE')]
===== sent Conversely, these principal divisors generate the abelian group of
cycles rationally equivalent to zero on $X$.
\end{lemma}

\begin{proof}
The first assertion follows from
Chow Homology, Lemma \ref{chow-lemma-rational-function}.
More precisely, let $W' \subset X \times \mathbf{P}^1$ be the closure
of the graph of $f$. Then $\text{div}_W(f) = [W'_0]_k - [W'_\infty]$
in $Z_k(W) \subset Z_k(X)$, see part (6) of
Chow Homology, Lemma \ref{chow-lemma-rational-function}.

\medskip\noindent
For the second, let $W' \subset X \times \mathbf{P}^1$ be a closed
subvariety of dimension $k + 1$ which dominates $\mathbf{P}^1$.
We will show that $[W'_0]_k ========

regex_match $W' \subset X \times \mathbf{P}^1$
nn candidate $W
nn candidate the closure
nn the closure
new_annotation [(15572, 15606, 'VAR'), (15610, 15621, 'TYPE')]
===== sent Conversely, these principal divisors generate the abelian group of
cycles rationally equivalent to zero on $X$.
\end{lemma}

\begin{proof}
The first assertion follows from
Chow Homology, Lemma \ref{chow-lemma-rational-function}.
More precisely, let $W' \subset X \times \mathbf{P}^1$ be the closure
of the graph of $f$. Then $\text{div}_W(f) = [W'_0]_k - [W'_\infty]$
in $Z_k(W) \subset Z_k(X)$, see part (6) of
Chow Homology, Lemma \ref{chow-lemma-rational-function}.

\medskip\noindent
For the second, let $W' \subset X \times \mathbf{P}^1$ be a closed
subvariety of dimension $k + 1$ which dominates $\mathbf{P}^1$.
We will show that $[W'_0]_k ========

regex_match $W' \subset X \times \mathbf{P}^1$
nn candidate $W
nn candidate X \times
nn candidate a closed
subvariety
nn a closed
subvariety
new_annotation [(15831, 15865, 'VAR'), (15869, 15888, 'TYPE')]
===== sent Let $W \subset X$ be the image of $W'$
under the projection to $X$. Then $W' \to W$ is proper and generically
finite\footnote{If $W' \to W$ is birational, then the result follows
from Chow Homology, Lemma \ref{chow-lemma-rational-function}.
Our task is to show that even if $W' \to W$
has degree $>1$ the basic rational equivalence
$[W'_0]_k \sim_{rat} [W'_\infty]_k$ comes from a principal divisor
on a subvariety of $X$.}. Let $f$ denote the projection $W' \to \mathbf{P}^1$
viewed as an element of $\mathbf{C}(W')^*$. Let
$g = \text{Nm}(f) \in \mathbf{C}(W)^*$ be the norm. ========

regex_match $W \subset X$
nn candidate $
nn candidate the image
nn the image
new_annotation [(16043, 16056, 'VAR'), (16060, 16069, 'TYPE')]
===== sent Let $W \subset X$ be the image of $W'$
under the projection to $X$. Then $W' \to W$ is proper and generically
finite\footnote{If $W' \to W$ is birational, then the result follows
from Chow Homology, Lemma \ref{chow-lemma-rational-function}.
Our task is to show that even if $W' \to W$
has degree $>1$ the basic rational equivalence
$[W'_0]_k \sim_{rat} [W'_\infty]_k$ comes from a principal divisor
on a subvariety of $X$.}. Let $f$ denote the projection $W' \to \mathbf{P}^1$
viewed as an element of $\mathbf{C}(W')^*$. Let
$g = \text{Nm}(f) \in \mathbf{C}(W)^*$ be the norm. ========

new_annotation []
===== sent Let $\alpha \sim_{rat} 0$ be a $k$-cycle on
$X$ rationally equivalent to $0$. Then the {pushforward}
of $\alpha$ is rationally equivalent to zero:
$f_* \alpha \sim_{rat} 0$. See Chapter I of \cite{F} or
Chow Homology, Lemma \ref{chow-lemma-proper-pushforward-rational-equivalence}.

\medskip\noindent
 ========

regex_match $\alpha \sim_{rat} 0$
nn candidate a $k$-cycle
nn a $k$-cycle
new_annotation [(17047, 17068, 'VAR'), (17072, 17083, 'TYPE')]
===== sent Set $r = \dim(X) - \dim(Y)$.
Let $\alpha \sim_{rat} 0$ be a $k$-cycle on
$Y$ rationally equivalent to $0$. Then the pullback
of $\alpha$ is rationally equivalent to zero:
$f^* \alpha \sim_{rat} 0$. See Chapter I of \cite{F} or
Chow Homology, Lemma \ref{chow-lemma-flat-pullback-rational-equivalence}.

\medskip\noindent
 ========

regex_match $\alpha \sim_{rat} 0$
nn candidate a $k$-cycle
nn a $k$-cycle
new_annotation [(17715, 17736, 'VAR'), (17740, 17751, 'TYPE')]
===== sent Z_k(Y) \ar[r] \ar[u]^{f^*} & A_k(Y) \ar[u]_{f^*}
}
$$
of groups of $k$-cycles.


\section{The short exact sequence for an open}
\label{section-ses}

\noindent
Let $X$ be a variety and let $U \subset X$ be an open subvariety.
Let $X \setminus U = \bigcup Z_i$ be the decomposition into irreducible
components\footnote{Since in this chapter we only consider Chow groups
of varieties, we are prohibited from taking $Z_k(X \setminus U)$
and ========

regex_match $X$
nn candidate a variety
nn a variety
new_annotation [(18259, 18262, 'VAR'), (18266, 18275, 'TYPE')]
===== sent Z_k(Y) \ar[r] \ar[u]^{f^*} & A_k(Y) \ar[u]_{f^*}
}
$$
of groups of $k$-cycles.


\section{The short exact sequence for an open}
\label{section-ses}

\noindent
Let $X$ be a variety and let $U \subset X$ be an open subvariety.
Let $X \setminus U = \bigcup Z_i$ be the decomposition into irreducible
components\footnote{Since in this chapter we only consider Chow groups
of varieties, we are prohibited from taking $Z_k(X \setminus U)$
and ========

regex_match $U \subset X$
nn candidate $U \subset X$
nn candidate an open subvariety
nn an open subvariety
new_annotation [(18284, 18297, 'VAR'), (18301, 18319, 'TYPE')]
===== sent Z_k(Y) \ar[r] \ar[u]^{f^*} & A_k(Y) \ar[u]_{f^*}
}
$$
of groups of $k$-cycles.


\section{The short exact sequence for an open}
\label{section-ses}

\noindent
Let $X$ be a variety and let $U \subset X$ be an open subvariety.
Let $X \setminus U = \bigcup Z_i$ be the decomposition into irreducible
components\footnote{Since in this chapter we only consider Chow groups
of varieties, we are prohibited from taking $Z_k(X \setminus U)$
and ========

regex_match $X \setminus U = \bigcup Z_i$
nn candidate $X
nn candidate the decomposition
nn the decomposition
new_annotation [(18325, 18354, 'VAR'), (18358, 18375, 'TYPE')]
===== sent \begin{lemma}
\label{lemma-dimension-product-varieties}
Let $X$ and $Y$ be varieties. ========

new_annotation []
===== sent \begin{lemma}
\label{lemma-pullback-by-regular-immersion}
Let $f : X \to Y$ be a morphism of varieties.
 ========

regex_match $f : X \to Y$
nn candidate X \to
nn candidate a morphism
nn a morphism
new_annotation [(21991, 22004, 'VAR'), (22008, 22018, 'TYPE')]
===== sent \end{proof}

\begin{lemma}
\label{lemma-diagonal-regular-immersion}
Let $X$ be a nonsingular variety. ========

regex_match $X$
nn candidate a nonsingular variety
nn a nonsingular variety
new_annotation [(23608, 23611, 'VAR'), (23615, 23636, 'TYPE')]
===== sent \begin{lemma}
\label{lemma-intersect-in-smooth}
Let $X$ be a nonsingular variety and let $W,V \subset X$
be closed subvarieties with $\dim(W) = s$ and $\dim(V) = r$. Then every
irreducible component $Z$ of $V \cap W$ has dimension $\geq r + s - \dim(X)$.
\end{lemma}

\begin{proof}
Since $V \cap W = \Delta^{-1}(V \times W)$ (scheme theoretically)
we conclude by Lemmas \ref{lemma-diagonal-regular-immersion} and
\ref{lemma-pullback-by-regular-immersion}.
\end{proof}

\noindent
This lemma suggests the following definition.

 ========

regex_match $X$
nn candidate a nonsingular variety
nn a nonsingular variety
new_annotation [(24090, 24093, 'VAR'), (24097, 24118, 'TYPE')]
===== sent \begin{lemma}
\label{lemma-intersect-in-smooth}
Let $X$ be a nonsingular variety and let $W,V \subset X$
be closed subvarieties with $\dim(W) = s$ and $\dim(V) = r$. Then every
irreducible component $Z$ of $V \cap W$ has dimension $\geq r + s - \dim(X)$.
\end{lemma}

\begin{proof}
Since $V \cap W = \Delta^{-1}(V \times W)$ (scheme theoretically)
we conclude by Lemmas \ref{lemma-diagonal-regular-immersion} and
\ref{lemma-pullback-by-regular-immersion}.
\end{proof}

\noindent
This lemma suggests the following definition.

 ========

new_annotation []
===== sent \begin{definition}
\label{definition-proper-intersection}
Let $X$ be a nonsingular variety.
 ========

regex_match $X$
nn candidate a nonsingular variety
nn a nonsingular variety
new_annotation [(24626, 24629, 'VAR'), (24633, 24654, 'TYPE')]
===== sent \begin{enumerate}
\item Let $W,V \subset X$ be closed subvarieties with
$\dim(W) = s$ and $\dim(V) = r$. We say that $W$ and $V$
{\it intersect properly} if $\dim(V \cap W) \leq r + s - \dim(X)$.
\item Let $\alpha = \sum n_i [W_i]$ be an $s$-cycle,
and $\beta = \sum_j m_j [V_j]$ be an $r$-cycle on $X$. We say
that $\alpha$ and $\beta$ {\it ========

new_annotation []
===== sent \begin{enumerate}
\item Let $W,V \subset X$ be closed subvarieties with
$\dim(W) = s$ and $\dim(V) = r$. We say that $W$ and $V$
{\it intersect properly} if $\dim(V \cap W) \leq r + s - \dim(X)$.
\item Let $\alpha = \sum n_i [W_i]$ be an $s$-cycle,
and $\beta = \sum_j m_j [V_j]$ be an $r$-cycle on $X$. We say
that $\alpha$ and $\beta$ {\it ========

regex_match $\alpha = \sum n_i [W_i]$
nn candidate an $s$-cycle
nn an $s$-cycle
new_annotation [(24862, 24887, 'VAR'), (24891, 24903, 'TYPE')]
===== sent \begin{lemma}
\label{lemma-tensor-coherent}
Let $X$ be a locally Noetherian scheme.
 ========

regex_match $X$
nn candidate a locally Noetherian scheme
nn a locally Noetherian scheme
new_annotation [(26144, 26147, 'VAR'), (26151, 26178, 'TYPE')]
===== sent Then $L \otimes_{\mathcal{O}_X}^\mathbf{L} K$
is pseudo-coherent by
Cohomology, Lemma \ref{cohomology-lemma-tensor-pseudo-coherent}.
\end{proof}

\begin{lemma}
\label{lemma-compute-tor-nonsingular}
Let $X$ be a nonsingular variety.
 ========

regex_match $X$
nn candidate a nonsingular variety
nn a nonsingular variety
new_annotation [(27705, 27708, 'VAR'), (27712, 27733, 'TYPE')]
===== sent Let $\mathcal{F}$, $\mathcal{G}$ be coherent $\mathcal{O}_X$-modules.
 ========

new_annotation []
===== sent \end{proof}

\noindent
Let $X$ be a nonsingular variety and $W, V \subset X$
be closed subvarieties with $\dim(W) = s$ and $\dim(V) = r$.
Assume $V$ and $W$ intersect properly.
 ========

regex_match $X$
nn candidate a nonsingular variety
nn a nonsingular variety
new_annotation [(28946, 28949, 'VAR'), (28953, 28974, 'TYPE')]
===== sent \medskip\noindent
In the case of transversal intersections, the intersection number is $1$.

\begin{lemma}
\label{lemma-transversal}
Let $X$ be a nonsingular variety. ========

regex_match $X$
nn candidate a nonsingular variety
nn a nonsingular variety
new_annotation [(30614, 30617, 'VAR'), (30621, 30642, 'TYPE')]
===== sent Let $V, W \subset X$ be
closed subvarieties which intersect properly. ========

new_annotation []
===== sent Let $Z$ be an irreducible
component of $V \cap W$ and assume that the multiplicity
(in the sense of Section \ref{section-cycle-of-closed}) of $Z$
in the closed subscheme ========

regex_match $Z$
nn candidate an irreducible
component
nn an irreducible
component
new_annotation [(30718, 30721, 'VAR'), (30725, 30749, 'TYPE')]
===== sent Then $e(X, V \cdot W, Z) = 1$ and $V$ and $W$ are smooth
in a general point of $Z$.
\end{lemma}

\begin{proof}
Let $(A, \mathfrak m, \kappa) =
(\mathcal{O}_{X, \xi}, \mathfrak m_\xi, \kappa(\xi))$ where $\xi \in Z$
is the generic point. ========

new_annotation []
===== sent Then $\dim(A) = \dim(X) - \dim(Z)$, see
Varieties, Lemma \ref{varieties-lemma-dimension-locally-algebraic}.
Let $I, J \subset A$ cut out the trace of $V$ and $W$
in $\Spec(A)$. Set $\overline{I} = I + \mathfrak m^2/\mathfrak m^2$.
Then $\dim_\kappa \overline{I} \leq \dim(X) - \dim(V)$ with equality
if and only if $A/I$ is regular (this follows from the lemma cited
above and the definition of regular rings, see
Algebra, Definition \ref{algebra-definition-regular-local}
and the discussion preceding it). ========

new_annotation []
===== sent Let $X$ be a nonsingular variety of dimension $4$.
Let $p \in X$ be a closed point. ========

regex_match $X$
nn candidate a nonsingular variety
nn a nonsingular variety
new_annotation [(33403, 33406, 'VAR'), (33410, 33431, 'TYPE')]
===== sent Let $X$ be a nonsingular variety of dimension $4$.
Let $p \in X$ be a closed point. ========

regex_match $p \in X$
nn candidate a closed point
nn a closed point
new_annotation [(33454, 33463, 'VAR'), (33467, 33481, 'TYPE')]
===== sent Let $V, W \subset X$
be closed subvarieties in $X$. ========

new_annotation []
===== sent \end{example}






\section{Algebraic multiplicities}
\label{section-multiplicities}

\noindent
Let $(A, \mathfrak m, \kappa)$ be a Noetherian local ring.
 ========

regex_match $(A, \mathfrak m, \kappa)$
nn candidate A
nn candidate a Noetherian local ring
nn a Noetherian local ring
new_annotation [(34608, 34634, 'VAR'), (34638, 34661, 'TYPE')]
===== sent Let $M$ be a finite $A$-module and let $I \subset A$ be an ideal
of definition (Algebra, Definition \ref{algebra-definition-ideal-definition}).
 ========

regex_match $M$
nn candidate I
nn I
new_annotation [(34667, 34670, 'VAR'), (34703, 34704, 'TYPE')]
===== sent Let $M$ be a finite $A$-module and let $I \subset A$ be an ideal
of definition (Algebra, Definition \ref{algebra-definition-ideal-definition}).
 ========

regex_match $I \subset A$
nn candidate I
nn candidate definition
nn definition
new_annotation [(34702, 34715, 'VAR'), (34731, 34741, 'TYPE')]
===== sent \begin{lemma}
\label{lemma-multiplicity-ses}
Let $A$ be a Noetherian local ring. ========

regex_match $A$
nn candidate a Noetherian local ring
nn a Noetherian local ring
new_annotation [(35774, 35777, 'VAR'), (35781, 35804, 'TYPE')]
===== sent Let $I \subset A$ be an ideal of
definition. ========

regex_match $I \subset A$
nn candidate $
nn candidate I
nn candidate an ideal
nn an ideal
new_annotation [(35810, 35823, 'VAR'), (35827, 35835, 'TYPE')]
===== sent Let $0 \to M' \to M \to M'' \to 0$ be a short exact sequence
of finite $A$-modules. ========

regex_match $0 \to M' \to M \to M'' \to 0$
nn candidate $0 \to M
nn candidate a short exact sequence
nn a short exact sequence
new_annotation [(35855, 35885, 'VAR'), (35889, 35911, 'TYPE')]
===== sent Let $d \geq ========

new_annotation []
===== sent \dim(\text{Supp}(M))$. Then
$$
e_I(M, d) = e_I(M', d) + e_I(M'', d)
$$
\end{lemma}

\begin{proof}
Immediate from the definitions and
Algebra, Lemma \ref{algebra-lemma-hilbert-ses-chi}.
\end{proof}

\begin{lemma}
\label{lemma-multiplicity-as-a-sum}
Let $A$ be a Noetherian local ring. ========

regex_match $A$
nn candidate a Noetherian local ring
nn a Noetherian local ring
new_annotation [(36199, 36202, 'VAR'), (36206, 36229, 'TYPE')]
===== sent Let $I \subset A$ be an ideal of
definition. ========

regex_match $I \subset A$
nn candidate $
nn candidate I
nn candidate an ideal
nn an ideal
new_annotation [(36235, 36248, 'VAR'), (36252, 36260, 'TYPE')]
===== sent Let $M$ be a finite $A$-module. ========

regex_match $M$
new_annotation []
===== sent Let $d \geq \dim(\text{Supp}(M))$.
Then
$$
e_I(M, d) =
\sum \text{length}_{A_\mathfrak p}(M_\mathfrak p) e_I(A/\mathfrak p, d)
$$
where the sum is over primes $\mathfrak p \subset A$ with
$\dim(A/\mathfrak p) = d$.
\end{lemma}

\begin{proof}
Both the left and side and the right hand side are additive in short
exact sequences of modules of dimension $\leq d$, see
Lemma \ref{lemma-multiplicity-ses} and
Algebra, Lemma \ref{algebra-lemma-length-additive}.
 ========

new_annotation []
===== sent \end{proof}

\begin{lemma}
\label{lemma-leading-coefficient}
Let $P$ be a polynomial of degree $r$ with leading coefficient $a$.
 ========

regex_match $P$
nn candidate a polynomial
nn a polynomial
new_annotation [(37042, 37045, 'VAR'), (37049, 37061, 'TYPE')]
===== sent Recall that if $R$ is a ring and
$f_1, \ldots, f_r \in R$, then $K_\bullet(f_1, \ldots, f_r)$
denotes the Koszul complex, see
More on Algebra, Section \ref{more-algebra-section-koszul}.

\begin{theorem}
\label{theorem-multiplicity-with-koszul}
\begin{reference}
\cite[Theorem 1 in part B of Chapter IV]{Serre_algebre_locale}
\end{reference}
Let $A$ be a Noetherian local ring. ========

regex_match $A$
nn candidate a Noetherian local ring
nn a Noetherian local ring
new_annotation [(38425, 38428, 'VAR'), (38432, 38455, 'TYPE')]
===== sent Let $I = (f_1, \ldots, f_r) \subset A$
be an ideal of definition. ========

new_annotation []
===== sent Let $M$ be a finite $A$-module. ========

regex_match $M$
new_annotation []
===== sent Using
additivity again we see that
$$
\sum (-1)^{p + q} \text{length}_A(E_1^{p, q}) =
\sum\nolimits_n \sum\nolimits_{p \leq t}
(-1)^n \text{length}_A(\text{gr}^p(K^n \otimes_A M))
$$
This is equal to
$$
\sum\nolimits_{n = -r, \ldots, 0} (-1)^n{r \choose |n|} \chi_{I, M}(t + n)
$$
by our choice of filtration above and the definition of $\chi_{I, M}$ in
Algebra, Section \ref{algebra-section-Noetherian-local}.
The lemma follows from Lemma \ref{lemma-leading-coefficient}
and the definition of $e_I(M, r)$.
\end{proof}

\begin{remark}[Trivial generalization]
\label{remark-trivial-generalization}
Let $(A, \mathfrak m, \kappa)$ be a Noetherian local ring.
 ========

regex_match $(A, \mathfrak m, \kappa)$
nn candidate A
nn candidate a Noetherian local ring
nn a Noetherian local ring
new_annotation [(41870, 41896, 'VAR'), (41900, 41923, 'TYPE')]
===== sent Let $M$ be a finite $A$-module. ========

regex_match $M$
new_annotation []
===== sent Let $I \subset A$ be an ideal.
 ========

regex_match $I \subset A$
nn candidate $
nn candidate I
nn candidate an ideal
nn an ideal
new_annotation [(41961, 41974, 'VAR'), (41978, 41986, 'TYPE')]
===== sent \begin{lemma}
\label{lemma-intersection-multiplicity-CM}
Let $X$ be a nonsingular variety and $W, V \subset X$ closed
subvarieties which intersect properly. ========

regex_match $X$
nn candidate a nonsingular variety
nn a nonsingular variety
new_annotation [(43603, 43606, 'VAR'), (43610, 43631, 'TYPE')]
===== sent Let $Z$ be an irreducible component
of $V \cap W$ with generic point ========

regex_match $Z$
nn candidate an irreducible component
nn an irreducible component
new_annotation [(43703, 43706, 'VAR'), (43710, 43734, 'TYPE')]
===== sent \end{proof}

\begin{lemma}
\label{lemma-one-ideal-ci}
Let $A$ be a Noetherian local ring. ========

regex_match $A$
nn candidate a Noetherian local ring
nn a Noetherian local ring
new_annotation [(45017, 45020, 'VAR'), (45024, 45047, 'TYPE')]
===== sent Let $I = (f_1, \ldots, f_r)$ be an ideal
generated by a regular sequence. ========

regex_match $I = (f_1, \ldots, f_r)$
nn candidate $
nn candidate I
nn candidate a regular sequence
nn a regular sequence
new_annotation [(45053, 45077, 'VAR'), (45103, 45121, 'TYPE')]
===== sent Let $M$ be a finite $A$-module. ========

regex_match $M$
new_annotation []
===== sent \end{proof}

\begin{lemma}
\label{lemma-multiplicity-with-lci}
Let $X$ be a nonsingular variety. ========

regex_match $X$
nn candidate a nonsingular variety
nn a nonsingular variety
new_annotation [(46414, 46417, 'VAR'), (46421, 46442, 'TYPE')]
===== sent Let $W,V \subset X$ be
closed subvarieties which intersect properly. ========

new_annotation []
===== sent Let $Z$ be an irreducible
component of $V \cap W$ with generic point $\xi$.
 ========

regex_match $Z$
nn candidate an irreducible
component
nn an irreducible
component
new_annotation [(46517, 46520, 'VAR'), (46524, 46548, 'TYPE')]
===== sent \end{proof}


\section{Intersection product using Tor formula}
\label{section-intersection-product}

\noindent
Let $X$ be a nonsingular variety. ========

regex_match $X$
nn candidate a nonsingular variety
nn a nonsingular variety
new_annotation [(50234, 50237, 'VAR'), (50241, 50262, 'TYPE')]
===== sent $$
where $W_i \cdot V_j$ is as defined in Section \ref{section-tor-formula}.
If $\beta = [V]$ where $V$ is a closed subvariety of dimension $s$,
then we sometimes write $\alpha \cdot \beta = \alpha \cdot V$.

\begin{lemma}
\label{lemma-rational-equivalence-and-intersection}
Let $X$ be a nonsingular variety. ========

regex_match $X$
nn candidate a nonsingular variety
nn a nonsingular variety
new_annotation [(50840, 50843, 'VAR'), (50847, 50868, 'TYPE')]
===== sent Let $a, b \in ========

new_annotation []
===== sent Let $k \geq 0$.
\begin{enumerate}
\item If $W \subset X \times \mathbf{P}^1$ is a closed subvariety
of dimension $k + 1$ which intersects $X \times a$ properly, then
\begin{enumerate}
\item $[W_a]_k = W \cdot X \times a$ as cycles on $X \times \mathbf{P}^1$, and
\item $[W_a]_k = \text{pr}_{X, *}(W \cdot X \times a)$ as cycles on $X$.
\end{enumerate}
\item Let $\alpha$ be a $(k + 1)$-cycle on $X \times \mathbf{P}^1$
which intersects $X \times a$ and $X \times b$ properly. ========

new_annotation []
===== sent Let $k \geq 0$.
\begin{enumerate}
\item If $W \subset X \times \mathbf{P}^1$ is a closed subvariety
of dimension $k + 1$ which intersects $X \times a$ properly, then
\begin{enumerate}
\item $[W_a]_k = W \cdot X \times a$ as cycles on $X \times \mathbf{P}^1$, and
\item $[W_a]_k = \text{pr}_{X, *}(W \cdot X \times a)$ as cycles on $X$.
\end{enumerate}
\item Let $\alpha$ be a $(k + 1)$-cycle on $X \times \mathbf{P}^1$
which intersects $X \times a$ and $X \times b$ properly. ========

regex_match $\alpha$
nn candidate a $(k + 1)$-cycle
nn a $(k + 1)$-cycle
new_annotation [(51287, 51295, 'VAR'), (51299, 51316, 'TYPE')]
===== sent \end{proof}

\noindent
For transversal intersections of closed subschemes the intersection
multiplicity is $1$.

\begin{lemma}
\label{lemma-transversal-subschemes}
Let $X$ be a nonsingular variety. ========

regex_match $X$
nn candidate a nonsingular variety
nn a nonsingular variety
new_annotation [(52465, 52468, 'VAR'), (52472, 52493, 'TYPE')]
===== sent Let $r, s \geq 0$ and let
$Y, Z \subset X$ be closed subschemes with $\dim(Y) \leq r$ and
$\dim(Z) \leq s$. Assume $[Y]_r = \sum n_i[Y_i]$ and
$[Z]_s = \sum m_j[Z_j]$ intersect properly.
 ========

new_annotation []
===== sent Let $T$ be an irreducible component of $Y_{i_0} \cap Z_{j_0}$
for some $i_0$ and $j_0$ and assume that the multiplicity
(in the sense of Section \ref{section-cycle-of-closed}) of $T$
in the closed subscheme $Y \cap Z$ is $1$.
Then
\begin{enumerate}
\item the coefficient of $T$ in $[Y]_r \cdot [Z]_s$ is $1$,
\item $Y$ and $Z$ are nonsingular at the generic point of $Z$,
\item $n_{i_0} = 1$, $m_{j_0} = 1$, and
\item $T$ is not contained in $Y_i$ or $Z_j$ for $i \not = i_0$ and
$j \not = j_0$.
\end{enumerate}
\end{lemma}

\begin{proof}
Set $n = \dim(X)$, $a = n - r$, $b = n - s$. Observe that
$\dim(T) = r + ========

regex_match $T$
nn candidate an irreducible component
nn an irreducible component
new_annotation [(52686, 52689, 'VAR'), (52693, 52717, 'TYPE')]
===== sent Let $(A, \mathfrak m, \kappa) =
(\mathcal{O}_{X, \xi}, \mathfrak m_\xi, \kappa(\xi))$ where $\xi \in T$
is the generic point. ========

new_annotation []
===== sent Then $\dim(A) = a + b$, see
Varieties, Lemma \ref{varieties-lemma-dimension-locally-algebraic}.
Let $I_0, I, J_0, J \subset A$ cut out the trace of
$Y_{i_0}$, $Y$, $Z_{j_0}$, $Z$ in $\Spec(A)$.
 ========

new_annotation []
===== sent Finally, the coefficient of $T$ in $[Y]_r \cdot [Z]_s$
is the coefficient of $T$ in $Y_{i_0} \cdot Z_{j_0}$ which is
is $1$ by Lemma \ref{lemma-transversal}.
\end{proof}



\section{Exterior product}
\label{section-exterior-product}

\noindent
Let $X$ and $Y$ be varieties.
 ========

new_annotation []
===== sent Let $V$, resp.\ $W$ be a closed subvariety of $X$, resp.\ $Y$.
The product $V\times ========

new_annotation []
===== sent \begin{lemma}
\label{lemma-exterior-product-rational-equivalence}
Let $X$ and $Y$ be varieties.
 ========

new_annotation []
===== sent Let $\alpha \in Z_r(X)$ and $\beta \in Z_s(Y)$.
 ========

new_annotation []
===== sent Let $W' \subset W_a$ be an irreducible component with generic point $\zeta$.
 ========

regex_match $W' \subset W_a$
nn candidate $W
nn candidate an irreducible component
nn an irreducible component
new_annotation [(57015, 57031, 'VAR'), (57035, 57059, 'TYPE')]
===== sent Let $\xi$ be the generic point of $V \times W'$. We have to show that
$$
\text{length}_{\mathcal{O}_{Y, \zeta}}(\mathcal{O}_{W_a, \zeta}) =
\text{length}_{\mathcal{O}_{X \times Y, \xi}}(
\mathcal{O}_{V \times W_a, \xi})
$$
In this formula we may replace
$\mathcal{O}_{Y, \zeta}$ by $\mathcal{O}_{W_a, \zeta}$ and
we may replace
$\mathcal{O}_{X \times Y, \zeta}$ by $\mathcal{O}_{V \times W_a, \zeta}$
(see Algebra, Lemma \ref{algebra-lemma-length-independent}).
 ========

regex_match $\xi$
nn candidate the generic point
nn the generic point
new_annotation [(57225, 57230, 'VAR'), (57234, 57251, 'TYPE')]
===== sent A_r(X) \otimes_\mathbf{Z} A_s(Y) \ar[r] &
A_{r + s}(X \times Y)
}
$$
for any pair of varieties $X$ and $Y$. For nonsingular varieties
we can think of the exterior product as an intersection product
of pullbacks.

\begin{lemma}
\label{lemma-exterior-product}
Let $X$ and $Y$ be nonsingular varieties.
 ========

new_annotation []
===== sent Let $\alpha \in Z_r(X)$ and $\beta \in Z_s(Y)$.
 ========

new_annotation []
===== sent Let $\xi$ be the generic point of $V \times W$.
 ========

regex_match $\xi$
nn candidate the generic point
nn the generic point
new_annotation [(59417, 59422, 'VAR'), (59426, 59443, 'TYPE')]
===== sent \end{proof}



\section{Reduction to the diagonal}
\label{section-reduction-diagonal}

\noindent
Let $X$ be a nonsingular variety. ========

regex_match $X$
nn candidate a nonsingular variety
nn a nonsingular variety
new_annotation [(60061, 60064, 'VAR'), (60068, 60089, 'TYPE')]
===== sent We will use $\Delta$
to denote either the diagonal morphism $\Delta : X \to X \times X$
or the image $\Delta \subset X \times X$.
Reduction to the diagonal is the statement that
intersection products on $X$ can be reduced to intersection products
of exterior products with the diagonal on $X \times X$.

\begin{lemma}
\label{lemma-tor-and-diagonal}
Let $X$ be a nonsingular variety.
 ========

regex_match $X$
nn candidate a nonsingular variety
nn a nonsingular variety
new_annotation [(60444, 60447, 'VAR'), (60451, 60472, 'TYPE')]
===== sent \end{proof}

\begin{lemma}
\label{lemma-reduction-diagonal}
Let $X$ be a nonsingular variety. ========

regex_match $X$
nn candidate a nonsingular variety
nn a nonsingular variety
new_annotation [(64579, 64582, 'VAR'), (64586, 64607, 'TYPE')]
===== sent Let $\alpha$, resp.\ $\beta$
be an $r$-cycle, resp.\ $s$-cycle on $X$. Assume $\alpha$ and $\beta$
intersect properly. ========

new_annotation []
===== sent Let $Z \subset V \cap W$ be an irreducible component
with generic point $\xi$. We have to show that the coefficient of
$Z$ in $\alpha \cdot \beta$ is the same as the coefficient of
$\Delta(Z)$ in $[\Delta] \cdot \alpha \times \beta$. The first is given
by the integer
$$
\sum (-1)^i
\text{length}_{\mathcal{O}_{X, \xi}}
\text{Tor}_i^{\mathcal{O}_X}(\mathcal{O}_V, \mathcal{O}_W)_\xi
$$
and the second by the integer
$$
\sum (-1)^i
\text{length}_{\mathcal{O}_{X \times Y, \Delta(\xi)}}
\text{Tor}_i^{\mathcal{O}_{X \times Y}}(
\mathcal{O}_\Delta, \mathcal{O}_{V \times W})_{\Delta(\xi)}
$$
However, by Lemma \ref{lemma-tor-and-diagonal} we have
$$
\text{Tor}_i^{\mathcal{O}_X}(\mathcal{O}_V, \mathcal{O}_W)_\xi \cong
\text{Tor}_i^{\mathcal{O}_{X \times Y}}(
\mathcal{O}_\Delta, \mathcal{O}_{V \times W})_{\Delta(\xi)}
$$
as $\mathcal{O}_{X \times X, \Delta(\xi)}$-modules. ========

regex_match $Z \subset V \cap W$
nn candidate $
nn candidate \subset V \cap
nn candidate an irreducible component
nn an irreducible component
new_annotation [(65538, 65558, 'VAR'), (65562, 65586, 'TYPE')]
===== sent This is one of the main results of \cite{Serre_algebre_locale}.
\end{reference}
Let $X$ be a nonsingular variety. ========

regex_match $X$
nn candidate a nonsingular variety
nn a nonsingular variety
new_annotation [(66807, 66810, 'VAR'), (66814, 66835, 'TYPE')]
===== sent Let $V \subset X$ and
$W \subset Y$ be closed subvarieties which intersect properly.
 ========

new_annotation []
===== sent Let $Z \subset V \cap W$ be an irreducible component.
 ========

regex_match $Z \subset V \cap W$
nn candidate $Z \subset V \cap
nn candidate an irreducible component
nn an irreducible component
new_annotation [(66926, 66946, 'VAR'), (66950, 66974, 'TYPE')]
===== sent \begin{lemma}
\label{lemma-tor-sheaf}
\begin{reference}
\cite[Chapter V]{Serre_algebre_locale}
\end{reference}
Let $X$ be a nonsingular variety. ========

regex_match $X$
nn candidate a nonsingular variety
nn a nonsingular variety
new_annotation [(67733, 67736, 'VAR'), (67740, 67761, 'TYPE')]
===== sent Let $\mathcal{F}$ and
$\mathcal{G}$ be coherent sheaves on $X$ with
$\dim(\text{Supp}(\mathcal{F})) \leq r$,
$\dim(\text{Supp}(\mathcal{G})) \leq s$, and
$\dim(\text{Supp}(\mathcal{F}) \cap \text{Supp}(\mathcal{G}) )
\leq r + s - \dim X$. ========

new_annotation []
===== sent If $\dim(V) = r$ and $\dim(W) = s$, then this equality is the
{\bf definition} of $V \cdot W$. On the other hand, if
$\dim(V) < r$ or $\dim(W) < s$, i.e., $[V]_r = 0$ or $[W]_s = 0$,
then we have to prove that $RHS(\mathcal{O}_V, \mathcal{O}_W) = 0$
\footnote{The reader can see that this is not a triviality by
taking $r = s = 1$ and $X$ a nonsingular surface and $V = W$
a closed point $x$ of $X$. In this case there are $3$ nonzero
$\text{Tor}$s of lengths $1, 2, 1$ at $x$.}.

\medskip\noindent
Let $Z \subset V \cap W$ be an irreducible component of dimension
$r + s - \dim(X)$. This is the maximal dimension of a component
and it suffices to show that the coefficient of $Z$ in $RHS$ is zero.
 ========

regex_match $Z \subset V \cap W$
nn candidate $Z \subset V \cap
nn candidate an irreducible component
nn an irreducible component
new_annotation [(71107, 71127, 'VAR'), (71131, 71155, 'TYPE')]
===== sent Let $\xi \in Z$ be the generic point. ========

regex_match $\xi \in Z$
nn candidate the generic point
nn the generic point
new_annotation [(71307, 71318, 'VAR'), (71322, 71339, 'TYPE')]
===== sent Hence vanishing by Lemma \ref{lemma-one-ideal-ci} because
the Hilbert function of $C$ with respect to $I$ has degree $\dim(C) < n$
by Algebra, Proposition \ref{algebra-proposition-dimension}.
\end{proof}

\begin{remark}
\label{remark-Serre-conjectures}
Let $(A, \mathfrak m, \kappa)$ be a regular local ring.
Let $M$ and $N$ be nonzero finite $A$-modules such that $M \otimes_A N$
is supported in $\{\mathfrak m\}$. Then
$$
\chi(M, N) = \sum (-1)^i \text{length}_A \text{Tor}_i^A(M, N)
$$
is finite. ========

regex_match $(A, \mathfrak m, \kappa)$
nn candidate A
nn candidate a regular local ring
nn a regular local ring
new_annotation [(72164, 72190, 'VAR'), (72194, 72214, 'TYPE')]
===== sent Hence vanishing by Lemma \ref{lemma-one-ideal-ci} because
the Hilbert function of $C$ with respect to $I$ has degree $\dim(C) < n$
by Algebra, Proposition \ref{algebra-proposition-dimension}.
\end{proof}

\begin{remark}
\label{remark-Serre-conjectures}
Let $(A, \mathfrak m, \kappa)$ be a regular local ring.
Let $M$ and $N$ be nonzero finite $A$-modules such that $M \otimes_A N$
is supported in $\{\mathfrak m\}$. Then
$$
\chi(M, N) = \sum (-1)^i \text{length}_A \text{Tor}_i^A(M, N)
$$
is finite. ========

new_annotation []
===== sent Let $r = \dim(\text{Supp}(M))$ and $s = \dim(\text{Supp}(N))$.
In \cite{Serre_algebre_locale} it is shown that $r + s \leq ========

new_annotation []
===== sent \begin{lemma}
\label{lemma-associative}
Let $X$ be a nonsingular variety. ========

regex_match $X$
nn candidate a nonsingular variety
nn a nonsingular variety
new_annotation [(73408, 73411, 'VAR'), (73415, 73436, 'TYPE')]
===== sent Let $U, V, W$ be closed
subvarieties. ========

new_annotation []
===== sent \begin{lemma}
\label{lemma-flat-pull-back-and-intersections-sheaves}
Let $f : X \to Y$ be a flat morphism of nonsingular varieties. ========

regex_match $f : X \to Y$
nn candidate X \to
nn candidate a flat morphism
nn a flat morphism
new_annotation [(75904, 75917, 'VAR'), (75921, 75936, 'TYPE')]
===== sent Let $\mathcal{F}$ and $\mathcal{G}$ be coherent
sheaves on $Y$ with $\dim(\text{Supp}(\mathcal{F})) \leq r$,
$\dim(\text{Supp}(\mathcal{G})) \leq s$, and
$\dim(\text{Supp}(\mathcal{F}) \cap \text{Supp}(\mathcal{G}) )
\leq r + s - \dim(Y)$. In this case the cycles
$[f^*\mathcal{F}]_{r + e}$ and $[f^*\mathcal{G}]_{s + e}$
intersect properly and
$$
f^*([\mathcal{F}]_r \cdot [\mathcal{G}]_s) =
[f^*\mathcal{F}]_{r + e} \cdot [f^*\mathcal{G}]_{s + e}
$$
\end{lemma}

\begin{proof}
The statement that $[f^*\mathcal{F}]_{r + e}$ and $[f^*\mathcal{G}]_{s + e}$
intersect properly is immediate from the assumption that $f$ has
relative dimension $e$. By
Lemmas \ref{lemma-tor-sheaf} and \ref{lemma-pullback}
it suffices to show that
$$
f^*\text{Tor}_i^{\mathcal{O}_Y}(\mathcal{F}, \mathcal{G}) =
\text{Tor}_i^{\mathcal{O}_X}(f^*\mathcal{F}, f^*\mathcal{G})
$$
as $\mathcal{O}_X$-modules. ========

new_annotation []
===== sent This follows from
Cohomology, Lemma \ref{cohomology-lemma-pullback-tensor-product}
and the fact that $f^*$ is exact, so $Lf^*\mathcal{F} = f^*\mathcal{F}$
and similarly for $\mathcal{G}$.
\end{proof}

\begin{lemma}
\label{lemma-flat-pullback-and-intersections}
Let $f : X \to Y$ be a flat morphism of nonsingular varieties.
 ========

regex_match $f : X \to Y$
nn candidate X \to
nn candidate a flat morphism
nn a flat morphism
new_annotation [(77139, 77152, 'VAR'), (77156, 77171, 'TYPE')]
===== sent Let $\alpha$ be a $r$-cycle on $Y$ and $\beta$ an $s$-cycle on $Y$.
Assume that $\alpha$ and $\beta$ intersect properly. ========

regex_match $\alpha$
nn candidate a $r$-cycle
nn a $r$-cycle
new_annotation [(77202, 77210, 'VAR'), (77214, 77225, 'TYPE')]
===== sent \end{reference}
Let $f : X \to Y$ be a flat proper morphism of nonsingular varieties.
 ========

regex_match $f : X \to Y$
nn candidate X \to
nn candidate a flat proper morphism
nn a flat proper morphism
new_annotation [(78285, 78298, 'VAR'), (78302, 78324, 'TYPE')]
===== sent Set $e = \dim(X) - \dim(Y)$. Let $\alpha$ be an $r$-cycle on $X$ and let
$\beta$ be a $s$-cycle on $Y$. Assume that $\alpha$ and $f^*(\beta)$ intersect
properly. ========

regex_match $\alpha$
nn candidate an $r$-cycle
nn an $r$-cycle
new_annotation [(78384, 78392, 'VAR'), (78396, 78408, 'TYPE')]
===== sent \medskip\noindent
Let $a$ be the dimension of the generic fibre of $V \to f(V)$.
If $a > 0$, then $f_*[V] = 0$. In particular $f_*\alpha$ and $\beta$
intersect properly. ========

regex_match $a$
nn candidate the dimension
nn the dimension
new_annotation [(79041, 79044, 'VAR'), (79048, 79061, 'TYPE')]
===== sent Let $Z \subset f(V) \cap ========

new_annotation []
===== sent Let $Z_i \subset V \cap f^{-1}(W)$,
$i = 1, \ldots, t$ be the irreducible components of $V \cap f^{-1}(W)$
dominating $Z$. By assumption each $Z_i$ has dimension
$r + s + e - \dim(X) = r + s - \dim(Y)$. Hence
$\dim(Z) \leq r + s - \dim(Y)$. ========

new_annotation []
===== sent \end{proof}

\begin{lemma}
\label{lemma-transfer}
Let $X \to P$ be a closed immersion of nonsingular varieties.
 ========

regex_match $X \to P$
nn candidate $X
nn candidate a closed immersion
nn a closed immersion
new_annotation [(82431, 82440, 'VAR'), (82444, 82462, 'TYPE')]
===== sent Let $C' \subset P \times \mathbf{P}^1$ be a closed subvariety of dimension
$r + 1$. Assume
\begin{enumerate}
\item the fibre $C = C'_0$ has dimension $r$, i.e., $C' \to \mathbf{P}^1$
is dominant,
\item $C'$ intersects $X \times \mathbf{P}^1$ properly,
\item $[C]_r$ intersects $X$ properly.
 ========

regex_match $C' \subset P \times \mathbf{P}^1$
nn candidate $C
nn candidate a closed subvariety
nn a closed subvariety
new_annotation [(82493, 82527, 'VAR'), (82531, 82550, 'TYPE')]
===== sent \end{lemma}

\begin{proof}
Let $\text{pr} : P \times \mathbf{P}^1 \to P$ be the projection.
 ========

regex_match $\text{pr} : P \times \mathbf{P}^1 \to P$
nn candidate P \times
nn candidate the projection
nn the projection
new_annotation [(83127, 83168, 'VAR'), (83172, 83186, 'TYPE')]
===== sent \medskip\noindent
Let $p \in \mathbf{P}(V)$ be a closed point. ========

regex_match $p \in \mathbf{P}(V)$
nn candidate a closed point
nn a closed point
new_annotation [(85260, 85281, 'VAR'), (85285, 85299, 'TYPE')]
===== sent \begin{lemma}
\label{lemma-projection-generically-finite}
Let $V$ be a vector space of dimension $n ========

regex_match $V$
nn candidate a vector space
nn a vector space
new_annotation [(85764, 85767, 'VAR'), (85771, 85785, 'TYPE')]
===== sent + 1$.
Let $X \subset \mathbf{P}(V)$ be a closed subscheme.
 ========

regex_match $X \subset \mathbf{P}(V)$
nn candidate $X
nn candidate a closed subscheme
nn a closed subscheme
new_annotation [(85812, 85837, 'VAR'), (85841, 85859, 'TYPE')]
===== sent \end{proof}

\begin{lemma}
\label{lemma-projection-generically-immersion}
Let $V$ be a vector space of dimension $n ========

regex_match $V$
nn candidate a vector space
nn a vector space
new_annotation [(86879, 86882, 'VAR'), (86886, 86900, 'TYPE')]
===== sent + 1$.
Let $X \subset \mathbf{P}(V)$ be a closed subvariety.
 ========

regex_match $X \subset \mathbf{P}(V)$
nn candidate $X
nn candidate a closed subvariety
nn a closed subvariety
new_annotation [(86927, 86952, 'VAR'), (86956, 86975, 'TYPE')]
===== sent Let $x \in X$ be a nonsingular point.
 ========

regex_match $x \in X$
nn candidate a nonsingular point
nn a nonsingular point
new_annotation [(86981, 86990, 'VAR'), (86994, 87013, 'TYPE')]
===== sent Let $U \subset \mathbf{P}(V) \setminus X$ be nonempty open and disjoint
from these images; such a $U$ exists because the images of $T$ and $T'$
in $\mathbf{P}(V)$ are constructible by
Morphisms, Lemma \ref{morphisms-lemma-chevalley}.
 ========

new_annotation []
===== sent \end{proof}

\begin{lemma}
\label{lemma-projection-injective}
Let $V$ be a vector space of dimension $n + 1$.
Let $Y, Z \subset \mathbf{P}(V)$ be closed subvarieties.
 ========

regex_match $V$
nn candidate a vector space
nn a vector space
new_annotation [(89497, 89500, 'VAR'), (89504, 89518, 'TYPE')]
===== sent \end{proof}

\begin{lemma}
\label{lemma-projection-injective}
Let $V$ be a vector space of dimension $n + 1$.
Let $Y, Z \subset \mathbf{P}(V)$ be closed subvarieties.
 ========

new_annotation []
===== sent Let $y \in Y'$, $z \in Z$ be closed points with $r_p(y) = r_p(z)$.
Then $p$ is on the line $\overline{yz}$ passing through $y$ and $z$.
Consider the finite type scheme
$$
T = \{(y, z, p) \mid y \in Y', z \in Z, p \in \overline{yz}\}
$$
and the morphism $T \to \mathbf{P}(V)$ given by $(y, z, p) \mapsto p$.
Observe that $T$ is irreducible and that $\dim(T) = \dim(Y) + \dim(Z) + 1$.
Hence the general fibre of $T \to \mathbf{P}(V)$ has dimension at most
$\dim(Y) + \dim(Z) + 1 - n$, more precisely, there exists a nonempty
open $U \subset \mathbf{P}(V) \setminus (Y \cup Z)$ over
which the fibre has dimension at most $\dim(Y) + \dim(Z) + 1 - n$
(Varieties, Lemma \ref{varieties-lemma-dimension-fibres-locally-algebraic}).
 ========

new_annotation []
===== sent Let $p \in U$ be a closed point and let $F \subset T$ be the fibre
of $T \to \mathbf{P}(V)$ over $p$. ========

regex_match $p \in U$
nn candidate a closed point
nn a closed point
new_annotation [(90616, 90625, 'VAR'), (90629, 90643, 'TYPE')]
===== sent Let $p \in U$ be a closed point and let $F \subset T$ be the fibre
of $T \to \mathbf{P}(V)$ over $p$. ========

regex_match $F \subset T$
nn candidate $F \subset
nn candidate the fibre
nn the fibre
new_annotation [(90652, 90665, 'VAR'), (90669, 90678, 'TYPE')]
===== sent Again by
Varieties, Lemma \ref{varieties-lemma-dimension-fibres-locally-algebraic}
the closure of the image of $F \to Y$ has dimension at most
$\dim(Y) + \dim(Z) + 1 - n$.
\end{proof}

\begin{lemma}
\label{lemma-find-lines}
Let $V$ be a vector space. ========

regex_match $V$
nn candidate a vector space
nn a vector space
new_annotation [(91050, 91053, 'VAR'), (91057, 91071, 'TYPE')]
===== sent Let $B \subset \mathbf{P}(V)$
be a closed subvariety of codimension $\geq 2$.
Let $p \in \mathbf{P}(V)$ be a closed point, $p ========

new_annotation []
===== sent Let $B \subset \mathbf{P}(V)$
be a closed subvariety of codimension $\geq 2$.
Let $p \in \mathbf{P}(V)$ be a closed point, $p ========

regex_match $p \in \mathbf{P}(V)$
nn candidate a closed point
nn a closed point
new_annotation [(91155, 91176, 'VAR'), (91180, 91194, 'TYPE')]
===== sent \end{proof}

\begin{lemma}
\label{lemma-doubly-transitive}
Let $V$ be a vector space. ========

regex_match $V$
nn candidate a vector space
nn a vector space
new_annotation [(91780, 91783, 'VAR'), (91787, 91801, 'TYPE')]
===== sent Let $G = \text{PGL}(V)$.
Then $G \times \mathbf{P}(V) \to \mathbf{P}(V)$ is
doubly transitive.
 ========

new_annotation []
===== sent \end{proof}

\begin{lemma}
\label{lemma-determinant}
Let $k$ be a field. ========

regex_match $k$
nn candidate a field
nn a field
new_annotation [(92109, 92112, 'VAR'), (92116, 92123, 'TYPE')]
===== sent Let $n \geq 1$ be an integer and let
$x_{ij}, 1 \leq i, j \leq n$ be variables. ========

regex_match $n \geq 1$
nn candidate $
nn candidate an integer
nn an integer
new_annotation [(92129, 92139, 'VAR'), (92143, 92153, 'TYPE')]
===== sent x_{n1} & \ldots & \ldots & x_{nn}
\end{matrix}
\right)
$$
is an irreducible element of the polynomial ring $k[x_{ij}]$.
\end{lemma}

\begin{proof}
Let $V$ be an $n$ dimensional vector space. ========

regex_match $V$
nn candidate an $n$ dimensional vector space
nn an $n$ dimensional vector space
new_annotation [(92502, 92505, 'VAR'), (92509, 92540, 'TYPE')]
===== sent Let $W$ be a vector space of dimension $n - 1$.
By elementary linear algebra, the morphism
$$
\Hom(W, V) \times ========

regex_match $W$
nn candidate a vector space
nn a vector space
new_annotation [(92669, 92672, 'VAR'), (92676, 92690, 'TYPE')]
===== sent \end{proof}

\noindent
Let $V$ be a vector space of dimension $n ========

regex_match $V$
nn candidate a vector space
nn a vector space
new_annotation [(92963, 92966, 'VAR'), (92970, 92984, 'TYPE')]
===== sent + 1$. Set $E = \text{End}(V)$.
Let $E^\vee = \Hom(E, \mathbf{C})$ be the dual vector space.
 ========

regex_match $E^\vee = \Hom(E, \mathbf{C})$
nn candidate the dual vector space
nn the dual vector space
new_annotation [(93036, 93066, 'VAR'), (93070, 93091, 'TYPE')]
===== sent $
Combining this with
Constructions, Lemma \ref{constructions-lemma-morphism-relative-proj}
we obtain
\begin{equation}
\label{equation-r-psi}
\mathbf{P} \times \mathbf{P}(V) \supset
U(\psi) \xrightarrow{r_\psi} \mathbf{P} \times \mathbf{P}(V)
\end{equation}
To understand this better we work out what happens on fibres over
$\mathbf{P}$. Let $g \in E$ be nonzero. ========

new_annotation []
===== sent Let $X, Y$ be closed subvarieties of $\mathbf{P}(V)$
which intersect properly such that $X \not ========

new_annotation []
===== sent = \mathbf{P}(V)$. There exists
a line $\ell \subset \mathbf{P}$ such that
\begin{enumerate}
\item $[\text{id}_V] \in \ell$,
\item $X \subset U_g$ for all $[g] \in \ell$,
\item $g(X)$ intersects $Y$ properly for all $[g] \in \ell$.
\end{enumerate}
\end{lemma}

\begin{proof}
Let $B \subset \mathbf{P}$ be the set of ``bad'' points, i.e., those
points $[g]$ that violate either (2) or (3). ========

regex_match $B \subset \mathbf{P}$
nn candidate $B \subset
nn candidate the set
nn the set
new_annotation [(95946, 95968, 'VAR'), (95972, 95979, 'TYPE')]
===== sent See \cite{Samuel}, \cite{ChevalleyI}, \cite{ChevalleyII}.
The key to this is Lemma \ref{lemma-moving}; the reader may find
this lemma in the form stated in
\cite[Example 11.4.1]{F} and find a proof in \cite{Roberts}.

\begin{lemma}
\label{lemma-moving}
\begin{reference}
See \cite{Roberts}.
\end{reference}
Let $X \subset \mathbf{P}^N$ be a nonsingular closed subvariety.
 ========

regex_match $X \subset \mathbf{P}^N$
nn candidate $X
nn candidate a nonsingular closed subvariety
nn a nonsingular closed subvariety
new_annotation [(99060, 99084, 'VAR'), (99088, 99119, 'TYPE')]
===== sent Let $n = \dim(X)$ and $0 \leq d, d' < n$. Let $Z \subset X$ be a closed
subvariety of dimension $d$ and $T_i \subset X$, $i \in I$ be a finite
collection of closed subvarieties of dimension $d'$. Then there exists
a subvariety $C \subset \mathbf{P}^N$ such that $C$ intersects $X$
properly and such that
$$
C \cdot X = Z + \sum\nolimits_{j \in J} m_j Z_j
$$
where $Z_j \subset X$ are irreducible of dimension $d$, distinct from $Z$, and
$$
\dim(Z_j \cap T_i) \leq ========

new_annotation []
===== sent Let $n = \dim(X)$ and $0 \leq d, d' < n$. Let $Z \subset X$ be a closed
subvariety of dimension $d$ and $T_i \subset X$, $i \in I$ be a finite
collection of closed subvarieties of dimension $d'$. Then there exists
a subvariety $C \subset \mathbf{P}^N$ such that $C$ intersects $X$
properly and such that
$$
C \cdot X = Z + \sum\nolimits_{j \in J} m_j Z_j
$$
where $Z_j \subset X$ are irreducible of dimension $d$, distinct from $Z$, and
$$
\dim(Z_j \cap T_i) \leq ========

regex_match $Z \subset X$
nn candidate $Z \subset
nn candidate a closed
subvariety
nn a closed
subvariety
new_annotation [(99167, 99180, 'VAR'), (99184, 99203, 'TYPE')]
===== sent \medskip\noindent
Let $C \subset \mathbf{P}(V_N)$ be the scheme theoretic closure of
$(r_{n + 1} \circ \ldots \circ r_N)^{-1}(\pi(Z))$. Because $\pi$
is \'etale at the point $x$ of $Z$, we see that the closed subscheme
$C \cap X$ contains $Z$ with multiplicity $1$ (local calculation omitted).
 ========

regex_match $C \subset \mathbf{P}(V_N)$
nn candidate the scheme theoretic closure
nn the scheme theoretic closure
new_annotation [(101304, 101331, 'VAR'), (101335, 101363, 'TYPE')]
===== sent Finally, let $V$ be an irreducible
component of $T_i \cap Z_j$ which is contained in $T_i \cap Z$. To finish
the proof it suffices to show that $V$ does not contain any of the
points $x_{it}$, because then $\dim(V) < \dim(Z \cap T_i)$.
To show this it suffices to show that $x_{it} \not \in Z_j$
for all $i, t, j$.

\medskip\noindent
Set $Z' = \pi(Z)$ and $Z'' = \pi^{-1}(Z')$, scheme theoretically. ========

regex_match $V$
nn candidate $V$
nn candidate an irreducible
component
nn an irreducible
component
new_annotation [(101999, 102002, 'VAR'), (102006, 102030, 'TYPE')]
===== sent \begin{lemma}
\label{lemma-move}
Let $C \subset \mathbf{P}^N$ be a closed subvariety.
Let $X \subset \mathbf{P}^N$ be subvariety and let $T_i \subset X$
be a finite collection of closed subvarieties.
 ========

regex_match $C \subset \mathbf{P}^N$
nn candidate a closed subvariety
nn a closed subvariety
new_annotation [(102992, 103016, 'VAR'), (103020, 103039, 'TYPE')]
===== sent \begin{lemma}
\label{lemma-move}
Let $C \subset \mathbf{P}^N$ be a closed subvariety.
Let $X \subset \mathbf{P}^N$ be subvariety and let $T_i \subset X$
be a finite collection of closed subvarieties.
 ========

new_annotation []
===== sent \begin{lemma}
\label{lemma-move}
Let $C \subset \mathbf{P}^N$ be a closed subvariety.
Let $X \subset \mathbf{P}^N$ be subvariety and let $T_i \subset X$
be a finite collection of closed subvarieties.
 ========

new_annotation []
===== sent Then there exists a closed subvariety
$C' \subset \mathbf{P}^N \times \mathbf{P}^1$ such that
\begin{enumerate}
\item $C' \to \mathbf{P}^1$ is dominant,
\item $C'_0 = C$ scheme theoretically,
\item $C'$ and $X \times \mathbf{P}^1$ intersect properly,
\item $C'_\infty$ properly intersects each of the given $T_i$.
\end{enumerate}
\end{lemma}

\begin{proof}
Write $\mathbf{P}^N = \mathbf{P}(V)$ so $\dim(V) = N + 1$. Let
$E = \text{End}(V)$. Let $E^\vee = \Hom(E, \mathbf{C})$. Set
$\mathbf{P} = \mathbf{P}(E^\vee)$ as in Lemma \ref{lemma-make-family}.
Choose a general line $\ell \subset \mathbf{P}$ passing through $\text{id}_V$.
Set $C' \subset \ell \times \mathbf{P}(V)$ equal to the
closed subscheme having fibre $r_g(C)$ over $[g] \in \ell$.
More precisely, $C'$ is the image of
$$
\ell \times C \subset \mathbf{P} \times \mathbf{P}(V)
$$
under the morphism (\ref{equation-r-psi}). ========

new_annotation []
===== sent \end{proof}

\begin{lemma}
\label{lemma-moving-move}
Let $X$ be a nonsingular projective variety. ========

regex_match $X$
nn candidate a nonsingular projective variety
nn a nonsingular projective variety
new_annotation [(104780, 104783, 'VAR'), (104787, 104819, 'TYPE')]
===== sent Let $\alpha$ be an
$r$-cycle and $\beta$ be an $s$-cycle on $X$. Then there exists
an $r$-cycle $\alpha'$ such that $\alpha' \sim_{rat} \alpha$ and
such that $\alpha'$ and $\beta$ intersect properly.
 ========

new_annotation []
===== sent Next, apply Lemma \ref{lemma-move}
to $C$, $X$, $T_i$ to find $C' \subset \mathbf{P}^N \times \mathbf{P}^1$.
Let $\gamma = C' \cdot X \times \mathbf{P}^1$ viewed as a cycle
on $X \times \mathbf{P}^1$. ========

new_annotation []
===== sent \begin{lemma}
\label{lemma-well-defined-special-case}
Let $X$ be a nonsingular variety. ========

regex_match $X$
nn candidate a nonsingular variety
nn a nonsingular variety
new_annotation [(106838, 106841, 'VAR'), (106845, 106866, 'TYPE')]
===== sent Let
$W \subset X \times \mathbf{P}^1$ be an $(s + 1)$-dimensional subvariety
dominating $\mathbf{P}^1$. Let $W_a$, resp.\ $W_b$ be the fibre of
$W \to \mathbf{P}^1$ over $a$, resp.\ $b$. Let $V$ be a $r$-dimensional
subvariety of $X$ such that $V$ intersects both $W_a$ and
$W_b$ properly. ========

new_annotation []
===== sent Let
$W \subset X \times \mathbf{P}^1$ be an $(s + 1)$-dimensional subvariety
dominating $\mathbf{P}^1$. Let $W_a$, resp.\ $W_b$ be the fibre of
$W \to \mathbf{P}^1$ over $a$, resp.\ $b$. Let $V$ be a $r$-dimensional
subvariety of $X$ such that $V$ intersects both $W_a$ and
$W_b$ properly. ========

regex_match $V$
nn candidate a $r$-dimensional
subvariety
nn a $r$-dimensional
subvariety
new_annotation [(107059, 107062, 'VAR'), (107066, 107094, 'TYPE')]
===== sent Let $Z_i$, $i \in I$ be the irreducible components of
$V \times \mathbf{P}^1 ========

new_annotation []
===== sent Let $U = \mathbf{P}^1 \setminus \{t_i, i \in I''\}$.
Note that $X \times a$ and $X \times b$ are contained in $X \times U$.
 ========

new_annotation []
===== sent Since we have the same for $b$
we conclude
\begin{align*}
V \cdot [W_a]
& =
\text{pr}_{X,*}(V \times \mathbf{P}^1 \cdot (W \cdot X\times a)) \\
& =
\text{pr}_{X, *}(\gamma \cdot X \times a) \\
& \sim_{rat} 
\text{pr}_{X, *}(\gamma \cdot X \times b) \\
& =
\text{pr}_{X,*}(V \times \mathbf{P}^1 \cdot (W \cdot X\times b)) \\
& =
V \cdot [W_b]
\end{align*}
The first and the last equality by the first paragraph of the proof,
the second and penultimate equalities were shown in this paragraph, and
the middle equivalence is
Lemma \ref{lemma-rational-equivalence-and-intersection}.
\end{proof}

\begin{theorem}
\label{theorem-well-defined}
Let $X$ be a nonsingular projective variety. ========

regex_match $X$
nn candidate a nonsingular projective variety
nn a nonsingular projective variety
new_annotation [(111680, 111683, 'VAR'), (111687, 111719, 'TYPE')]
===== sent Let $\alpha$, resp.\ $\beta$
be an $r$, resp.\ $s$ cycle on $X$. Assume that $\alpha$ and $\beta$
intersect properly so that $\alpha \cdot \beta$ is defined. ========

new_annotation []
===== sent The condition $\alpha \sim_{rat} 0$ means there
are finitely many $(r + 1)$-dimensional closed subvarieties
$W_i \subset X \times \mathbf{P}^1$ such that
$$
\alpha = \sum [W_{i, a_i}]_r - [W_{i, b_i}]_r
$$
for some pairs of points $a_i, b_i$ of $\mathbf{P}^1$.
Let $W_{i, a_i}^t$ and $W_{i, b_i}^t$ be the irreducible components
of $W_{i, a_i}$ and $W_{i, b_i}$.
We will use induction on the maximum $d$ of the integers
$$
\dim(Z \cap W_{i, a_i}^t),\quad ========

new_annotation []
===== sent Let $n = \dim(X)$ and $0 \leq d, d' < n$. Let
$X^{reg} \subset X$ be the open subset of nonsingular points. ========

new_annotation []
===== sent Then there exists a subvariety $C \subset \mathbf{P}^N$
such that $C$ intersects $X$ properly and such that
$$
(C \cdot X)|_{X^{reg}} = Z + \sum\nolimits_{j \in J} m_j Z_j
$$
where $Z_j \subset X^{reg}$ are irreducible of dimension $d$, distinct
from $Z$, and
$$
\dim(Z_j \cap T_i) \leq \dim(Z \cap T_i)
$$
with strict inequality if $Z$ does not intersect $T_i$ properly in $X^{reg}$.
\end{remark}



\section{Chow rings}
\label{section-chow-rings}

\noindent
Let $X$ be a nonsingular projective variety. ========

regex_match $X$
nn candidate a nonsingular projective variety
nn a nonsingular projective variety
new_annotation [(116629, 116632, 'VAR'), (116636, 116668, 'TYPE')]
===== sent Let $\alpha \in Z_r(X)$ and $\beta \in Z_s(X)$.
If $\alpha$ and $\beta$ intersect properly, we use the
definition given in Section \ref{section-intersection-product}.
If not, then we choose $\alpha \sim_{rat} \alpha'$ as in
Lemma \ref{lemma-moving-move} and we set
$$
\alpha \cdot \beta =
\text{class of }\alpha' \cdot \beta \in A_{r + s - \dim(X)}(X)
$$
This is well defined and passes through rational equivalence by
Theorem \ref{theorem-well-defined}. The intersection product
on $A_*(X)$ is commutative (this is clear), associative
(Lemma \ref{lemma-associative}) and has a unit $[X] \in A_{\dim(X)}(X)$.

\medskip\noindent
Often it is convenient to use $A^c(X) = A_{\dim X - c}(X)$ to denote the
group of codimension $c$ cycles modulo rational equivalence.
 ========

new_annotation []
===== sent The intersection product defines a product
$$
A^k(X) \times A^l(X) \longrightarrow A^{k+l}(X)
$$
which is commutative, associative, and has a unit $1 = [X] \in A^0(X)$.


\section{Pullback for a general morphism}
\label{section-general-pullback}

\noindent
Let $f : X \to Y$ be a morphism of nonsingular projective varieties.
 ========

regex_match $f : X \to Y$
nn candidate X \to
nn candidate a morphism
nn a morphism
new_annotation [(117859, 117872, 'VAR'), (117876, 117886, 'TYPE')]
===== sent \begin{lemma}
\label{lemma-pullback-and-intersection-product}
Let $f : X \to Y$ be a morphism of nonsingular projective varieties.
 ========

regex_match $f : X \to Y$
nn candidate X \to
nn candidate a morphism
nn a morphism
new_annotation [(118452, 118465, 'VAR'), (118469, 118479, 'TYPE')]
===== sent Suppose that $X$ and $Y$ be nonsingular
projective varieties, and let $f : X \to Y$ be a morphism.
 ========

regex_match $f : X \to Y$
nn candidate X \to
nn candidate a morphism
nn a morphism
new_annotation [(121750, 121763, 'VAR'), (121767, 121777, 'TYPE')]
===== sent Let $f^{-1}(Z)$
be the scheme theoretic inverse image:
$$
\xymatrix{
f^{-1}(Z) \ar[r] \ar[d] & Z \ar[d] \\
X \ar[r] & Y
}
$$
is a fibre product diagram of schemes. ========

new_annotation []
file:  spaces-simplicial.tex
===== sent \medskip\noindent
Let $X$ be a simplicial space. ========

regex_match $X$
nn candidate a simplicial space
nn a simplicial space
new_annotation [(1804, 1807, 'VAR'), (1811, 1829, 'TYPE')]
===== sent \begin{lemma}
\label{lemma-simplicial-site}
Let $X$ be a simplicial space. ========

regex_match $X$
nn candidate a simplicial space
nn a simplicial space
new_annotation [(2935, 2938, 'VAR'), (2942, 2960, 'TYPE')]
===== sent Let $X$ be a simplicial space. ========

regex_match $X$
nn candidate a simplicial space
nn a simplicial space
new_annotation [(3068, 3071, 'VAR'), (3075, 3093, 'TYPE')]
===== sent Let $\mathcal{F}$ be a sheaf on $X_{Zar}$.
It is clear from the definition of coverings, that the restriction
of $\mathcal{F}$ to the opens of $X_n$ defines a sheaf $\mathcal{F}_n$
on the topological space $X_n$. For every $\varphi : [m] \to [n]$ the
restriction maps of $\mathcal{F}$ for pairs $U \subset X_n$, $V \subset X_m$
with $X(\varphi)(U) \subset V$, define an $X(\varphi)$-map
$\mathcal{F}(\varphi) : \mathcal{F}_m \to \mathcal{F}_n$, see
Sheaves, Definition \ref{sheaves-definition-f-map}.
Moreover, given $\varphi : [m] \to [n]$ and $\psi : [l] \to [m]$
we have
$$
\mathcal{F}(\varphi) \circ \mathcal{F}(\psi) =
\mathcal{F}(\varphi \circ \psi)
$$
(LHS uses composition of $f$-maps, see
Sheaves, Definition \ref{sheaves-definition-composition-f-maps}).
 ========

regex_match $\mathcal{F}$
nn candidate a sheaf
nn a sheaf
new_annotation [(3099, 3112, 'VAR'), (3116, 3123, 'TYPE')]
===== sent Clearly, the converse is true as well: if we have a system
$(\{\mathcal{F}_n\}_{n \geq 0},
\{\mathcal{F}(\varphi)\}_{\varphi \in \text{Arrows}(\Delta)})$
as above, satisfying the displayed equalities,
then we obtain a sheaf on $X_{Zar}$.

\begin{lemma}
\label{lemma-describe-sheaves-simplicial-site}
Let $X$ be a simplicial space. ========

regex_match $X$
nn candidate a simplicial space
nn a simplicial space
new_annotation [(4163, 4166, 'VAR'), (4170, 4188, 'TYPE')]
===== sent \end{proof}

\begin{lemma}
\label{lemma-simplicial-space-site-functorial}
Let $f : Y \to X$ be a morphism of simplicial spaces.
 ========

regex_match $f : Y \to X$
nn candidate : Y \to X$
nn candidate a morphism
nn a morphism
new_annotation [(4506, 4519, 'VAR'), (4523, 4533, 'TYPE')]
===== sent Let $V \subset Y_n$ be an open subset. ========

regex_match $V \subset Y_n$
nn candidate $V
nn candidate an open subset
nn an open subset
new_annotation [(5095, 5110, 'VAR'), (5114, 5128, 'TYPE')]
===== sent Let $(U, \varphi)$ be an object. ========

regex_match $(U, \varphi)$
nn candidate an object
nn an object
new_annotation [(5759, 5773, 'VAR'), (5777, 5786, 'TYPE')]
===== sent \end{proof}

\begin{lemma}
\label{lemma-describe-functoriality}
Let $f : Y \to X$ be a morphism of simplicial spaces. ========

regex_match $f : Y \to X$
nn candidate a morphism
nn a morphism
new_annotation [(6655, 6668, 'VAR'), (6672, 6682, 'TYPE')]
===== sent Since $f_{Zar}^{-1}\mathcal{F} = u_s\mathcal{F}$ is the sheafification
of $u_p\mathcal{F}$ and since sheafification uses only coverings and
since coverings in $Y_{Zar}$ use only inclusions between opens on the
same $Y_n$, the result follows from the fact that $f_n^{-1}\mathcal{F}_n$
is (correspondingly) the sheafification of $f_{n, p}\mathcal{F}_n$, see
Sheaves, Section \ref{sheaves-section-presheaves-functorial}.
\end{proof}

\noindent
Let $X$ be a topological space. ========

regex_match $X$
nn candidate a topological space
nn a topological space
new_annotation [(8163, 8166, 'VAR'), (8170, 8189, 'TYPE')]
===== sent We identify the topos $\Sh(X_{Zar})$ with the category
of sheaves on $X$.

\begin{lemma}
\label{lemma-restriction-to-components}
Let $X$ be a simplicial space. ========

regex_match $X$
nn candidate a simplicial space
nn a simplicial space
new_annotation [(8504, 8507, 'VAR'), (8511, 8529, 'TYPE')]
===== sent Hence the lemma follows from the discussion in
Sites, Sections \ref{sites-section-cocontinuous-functors} and
\ref{sites-section-cocontinuous-morphism-topoi}
and
Modules on Sites, Section \ref{sites-modules-section-exactness-lower-shriek}.
More precisely,
Sites, Lemmas \ref{sites-lemma-cocontinuous-morphism-topoi},
\ref{sites-lemma-when-shriek}, and
\ref{sites-lemma-preserve-equalizers}
and
Modules on Sites, Lemmas
\ref{sites-modules-lemma-g-shriek-adjoint} and
\ref{sites-modules-lemma-exactness-lower-shriek}.
\end{proof}

\begin{lemma}
\label{lemma-restriction-injective-to-component}
Let $X$ be a simplicial space. ========

regex_match $X$
nn candidate a simplicial space
nn a simplicial space
new_annotation [(9922, 9925, 'VAR'), (9929, 9947, 'TYPE')]
===== sent If $\mathcal{I}$ is an injective abelian
sheaf on $X_{Zar}$, then $\mathcal{I}_n$ is an injective abelian sheaf
on $X_n$.
\end{lemma}

\begin{proof}
This follows from
Homology, Lemma \ref{homology-lemma-adjoint-preserve-injectives}
and
Lemma \ref{lemma-restriction-to-components}.
\end{proof}

\begin{lemma}
\label{lemma-restriction-to-components-functorial}
Let $f : Y \to X$ be a morphism of simplicial spaces. ========

regex_match $f : Y \to X$
nn candidate : Y \to X$
nn candidate a morphism
nn a morphism
new_annotation [(10312, 10325, 'VAR'), (10329, 10339, 'TYPE')]
===== sent \end{lemma}

\begin{proof}
Direct from the description of pullback functors in
Lemmas \ref{lemma-describe-functoriality} and
\ref{lemma-restriction-to-components}.
\end{proof}

\begin{lemma}
\label{lemma-augmentation}
Let $Y$ be a simplicial space and let $a : Y \to X$ be an augmentation
(Simplicial, Definition \ref{simplicial-definition-augmentation}).
 ========

regex_match $Y$
nn candidate a simplicial space
nn a simplicial space
new_annotation [(10738, 10741, 'VAR'), (10745, 10763, 'TYPE')]
===== sent \end{lemma}

\begin{proof}
Direct from the description of pullback functors in
Lemmas \ref{lemma-describe-functoriality} and
\ref{lemma-restriction-to-components}.
\end{proof}

\begin{lemma}
\label{lemma-augmentation}
Let $Y$ be a simplicial space and let $a : Y \to X$ be an augmentation
(Simplicial, Definition \ref{simplicial-definition-augmentation}).
 ========

regex_match $a : Y \to X$
nn candidate a : Y \to X$
nn candidate an augmentation
nn an augmentation
new_annotation [(10772, 10785, 'VAR'), (10789, 10804, 'TYPE')]
===== sent Let $a_n : Y_n \to X$ be the corresponding morphisms of topological spaces.
 ========

regex_match $a_n : Y_n \to X$
nn candidate \to
nn candidate the corresponding morphisms
nn the corresponding morphisms
new_annotation [(10876, 10893, 'VAR'), (10897, 10924, 'TYPE')]
===== sent The last fact is immediate from the exactness of
the functors $a_n^{-1}$.

\medskip\noindent
Let $\mathcal{F}$ be an object of $\Sh(X)$ and let $\mathcal{G}$
be an object of $\Sh(Y_{Zar})$. Given
$\beta : a^{-1}\mathcal{F} \to \mathcal{G}$ we can look at the
components $\beta_n : a_n^{-1}\mathcal{F} \to \mathcal{G}_n$.
These maps are adjoint to maps
$\beta_n : \mathcal{F} \to a_{n, *}\mathcal{G}_n$.
Compatibility with the simplicial structure shows that
$\beta_0$ maps into $a_*\mathcal{G}$.
Conversely, suppose given a map $\alpha : \mathcal{F} \to a_*\mathcal{G}$.
For any $n$ choose a $\varphi : [0] \to [n]$. ========

regex_match $\mathcal{F}$
nn candidate an object
nn an object
new_annotation [(12261, 12274, 'VAR'), (12278, 12287, 'TYPE')]
===== sent The last fact is immediate from the exactness of
the functors $a_n^{-1}$.

\medskip\noindent
Let $\mathcal{F}$ be an object of $\Sh(X)$ and let $\mathcal{G}$
be an object of $\Sh(Y_{Zar})$. Given
$\beta : a^{-1}\mathcal{F} \to \mathcal{G}$ we can look at the
components $\beta_n : a_n^{-1}\mathcal{F} \to \mathcal{G}_n$.
These maps are adjoint to maps
$\beta_n : \mathcal{F} \to a_{n, *}\mathcal{G}_n$.
Compatibility with the simplicial structure shows that
$\beta_0$ maps into $a_*\mathcal{G}$.
Conversely, suppose given a map $\alpha : \mathcal{F} \to a_*\mathcal{G}$.
For any $n$ choose a $\varphi : [0] \to [n]$. ========

new_annotation []
===== sent \end{proof}

\begin{lemma}
\label{lemma-simplicial-resolution-Z}
Let $X$ be a simplicial topological space. ========

regex_match $X$
nn candidate a simplicial topological space
nn a simplicial topological space
new_annotation [(13531, 13534, 'VAR'), (13538, 13568, 'TYPE')]
===== sent The complex of
abelian presheaves on $X_{Zar}$
$$
\ldots \to \mathbf{Z}_{X_2} \to \mathbf{Z}_{X_1} \to \mathbf{Z}_{X_0}
$$
with boundary $\sum (-1)^i d^n_i$ is a resolution
of the constant presheaf $\mathbf{Z}$.
\end{lemma}

\begin{proof}
Let $U \subset X_m$ be an object of $X_{Zar}$. Then the value of
the complex above on $U$ is the complex of abelian groups
$$
\ldots \to
\mathbf{Z}[\Mor_\Delta([2], [m])] \to
\mathbf{Z}[\Mor_\Delta([1], [m])] \to
\mathbf{Z}[\Mor_\Delta([0], [m])]
$$
 ========

regex_match $U \subset X_m$
nn candidate $U \subset X_m$
nn candidate an object
nn an object
new_annotation [(13813, 13828, 'VAR'), (13832, 13841, 'TYPE')]
===== sent This complex is acyclic in positive degrees
and equal to $\mathbf{Z}$ in degree $0$.
\end{proof}

\begin{lemma}
\label{lemma-simplicial-sheaf-cohomology}
Let $X$ be a simplicial topological space. ========

regex_match $X$
nn candidate a simplicial topological space
nn a simplicial topological space
new_annotation [(14795, 14798, 'VAR'), (14802, 14832, 'TYPE')]
===== sent Let $\mathcal{F}$ be an abelian
sheaf on $X$. There is a spectral sequence $(E_r, d_r)_{r \geq 0}$ with
$$
E_1^{p, q} = H^q(X_p, \mathcal{F}_p)
$$
converging to $H^{p + q}(X_{Zar}, \mathcal{F})$.
This spectral sequence is functorial in $\mathcal{F}$.
\end{lemma}

\begin{proof}
Let $\mathcal{F} \to \mathcal{I}^\bullet$ be an injective resolution.
 ========

regex_match $\mathcal{F}$
nn candidate an abelian
sheaf
nn an abelian
sheaf
new_annotation [(14838, 14851, 'VAR'), (14855, 14871, 'TYPE')]
===== sent Let $\mathcal{F}$ be an abelian
sheaf on $X$. There is a spectral sequence $(E_r, d_r)_{r \geq 0}$ with
$$
E_1^{p, q} = H^q(X_p, \mathcal{F}_p)
$$
converging to $H^{p + q}(X_{Zar}, \mathcal{F})$.
This spectral sequence is functorial in $\mathcal{F}$.
\end{lemma}

\begin{proof}
Let $\mathcal{F} \to \mathcal{I}^\bullet$ be an injective resolution.
 ========

regex_match $\mathcal{F} \to \mathcal{I}^\bullet$
nn candidate an injective resolution
nn an injective resolution
new_annotation [(15116, 15153, 'VAR'), (15157, 15180, 'TYPE')]
===== sent Hence the cohomology of the columns computes the groups
$H^q(X_p, \mathcal{F}_p)$. We conclude by applying
Homology, Lemmas \ref{homology-lemma-first-quadrant-ss} and
\ref{homology-lemma-double-complex-gives-resolution}.
\end{proof}

\begin{lemma}
\label{lemma-augmentation-pushforward-higher-direct-image}
Let $X$ be a simplicial space and let $a : X \to Y$
be an augmentation. ========

regex_match $X$
nn candidate a simplicial space
nn a simplicial space
new_annotation [(16497, 16500, 'VAR'), (16504, 16522, 'TYPE')]
===== sent Hence the cohomology of the columns computes the groups
$H^q(X_p, \mathcal{F}_p)$. We conclude by applying
Homology, Lemmas \ref{homology-lemma-first-quadrant-ss} and
\ref{homology-lemma-double-complex-gives-resolution}.
\end{proof}

\begin{lemma}
\label{lemma-augmentation-pushforward-higher-direct-image}
Let $X$ be a simplicial space and let $a : X \to Y$
be an augmentation. ========

new_annotation []
===== sent Let $\mathcal{F}$ be an abelian sheaf
on $X_{Zar}$. ========

regex_match $\mathcal{F}$
nn candidate an abelian sheaf
nn an abelian sheaf
new_annotation [(16569, 16582, 'VAR'), (16586, 16602, 'TYPE')]
===== sent V = a_0^{-1}(V)$ and $X_1 \times_Y V = a_1^{-1}(V)$,
\item[(2d)] $H^0((X \times_Y V)_{Zar}, \mathcal{F}|_{(X \times_Y V)_{Zar}})$
is the equalizer of the two maps
$H^0(X_0 \times_Y V, \mathcal{F}_0) \to H^0(X_1 \times_Y V, \mathcal{F}_1)$
for example by Lemma \ref{lemma-simplicial-sheaf-cohomology}.
\end{enumerate}
Part (1) follows after one defines an exact left adjoint
$j_! : \textit{Ab}((X \times_Y V)_{Zar}) \to \textit{Ab}(X_{Zar})$
(extension by zero) to restriction
$\textit{Ab}(X_{Zar}) \to \textit{Ab}((X \times_Y V)_{Zar})$
and using Homology, Lemma \ref{homology-lemma-adjoint-preserve-injectives}.
\end{proof}

\noindent
Let $X$ be a topological space. Denote $X_\bullet$ the constant simplicial
topological space with value $X$. By
Lemma \ref{lemma-describe-sheaves-simplicial-site}
a sheaf on $X_{\bullet, Zar}$ is the same
thing as a cosimplicial object in the category of sheaves on $X$.

\begin{lemma}
\label{lemma-constant-simplicial-space}
Let $X$ be a topological space. ========

regex_match $X$
nn candidate a topological space
nn a topological space
new_annotation [(18407, 18410, 'VAR'), (18414, 18433, 'TYPE')]
===== sent V = a_0^{-1}(V)$ and $X_1 \times_Y V = a_1^{-1}(V)$,
\item[(2d)] $H^0((X \times_Y V)_{Zar}, \mathcal{F}|_{(X \times_Y V)_{Zar}})$
is the equalizer of the two maps
$H^0(X_0 \times_Y V, \mathcal{F}_0) \to H^0(X_1 \times_Y V, \mathcal{F}_1)$
for example by Lemma \ref{lemma-simplicial-sheaf-cohomology}.
\end{enumerate}
Part (1) follows after one defines an exact left adjoint
$j_! : \textit{Ab}((X \times_Y V)_{Zar}) \to \textit{Ab}(X_{Zar})$
(extension by zero) to restriction
$\textit{Ab}(X_{Zar}) \to \textit{Ab}((X \times_Y V)_{Zar})$
and using Homology, Lemma \ref{homology-lemma-adjoint-preserve-injectives}.
\end{proof}

\noindent
Let $X$ be a topological space. Denote $X_\bullet$ the constant simplicial
topological space with value $X$. By
Lemma \ref{lemma-describe-sheaves-simplicial-site}
a sheaf on $X_{\bullet, Zar}$ is the same
thing as a cosimplicial object in the category of sheaves on $X$.

\begin{lemma}
\label{lemma-constant-simplicial-space}
Let $X$ be a topological space. ========

regex_match $X$
nn candidate a topological space
nn a topological space
new_annotation [(18733, 18736, 'VAR'), (18740, 18759, 'TYPE')]
===== sent Let $X_\bullet$ be the constant
simplicial topological space with value $X$. ========

regex_match $X_\bullet$
nn candidate the constant
simplicial topological space
nn the constant
simplicial topological space
new_annotation [(18765, 18776, 'VAR'), (18780, 18821, 'TYPE')]
===== sent \medskip\noindent
{\bf Case A.}
Let $\mathcal{C}$ be a simplicial object in the category whose objects
are sites and whose morphisms are morphisms of sites. ========

regex_match $\mathcal{C}$
nn candidate a simplicial object
nn a simplicial object
new_annotation [(21264, 21277, 'VAR'), (21281, 21300, 'TYPE')]
===== sent This means that
for every morphism $\varphi : [m] \to [n]$ of $\Delta$ we have a morphism
of sites $f_\varphi : \mathcal{C}_n \to \mathcal{C}_m$. This morphism is
given by a continuous functor in the opposite direction which we will denote
$u_\varphi : \mathcal{C}_m \to \mathcal{C}_n$.

\begin{lemma}
\label{lemma-simplicial-site-site}
Let $\mathcal{C}$ be a simplicial object in the category of sites.
 ========

regex_match $\mathcal{C}$
nn candidate a simplicial object
nn a simplicial object
new_annotation [(21726, 21739, 'VAR'), (21743, 21762, 'TYPE')]
===== sent = u_{\varphi \circ \psi}$.

\medskip\noindent
Let $\{(\text{id}, f_i) :  U_i \to U\}$ be a covering as in (3)
and ========

regex_match $\{(\text{id}, f_i) :  U_i \to U\}$
nn candidate a covering
nn a covering
new_annotation [(22619, 22654, 'VAR'), (22658, 22668, 'TYPE')]
===== sent let $(\varphi, g) : W \to U$ be a morphism with
$W \in \Ob(\mathcal{C}_m)$. We claim that
$$
W \times_{(\varphi, g), U, (\text{id}, f_i)} U_i =
W \times_{g, u_\varphi(U), u_\varphi(f_i)} u_\varphi(U_i)
$$
in the category $\mathcal{C}_{total}$. ========

regex_match $(\varphi, g) : W \to U$
nn candidate W \to
nn candidate a morphism
nn a morphism
new_annotation [(22687, 22711, 'VAR'), (22715, 22725, 'TYPE')]
===== sent \end{proof}

\noindent
{\bf Case B.}
Let $\mathcal{C}$ be a simplicial object in the category whose objects are
sites and whose morphisms are cocontinuous functors. ========

regex_match $\mathcal{C}$
nn candidate a simplicial object
nn a simplicial object
new_annotation [(23309, 23322, 'VAR'), (23326, 23345, 'TYPE')]
===== sent This means that for
every morphism $\varphi : [m] \to [n]$ of $\Delta$ we have a cocontinuous
functor denoted $u_\varphi : \mathcal{C}_n \to \mathcal{C}_m$. The associated
morphism of topoi is denoted
$f_\varphi : \Sh(\mathcal{C}_n) \to \Sh(\mathcal{C}_m)$.

\begin{lemma}
\label{lemma-simplicial-cocontinuous-site}
Let $\mathcal{C}$ be a simplicial object in the category whose objects are
sites and whose morphisms are cocontinuous functors. ========

regex_match $\mathcal{C}$
nn candidate a simplicial object
nn a simplicial object
new_annotation [(23753, 23766, 'VAR'), (23770, 23789, 'TYPE')]
===== sent = u_{\varphi \circ \psi}$.

\medskip\noindent
Let $\{(\text{id}, f_i) :  U_i \to ========

new_annotation []
===== sent U\}$ be a covering as in (3)
and let $(\varphi, g) : W \to U$ be a morphism with
$W \in \Ob(\mathcal{C}_m)$. We claim that
$$
W \times_{(\varphi, g), U, (\text{id}, f_i)} U_i =
W \times_{g, U, f_i} U_i
$$
in the category $\mathcal{C}_{total}$ where the right hand side
is the object of $\mathcal{C}_m$ defined in
Sites, Remark \ref{sites-remark-cartesian-cocontinuous}
which exists by property $P$. Compatibility of this type of fibre product
with compositions of functors implies the claim (details omitted).
 ========

regex_match $(\varphi, g) : W \to U$
nn candidate W \to
nn candidate a morphism
nn a morphism
new_annotation [(24926, 24950, 'VAR'), (24954, 24964, 'TYPE')]
===== sent We let $\mathcal{C}_{total}$ denote the
site defined in
Lemma \ref{lemma-simplicial-site-site} (case A) or
Lemma \ref{lemma-simplicial-cocontinuous-site} (case B).
 ========

new_annotation []
===== sent \end{situation}

\noindent
Let $\mathcal{C}$ be as in Situation \ref{situation-simplicial-site}.
Let $\mathcal{F}$ be a sheaf on $\mathcal{C}_{total}$.
It is clear from the definition of coverings, that the restriction
of $\mathcal{F}$ to the objects of $\mathcal{C}_n$ defines a sheaf
$\mathcal{F}_n$ on the site $\mathcal{C}_n$. For every
$\varphi : [m] ========

new_annotation []
===== sent \end{situation}

\noindent
Let $\mathcal{C}$ be as in Situation \ref{situation-simplicial-site}.
Let $\mathcal{F}$ be a sheaf on $\mathcal{C}_{total}$.
It is clear from the definition of coverings, that the restriction
of $\mathcal{F}$ to the objects of $\mathcal{C}_n$ defines a sheaf
$\mathcal{F}_n$ on the site $\mathcal{C}_n$. For every
$\varphi : [m] ========

regex_match $\mathcal{F}$
nn candidate a sheaf
nn a sheaf
new_annotation [(26889, 26902, 'VAR'), (26906, 26913, 'TYPE')]
===== sent Let $\mathcal{G}$ be a sheaf on $\mathcal{C}_n$.
Consider the sheaf $\mathcal{H}$ on $\mathcal{C}_{total}$
whose degree $m$ part is the sheaf
$$
\mathcal{H}_m = \coprod\nolimits_{\varphi : [n] \to [m]}
f_\varphi^{-1}\mathcal{G}
$$
given in part (3) of the statement of the lemma.
 ========

regex_match $\mathcal{G}$
nn candidate a sheaf
nn a sheaf
new_annotation [(31767, 31780, 'VAR'), (31784, 31791, 'TYPE')]
===== sent In Situation \ref{situation-simplicial-site} let $a_0$ be an
augmentation towards a site $\mathcal{D}$ as in
Remark \ref{remark-augmentation-site}. ========

new_annotation []
===== sent Then $a_0$ induces
\begin{enumerate}
\item a morphism of topoi $a_n : \Sh(\mathcal{C}_n) \to \Sh(\mathcal{D})$
for all $n \geq 0$,
\item a morphism of topoi $a : \Sh(\mathcal{C}_{total}) \to \Sh(\mathcal{D})$
\end{enumerate}
such that
\begin{enumerate}
\item for all $\varphi : [m] \to [n]$ we have $a_m \circ f_\varphi = a_n$,
\item if $g_n : \Sh(\mathcal{C}_n) \to \Sh(\mathcal{C}_{total})$
is as in Lemma \ref{lemma-restriction-to-components-site}, then
$a \circ g_n = a_n$, and
\item $a_*\mathcal{F}$ for $\mathcal{F} \in \Sh(\mathcal{C}_{total})$
is the equalizer of the two maps
$a_{0, *}\mathcal{F}_0 \to a_{1, *}\mathcal{F}_1$.
\end{enumerate}
\end{lemma}

\begin{proof}
Case A. Let $u_n : \mathcal{D} \to \mathcal{C}_n$ be the common
value of the functors $u_\varphi \circ u_0$ for $\varphi : [0] ========

regex_match $u_n : \mathcal{D} \to \mathcal{C}_n$
nn candidate the common
value
nn the common
value
new_annotation [(36582, 36619, 'VAR'), (36623, 36639, 'TYPE')]
===== sent \to [n]$.
Then $u_n$ corresponds to a morphism of sites
$a_n : \mathcal{C}_n \to \mathcal{D}$, see
Sites, Lemma \ref{sites-lemma-composition-morphisms-sites}.
The same lemma shows that for all $\varphi : [m] \to [n]$ we have
$a_m \circ f_\varphi = a_n$.

\medskip\noindent
Case B. Let $u_n : \mathcal{C}_n \to \mathcal{D}$ be the common
value of the functors $u_0 \circ u_\varphi$ for $\varphi : [0] ========

regex_match $u_n : \mathcal{C}_n \to \mathcal{D}$
nn candidate the common
value
nn the common
value
new_annotation [(36982, 37019, 'VAR'), (37023, 37039, 'TYPE')]
===== sent Finally, for $a_* : \Sh(\mathcal{C}_{total}) \to \Sh(\mathcal{D})$
we take the functor which to a sheaf $\mathcal{F}$ on $\Sh(\mathcal{D})$
associates
$$
\xymatrix{
a_*\mathcal{F} \ar@{=}[r] &
\text{Equalizer}(a_{0, *}\mathcal{F}_0
\ar@<1ex>[r] \ar@<-1ex>[r] &
a_{1, *}\mathcal{F}_1)
}
$$
Here the two maps come from the two maps $\varphi : [0] \to [1]$
via
$$
a_{0, *}\mathcal{F}_0 \to
a_{0, *}f_{\varphi, *} f_\varphi^{-1}\mathcal{F}_0
\xrightarrow{\mathcal{F}(\varphi)}
a_{0, *}f_{\varphi, *} \mathcal{F}_0 = a_{1, *}\mathcal{F}_1
$$
where the first arrow comes from $1 \to f_{\varphi, *} f_\varphi^{-1}$.
Let $\mathcal{G}_\bullet$ denote the constant simplicial sheaf
with value $\mathcal{G}$ and let $a_{\bullet, *}\mathcal{F}$
denote the simplicial sheaf having $a_{n, *}\mathcal{F}_n$ in degree $n$.
By the usual adjuntion for the morphisms of topoi $a_n$ we see that
a map $a^{-1}\mathcal{G} \to \mathcal{F}$
is the same thing as a map
$$
\mathcal{G}_\bullet \longrightarrow a_{\bullet, *}\mathcal{F}
$$
of simplicial sheaves.
 ========

new_annotation []
===== sent Finally, for $a_* : \Sh(\mathcal{C}_{total}) \to \Sh(\mathcal{D})$
we take the functor which to a sheaf $\mathcal{F}$ on $\Sh(\mathcal{D})$
associates
$$
\xymatrix{
a_*\mathcal{F} \ar@{=}[r] &
\text{Equalizer}(a_{0, *}\mathcal{F}_0
\ar@<1ex>[r] \ar@<-1ex>[r] &
a_{1, *}\mathcal{F}_1)
}
$$
Here the two maps come from the two maps $\varphi : [0] \to [1]$
via
$$
a_{0, *}\mathcal{F}_0 \to
a_{0, *}f_{\varphi, *} f_\varphi^{-1}\mathcal{F}_0
\xrightarrow{\mathcal{F}(\varphi)}
a_{0, *}f_{\varphi, *} \mathcal{F}_0 = a_{1, *}\mathcal{F}_1
$$
where the first arrow comes from $1 \to f_{\varphi, *} f_\varphi^{-1}$.
Let $\mathcal{G}_\bullet$ denote the constant simplicial sheaf
with value $\mathcal{G}$ and let $a_{\bullet, *}\mathcal{F}$
denote the simplicial sheaf having $a_{n, *}\mathcal{F}_n$ in degree $n$.
By the usual adjuntion for the morphisms of topoi $a_n$ we see that
a map $a^{-1}\mathcal{G} \to \mathcal{F}$
is the same thing as a map
$$
\mathcal{G}_\bullet \longrightarrow a_{\bullet, *}\mathcal{F}
$$
of simplicial sheaves.
 ========

new_annotation []
===== sent \begin{remark}
\label{remark-morphism-simplicial-sites}
Let $\mathcal{C}_n, f_\varphi, u_\varphi$ and
$\mathcal{C}'_n, f'_\varphi, u'_\varphi$ be as in
Situation \ref{situation-simplicial-site}. A
{\it morphism $h$ between simplicial sites} will mean
\begin{enumerate}
\item[(A)] Morphisms of sites
$h_n : \mathcal{C}_n \to \mathcal{C}'_n$
such that $f'_\varphi \circ h_n = h_m \circ f_\varphi$
as morphisms of sites for all $\varphi : [m] \to [n]$.
\item[(B)] Cocontinuous functors
$v_n : \mathcal{C}_n \to \mathcal{C}'_n$
inducing morphisms of topoi $h_n : \Sh(\mathcal{C}_n) \to \Sh(\mathcal{C}'_n)$
such that $u'_\varphi \circ v_n = v_m \circ u_\varphi$
as functors for all $\varphi : [m] \to [n]$.
\end{enumerate}
 ========

new_annotation []
===== sent In both cases we have
$f'_\varphi \circ h_n = h_m \circ f_\varphi$
as morphisms of topoi, see
Sites, Lemma \ref{sites-lemma-composition-cocontinuous}
for case B and Sites,
Definition \ref{sites-definition-composition-morphisms-sites}
for case A.
\end{remark}

\begin{lemma}
\label{lemma-morphism-simplicial-sites}
Let $\mathcal{C}_n, f_\varphi, u_\varphi$ and
$\mathcal{C}'_n, f'_\varphi, u'_\varphi$ be as in
Situation \ref{situation-simplicial-site}.
Let $h$ be a morphism between simplicial sites as in
Remark \ref{remark-morphism-simplicial-sites}.
Then we obtain a morphism of topoi
$$
h_{total} : \Sh(\mathcal{C}_{total}) \to \Sh(\mathcal{C}'_{total})
$$
and commutative diagrams
$$
\xymatrix{
\Sh(\mathcal{C}_n) \ar[d]_{g_n} \ar[r]_{h_n} &
\Sh(\mathcal{C}'_n) \ar[d]^{g'_n} \\
 ========

new_annotation []
===== sent In both cases we have
$f'_\varphi \circ h_n = h_m \circ f_\varphi$
as morphisms of topoi, see
Sites, Lemma \ref{sites-lemma-composition-cocontinuous}
for case B and Sites,
Definition \ref{sites-definition-composition-morphisms-sites}
for case A.
\end{remark}

\begin{lemma}
\label{lemma-morphism-simplicial-sites}
Let $\mathcal{C}_n, f_\varphi, u_\varphi$ and
$\mathcal{C}'_n, f'_\varphi, u'_\varphi$ be as in
Situation \ref{situation-simplicial-site}.
Let $h$ be a morphism between simplicial sites as in
Remark \ref{remark-morphism-simplicial-sites}.
Then we obtain a morphism of topoi
$$
h_{total} : \Sh(\mathcal{C}_{total}) \to \Sh(\mathcal{C}'_{total})
$$
and commutative diagrams
$$
\xymatrix{
\Sh(\mathcal{C}_n) \ar[d]_{g_n} \ar[r]_{h_n} &
\Sh(\mathcal{C}'_n) \ar[d]^{g'_n} \\
 ========

regex_match $h$
nn candidate a morphism
nn a morphism
new_annotation [(40951, 40954, 'VAR'), (40958, 40968, 'TYPE')]
===== sent Let $h_{total}$ be the morphism of topoi associated to $v_{total}$.
The commutativity of the displayed diagram of the lemma follows
immediately from Sites, Lemma ========

regex_match $h_{total}$
nn candidate the morphism
nn the morphism
new_annotation [(43551, 43562, 'VAR'), (43566, 43578, 'TYPE')]
===== sent With notation and hypotheses as in Lemma \ref{lemma-morphism-simplicial-sites}.
For $K \in D(\mathcal{C}_{total})$ we have
$(g'_n)^{-1}Rh_{total, *}K = Rh_{n, *}g_n^{-1}K$.
\end{lemma}

\begin{proof}
Let $\mathcal{I}^\bullet$ be a K-injective complex on $\mathcal{C}_{total}$
representing $K$. ========

regex_match $\mathcal{I}^\bullet$
nn candidate a K-injective complex
nn a K-injective complex
new_annotation [(44563, 44584, 'VAR'), (44588, 44609, 'TYPE')]
===== sent \end{proof}

\begin{remark}
\label{remark-morphism-augmentation-simplicial-sites}
Let $\mathcal{C}_n, f_\varphi, u_\varphi$ and
$\mathcal{C}'_n, f'_\varphi, u'_\varphi$ be as in
Situation \ref{situation-simplicial-site}.
Let $a_0$, resp.\ $a'_0$ be an augmentation
towards a site $\mathcal{D}$, resp.\ $\mathcal{D}'$
as in Remark \ref{remark-augmentation-site}.
Let $h$ be a morphism between simplicial sites as in
Remark \ref{remark-morphism-simplicial-sites}.
We say a morphism of topoi $h_{-1} : \Sh(\mathcal{D}) \to \Sh(\mathcal{D}')$
is {\it compatible with $h$, $a_0$, $a'_0$} if
\begin{enumerate}
 ========

new_annotation []
===== sent \end{proof}

\begin{remark}
\label{remark-morphism-augmentation-simplicial-sites}
Let $\mathcal{C}_n, f_\varphi, u_\varphi$ and
$\mathcal{C}'_n, f'_\varphi, u'_\varphi$ be as in
Situation \ref{situation-simplicial-site}.
Let $a_0$, resp.\ $a'_0$ be an augmentation
towards a site $\mathcal{D}$, resp.\ $\mathcal{D}'$
as in Remark \ref{remark-augmentation-site}.
Let $h$ be a morphism between simplicial sites as in
Remark \ref{remark-morphism-simplicial-sites}.
We say a morphism of topoi $h_{-1} : \Sh(\mathcal{D}) \to \Sh(\mathcal{D}')$
is {\it compatible with $h$, $a_0$, $a'_0$} if
\begin{enumerate}
 ========

new_annotation []
===== sent \end{proof}

\begin{remark}
\label{remark-morphism-augmentation-simplicial-sites}
Let $\mathcal{C}_n, f_\varphi, u_\varphi$ and
$\mathcal{C}'_n, f'_\varphi, u'_\varphi$ be as in
Situation \ref{situation-simplicial-site}.
Let $a_0$, resp.\ $a'_0$ be an augmentation
towards a site $\mathcal{D}$, resp.\ $\mathcal{D}'$
as in Remark \ref{remark-augmentation-site}.
Let $h$ be a morphism between simplicial sites as in
Remark \ref{remark-morphism-simplicial-sites}.
We say a morphism of topoi $h_{-1} : \Sh(\mathcal{D}) \to \Sh(\mathcal{D}')$
is {\it compatible with $h$, $a_0$, $a'_0$} if
\begin{enumerate}
 ========

regex_match $h$
nn candidate a morphism
nn a morphism
new_annotation [(45365, 45368, 'VAR'), (45372, 45382, 'TYPE')]
===== sent \end{enumerate}
In both cases we have $a'_0 \circ h_0 = h_{-1} \circ a_0$
as morphisms of topoi, see
Sites, Lemma \ref{sites-lemma-composition-cocontinuous}
for case B and Sites,
Definition \ref{sites-definition-composition-morphisms-sites}
for case A.
\end{remark}

\begin{lemma}
\label{lemma-morphism-augmentation-simplicial-sites}
Let $\mathcal{C}_n, f_\varphi, u_\varphi, \mathcal{D}, a_0$,
$\mathcal{C}'_n, f'_\varphi, u'_\varphi, \mathcal{D}', a'_0$, and
$h_n$, $n \geq -1$ be as in
Remark \ref{remark-morphism-augmentation-simplicial-sites}.
Then we obtain a commutative diagram
$$
\xymatrix{
\Sh(\mathcal{C}_{total}) \ar[d]_a \ar[r]_{h_{total}} &
\Sh(\mathcal{C}'_{total}) \ar[d]^{a'} \\
\Sh(\mathcal{D}) \ar[r]^{h_{-1}} &
\Sh(\mathcal{D}')
}
$$
\end{lemma}

\begin{proof}
The morphism $h$ is defined in Lemma \ref{lemma-morphism-simplicial-sites}.
 ========

new_annotation []
===== sent \begin{lemma}
\label{lemma-restriction-module-to-components-site}
In Situation \ref{situation-simplicial-site}. Let $\mathcal{O}$
be a sheaf of rings on $\mathcal{C}_{total}$.
There is a canonical morphism of ringed topoi
$g_n : (\Sh(\mathcal{C}_n), \mathcal{O}_n) \to
(\Sh(\mathcal{C}_{total}), \mathcal{O})$
agreeing with the morphism $g_n$ of
Lemma \ref{lemma-restriction-to-components-site} on underlying topoi.
 ========

new_annotation []
===== sent In Situation \ref{situation-simplicial-site}.
Let $\mathcal{O}$ be a sheaf of rings on $\mathcal{C}_{total}$.
If $\mathcal{I}$ is injective in $\textit{Mod}(\mathcal{O})$, then
$\mathcal{I}_n$ is a limp sheaf on $\mathcal{C}_n$.
\end{lemma}

\begin{proof}
This follows from
Cohomology on Sites, Lemma \ref{sites-cohomology-lemma-pullback-injective-limp}
applied to the inclusion functor $\mathcal{C}_n \to \mathcal{C}_{total}$
and its properties proven in Lemma \ref{lemma-restriction-to-components-site}.
\end{proof}

\begin{lemma}
\label{lemma-exactness-g-shriek-modules}
 ========

regex_match $\mathcal{O}$
nn candidate a sheaf
nn a sheaf
new_annotation [(50474, 50487, 'VAR'), (50491, 50498, 'TYPE')]
===== sent \end{proof}

\begin{lemma}
\label{lemma-restriction-injective-to-component-site-module}
In Situation \ref{situation-simplicial-site}.
Let $\mathcal{O}$ be a sheaf of rings on $\mathcal{C}_{total}$
such that $f_\varphi^{-1}\mathcal{O}_n \to \mathcal{O}_m$
is flat for all $\varphi : [n] \to [m]$.
If $\mathcal{I}$ is injective in $\textit{Mod}(\mathcal{O})$, then
$\mathcal{I}_n$ is injective in $\textit{Mod}(\mathcal{O}_n)$.
\end{lemma}

\begin{proof}
 ========

regex_match $\mathcal{O}$
nn candidate a sheaf
nn a sheaf
new_annotation [(52089, 52102, 'VAR'), (52106, 52113, 'TYPE')]
===== sent This follows from
Homology, Lemma \ref{homology-lemma-adjoint-preserve-injectives}
and
Lemma \ref{lemma-exactness-g-shriek-modules}.
\end{proof}







\section{Morphisms of ringed simplicial sites}
\label{section-morphism-simplicial-sites-modules}

\noindent
We continue the discussion of Section \ref{section-morphism-simplicial-sites}.

\begin{remark}
\label{remark-morphism-simplicial-sites-modules}
Let $\mathcal{C}_n, f_\varphi, u_\varphi$ and
$\mathcal{C}'_n, f'_\varphi, u'_\varphi$ be as in
Situation \ref{situation-simplicial-site}.
Let $\mathcal{O}$ and $\mathcal{O}'$
be a sheaf of rings on $\mathcal{C}_{total}$ and $\mathcal{C}'_{total}$.
We will say that $(h, h^\sharp)$ is a
{\it morphism between ringed simplicial sites}
if $h$ is a morphism between simplicial sites as in
Remark \ref{remark-morphism-simplicial-sites}
and $h^\sharp : h_{total}^{-1}\mathcal{O}' \to \mathcal{O}$
or equivalently $h^\sharp : \mathcal{O}' \to h_{total, *}\mathcal{O}$
is a homomorphism of sheaves of rings.
 ========

new_annotation []
===== sent This follows from
Homology, Lemma \ref{homology-lemma-adjoint-preserve-injectives}
and
Lemma \ref{lemma-exactness-g-shriek-modules}.
\end{proof}







\section{Morphisms of ringed simplicial sites}
\label{section-morphism-simplicial-sites-modules}

\noindent
We continue the discussion of Section \ref{section-morphism-simplicial-sites}.

\begin{remark}
\label{remark-morphism-simplicial-sites-modules}
Let $\mathcal{C}_n, f_\varphi, u_\varphi$ and
$\mathcal{C}'_n, f'_\varphi, u'_\varphi$ be as in
Situation \ref{situation-simplicial-site}.
Let $\mathcal{O}$ and $\mathcal{O}'$
be a sheaf of rings on $\mathcal{C}_{total}$ and $\mathcal{C}'_{total}$.
We will say that $(h, h^\sharp)$ is a
{\it morphism between ringed simplicial sites}
if $h$ is a morphism between simplicial sites as in
Remark \ref{remark-morphism-simplicial-sites}
and $h^\sharp : h_{total}^{-1}\mathcal{O}' \to \mathcal{O}$
or equivalently $h^\sharp : \mathcal{O}' \to h_{total, *}\mathcal{O}$
is a homomorphism of sheaves of rings.
 ========

new_annotation []
===== sent \end{remark}

\begin{lemma}
\label{lemma-morphism-simplicial-sites-modules}
Let $\mathcal{C}_n, f_\varphi, u_\varphi$ and
$\mathcal{C}'_n, f'_\varphi, u'_\varphi$ be as in
Situation \ref{situation-simplicial-site}.
Let $\mathcal{O}$ and $\mathcal{O}'$
be a sheaf of rings on $\mathcal{C}_{total}$ and $\mathcal{C}'_{total}$.
Let $(h, h^\sharp)$ be a morphism between simplicial sites as in
Remark \ref{remark-morphism-simplicial-sites-modules}.
Then we obtain a morphism of ringed topoi
$$
h_{total} :
(\Sh(\mathcal{C}_{total}, \mathcal{O})
\to
(\Sh(\mathcal{C}'_{total}), \mathcal{O}')
$$
and commutative diagrams
$$
\xymatrix{
(\Sh(\mathcal{C}_n), \mathcal{O}_n) \ar[d]_{g_n} \ar[r]_{h_n} &
(\Sh(\mathcal{C}'_n), \mathcal{O}'_n) \ar[d]^{g'_n} \\
(\Sh(\mathcal{C}_{total}), \mathcal{O}) \ar[r]^{h_{total}} &
(\Sh(\mathcal{C}'_{total}), \mathcal{O}')
}
$$
of ringed topoi where $g_n$ and $g'_n$ are as in
Lemma \ref{lemma-restriction-module-to-components-site}.
 ========

new_annotation []
===== sent \end{remark}

\begin{lemma}
\label{lemma-morphism-simplicial-sites-modules}
Let $\mathcal{C}_n, f_\varphi, u_\varphi$ and
$\mathcal{C}'_n, f'_\varphi, u'_\varphi$ be as in
Situation \ref{situation-simplicial-site}.
Let $\mathcal{O}$ and $\mathcal{O}'$
be a sheaf of rings on $\mathcal{C}_{total}$ and $\mathcal{C}'_{total}$.
Let $(h, h^\sharp)$ be a morphism between simplicial sites as in
Remark \ref{remark-morphism-simplicial-sites-modules}.
Then we obtain a morphism of ringed topoi
$$
h_{total} :
(\Sh(\mathcal{C}_{total}, \mathcal{O})
\to
(\Sh(\mathcal{C}'_{total}), \mathcal{O}')
$$
and commutative diagrams
$$
\xymatrix{
(\Sh(\mathcal{C}_n), \mathcal{O}_n) \ar[d]_{g_n} \ar[r]_{h_n} &
(\Sh(\mathcal{C}'_n), \mathcal{O}'_n) \ar[d]^{g'_n} \\
(\Sh(\mathcal{C}_{total}), \mathcal{O}) \ar[r]^{h_{total}} &
(\Sh(\mathcal{C}'_{total}), \mathcal{O}')
}
$$
of ringed topoi where $g_n$ and $g'_n$ are as in
Lemma \ref{lemma-restriction-module-to-components-site}.
 ========

new_annotation []
===== sent \end{remark}

\begin{lemma}
\label{lemma-morphism-simplicial-sites-modules}
Let $\mathcal{C}_n, f_\varphi, u_\varphi$ and
$\mathcal{C}'_n, f'_\varphi, u'_\varphi$ be as in
Situation \ref{situation-simplicial-site}.
Let $\mathcal{O}$ and $\mathcal{O}'$
be a sheaf of rings on $\mathcal{C}_{total}$ and $\mathcal{C}'_{total}$.
Let $(h, h^\sharp)$ be a morphism between simplicial sites as in
Remark \ref{remark-morphism-simplicial-sites-modules}.
Then we obtain a morphism of ringed topoi
$$
h_{total} :
(\Sh(\mathcal{C}_{total}, \mathcal{O})
\to
(\Sh(\mathcal{C}'_{total}), \mathcal{O}')
$$
and commutative diagrams
$$
\xymatrix{
(\Sh(\mathcal{C}_n), \mathcal{O}_n) \ar[d]_{g_n} \ar[r]_{h_n} &
(\Sh(\mathcal{C}'_n), \mathcal{O}'_n) \ar[d]^{g'_n} \\
(\Sh(\mathcal{C}_{total}), \mathcal{O}) \ar[r]^{h_{total}} &
(\Sh(\mathcal{C}'_{total}), \mathcal{O}')
}
$$
of ringed topoi where $g_n$ and $g'_n$ are as in
Lemma \ref{lemma-restriction-module-to-components-site}.
 ========

regex_match $(h, h^\sharp)$
nn candidate a morphism
nn a morphism
new_annotation [(53738, 53753, 'VAR'), (53757, 53767, 'TYPE')]
===== sent \end{proof}








\section{Cohomology on simplicial sites}
\label{section-cohomology-simplicial-sites}

\noindent
Let $\mathcal{C}$ be as in Situation \ref{situation-simplicial-site}.
In statement of the following lemmas we will let
$g_n : \Sh(\mathcal{C}_n) \to \Sh(\mathcal{C}_{total})$ be the
morphism of topoi of
Lemma \ref{lemma-restriction-to-components-site}. If $\varphi : [m] \to [n]$
is a morphism of $\Delta$, then the diagram of topoi
$$
\xymatrix{
\Sh(\mathcal{C}_n) \ar[rd]_{g_n} \ar[rr]_{f_\varphi} & &
\Sh(\mathcal{C}_m) \ar[ld]^{g_m} \\
 ========

new_annotation []
===== sent This complex is acyclic in positive degrees
and equal to $\mathbf{Z}$ in degree $0$.
\end{proof}

\begin{lemma}
\label{lemma-cech-complex}
In Situation \ref{situation-simplicial-site}. Let $\mathcal{F}$ be an abelian
sheaf on $\mathcal{C}_{total}$ there is a canonical complex
$$
0 \to \Gamma(\mathcal{C}_{total}, \mathcal{F}) \to
\Gamma(\mathcal{C}_0, \mathcal{F}_0) \to
\Gamma(\mathcal{C}_1, \mathcal{F}_1) \to
\Gamma(\mathcal{C}_2, \mathcal{F}_2) \to \ldots
$$
which is exact in degrees $-1, 0$ and exact everywhere
if $\mathcal{F}$ is injective.
\end{lemma}

\begin{proof}
 ========

regex_match $\mathcal{F}$
nn candidate an abelian
sheaf
nn an abelian
sheaf
new_annotation [(59146, 59159, 'VAR'), (59163, 59179, 'TYPE')]
===== sent In Situation \ref{situation-simplicial-site}. For $K$ in
$D^+(\mathcal{C}_{total})$ there is a spectral sequence
$(E_r, d_r)_{r \geq 0}$ with
$$
E_1^{p, q} = H^q(\mathcal{C}_p, K_p),\quad
d_1^{p, q} : E_1^{p, q} \to E_1^{p + 1, q}
$$
converging to $H^{p + q}(\mathcal{C}_{total}, K)$.
This spectral sequence is functorial in $K$.
\end{lemma}

\begin{proof}
Let $\mathcal{I}^\bullet$ be a bounded below complex of injectives
representing $K$. Consider the double complex with terms
$$
A^{p, q} = \Gamma(\mathcal{C}_p, \mathcal{I}^q_p)
$$
where the horizontal arrows come from Lemma \ref{lemma-cech-complex}
and the vertical arrows from the differentials of the
complex $\mathcal{I}^\bullet$. The rows of the double complex are exact
in positive degrees and evaluate to
$\Gamma(\mathcal{C}_{total}, \mathcal{I}^q)$ in degree $0$.
On the other hand, since restriction to $\mathcal{C}_p$ is exact
(Lemma \ref{lemma-restriction-to-components-site})
the complex $\mathcal{I}_p^\bullet$ represents $K_p$ in
$D(\mathcal{C}_p)$. The sheaves $\mathcal{I}_p^q$ are injective
abelian sheaves on $\mathcal{C}_p$
(Lemma \ref{lemma-restriction-injective-to-component-site}).
 ========

regex_match $\mathcal{I}^\bullet$
nn candidate complex
nn complex
new_annotation [(60315, 60336, 'VAR'), (60356, 60363, 'TYPE')]
===== sent Hence the cohomology of the columns computes the groups
$H^q(\mathcal{C}_p, K_p)$. We conclude by applying
Homology, Lemmas \ref{homology-lemma-first-quadrant-ss} and
\ref{homology-lemma-double-complex-gives-resolution}.
\end{proof}

\begin{lemma}
\label{lemma-sanity-check}
Let $\mathcal{C}$ be as in Situation \ref{situation-simplicial-site}.
Let $U \in \Ob(\mathcal{C}_n)$. Let
$\mathcal{F} \in \textit{Ab}(\mathcal{C}_{total})$.
 ========

new_annotation []
===== sent Hence the cohomology of the columns computes the groups
$H^q(\mathcal{C}_p, K_p)$. We conclude by applying
Homology, Lemmas \ref{homology-lemma-first-quadrant-ss} and
\ref{homology-lemma-double-complex-gives-resolution}.
\end{proof}

\begin{lemma}
\label{lemma-sanity-check}
Let $\mathcal{C}$ be as in Situation \ref{situation-simplicial-site}.
Let $U \in \Ob(\mathcal{C}_n)$. Let
$\mathcal{F} \in \textit{Ab}(\mathcal{C}_{total})$.
 ========

new_annotation []
===== sent \end{proof}






\section{Cohomology and augmentations of simplicial sites}
\label{section-cohomology-augmentation-simplicial-sites}

\noindent
Consider a simplicial site $\mathcal{C}$ as in
Situation \ref{situation-simplicial-site}.
Let $a_0$ be an augmentation towards a site $\mathcal{D}$ as in
Remark \ref{remark-augmentation-site}.
 ========

regex_match $a_0$
nn candidate an augmentation
nn an augmentation
new_annotation [(62313, 62318, 'VAR'), (62322, 62337, 'TYPE')]
===== sent \end{proof}

\begin{lemma}
\label{lemma-augmentation-spectral-sequence}
In Situation \ref{situation-simplicial-site} let
$a_0$ be an augmentation towards a site $\mathcal{D}$
as in Remark \ref{remark-augmentation-site}.
For any $K$ in $D^+(\mathcal{C}_{total})$ there is a spectral
sequence 
$(E_r, d_r)_{r \geq 0}$ with
$$
E_1^{p, q} = R^qa_{p, *} K_p,\quad
d_1^{p, q} : E_1^{p, q} \to E_1^{p + 1, q}
$$
converging to $R^{p + q}a_*K$. This spectral sequence is functorial in $K$.
\end{lemma}

\begin{proof}
Let $\mathcal{I}^\bullet$ be a bounded below complex of injectives
representing $K$. Consider the double complex with terms
$$
A^{p, q} = a_{p, *}\mathcal{I}^q_p
$$
where the horizontal arrows come from
Lemma \ref{lemma-augmentation-cech-complex}
and the vertical arrows from the differentials of the
complex $\mathcal{I}^\bullet$. The rows of the double complex are exact
in positive degrees and evaluate to $a_*\mathcal{I}^q$ in degree ========

regex_match $\mathcal{I}^\bullet$
nn candidate complex
nn complex
new_annotation [(67145, 67166, 'VAR'), (67186, 67193, 'TYPE')]
===== sent In Situation \ref{situation-simplicial-site} let $\mathcal{O}$
be a sheaf of rings on $\mathcal{C}_{total}$.
In statement of the following lemmas we will let
$g_n : (\Sh(\mathcal{C}_n), \mathcal{O}_n) \to
(\Sh(\mathcal{C}_{total}), \mathcal{O})$
be the morphism of ringed topoi of
Lemma \ref{lemma-restriction-module-to-components-site}.
If $\varphi : [m] \to [n]$ is a morphism of $\Delta$, then the diagram
of ringed topoi
$$
\xymatrix{
(\Sh(\mathcal{C}_n), \mathcal{O}_n) \ar[rd]_{g_n} \ar[rr]_{f_\varphi} & &
(\Sh(\mathcal{C}_m), \mathcal{O}_m) \ar[ld]^{g_m} \\
& (\Sh(\mathcal{C}_{total}), \mathcal{O})
}
$$
is not commutative, but there is a $2$-morphism $g_n \to g_m \circ f_\varphi$
coming from the maps
$\mathcal{F}(\varphi) : f_\varphi^{-1}\mathcal{F}_m \to \mathcal{F}_n$.
See Sites, Section \ref{sites-section-2-category}.

\begin{lemma}
 ========

new_annotation []
===== sent \label{lemma-simplicial-resolution-ringed}
In Situation \ref{situation-simplicial-site} let $\mathcal{O}$
be a sheaf of rings on $\mathcal{C}_{total}$. There is a complex
$$
\ldots \to g_{2!}\mathcal{O}_2 \to g_{1!}\mathcal{O}_1 \to g_{0!}\mathcal{O}_0
$$
of $\mathcal{O}$-modules which forms a resolution of
$\mathcal{O}$.
Here $g_{n!}$ is as in Lemma \ref{lemma-restriction-module-to-components-site}.
\end{lemma}

\begin{proof}
We will use the description of $g_{n!}$ given in
Lemma \ref{lemma-restriction-to-components-site}.
As maps of the complex we take $\sum (-1)^i d^n_i$ where
$d^n_i : g_{n!}\mathcal{O}_n \to g_{n - 1!}\mathcal{O}_{n - 1}$
is the adjoint to the map
$\mathcal{O}_n \to \bigoplus_{[n - 1] \to [n]} \mathcal{O}_n =
g_n^*g_{n - 1!}\mathcal{O}_{n - 1}$
corresponding to the factor labeled with $\delta^n_i : [n - 1] \to [n]$.
 ========

new_annotation []
===== sent This complex is acyclic in positive degrees
and equal to $\mathcal{O}_m$ in degree $0$.
\end{proof}

\begin{lemma}
\label{lemma-cech-complex-modules}
In Situation \ref{situation-simplicial-site} let $\mathcal{O}$
be a sheaf of rings. ========

new_annotation []
===== sent Let $\mathcal{F}$ be a
sheaf of $\mathcal{O}$-modules. ========

new_annotation []
===== sent let $\mathcal{O}$
be a sheaf of rings. ========

new_annotation []
===== sent For $K$ in $D^+(\mathcal{O})$
there is a spectral sequence $(E_r, d_r)_{r \geq 0}$ with
$$
E_1^{p, q} = H^q(\mathcal{C}_p, K_p),\quad
d_1^{p, q} : E_1^{p, q} \to E_1^{p + 1, q}
$$
converging to $H^{p + q}(\mathcal{C}_{total}, K)$.
This spectral sequence is functorial in $K$.
\end{lemma}

\begin{proof}
Let $\mathcal{I}^\bullet$ be a bounded below complex of injective
$\mathcal{O}$-modules representing $K$. Consider the double complex with terms
$$
A^{p, q} = \Gamma(\mathcal{C}_p, \mathcal{I}^q_p)
$$
where the horizontal arrows come from
Lemma \ref{lemma-cech-complex-modules}
and the vertical arrows from the differentials of the
complex $\mathcal{I}^\bullet$. ========

regex_match $\mathcal{I}^\bullet$
nn candidate complex
nn complex
new_annotation [(72425, 72446, 'VAR'), (72466, 72473, 'TYPE')]
===== sent Hence the cohomology of the columns computes the groups
$H^q(\mathcal{C}_p, K_p)$ by Leray's acyclicity lemma
(Derived Categories, Lemma \ref{derived-lemma-leray-acyclicity})
and
Cohomology on Sites, Lemma \ref{sites-cohomology-lemma-limp-acyclic}.
We conclude by applying
Homology, Lemma \ref{homology-lemma-first-quadrant-ss}.
\end{proof}

\begin{lemma}
\label{lemma-sanity-check-modules}
In Situation \ref{situation-simplicial-site} let $\mathcal{O}$
be a sheaf of rings. ========

new_annotation []
===== sent Let $U \in \Ob(\mathcal{C}_n)$. Let
$\mathcal{F} \in \textit{Mod}(\mathcal{O})$.
 ========

new_annotation []
===== sent \medskip\noindent
Consider a simplicial site $\mathcal{C}$ as in
Situation \ref{situation-simplicial-site}.
Let $a_0$ be an augmentation towards a site $\mathcal{D}$ as in
Remark \ref{remark-augmentation-site}.
Let $\mathcal{O}$ be a sheaf of rings on $\mathcal{C}_{total}$.
Let $\mathcal{O}_\mathcal{D}$ be a sheaf of rings on $\mathcal{D}$.
Suppose we are given a morphism
$$
a^\sharp : \mathcal{O}_\mathcal{D} \longrightarrow a_*\mathcal{O}
$$
where $a$ is as in Lemma \ref{lemma-augmentation-site}.
 ========

regex_match $a_0$
nn candidate an augmentation
nn an augmentation
new_annotation [(75237, 75242, 'VAR'), (75246, 75261, 'TYPE')]
===== sent \medskip\noindent
Consider a simplicial site $\mathcal{C}$ as in
Situation \ref{situation-simplicial-site}.
Let $a_0$ be an augmentation towards a site $\mathcal{D}$ as in
Remark \ref{remark-augmentation-site}.
Let $\mathcal{O}$ be a sheaf of rings on $\mathcal{C}_{total}$.
Let $\mathcal{O}_\mathcal{D}$ be a sheaf of rings on $\mathcal{D}$.
Suppose we are given a morphism
$$
a^\sharp : \mathcal{O}_\mathcal{D} \longrightarrow a_*\mathcal{O}
$$
where $a$ is as in Lemma \ref{lemma-augmentation-site}.
 ========

regex_match $\mathcal{O}$
nn candidate a sheaf
nn a sheaf
new_annotation [(75340, 75353, 'VAR'), (75357, 75364, 'TYPE')]
===== sent \medskip\noindent
Consider a simplicial site $\mathcal{C}$ as in
Situation \ref{situation-simplicial-site}.
Let $a_0$ be an augmentation towards a site $\mathcal{D}$ as in
Remark \ref{remark-augmentation-site}.
Let $\mathcal{O}$ be a sheaf of rings on $\mathcal{C}_{total}$.
Let $\mathcal{O}_\mathcal{D}$ be a sheaf of rings on $\mathcal{D}$.
Suppose we are given a morphism
$$
a^\sharp : \mathcal{O}_\mathcal{D} \longrightarrow a_*\mathcal{O}
$$
where $a$ is as in Lemma \ref{lemma-augmentation-site}.
 ========

regex_match $\mathcal{O}_\mathcal{D}$
nn candidate a sheaf
nn a sheaf
new_annotation [(75404, 75429, 'VAR'), (75433, 75440, 'TYPE')]
===== sent Let $\mathcal{F} \to \mathcal{G}$
be an injective map of $a^{-1}\mathcal{O}_\mathcal{D}$-modules.
 ========

new_annotation []
===== sent With notation as above for any $K$ in $D^+(\mathcal{O})$ there is a spectral
sequence $(E_r, d_r)_{r \geq 0}$ in $\textit{Mod}(\mathcal{O}_\mathcal{D})$
with
$$
E_1^{p, q} = R^qa_{p, *} K_p
$$
converging to $R^{p + q}a_*K$. This spectral sequence is functorial in $K$.
\end{lemma}

\begin{proof}
Let $\mathcal{I}^\bullet$ be a bounded below complex of injective
$\mathcal{O}$-modules representing $K$. Consider the double complex with terms
$$
A^{p, q} = a_{p, *}\mathcal{I}^q_p
$$
where the horizontal arrows come from
Lemma \ref{lemma-augmentation-cech-complex-modules}
and the vertical arrows from the differentials of the
complex $\mathcal{I}^\bullet$. The lemma
says rows of the double complex are exact
in positive degrees and evaluate to
$a_*\mathcal{I}^q$ in degree $0$.
 ========

regex_match $\mathcal{I}^\bullet$
nn candidate complex
nn complex
new_annotation [(81053, 81074, 'VAR'), (81094, 81101, 'TYPE')]
===== sent \end{enumerate}
Let $\mathcal{O}$ be a sheaf of rings on $\mathcal{C}_{total}$ and
$\mathcal{O}_\mathcal{D}$ a sheaf of rings on $\mathcal{D}$
and $a^\sharp : \mathcal{O}_\mathcal{D} \to a_*\mathcal{O}$ a
morphism as in
Section \ref{section-cohomology-augmentation-ringed-simplicial-sites}.
\begin{enumerate}
\item[(3)] The pullback $a^*\mathcal{F}$ of a sheaf of
$\mathcal{O}_\mathcal{D}$-modules is cartesian.
 ========

regex_match $\mathcal{O}$
nn candidate a sheaf
nn a sheaf
new_annotation [(86005, 86018, 'VAR'), (86022, 86029, 'TYPE')]
===== sent let $\mathcal{O}$ be a sheaf of rings on $\mathcal{C}_{total}$.
The category of cartesian $\mathcal{O}$-modules
is equivalent to the category of pairs $(\mathcal{F}, \alpha)$
where $\mathcal{F}$ is a $\mathcal{O}_0$-module
and
$$
\alpha :
(f_{\delta_1^1})^*\mathcal{F}
\longrightarrow (f_{\delta_0^1})^*\mathcal{F}
$$
is an isomorphism of $\mathcal{O}_1$-modules such that
$(f_{\delta^2_1})^*\alpha =
(f_{\delta^2_0})^*\alpha \circ (f_{\delta^2_2})^*\alpha$
as $\mathcal{O}_2$-module maps.
 ========

regex_match $\mathcal{O}$
nn candidate a sheaf
nn a sheaf
new_annotation [(90131, 90144, 'VAR'), (90148, 90155, 'TYPE')]
===== sent \item Let $\mathcal{O}$ be a sheaf of rings on $\mathcal{C}_{total}$
such that the morphisms
$$
f_{\delta^n_j} : (\Sh(\mathcal{C}_n), \mathcal{O}_n)
\to (\Sh(\mathcal{C}_{n - 1}), \mathcal{O}_{n - 1})
$$
are flat. ========

regex_match $\mathcal{O}$
nn candidate a sheaf
nn a sheaf
new_annotation [(91136, 91149, 'VAR'), (91153, 91160, 'TYPE')]
===== sent \end{remark}

\begin{lemma}
\label{lemma-derived-cartesian-modules}
In Situation \ref{situation-simplicial-site}.
\begin{enumerate}
\item An object $K$ of $D(\mathcal{C}_{total})$ is cartesian if and only
if $H^q(K)$ is a cartesian abelian sheaf for all $q$.
\item Let $\mathcal{O}$ be a sheaf
of rings on $\mathcal{C}_{total}$ such that the morphisms
$f_{\delta^n_j} : (\Sh(\mathcal{C}_n), \mathcal{O}_n)
\to (\Sh(\mathcal{C}_{n - 1}), \mathcal{O}_{n - 1})$ are flat.
 ========

regex_match $\mathcal{O}$
nn candidate a sheaf
nn a sheaf
new_annotation [(93635, 93648, 'VAR'), (93652, 93659, 'TYPE')]
===== sent Part (2) follows from the characterization in
Lemma \ref{lemma-check-cartesian-module}
and the fact that $L(f_{\delta^n_j})^* = (f_{\delta^n_j})^*$
by flatness.
\end{proof}

\begin{lemma}
\label{lemma-derived-cartesian-shriek}
In Situation \ref{situation-simplicial-site}.
\begin{enumerate}
\item An object $K$ of $D(\mathcal{C}_{total})$ is cartesian if and only
the canonical map
$$
g_{n!}K_n \longrightarrow
g_{n!}\mathbf{Z} \otimes^\mathbf{L}_\mathbf{Z} K
$$
is an isomorphism for all $n$.
\item Let $\mathcal{O}$ be a sheaf of rings on $\mathcal{C}_{total}$
such that the morphisms $f_\varphi^{-1}\mathcal{O}_n \to \mathcal{O}_m$
are flat for all $\varphi : [n] \to [m]$. Then an object $K$ of
$D(\mathcal{O})$ is cartesian if and only the canonical map
$$
g_{n!}K_n \longrightarrow
g_{n!}\mathcal{O}_n \otimes^\mathbf{L}_\mathcal{O} K
$$
is an isomorphism for all $n$.
\end{enumerate}
\end{lemma}

\begin{proof}
Proof of (1). ========

regex_match $\mathcal{O}$
nn candidate a sheaf
nn a sheaf
new_annotation [(94584, 94597, 'VAR'), (94601, 94608, 'TYPE')]
===== sent let $\mathcal{O}$ be a sheaf of rings on $\mathcal{C}_{total}$.
Let $\mathcal{F}$ be a sheaf of $\mathcal{O}$-modules.
 ========

regex_match $\mathcal{O}$
nn candidate a sheaf
nn a sheaf
new_annotation [(96627, 96640, 'VAR'), (96644, 96651, 'TYPE')]
===== sent let $\mathcal{O}$ be a sheaf of rings on $\mathcal{C}_{total}$.
Let $\mathcal{F}$ be a sheaf of $\mathcal{O}$-modules.
 ========

regex_match $\mathcal{F}$
nn candidate a sheaf
nn a sheaf
new_annotation [(96691, 96704, 'VAR'), (96708, 96715, 'TYPE')]
===== sent Since pullbacks of
quasi-coherent modules are quasi-coherent
(Modules on Sites, Lemma \ref{sites-modules-lemma-local-pullback})
we see that $\mathcal{F}_n$ is a quasi-coherent $\mathcal{O}_n$-module
for all $n$. To show that $\mathcal{F}$ is cartesian, let $U$
be an object of $\mathcal{C}_n$ for some $n$. Let us view $U$
as an object of $\mathcal{C}_{total}$. Because $\mathcal{F}$
is quasi-coherent there exists a covering $\{U_i \to U\}$
and for each $i$ a presentation
$$
\bigoplus\nolimits_{j \in J_i} \mathcal{O}_{\mathcal{C}_{total}/U_i} \to
\bigoplus\nolimits_{k \in K_i} \mathcal{O}_{\mathcal{C}_{total}/U_i} \to
\mathcal{F}|_{\mathcal{C}_{total}/U_i} \to 0
$$
Observe that $\{U_i \to U\}$ is a covering of $\mathcal{C}_n$ by
the construction of the site $\mathcal{C}_{total}$.
 ========

new_annotation []
===== sent Next, let $V$ be an object of $\mathcal{C}_m$ for some $m$ and let
$V \to U$ be a morphism of $\mathcal{C}_{total}$ lying over
$\varphi : [n] \to [m]$. ========

regex_match $V$
nn candidate $V$
nn candidate an object
nn an object
new_annotation [(97849, 97852, 'VAR'), (97856, 97865, 'TYPE')]
===== sent \begin{lemma}
\label{lemma-downstairs}
Let $f : (\Sh(\mathcal{C}), \mathcal{O}_\mathcal{C}) \to
(\Sh(\mathcal{D}), \mathcal{O}_\mathcal{D})$ be a morphism of ringed topoi.
 ========

regex_match $f : (\Sh(\mathcal{C}), \mathcal{O}_\mathcal{C}) \to
(\Sh(\mathcal{D}), \mathcal{O}_\mathcal{D})$
nn candidate a morphism
nn a morphism
new_annotation [(100379, 100476, 'VAR'), (100480, 100490, 'TYPE')]
===== sent The final part follows formally from
fact that $Lf^*$ and $Rf_*$ are adjoint; compare with
Categories, Lemma \ref{categories-lemma-adjoint-fully-faithful}.
\end{proof}

\begin{lemma}
\label{lemma-upstairs}
Let $f : (\Sh(\mathcal{C}), \mathcal{O}_\mathcal{C}) \to
(\Sh(\mathcal{D}), \mathcal{O}_\mathcal{D})$ be a morphism of ringed topoi.
 ========

regex_match $f : (\Sh(\mathcal{C}), \mathcal{O}_\mathcal{C}) \to
(\Sh(\mathcal{D}), \mathcal{O}_\mathcal{D})$
nn candidate (\Sh(\mathcal{C
nn candidate a morphism
nn a morphism
new_annotation [(101673, 101770, 'VAR'), (101774, 101784, 'TYPE')]
===== sent The final part follows formally from
fact that $Lf^*$ and $Rf_*$ are adjoint; compare with
Categories, Lemma \ref{categories-lemma-adjoint-fully-faithful}.
\end{proof}

\begin{lemma}
\label{lemma-bounded-in-image-upstairs}
Let $f : (\Sh(\mathcal{C}), \mathcal{O}_\mathcal{C}) \to
(\Sh(\mathcal{D}), \mathcal{O}_\mathcal{D})$ be a morphism of ringed topoi.
 ========

regex_match $f : (\Sh(\mathcal{C}), \mathcal{O}_\mathcal{C}) \to
(\Sh(\mathcal{D}), \mathcal{O}_\mathcal{D})$
nn candidate a morphism
nn a morphism
new_annotation [(102986, 103083, 'VAR'), (103087, 103097, 'TYPE')]
===== sent Let $K$ be an object of ========

regex_match $K$
nn candidate an object
nn an object
new_annotation [(103119, 103122, 'VAR'), (103126, 103135, 'TYPE')]
===== sent \end{proof}

\begin{lemma}
\label{lemma-bounded-in-image-downstairs}
Let $f : (\Sh(\mathcal{C}), \mathcal{O}_\mathcal{C}) \to
(\Sh(\mathcal{D}), \mathcal{O}_\mathcal{D})$ be a morphism of ringed topoi.
 ========

regex_match $f : (\Sh(\mathcal{C}), \mathcal{O}_\mathcal{C}) \to
(\Sh(\mathcal{D}), \mathcal{O}_\mathcal{D})$
nn candidate a morphism
nn a morphism
new_annotation [(103889, 103986, 'VAR'), (103990, 104000, 'TYPE')]
===== sent Let $K$ be an object of ========

regex_match $K$
nn candidate an object
nn an object
new_annotation [(104022, 104025, 'VAR'), (104029, 104038, 'TYPE')]
===== sent \end{proof}

\begin{lemma}
\label{lemma-equivalence-bounded}
Let $f : (\Sh(\mathcal{C}), \mathcal{O}) \to (\Sh(\mathcal{C}'), \mathcal{O}')$
be a morphism of ringed topoi.
 ========

new_annotation []
===== sent Let $\mathcal{A} \subset \textit{Mod}(\mathcal{O})$
and $\mathcal{A}' \subset \textit{Mod}(\mathcal{O}')$
be weak Serre subcategories. ========

new_annotation []
===== sent Let $\mathcal{F} \in \Ob(\mathcal{A})$. Then we can write
$\mathcal{F} = f^*\mathcal{F}'$. Then
$Rf_*\mathcal{F} = Rf_* f^*\mathcal{F}' = \mathcal{F}'$.
In particular, we have $R^pf_*\mathcal{F} = 0$ for $p > 0$
and $f_*\mathcal{F} \in \Ob(\mathcal{A}')$.
 ========

new_annotation []
===== sent This is analogous to \cite[Theorem 2.2.3]{six-I}.
\end{reference}
Let $f : (\Sh(\mathcal{C}), \mathcal{O}) \to (\Sh(\mathcal{C}'), \mathcal{O}')$
be a morphism of ringed topoi.
 ========

new_annotation []
===== sent Let $\mathcal{A} \subset \textit{Mod}(\mathcal{O})$
and $\mathcal{A}' \subset \textit{Mod}(\mathcal{O}')$
be weak Serre subcategories. ========

new_annotation []
===== sent This is analogous to \cite[Theorem 2.2.3]{six-I}.
\end{reference}
Let $f : (\mathcal{C}, \mathcal{O}) \to (\mathcal{C}', \mathcal{O}')$
be a morphism of ringed sites.
 ========

new_annotation []
===== sent Let $\mathcal{A} \subset \textit{Mod}(\mathcal{O})$
and $\mathcal{A}' \subset \textit{Mod}(\mathcal{O}')$
be weak Serre subcategories. ========

new_annotation []
===== sent Assume
\begin{enumerate}
\item $f$ is flat,
\item $f^*$ induces an equivalence of categories
$\mathcal{A}' \to \mathcal{A}$,
\item $\mathcal{F}' \to Rf_*f^*\mathcal{F}'$ is an isomorphism
for $\mathcal{F}' \in \Ob(\mathcal{A}')$,
\item $\mathcal{C}, \mathcal{O}, \mathcal{A}$ satisfy the
assumption of
Cohomology on Sites, Situation \ref{sites-cohomology-situation-olsson-laszlo},
\item $f : (\mathcal{C}, \mathcal{O}) \to (\mathcal{C}', \mathcal{O}')$
and $\mathcal{A}$ satisfy the assumption of
Cohomology on Sites, Situation
\ref{sites-cohomology-situation-olsson-laszlo-prime}.
\end{enumerate}
Then $f^* : D_{\mathcal{A}'}(\mathcal{O}') \to D_\mathcal{A}(\mathcal{O})$
is an equivalence of categories with quasi-inverse given by
$Rf_* : D_\mathcal{A}(\mathcal{O}) \to D_{\mathcal{A}'}(\mathcal{O}')$.
\end{lemma}

\begin{proof}
The proof of this lemma is exactly the same as the proof
of Lemma \ref{lemma-equivalence-unbounded-one}
except the reference to
Cohomology on Sites, Lemma
\ref{sites-cohomology-lemma-olsson-laszlo-map-version-one}
is replaced by a reference to
Cohomology on Sites, Lemma
\ref{sites-cohomology-lemma-olsson-laszlo-map-version-two}.
\end{proof}


\noindent
Let $\mathcal{C}$ be a category. ========

regex_match $\mathcal{C}$
nn candidate a category
nn a category
new_annotation [(110270, 110283, 'VAR'), (110287, 110297, 'TYPE')]
===== sent Let $\mathcal{O}$ be a sheaf of rings for the
$\tau$-topology. ========

regex_match $\mathcal{O}$
nn candidate a sheaf
nn a sheaf
new_annotation [(110877, 110890, 'VAR'), (110894, 110901, 'TYPE')]
===== sent Let $\mathcal{B} \subset \Ob(\mathcal{C})$ be a subset.
 ========

regex_match $\mathcal{B} \subset \Ob(\mathcal{C})$
nn candidate $\mathcal{B
nn candidate a subset
nn a subset
new_annotation [(111348, 111386, 'VAR'), (111390, 111398, 'TYPE')]
===== sent Let $\mathcal{A} \subset \textit{PMod}(\mathcal{O})$
be a full subcategory. ========

new_annotation []
===== sent \end{proof}

\begin{lemma}
\label{lemma-abelian-postnikov}
In Situation \ref{situation-simplicial-site}. Let $K$ be
an object of $D(\mathcal{C}_{total})$. Set
$$
X_n = (g_{n!}\mathbf{Z})
\otimes^\mathbf{L}_\mathbf{Z} K
\quad\text{and}\quad
Y_n =
(g_{n!}\mathbf{Z} \to \ldots \to g_{0!}\mathbf{Z})[-n]
\otimes^\mathbf{L}_\mathbf{Z} K
$$
as objects of $D(\mathcal{C}_{total})$ where the maps are
as in Lemma \ref{lemma-simplicial-resolution-Z-site}.
With the evident canonical maps $Y_n \to X_n$ and
$Y_0 \to Y_1[1] \to Y_2[2] \to \ldots$ we have
\begin{enumerate}
\item the distinguished triangles $Y_n \to X_n \to Y_{n - 1} \to Y_n[1]$
define a Postnikov system
(Derived Categories, Definition \ref{derived-definition-postnikov-system})
for $\ldots ========

new_annotation []
===== sent 1$.
\end{enumerate}
Then any map $\{K_n \to K'_n\}$ between the associated simplicial systems 
of $K$ and $K'$ comes from a map $K \to K'$ in $D(\mathcal{C}_{total})$.
\end{lemma}

\begin{proof}
Let $\{K_n \to K'_n\}_{n \geq 0}$
be a morphism of simplicial systems of the derived category.
 ========

new_annotation []
===== sent The (slightly) easier
case of abelian sheaves is discussed in Section \ref{section-glueing}.

\begin{definition}
\label{definition-cartesian-derived-modules}
In Situation \ref{situation-simplicial-site}. Let $\mathcal{O}$
be a sheaf of rings on $\mathcal{C}_{total}$. A
{\it simplicial system of the derived category of modules}
consists of the following data
\begin{enumerate}
\item for every $n$ an object $K_n$ of $D(\mathcal{O}_n)$,
\item for every $\varphi : [m] \to [n]$ a map
$K_\varphi : Lf_\varphi^*K_m \to K_n$ in $D(\mathcal{O}_n)$
\end{enumerate}
subject to the condition that
$$
K_{\varphi \circ \psi} = K_\varphi \circ Lf_\varphi^*K_\psi :
 ========

new_annotation []
===== sent let $\mathcal{O}$ be a
sheaf of rings on $\mathcal{C}_{total}$.
If $K \in D(\mathcal{O})$ is an object, then $(K_n, K(\varphi))$
is a simplicial system of the derived category of modules.
 ========

new_annotation []
===== sent let $\mathcal{O}$
be a sheaf of rings on $\mathcal{C}_{total}$. Let $K$ be
an object of $D(\mathcal{C}_{total})$. Set
$$
X_n = (g_{n!}\mathcal{O}_n)
\otimes^\mathbf{L}_\mathcal{O} K
\quad\text{and}\quad
Y_n =
(g_{n!}\mathcal{O}_n \to \ldots \to g_{0!}\mathcal{O}_0)[-n]
\otimes^\mathbf{L}_\mathcal{O} K
$$
as objects of $D(\mathcal{O})$ where the maps are
as in Lemma \ref{lemma-simplicial-resolution-Z-site}.
With the evident canonical maps $Y_n \to X_n$ and
$Y_0 \to Y_1[1] \to Y_2[2] \to \ldots$ we have
\begin{enumerate}
\item the distinguished triangles $Y_n \to X_n \to Y_{n - 1} \to Y_n[1]$
define a Postnikov system
(Derived Categories, Definition \ref{derived-definition-postnikov-system})
for $\ldots ========

new_annotation []
===== sent let $\mathcal{O}$
be a sheaf of rings on $\mathcal{C}_{total}$. Let $K$ be
an object of $D(\mathcal{C}_{total})$. Set
$$
X_n = (g_{n!}\mathcal{O}_n)
\otimes^\mathbf{L}_\mathcal{O} K
\quad\text{and}\quad
Y_n =
(g_{n!}\mathcal{O}_n \to \ldots \to g_{0!}\mathcal{O}_0)[-n]
\otimes^\mathbf{L}_\mathcal{O} K
$$
as objects of $D(\mathcal{O})$ where the maps are
as in Lemma \ref{lemma-simplicial-resolution-Z-site}.
With the evident canonical maps $Y_n \to X_n$ and
$Y_0 \to Y_1[1] \to Y_2[2] \to \ldots$ we have
\begin{enumerate}
\item the distinguished triangles $Y_n \to X_n \to Y_{n - 1} \to Y_n[1]$
define a Postnikov system
(Derived Categories, Definition \ref{derived-definition-postnikov-system})
for $\ldots ========

new_annotation []
===== sent In Situation \ref{situation-simplicial-site} let $\mathcal{O}$ be
a sheaf of rings on $\mathcal{C}_{total}$.
 ========

new_annotation []
===== sent In Situation \ref{situation-simplicial-site} let $\mathcal{O}$ be
a sheaf of rings on $\mathcal{C}_{total}$.
If $K, K' \in D(\mathcal{O})$.
Assume
\begin{enumerate}
\item $f_\varphi^{-1}\mathcal{O}_n \to \mathcal{O}_m$ is flat for
$\varphi : [m] \to [n]$,
\item $K$ is cartesian,
\item $\Hom(K_i[i - 1], K'_i) = 0$ for $i > ========

new_annotation []
===== sent let $\mathcal{O}$ be
a sheaf of rings on $\mathcal{C}_{total}$. Let
$(K_n, K_\varphi)$ be a simplicial system of the derived category
of modules. ========

new_annotation []
===== sent \end{proof}







\section{The site associated to a semi-representable object}
\label{section-semi-representable}

\noindent
Let $\mathcal{C}$ be a site. ========

regex_match $\mathcal{C}$
nn candidate a site
nn a site
new_annotation [(136055, 136068, 'VAR'), (136072, 136078, 'TYPE')]
===== sent \medskip\noindent
Let $\mathcal{C}$ be a site. ========

regex_match $\mathcal{C}$
nn candidate a site
nn a site
new_annotation [(137353, 137366, 'VAR'), (137370, 137376, 'TYPE')]
===== sent Let $K = \{U_i\}_{i \in I}$ be an object of
$\text{SR}(\mathcal{C})$. There is a continuous and cocontinuous
localization functor $j : \mathcal{C}/K \to \mathcal{C}$ which is
the product of the localization functors
$j_i : \mathcal{C}/V_i \to \mathcal{C}$.
We obtain functors $j_!$, $j^{-1}$, $j_*$ exactly
as in Sites, Section \ref{sites-section-localize}.
In terms of of the product decomposition
$\Sh(\mathcal{C}/K) = \prod\nolimits_{i \in I} \Sh(\mathcal{C}/U_i)$
we have
$$
\begin{matrix}
j_! & : &
(\mathcal{F}_i)_{i \in I} &
\longmapsto &
\coprod j_{i, !}\mathcal{F}_i \\
 ========

regex_match $K = \{U_i\}_{i \in I}$
nn candidate $
nn candidate an object
nn an object
new_annotation [(137382, 137405, 'VAR'), (137409, 137418, 'TYPE')]
===== sent \medskip\noindent
Let $f : K \to L$ be a morphism of $\text{SR}(\mathcal{C})$.
Then we obtain a continuous and cocontinuous functor
$$
v : \mathcal{C}/K \longrightarrow \mathcal{C}/L
$$
by applying the construction of Sites, Lemma \ref{sites-lemma-relocalize}
to the components. ========

regex_match $f : K \to L$
nn candidate a morphism
nn a morphism
new_annotation [(138183, 138196, 'VAR'), (138200, 138210, 'TYPE')]
===== sent U_i \to V_{\alpha(i)}$.

\begin{lemma}
\label{lemma-has-P}
Let $\mathcal{C}$ be a site.
 ========

regex_match $\mathcal{C}$
nn candidate a site
nn a site
new_annotation [(139706, 139719, 'VAR'), (139723, 139729, 'TYPE')]
===== sent \end{proof}

\begin{lemma}
\label{lemma-push-pull-localization}
Let $\mathcal{C}$ be a site and $K$ in $\text{SR}(\mathcal{C})$.
 ========

regex_match $\mathcal{C}$
nn candidate a site
nn a site
new_annotation [(140714, 140727, 'VAR'), (140731, 140737, 'TYPE')]
===== sent \end{proof}

\begin{lemma}
\label{lemma-localize-compare}
Let $\mathcal{C}$ be a site.
 ========

regex_match $\mathcal{C}$
nn candidate a site
nn a site
new_annotation [(141685, 141698, 'VAR'), (141702, 141708, 'TYPE')]
===== sent \end{proof}

\begin{lemma}
\label{lemma-localize-injective}
Let $\mathcal{C}$ be a site. ========

regex_match $\mathcal{C}$
nn candidate a site
nn a site
new_annotation [(143291, 143304, 'VAR'), (143308, 143314, 'TYPE')]
===== sent \end{proof}

\begin{remark}[Variant for over an object]
\label{remark-semi-representable-over-object}
Let $\mathcal{C}$ be a site. ========

regex_match $\mathcal{C}$
nn candidate a site
nn a site
new_annotation [(143858, 143871, 'VAR'), (143875, 143881, 'TYPE')]
===== sent Let $X \in \Ob(\mathcal{C})$.
The category $\text{SR}(\mathcal{C}, X)$
of {\it semi-representable objects over $X$}
is defined by the formula
$\text{SR}(\mathcal{C}, X) = \text{SR}(\mathcal{C}/X)$.
See Hypercoverings, Definition \ref{hypercovering-definition-SR}.
 ========

new_annotation []
===== sent L$ in $\text{SR}(\mathcal{C}, X)$.
\end{enumerate}
All results of this section hold in this situation by replacing
$\mathcal{C}$ everywhere by $\mathcal{C}/X$.
\end{remark}

\begin{remark}[Ringed variant]
\label{remark-semi-representable-ringed}
Let $\mathcal{C}$ be a site. ========

regex_match $\mathcal{C}$
nn candidate a site
nn a site
new_annotation [(144830, 144843, 'VAR'), (144847, 144853, 'TYPE')]
===== sent Let $\mathcal{O}_\mathcal{C}$
be a sheaf of rings on $\mathcal{C}$. In this case, for any
semi-representable object $K$ of $\mathcal{C}$ the site
$\mathcal{C}/K$ is a ringed site with sheaf
of rings $\mathcal{O}_K = j^{-1}\mathcal{O}_\mathcal{C}$.
The constructions above give
\begin{enumerate}
\item a ringed site $(\mathcal{C}/K, \mathcal{O}_K)$
for $K$ in $\text{SR}(\mathcal{C})$,
\item a decomposition
$\textit{Mod}(\mathcal{O}_K) =
\prod \textit{Mod}(\mathcal{O}_{U_i})$ if $K = \{U_i\}$,
\item a localization morphism
$j : (\Sh(\mathcal{C}/K), \mathcal{O}_K) \to
(\Sh(\mathcal{C}), \mathcal{O}_\mathcal{C})$
of ringed topoi,
\item a morphism
$f : (\Sh(\mathcal{C}/K), \mathcal{O}_K) \to
(\Sh(\mathcal{C}/L), \mathcal{O}_L)$ of ringed topoi
for $f : K \to L$ in $\text{SR}(\mathcal{C})$.
\end{enumerate}
Many of the results above hold in this setting. ========

new_annotation []
===== sent \end{remark}

\begin{remark}[Ringed variant over an object]
\label{remark-semi-representable-ringed-over-object}
Let $\mathcal{C}$ be a site. ========

regex_match $\mathcal{C}$
nn candidate a site
nn a site
new_annotation [(146477, 146490, 'VAR'), (146494, 146500, 'TYPE')]
===== sent Let $\mathcal{O}_\mathcal{C}$
be a sheaf of rings on $\mathcal{C}$. Let $X \in ========

new_annotation []
===== sent Let $\mathcal{O}_\mathcal{C}$
be a sheaf of rings on $\mathcal{C}$. Let $X \in ========

new_annotation []
===== sent \end{remark}





\section{The site associate to a simplicial semi-representable object}
\label{section-simplicial-semi-representable}

\noindent
Let $\mathcal{C}$ be a site. ========

regex_match $\mathcal{C}$
nn candidate a site
nn a site
new_annotation [(147618, 147631, 'VAR'), (147635, 147641, 'TYPE')]
===== sent Let $K$ be a simplicial object of
$\text{SR}(\mathcal{C})$. As usual, set $K_n = K([n])$ and denote
$K(\varphi) : K_n \to K_m$ the morphism associated to $\varphi : [m] \to [n]$.
 ========

regex_match $K$
nn candidate a simplicial object
nn a simplicial object
new_annotation [(147647, 147650, 'VAR'), (147654, 147673, 'TYPE')]
===== sent \begin{lemma}
\label{lemma-augmentation-simplicial-semi-representable}
Let $\mathcal{C}$ be a site. ========

regex_match $\mathcal{C}$
nn candidate a site
nn a site
new_annotation [(149784, 149797, 'VAR'), (149801, 149807, 'TYPE')]
===== sent Let $K$ be a simplicial object of
$\text{SR}(\mathcal{C})$. The localization functor
$j_0 : \mathcal{C}/K_0 \to \mathcal{C}$ defines an augmentation
$a_0 : \Sh(\mathcal{C}/K_0) \to \Sh(\mathcal{C})$, as in case (B) of
Remark \ref{remark-augmentation-site}.
The corresponding morphisms of topoi
$$
a_n : \Sh(\mathcal{C}/K_n) \longrightarrow \Sh(\mathcal{C}),\quad
a : \Sh((\mathcal{C}/K)_{total}) \longrightarrow \Sh(\mathcal{C})
$$
of Lemma \ref{lemma-augmentation-site}
are equal to the morphisms of topoi associated to the
continuous and cocontinuous localization functors
$j_n : \mathcal{C}/K_n \to \mathcal{C}$ and
$j_{total} : (\mathcal{C}/K)_{total} \to \mathcal{C}$.
\end{lemma}

\begin{proof}
This is immediate from working through the definitions.
 ========

regex_match $K$
nn candidate a simplicial object
nn a simplicial object
new_annotation [(149813, 149816, 'VAR'), (149820, 149839, 'TYPE')]
===== sent The functor $a^{Sh}_!$ is constructed in
Sites, Lemma \ref{sites-lemma-when-shriek}
and the functor $a_!$ is constructed in
Modules on Sites, Lemma
\ref{sites-modules-lemma-g-shriek-adjoint}.
\end{proof}

\begin{lemma}
\label{lemma-sanity-check-simplicial-semi-representable}
Let $\mathcal{C}$ be a site. ========

regex_match $\mathcal{C}$
nn candidate a site
nn a site
new_annotation [(152105, 152118, 'VAR'), (152122, 152128, 'TYPE')]
===== sent Let $K$ be a simplicial object of
$\text{SR}(\mathcal{C})$. Let $U/U_{n, i}$ be an object of
$\mathcal{C}/K_n$. Let
$\mathcal{F} \in \textit{Ab}((\mathcal{C}/K)_{total})$.
 ========

regex_match $K$
nn candidate a simplicial object
nn a simplicial object
new_annotation [(152134, 152137, 'VAR'), (152141, 152160, 'TYPE')]
===== sent Let $K$ be a simplicial object of
$\text{SR}(\mathcal{C})$. Let $U/U_{n, i}$ be an object of
$\mathcal{C}/K_n$. Let
$\mathcal{F} \in \textit{Ab}((\mathcal{C}/K)_{total})$.
 ========

regex_match $U/U_{n, i}$
nn candidate $U/U_{n
nn candidate an object
nn an object
new_annotation [(152194, 152206, 'VAR'), (152210, 152219, 'TYPE')]
===== sent Then
$$
H^p(U, \mathcal{F}) = H^p(U, \mathcal{F}_{n, i})
$$
where
\begin{enumerate}
\item on the left hand side $U$ is viewed as an object of
$\mathcal{C}_{total}$, and
\item on the right hand side $\mathcal{F}_{n, i}$ is the $i$th
component of the sheaf $\mathcal{F}_n$ on $\mathcal{C}/K_n$
in the decomposition $\Sh(\mathcal{C}/K_n) = \prod \Sh(\mathcal{C}/U_{n, i})$
of Section \ref{section-semi-representable}.
\end{enumerate}
\end{lemma}

\begin{proof}
This follows immediately from Lemma \ref{lemma-sanity-check}
and the product decompositions of Section \ref{section-semi-representable}.
\end{proof}

\begin{remark}[Variant for over an object]
\label{remark-augmentation-over-object}
Let $\mathcal{C}$ be a site. ========

regex_match $\mathcal{C}$
nn candidate a site
nn a site
new_annotation [(152997, 153010, 'VAR'), (153014, 153020, 'TYPE')]
===== sent Let $X \in \Ob(\mathcal{C})$.
 ========

new_annotation []
===== sent To prove this one replaces
the site $\mathcal{C}$ everywhere by $\mathcal{C}/X$.
\end{remark}

\begin{remark}[Ringed variant]
\label{remark-augmentation-ringed}
Let $\mathcal{C}$ be a site. ========

regex_match $\mathcal{C}$
nn candidate a site
nn a site
new_annotation [(154239, 154252, 'VAR'), (154256, 154262, 'TYPE')]
===== sent Let $\mathcal{O}_\mathcal{C}$ be a sheaf of rings.
 ========

regex_match $\mathcal{O}_\mathcal{C}$
nn candidate a sheaf
nn a sheaf
new_annotation [(154268, 154293, 'VAR'), (154297, 154304, 'TYPE')]
===== sent This in turn implies that for
the flat morphism of ringed topoi
$g_n : (\Sh(\mathcal{C}/K_n), \mathcal{O}_n) \to
(\Sh((\mathcal{C}/K)_{total}), \mathcal{O})$
the functor $g_{n!} : \textit{Mod}(\mathcal{O}_n) \to
\textit{Mod}(\mathcal{O})$ left adjoint to $g_n^*$ is exact, see
Lemma \ref{lemma-exactness-g-shriek-modules}.
\end{remark}

\begin{remark}[Ringed variant over an object]
\label{remark-augmentation-ringed-over-object}
Let $\mathcal{C}$ be a site. ========

regex_match $\mathcal{C}$
nn candidate a site
nn a site
new_annotation [(156552, 156565, 'VAR'), (156569, 156575, 'TYPE')]
===== sent Let $\mathcal{O}_\mathcal{C}$ be a sheaf of rings.
 ========

regex_match $\mathcal{O}_\mathcal{C}$
nn candidate a sheaf
nn a sheaf
new_annotation [(156581, 156606, 'VAR'), (156610, 156617, 'TYPE')]
===== sent Let $X \in \Ob(\mathcal{C})$ and denote
$\mathcal{O}_X = \mathcal{O}_\mathcal{C}|_{\mathcal{C}/X}$.
 ========

new_annotation []
===== sent \end{remark}






\section{Cohomological descent for hypercoverings}
\label{section-cohomological-descent-hypercoverings}

\noindent
Let $\mathcal{C}$ be a site. ========

regex_match $\mathcal{C}$
nn candidate a site
nn a site
new_annotation [(157616, 157629, 'VAR'), (157633, 157639, 'TYPE')]
===== sent We let $K$ be a hypercovering
as defined in Hypercoverings, Definition
\ref{hypercovering-definition-hypercovering-variant}. We will study
the augmentation
$$
a : \Sh((\mathcal{C}/K)_{total}) \longrightarrow \Sh(\mathcal{C})
$$
of Section \ref{section-simplicial-semi-representable}.

\begin{lemma}
\label{lemma-hypercovering-descent-sheaves}
Let $\mathcal{C}$ be a site with equalizers and fibre products.
 ========

regex_match $K$
nn candidate a hypercovering
nn a hypercovering
new_annotation [(157723, 157726, 'VAR'), (157730, 157745, 'TYPE')]
===== sent We let $K$ be a hypercovering
as defined in Hypercoverings, Definition
\ref{hypercovering-definition-hypercovering-variant}. We will study
the augmentation
$$
a : \Sh((\mathcal{C}/K)_{total}) \longrightarrow \Sh(\mathcal{C})
$$
of Section \ref{section-simplicial-semi-representable}.

\begin{lemma}
\label{lemma-hypercovering-descent-sheaves}
Let $\mathcal{C}$ be a site with equalizers and fibre products.
 ========

regex_match $\mathcal{C}$
nn candidate a site
nn a site
new_annotation [(158063, 158076, 'VAR'), (158080, 158086, 'TYPE')]
===== sent Let $K$ be a hypercovering. ========

regex_match $K$
nn candidate a hypercovering
nn a hypercovering
new_annotation [(158127, 158130, 'VAR'), (158134, 158149, 'TYPE')]
===== sent \medskip\noindent
Let $\mathcal{F}$ be a sheaf on $\mathcal{C}$.
Recall that $a_*a^{-1}\mathcal{F}$ is the equalizer
of the two maps $a_{0, *}a_0^{-1}\mathcal{F} \to a_{1, *}a_1^{-1}\mathcal{F}$,
see Lemma \ref{lemma-comparison}.
 ========

regex_match $\mathcal{F}$
nn candidate a sheaf
nn a sheaf
new_annotation [(159206, 159219, 'VAR'), (159223, 159230, 'TYPE')]
===== sent Thus it suffices to prove
that $\SheafHom(-, \mathcal{F})$ transforms coequalizers
into equalizers which is immediate from the construction
in Sites, Section \ref{sites-section-glueing-sheaves}.

\medskip\noindent
Let $\mathcal{G}$ be a cartesian sheaf on $(\mathcal{C}/K)_{total}$.
We will show that $\mathcal{G} = a^{-1}\mathcal{F}$ for some sheaf
$\mathcal{F}$ on $\mathcal{C}$. ========

regex_match $\mathcal{G}$
nn candidate a cartesian sheaf
nn a cartesian sheaf
new_annotation [(160067, 160080, 'VAR'), (160084, 160101, 'TYPE')]
===== sent all $n$.

\medskip\noindent
Let $X$ be the final object of $\mathcal{C}$.
Then $\{U_0 \to X\}$ is a covering,
$\{U_1 \to U_0 \times U_0\}$ is a covering, and
$\{U_2 \to (\text{cosq}_1 \text{sk}_1 ========

regex_match $X$
nn candidate the final object
nn the final object
new_annotation [(162640, 162643, 'VAR'), (162647, 162663, 'TYPE')]
===== sent The proof is finished by an application of
Sites, Lemma \ref{sites-lemma-mapping-property-glue}
to the covering $\{U_0 \to X\}$.
\end{proof}

\begin{lemma}
\label{lemma-hypercovering-cech-complex}
Let $\mathcal{C}$ be a site with equalizers and fibre products.
 ========

regex_match $\mathcal{C}$
nn candidate a site
nn a site
new_annotation [(165430, 165443, 'VAR'), (165447, 165453, 'TYPE')]
===== sent Let $K$ be a hypercovering. ========

regex_match $K$
nn candidate a hypercovering
nn a hypercovering
new_annotation [(165494, 165497, 'VAR'), (165501, 165516, 'TYPE')]
===== sent Thus the equality of complexes comes 
from the canonical identifications
$\SheafHom'(\mathcal{G}, \mathcal{F}) =
\SheafHom(\mathbf{Z}_\mathcal{G}, \mathcal{F})$ for
$\mathcal{G}$ in $\Sh(\mathcal{C})$.
\end{proof}

\begin{lemma}
\label{lemma-hypercovering-descent-bounded-abelian}
Let $\mathcal{C}$ be a site with equalizers and fibre products.
 ========

regex_match $\mathcal{C}$
nn candidate a site
nn a site
new_annotation [(166544, 166557, 'VAR'), (166561, 166567, 'TYPE')]
===== sent Let $K$ be a hypercovering. ========

regex_match $K$
nn candidate a hypercovering
nn a hypercovering
new_annotation [(166608, 166611, 'VAR'), (166615, 166630, 'TYPE')]
===== sent First, let $\mathcal{I}$ be an injective abelian sheaf on $\mathcal{C}$.
Then the spectral sequence of
Lemma \ref{lemma-augmentation-spectral-sequence}
for the sheaf $a^{-1}\mathcal{I}$ degenerates as
$(a^{-1}\mathcal{I})_p = a_p^{-1}\mathcal{I}$
is injective by Lemma \ref{lemma-localize-injective}.
 ========

regex_match $\mathcal{I}$
nn candidate $\mathcal{I}$
nn candidate an injective abelian sheaf
nn an injective abelian sheaf
new_annotation [(166762, 166775, 'VAR'), (166779, 166805, 'TYPE')]
===== sent We conclude that
$R^pa_*a^{-1}\mathcal{I} = 0$ for $p > 0$.
On the other hand, we have $\mathcal{I} = a_*a^{-1}\mathcal{I}$
by Lemma \ref{lemma-hypercovering-descent-sheaves}.

\medskip\noindent
Next, let $E$ be as in the statement of the lemma.
 ========

new_annotation []
===== sent \end{proof}

\begin{lemma}
\label{lemma-compare-cohomology-hypercovering}
Let $\mathcal{C}$ be a site with equalizers and fibre products.
 ========

regex_match $\mathcal{C}$
nn candidate a site
nn a site
new_annotation [(168427, 168440, 'VAR'), (168444, 168450, 'TYPE')]
===== sent Let $K$ be a hypercovering.
 ========

regex_match $K$
nn candidate a hypercovering
nn a hypercovering
new_annotation [(168491, 168494, 'VAR'), (168498, 168513, 'TYPE')]
===== sent Then we have a canonical isomorphism
$$
R\Gamma(\mathcal{C}, E) =
R\Gamma((\mathcal{C}/K)_{total}, a^{-1}E)
$$
for $E \in D^+(\mathcal{C})$.
\end{lemma}

\begin{proof}
This follows from Lemma \ref{lemma-hypercovering-descent-bounded-abelian}
because $R\Gamma((\mathcal{C}/K)_{total}, -) =
R\Gamma(\mathcal{C}, -) \circ Ra_*$ by
Cohomology on Sites, Remark \ref{sites-cohomology-remark-before-Leray}.
\end{proof}

\begin{lemma}
\label{lemma-hypercovering-equivalence-bounded}
Let $\mathcal{C}$ be a site with equalizers and fibre products.
 ========

regex_match $\mathcal{C}$
nn candidate a site
nn a site
new_annotation [(168994, 169007, 'VAR'), (169011, 169017, 'TYPE')]
===== sent Let $K$ be a hypercovering.
 ========

regex_match $K$
nn candidate a hypercovering
nn a hypercovering
new_annotation [(169058, 169061, 'VAR'), (169065, 169080, 'TYPE')]
===== sent Let $\mathcal{A} \subset \textit{Ab}((\mathcal{C}/K)_{total})$
denote the weak Serre subcategory of cartesian abelian sheaves.
 ========

new_annotation []
===== sent \begin{remark}
\label{remark-compare-cohomology-hypercovering-presheaf}
Let $\mathcal{C}$ be a site. ========

regex_match $\mathcal{C}$
nn candidate a site
nn a site
new_annotation [(169863, 169876, 'VAR'), (169880, 169886, 'TYPE')]
===== sent Let $\mathcal{G}$ be a presheaf of sets on
$\mathcal{C}$. If $\mathcal{C}$ has equalizers and fibre products, then
we've defined the notion of a hypercovering of $\mathcal{G}$ in
Hypercoverings, Definition \ref{hypercovering-definition-hypercovering-variant}.
We claim that all the results in this section have a
valid counterpart in this setting.
 ========

regex_match $\mathcal{G}$
nn candidate a presheaf
nn a presheaf
new_annotation [(169892, 169905, 'VAR'), (169909, 169919, 'TYPE')]
===== sent This is
Lemma \ref{lemma-compare-cohomology-hypercovering}.
Let $R\Gamma(\mathcal{G}, -) : D(\mathcal{C}) \to D(\textit{Ab})$
be defined as the derived functor of the functor
$H^0(\mathcal{G}, -) = H^0(\mathcal{G}^\#, -)$
discussed in Hypercoverings, Section
\ref{hypercovering-section-hypercoverings-verdier} and
Cohomology on Sites, Section \ref{sites-cohomology-section-limp}.
We have
$$
R\Gamma(\mathcal{G}, E) = R\Gamma(\mathcal{C}/\mathcal{G}, j^{-1}E)
$$
by the analogue of Cohomology on Sites, Lemma
\ref{sites-cohomology-lemma-cohomology-of-open}
for the localization fuctor $j : \mathcal{C}/\mathcal{G} \to \mathcal{C}$.
Putting everything together we obtain
$$
R\Gamma(\mathcal{G}, E) =
R\Gamma((\mathcal{C}/K)_{total}, a^{-1}j^{-1}E) =
R\Gamma((\mathcal{C}/K)_{total}, g^{-1}E)
$$
for $E \in D^+(\mathcal{C})$ where
$g : \Sh((\mathcal{C}/K)_{total}) \to \Sh(\mathcal{C})$
is the composition of $a$ and $j$.
\end{remark}







\section{Cohomological descent for hypercoverings: modules}
\label{section-cohomological-descent-hypercoverings-modules}

\noindent
Let $\mathcal{C}$ be a site. ========

new_annotation []
===== sent This is
Lemma \ref{lemma-compare-cohomology-hypercovering}.
Let $R\Gamma(\mathcal{G}, -) : D(\mathcal{C}) \to D(\textit{Ab})$
be defined as the derived functor of the functor
$H^0(\mathcal{G}, -) = H^0(\mathcal{G}^\#, -)$
discussed in Hypercoverings, Section
\ref{hypercovering-section-hypercoverings-verdier} and
Cohomology on Sites, Section \ref{sites-cohomology-section-limp}.
We have
$$
R\Gamma(\mathcal{G}, E) = R\Gamma(\mathcal{C}/\mathcal{G}, j^{-1}E)
$$
by the analogue of Cohomology on Sites, Lemma
\ref{sites-cohomology-lemma-cohomology-of-open}
for the localization fuctor $j : \mathcal{C}/\mathcal{G} \to \mathcal{C}$.
Putting everything together we obtain
$$
R\Gamma(\mathcal{G}, E) =
R\Gamma((\mathcal{C}/K)_{total}, a^{-1}j^{-1}E) =
R\Gamma((\mathcal{C}/K)_{total}, g^{-1}E)
$$
for $E \in D^+(\mathcal{C})$ where
$g : \Sh((\mathcal{C}/K)_{total}) \to \Sh(\mathcal{C})$
is the composition of $a$ and $j$.
\end{remark}







\section{Cohomological descent for hypercoverings: modules}
\label{section-cohomological-descent-hypercoverings-modules}

\noindent
Let $\mathcal{C}$ be a site. ========

regex_match $\mathcal{C}$
nn candidate a site
nn a site
new_annotation [(172360, 172373, 'VAR'), (172377, 172383, 'TYPE')]
===== sent Let $\mathcal{O}_\mathcal{C}$
be a sheaf of rings. ========

new_annotation []
===== sent Assume $\mathcal{C}$
has equalizers and fibre products and let $K$ be a hypercovering
as defined in Hypercoverings, Definition
\ref{hypercovering-definition-hypercovering-variant}. We will study
cohomological descent for the augmentation
$$
a :
(\Sh((\mathcal{C}/K)_{total}), \mathcal{O})
\longrightarrow
(\Sh(\mathcal{C}), \mathcal{O}_\mathcal{C})
$$
of Remark \ref{remark-augmentation-ringed}.

\begin{lemma}
\label{lemma-hypercovering-descent-modules}
Let $\mathcal{C}$ be a site with equalizers and fibre products.
 ========

regex_match $K$
nn candidate a hypercovering
nn a hypercovering
new_annotation [(172499, 172502, 'VAR'), (172506, 172521, 'TYPE')]
===== sent Assume $\mathcal{C}$
has equalizers and fibre products and let $K$ be a hypercovering
as defined in Hypercoverings, Definition
\ref{hypercovering-definition-hypercovering-variant}. We will study
cohomological descent for the augmentation
$$
a :
(\Sh((\mathcal{C}/K)_{total}), \mathcal{O})
\longrightarrow
(\Sh(\mathcal{C}), \mathcal{O}_\mathcal{C})
$$
of Remark \ref{remark-augmentation-ringed}.

\begin{lemma}
\label{lemma-hypercovering-descent-modules}
Let $\mathcal{C}$ be a site with equalizers and fibre products.
 ========

regex_match $\mathcal{C}$
nn candidate a site
nn a site
new_annotation [(172895, 172908, 'VAR'), (172912, 172918, 'TYPE')]
===== sent Let $\mathcal{O}_\mathcal{C}$ be a sheaf of rings.
 ========

regex_match $\mathcal{O}_\mathcal{C}$
nn candidate a sheaf
nn a sheaf
new_annotation [(172959, 172984, 'VAR'), (172988, 172995, 'TYPE')]
===== sent Let $K$ be a hypercovering. ========

regex_match $K$
nn candidate a hypercovering
nn a hypercovering
new_annotation [(173010, 173013, 'VAR'), (173017, 173032, 'TYPE')]
===== sent Since $a^{-1}\mathcal{O}_\mathcal{C} = \mathcal{O}$ we have
$a^* = a^{-1}$. Hence the lemma follows
immediately from Lemma \ref{lemma-hypercovering-descent-sheaves}.
\end{proof}

\begin{lemma}
\label{lemma-hypercovering-descent-bounded-modules}
Let $\mathcal{C}$ be a site with equalizers and fibre products.
 ========

regex_match $\mathcal{C}$
nn candidate a site
nn a site
new_annotation [(173535, 173548, 'VAR'), (173552, 173558, 'TYPE')]
===== sent Let $\mathcal{O}_\mathcal{C}$ be a sheaf of rings.
 ========

regex_match $\mathcal{O}_\mathcal{C}$
nn candidate a sheaf
nn a sheaf
new_annotation [(173599, 173624, 'VAR'), (173628, 173635, 'TYPE')]
===== sent Let $K$ be a hypercovering. ========

regex_match $K$
nn candidate a hypercovering
nn a hypercovering
new_annotation [(173650, 173653, 'VAR'), (173657, 173672, 'TYPE')]
===== sent Since $a^{-1}\mathcal{O}_\mathcal{C} = \mathcal{O}$ we have
$La^* = a^* = a^{-1}$. Moreover $Ra_*$ agrees with
$Ra_*$ on abelian sheaves, see
Cohomology on Sites, Lemma
\ref{sites-cohomology-lemma-modules-abelian-unbounded}.
Hence the lemma follows
immediately from Lemma \ref{lemma-hypercovering-descent-bounded-abelian}.
\end{proof}

\begin{lemma}
\label{lemma-compare-cohomology-hypercovering-modules}
Let $\mathcal{C}$ be a site with equalizers and fibre products.
 ========

regex_match $\mathcal{C}$
nn candidate a site
nn a site
new_annotation [(174212, 174225, 'VAR'), (174229, 174235, 'TYPE')]
===== sent Let $\mathcal{O}_\mathcal{C}$ be a sheaf of rings.
 ========

regex_match $\mathcal{O}_\mathcal{C}$
nn candidate a sheaf
nn a sheaf
new_annotation [(174276, 174301, 'VAR'), (174305, 174312, 'TYPE')]
===== sent Let $K$ be a hypercovering.
 ========

regex_match $K$
nn candidate a hypercovering
nn a hypercovering
new_annotation [(174327, 174330, 'VAR'), (174334, 174349, 'TYPE')]
===== sent Then we have a canonical isomorphism
$$
R\Gamma(\mathcal{C}, E) =
R\Gamma((\mathcal{C}/K)_{total}, La^*E)
$$
for $E \in D^+(\mathcal{O}_\mathcal{C})$.
\end{lemma}

\begin{proof}
This follows from Lemma \ref{lemma-hypercovering-descent-bounded-modules}
because $R\Gamma((\mathcal{C}/K)_{total}, -) =
R\Gamma(\mathcal{C}, -) \circ Ra_*$ by
Cohomology on Sites, Remark \ref{sites-cohomology-remark-before-Leray}
or by
Cohomology on Sites, Lemma \ref{sites-cohomology-lemma-Leray-unbounded}.
\end{proof}

\begin{lemma}
\label{lemma-hypercovering-equivalence-bounded-modules}
Let $\mathcal{C}$ be a site with equalizers and fibre products.
 ========

regex_match $\mathcal{C}$
nn candidate a site
nn a site
new_annotation [(174926, 174939, 'VAR'), (174943, 174949, 'TYPE')]
===== sent Let $\mathcal{O}_\mathcal{C}$ be a sheaf of rings.
 ========

regex_match $\mathcal{O}_\mathcal{C}$
nn candidate a sheaf
nn a sheaf
new_annotation [(174990, 175015, 'VAR'), (175019, 175026, 'TYPE')]
===== sent Let $K$ be a hypercovering.
 ========

regex_match $K$
nn candidate a hypercovering
nn a hypercovering
new_annotation [(175041, 175044, 'VAR'), (175048, 175063, 'TYPE')]
===== sent Let $\mathcal{A} \subset \textit{Mod}(\mathcal{O})$
denote the weak Serre subcategory of cartesian $\mathcal{O}$-modules.
 ========

new_annotation []
===== sent Use
Lemmas \ref{lemma-equivalence-bounded},
\ref{lemma-hypercovering-descent-modules}, and
\ref{lemma-hypercovering-descent-bounded-modules}.
\end{proof}






\section{Cohomological descent for hypercoverings of an object}
\label{section-cohomological-descent-hypercoverings-X}

\noindent
In this section we assume $\mathcal{C}$ has fibre products
and $X \in \Ob(\mathcal{C})$. We let $K$ be a hypercovering of $X$
as defined in
Hypercoverings, Definition \ref{hypercovering-definition-hypercovering}.
We will study the augmentation
$$
a : \Sh((\mathcal{C}/K)_{total}) \longrightarrow \Sh(\mathcal{C}/X)
$$
of Remark \ref{remark-augmentation-over-object}.
Observe that $\mathcal{C}/X$ is a site which has equalizers
and fibre products and that $K$ is a
hypercovering for the site $\mathcal{C}/X$\footnote{The converse may not
be the case, i.e., if $K$ is a simplicial object of
$\text{SR}(\mathcal{C}, X) = \text{SR}(\mathcal{C}/X)$
which defines a hypercovering for the site $\mathcal{C}/X$ as in
Hypercoverings, Definition \ref{hypercovering-definition-hypercovering-variant},
then it may not be true that $K$ defines a hypercovering of $X$.
For example, if $K_0 = \{U_{0, i}\}_{i \in I_0}$
then the latter condition guarantees
$\{U_{0, i} \to X\}$ is a covering of $\mathcal{C}$
whereas the former condition only requires
$\coprod h_{U_{0, i}}^\# \to h_X^\#$ to be a surjective map
of sheaves.} ========

regex_match $K$
nn candidate a hypercovering
nn a hypercovering
new_annotation [(176026, 176029, 'VAR'), (176033, 176048, 'TYPE')]
===== sent \begin{lemma}
\label{lemma-hypercovering-X-descent-sheaves}
Let $\mathcal{C}$ be a site with fibre products and $X \in \Ob(\mathcal{C})$.
Let $K$ be a hypercovering of $X$. Then
\begin{enumerate}
\item $a^{-1} : \Sh(\mathcal{C}/X) \to \Sh((\mathcal{C}/K)_{total})$
is fully faithful with essential image the cartesian sheaves of sets,
\item $a^{-1} : \textit{Ab}(\mathcal{C}/X) \to
\textit{Ab}((\mathcal{C}/K)_{total})$
is fully faithful with essential image the cartesian sheaves
of abelian groups.
 ========

regex_match $\mathcal{C}$
nn candidate a site
nn a site
new_annotation [(177356, 177369, 'VAR'), (177373, 177379, 'TYPE')]
===== sent \begin{lemma}
\label{lemma-hypercovering-X-descent-sheaves}
Let $\mathcal{C}$ be a site with fibre products and $X \in \Ob(\mathcal{C})$.
Let $K$ be a hypercovering of $X$. Then
\begin{enumerate}
\item $a^{-1} : \Sh(\mathcal{C}/X) \to \Sh((\mathcal{C}/K)_{total})$
is fully faithful with essential image the cartesian sheaves of sets,
\item $a^{-1} : \textit{Ab}(\mathcal{C}/X) \to
\textit{Ab}((\mathcal{C}/K)_{total})$
is fully faithful with essential image the cartesian sheaves
of abelian groups.
 ========

regex_match $K$
nn candidate a hypercovering
nn a hypercovering
new_annotation [(177434, 177437, 'VAR'), (177441, 177456, 'TYPE')]
===== sent Via Remarks \ref{remark-semi-representable-over-object} and
\ref{remark-augmentation-over-object} and the discussion in
the introduction to this section
this follows from Lemma \ref{lemma-hypercovering-descent-sheaves}.
\end{proof}

\begin{lemma}
\label{lemma-hypercovering-X-descent-bounded-abelian}
Let $\mathcal{C}$ be a site with fibre product and $X \in \Ob(\mathcal{C})$.
Let $K$ be a hypercovering of $X$. For
$E \in D^+(\mathcal{C}/X)$ the map
$$
E \longrightarrow Ra_*a^{-1}E
$$
is an isomorphism.
 ========

regex_match $\mathcal{C}$
nn candidate a site
nn a site
new_annotation [(178196, 178209, 'VAR'), (178213, 178219, 'TYPE')]
===== sent Via Remarks \ref{remark-semi-representable-over-object} and
\ref{remark-augmentation-over-object} and the discussion in
the introduction to this section
this follows from Lemma \ref{lemma-hypercovering-descent-sheaves}.
\end{proof}

\begin{lemma}
\label{lemma-hypercovering-X-descent-bounded-abelian}
Let $\mathcal{C}$ be a site with fibre product and $X \in \Ob(\mathcal{C})$.
Let $K$ be a hypercovering of $X$. For
$E \in D^+(\mathcal{C}/X)$ the map
$$
E \longrightarrow Ra_*a^{-1}E
$$
is an isomorphism.
 ========

regex_match $K$
nn candidate a hypercovering
nn a hypercovering
new_annotation [(178273, 178276, 'VAR'), (178280, 178295, 'TYPE')]
===== sent Via Remarks \ref{remark-semi-representable-over-object} and
\ref{remark-augmentation-over-object} and the discussion in
the introduction to this section
this follows from Lemma \ref{lemma-hypercovering-descent-bounded-abelian}.
\end{proof}

\begin{lemma}
\label{lemma-compare-cohomology-hypercovering-X}
Let $\mathcal{C}$ be a site with fibre products and $X \in \Ob(\mathcal{C})$.
Let $K$ be a hypercovering of $X$.
 ========

regex_match $\mathcal{C}$
nn candidate a site
nn a site
new_annotation [(178733, 178746, 'VAR'), (178750, 178756, 'TYPE')]
===== sent Via Remarks \ref{remark-semi-representable-over-object} and
\ref{remark-augmentation-over-object} and the discussion in
the introduction to this section
this follows from Lemma \ref{lemma-hypercovering-descent-bounded-abelian}.
\end{proof}

\begin{lemma}
\label{lemma-compare-cohomology-hypercovering-X}
Let $\mathcal{C}$ be a site with fibre products and $X \in \Ob(\mathcal{C})$.
Let $K$ be a hypercovering of $X$.
 ========

regex_match $K$
nn candidate a hypercovering
nn a hypercovering
new_annotation [(178811, 178814, 'VAR'), (178818, 178833, 'TYPE')]
===== sent a^{-1}E)
$$
for $E \in D^+(\mathcal{C}/X)$.
\end{lemma}

\begin{proof}
Via Remarks \ref{remark-semi-representable-over-object} and
\ref{remark-augmentation-over-object}
this follows from Lemma \ref{lemma-compare-cohomology-hypercovering}.
\end{proof}

\begin{lemma}
\label{lemma-hypercovering-X-equivalence-bounded}
Let $\mathcal{C}$ be a site with fibre products and $X \in \Ob(\mathcal{C})$.
Let $K$ be a hypercovering of $X$.
Let $\mathcal{A} \subset \textit{Ab}((\mathcal{C}/K)_{total})$
denote the weak Serre subcategory of cartesian abelian sheaves.
 ========

regex_match $\mathcal{C}$
nn candidate a site
nn a site
new_annotation [(179251, 179264, 'VAR'), (179268, 179274, 'TYPE')]
===== sent a^{-1}E)
$$
for $E \in D^+(\mathcal{C}/X)$.
\end{lemma}

\begin{proof}
Via Remarks \ref{remark-semi-representable-over-object} and
\ref{remark-augmentation-over-object}
this follows from Lemma \ref{lemma-compare-cohomology-hypercovering}.
\end{proof}

\begin{lemma}
\label{lemma-hypercovering-X-equivalence-bounded}
Let $\mathcal{C}$ be a site with fibre products and $X \in \Ob(\mathcal{C})$.
Let $K$ be a hypercovering of $X$.
Let $\mathcal{A} \subset \textit{Ab}((\mathcal{C}/K)_{total})$
denote the weak Serre subcategory of cartesian abelian sheaves.
 ========

regex_match $K$
nn candidate a hypercovering
nn a hypercovering
new_annotation [(179329, 179332, 'VAR'), (179336, 179351, 'TYPE')]
===== sent a^{-1}E)
$$
for $E \in D^+(\mathcal{C}/X)$.
\end{lemma}

\begin{proof}
Via Remarks \ref{remark-semi-representable-over-object} and
\ref{remark-augmentation-over-object}
this follows from Lemma \ref{lemma-compare-cohomology-hypercovering}.
\end{proof}

\begin{lemma}
\label{lemma-hypercovering-X-equivalence-bounded}
Let $\mathcal{C}$ be a site with fibre products and $X \in \Ob(\mathcal{C})$.
Let $K$ be a hypercovering of $X$.
Let $\mathcal{A} \subset \textit{Ab}((\mathcal{C}/K)_{total})$
denote the weak Serre subcategory of cartesian abelian sheaves.
 ========

new_annotation []
===== sent In this section we assume $\mathcal{C}$ has fibre products
and $X \in \Ob(\mathcal{C})$. We let $K$ be a hypercovering of $X$
as defined in
Hypercoverings, Definition \ref{hypercovering-definition-hypercovering}.
Let $\mathcal{O}_\mathcal{C}$ be a sheaf of rings on $\mathcal{C}$.
Set $\mathcal{O}_X = \mathcal{O}_\mathcal{C}|_{\mathcal{C}/X}$.
We will study the augmentation
$$
a :
(\Sh((\mathcal{C}/K)_{total}), \mathcal{O})
\longrightarrow
(\Sh(\mathcal{C}/X), \mathcal{O}_X)
$$
of Remark \ref{remark-augmentation-ringed-over-object}.
Observe that $\mathcal{C}/X$ is a site which has equalizers
and fibre products and that $K$ is a
hypercovering for the site $\mathcal{C}/X$.
Therefore the results in this section are immediate consequences
of the corresponding results in
Section \ref{section-cohomological-descent-hypercoverings-modules}.

\begin{lemma}
\label{lemma-hypercovering-X-descent-modules}
Let $\mathcal{C}$ be a site with fibre products and $X \in \Ob(\mathcal{C})$.
Let $\mathcal{O}_\mathcal{C}$ be a sheaf of rings.
 ========

regex_match $K$
nn candidate a hypercovering
nn a hypercovering
new_annotation [(180104, 180107, 'VAR'), (180111, 180126, 'TYPE')]
===== sent In this section we assume $\mathcal{C}$ has fibre products
and $X \in \Ob(\mathcal{C})$. We let $K$ be a hypercovering of $X$
as defined in
Hypercoverings, Definition \ref{hypercovering-definition-hypercovering}.
Let $\mathcal{O}_\mathcal{C}$ be a sheaf of rings on $\mathcal{C}$.
Set $\mathcal{O}_X = \mathcal{O}_\mathcal{C}|_{\mathcal{C}/X}$.
We will study the augmentation
$$
a :
(\Sh((\mathcal{C}/K)_{total}), \mathcal{O})
\longrightarrow
(\Sh(\mathcal{C}/X), \mathcal{O}_X)
$$
of Remark \ref{remark-augmentation-ringed-over-object}.
Observe that $\mathcal{C}/X$ is a site which has equalizers
and fibre products and that $K$ is a
hypercovering for the site $\mathcal{C}/X$.
Therefore the results in this section are immediate consequences
of the corresponding results in
Section \ref{section-cohomological-descent-hypercoverings-modules}.

\begin{lemma}
\label{lemma-hypercovering-X-descent-modules}
Let $\mathcal{C}$ be a site with fibre products and $X \in \Ob(\mathcal{C})$.
Let $\mathcal{O}_\mathcal{C}$ be a sheaf of rings.
 ========

regex_match $\mathcal{O}_\mathcal{C}$
nn candidate a sheaf
nn a sheaf
new_annotation [(180225, 180250, 'VAR'), (180254, 180261, 'TYPE')]
===== sent In this section we assume $\mathcal{C}$ has fibre products
and $X \in \Ob(\mathcal{C})$. We let $K$ be a hypercovering of $X$
as defined in
Hypercoverings, Definition \ref{hypercovering-definition-hypercovering}.
Let $\mathcal{O}_\mathcal{C}$ be a sheaf of rings on $\mathcal{C}$.
Set $\mathcal{O}_X = \mathcal{O}_\mathcal{C}|_{\mathcal{C}/X}$.
We will study the augmentation
$$
a :
(\Sh((\mathcal{C}/K)_{total}), \mathcal{O})
\longrightarrow
(\Sh(\mathcal{C}/X), \mathcal{O}_X)
$$
of Remark \ref{remark-augmentation-ringed-over-object}.
Observe that $\mathcal{C}/X$ is a site which has equalizers
and fibre products and that $K$ is a
hypercovering for the site $\mathcal{C}/X$.
Therefore the results in this section are immediate consequences
of the corresponding results in
Section \ref{section-cohomological-descent-hypercoverings-modules}.

\begin{lemma}
\label{lemma-hypercovering-X-descent-modules}
Let $\mathcal{C}$ be a site with fibre products and $X \in \Ob(\mathcal{C})$.
Let $\mathcal{O}_\mathcal{C}$ be a sheaf of rings.
 ========

regex_match $\mathcal{C}$
nn candidate a site
nn a site
new_annotation [(180917, 180930, 'VAR'), (180934, 180940, 'TYPE')]
===== sent In this section we assume $\mathcal{C}$ has fibre products
and $X \in \Ob(\mathcal{C})$. We let $K$ be a hypercovering of $X$
as defined in
Hypercoverings, Definition \ref{hypercovering-definition-hypercovering}.
Let $\mathcal{O}_\mathcal{C}$ be a sheaf of rings on $\mathcal{C}$.
Set $\mathcal{O}_X = \mathcal{O}_\mathcal{C}|_{\mathcal{C}/X}$.
We will study the augmentation
$$
a :
(\Sh((\mathcal{C}/K)_{total}), \mathcal{O})
\longrightarrow
(\Sh(\mathcal{C}/X), \mathcal{O}_X)
$$
of Remark \ref{remark-augmentation-ringed-over-object}.
Observe that $\mathcal{C}/X$ is a site which has equalizers
and fibre products and that $K$ is a
hypercovering for the site $\mathcal{C}/X$.
Therefore the results in this section are immediate consequences
of the corresponding results in
Section \ref{section-cohomological-descent-hypercoverings-modules}.

\begin{lemma}
\label{lemma-hypercovering-X-descent-modules}
Let $\mathcal{C}$ be a site with fibre products and $X \in \Ob(\mathcal{C})$.
Let $\mathcal{O}_\mathcal{C}$ be a sheaf of rings.
 ========

regex_match $\mathcal{O}_\mathcal{C}$
nn candidate a sheaf
nn a sheaf
new_annotation [(180995, 181020, 'VAR'), (181024, 181031, 'TYPE')]
===== sent Let $K$ be a hypercovering of $X$. ========

regex_match $K$
nn candidate a hypercovering
nn a hypercovering
new_annotation [(181046, 181049, 'VAR'), (181053, 181068, 'TYPE')]
===== sent Via Remarks \ref{remark-semi-representable-ringed-over-object} and
\ref{remark-augmentation-ringed-over-object} and the discussion in
the introduction to this section
this follows from Lemma \ref{lemma-hypercovering-descent-modules}.
\end{proof}

\begin{lemma}
\label{lemma-hypercovering-X-descent-bounded-modules}
Let $\mathcal{C}$ be a site with fibre products and $X \in \Ob(\mathcal{C})$.
Let $\mathcal{O}_\mathcal{C}$ be a sheaf of rings.
 ========

regex_match $\mathcal{C}$
nn candidate a site
nn a site
new_annotation [(181638, 181651, 'VAR'), (181655, 181661, 'TYPE')]
===== sent Via Remarks \ref{remark-semi-representable-ringed-over-object} and
\ref{remark-augmentation-ringed-over-object} and the discussion in
the introduction to this section
this follows from Lemma \ref{lemma-hypercovering-descent-modules}.
\end{proof}

\begin{lemma}
\label{lemma-hypercovering-X-descent-bounded-modules}
Let $\mathcal{C}$ be a site with fibre products and $X \in \Ob(\mathcal{C})$.
Let $\mathcal{O}_\mathcal{C}$ be a sheaf of rings.
 ========

regex_match $\mathcal{O}_\mathcal{C}$
nn candidate a sheaf
nn a sheaf
new_annotation [(181716, 181741, 'VAR'), (181745, 181752, 'TYPE')]
===== sent Let $K$ be a hypercovering of $X$. For
$E \in D^+(\mathcal{O}_X)$ the map
$$
E \longrightarrow Ra_*La^*E
$$
is an isomorphism.
 ========

regex_match $K$
nn candidate a hypercovering
nn a hypercovering
new_annotation [(181767, 181770, 'VAR'), (181774, 181789, 'TYPE')]
===== sent Via Remarks \ref{remark-semi-representable-ringed-over-object} and
\ref{remark-augmentation-ringed-over-object} and the discussion in
the introduction to this section
this follows from Lemma \ref{lemma-hypercovering-descent-bounded-modules}.
\end{proof}

\begin{lemma}
\label{lemma-compare-cohomology-hypercovering-X-modules}
Let $\mathcal{C}$ be a site with fibre products and $X \in \Ob(\mathcal{C})$.
Let $\mathcal{O}_\mathcal{C}$ be a sheaf of rings.
 ========

regex_match $\mathcal{C}$
nn candidate a site
nn a site
new_annotation [(182247, 182260, 'VAR'), (182264, 182270, 'TYPE')]
===== sent Via Remarks \ref{remark-semi-representable-ringed-over-object} and
\ref{remark-augmentation-ringed-over-object} and the discussion in
the introduction to this section
this follows from Lemma \ref{lemma-hypercovering-descent-bounded-modules}.
\end{proof}

\begin{lemma}
\label{lemma-compare-cohomology-hypercovering-X-modules}
Let $\mathcal{C}$ be a site with fibre products and $X \in \Ob(\mathcal{C})$.
Let $\mathcal{O}_\mathcal{C}$ be a sheaf of rings.
 ========

regex_match $\mathcal{O}_\mathcal{C}$
nn candidate a sheaf
nn a sheaf
new_annotation [(182325, 182350, 'VAR'), (182354, 182361, 'TYPE')]
===== sent Let $K$ be a hypercovering of $X$.
 ========

regex_match $K$
nn candidate a hypercovering
nn a hypercovering
new_annotation [(182376, 182379, 'VAR'), (182383, 182398, 'TYPE')]
===== sent Then we have a canonical isomorphism
$$
R\Gamma(X, E) = R\Gamma((\mathcal{C}/K)_{total}, La^*E)
$$
for $E \in D^+(\mathcal{O}_\mathcal{C})$.
\end{lemma}

\begin{proof}
Via Remarks \ref{remark-semi-representable-ringed-over-object} and
\ref{remark-augmentation-ringed-over-object} and the discussion in
the introduction to this section
this follows from Lemma \ref{lemma-compare-cohomology-hypercovering-modules}.
\end{proof}

\begin{lemma}
\label{lemma-hypercovering-X-equivalence-bounded-modules}
Let $\mathcal{C}$ be a site with fibre products and $X \in \Ob(\mathcal{C})$.
Let $\mathcal{O}_\mathcal{C}$ be a sheaf of rings.
 ========

regex_match $\mathcal{C}$
nn candidate a site
nn a site
new_annotation [(182909, 182922, 'VAR'), (182926, 182932, 'TYPE')]
===== sent Then we have a canonical isomorphism
$$
R\Gamma(X, E) = R\Gamma((\mathcal{C}/K)_{total}, La^*E)
$$
for $E \in D^+(\mathcal{O}_\mathcal{C})$.
\end{lemma}

\begin{proof}
Via Remarks \ref{remark-semi-representable-ringed-over-object} and
\ref{remark-augmentation-ringed-over-object} and the discussion in
the introduction to this section
this follows from Lemma \ref{lemma-compare-cohomology-hypercovering-modules}.
\end{proof}

\begin{lemma}
\label{lemma-hypercovering-X-equivalence-bounded-modules}
Let $\mathcal{C}$ be a site with fibre products and $X \in \Ob(\mathcal{C})$.
Let $\mathcal{O}_\mathcal{C}$ be a sheaf of rings.
 ========

regex_match $\mathcal{O}_\mathcal{C}$
nn candidate a sheaf
nn a sheaf
new_annotation [(182987, 183012, 'VAR'), (183016, 183023, 'TYPE')]
===== sent Let $K$ be a hypercovering of $X$.
Let $\mathcal{A} \subset \textit{Mod}(\mathcal{O})$
denote the weak Serre subcategory of cartesian $\mathcal{O}$-modules.
 ========

regex_match $K$
nn candidate a hypercovering
nn a hypercovering
new_annotation [(183038, 183041, 'VAR'), (183045, 183060, 'TYPE')]
===== sent Let $K$ be a hypercovering of $X$.
Let $\mathcal{A} \subset \textit{Mod}(\mathcal{O})$
denote the weak Serre subcategory of cartesian $\mathcal{O}$-modules.
 ========

new_annotation []
===== sent \noindent
Let $\mathcal{C}$ be a site with fibre products and
let $X \in ========

regex_match $\mathcal{C}$
nn candidate a site
nn a site
new_annotation [(183734, 183747, 'VAR'), (183751, 183757, 'TYPE')]
===== sent \noindent
Let $\mathcal{C}$ be a site with fibre products and
let $X \in ========

new_annotation []
===== sent Let $U$ be a simplicial object of $\mathcal{C}$.
As usual we denote $U_n = U([n])$ and $f_\varphi : U_n \to U_m$
the morphism $f_\varphi = U(\varphi)$ corresponding to
$\varphi : [m] \to [n]$.
Assume we have an augmentation
$$
a : U \to X
$$
From this we obtain a simplicial site $(\mathcal{C}/U)_{total}$
and an augmentation morphism
$$
a : \Sh((\mathcal{C}/U)_{total}) \longrightarrow \Sh(\mathcal{C}/X)
$$
by thinking of $U$ as a simiplical semi-representable
object of $\mathcal{C}/X$ whose degree $n$ part is the singleton
element $\{U_n/X\}$ and applying the constructions in
Remark \ref{remark-augmentation-over-object}.

\medskip\noindent
An object of the site $(\mathcal{C}/U)_{total}$ is given by
a $V/U_n$ and a morphism $(\varphi, f) : V/U_n \to W/U_m$ is given
by a morphism $\varphi : [m] \to [n]$ in $\Delta$ and a morphism
$f : V \to W$ such that the diagram
$$
\xymatrix{
V \ar[r]_f \ar[d] & W \ar[d] \\
U_n \ar[r]^{f_\varphi} & U_m
}
$$
is commutative. ========

regex_match $U$
nn candidate a simplicial object
nn a simplicial object
new_annotation [(184001, 184004, 'VAR'), (184008, 184027, 'TYPE')]
===== sent only if
the simplicial semi-representable object $\{U_n\}$ is a hypercovering of $X$
in the sense of Section \ref{section-cohomological-descent-hypercoverings-X}.

\begin{lemma}
\label{lemma-hypercovering-X-simple-descent-sheaves}
Let $\mathcal{C}$ be a site with fibre product and $X \in \Ob(\mathcal{C})$.
Let $a : U \to X$ be a hypercovering of $X$ in $\mathcal{C}$ as defined above.
 ========

regex_match $\mathcal{C}$
nn candidate a site
nn a site
new_annotation [(185905, 185918, 'VAR'), (185922, 185928, 'TYPE')]
===== sent only if
the simplicial semi-representable object $\{U_n\}$ is a hypercovering of $X$
in the sense of Section \ref{section-cohomological-descent-hypercoverings-X}.

\begin{lemma}
\label{lemma-hypercovering-X-simple-descent-sheaves}
Let $\mathcal{C}$ be a site with fibre product and $X \in \Ob(\mathcal{C})$.
Let $a : U \to X$ be a hypercovering of $X$ in $\mathcal{C}$ as defined above.
 ========

regex_match $a : U \to X$
nn candidate a : U \to
nn candidate a hypercovering
nn a hypercovering
new_annotation [(185982, 185995, 'VAR'), (185999, 186014, 'TYPE')]
===== sent \end{lemma}

\begin{proof}
This is a special case of
Lemma \ref{lemma-hypercovering-X-descent-sheaves}.
\end{proof}

\begin{lemma}
\label{lemma-hypercovering-X-simple-descent-bounded-abelian}
Let $\mathcal{C}$ be a site with fibre product and $X \in \Ob(\mathcal{C})$.
Let $a : U \to X$ be a hypercovering of $X$ in $\mathcal{C}$ as defined above.
 ========

regex_match $\mathcal{C}$
nn candidate a site
nn a site
new_annotation [(186652, 186665, 'VAR'), (186669, 186675, 'TYPE')]
===== sent \end{lemma}

\begin{proof}
This is a special case of
Lemma \ref{lemma-hypercovering-X-descent-sheaves}.
\end{proof}

\begin{lemma}
\label{lemma-hypercovering-X-simple-descent-bounded-abelian}
Let $\mathcal{C}$ be a site with fibre product and $X \in \Ob(\mathcal{C})$.
Let $a : U \to X$ be a hypercovering of $X$ in $\mathcal{C}$ as defined above.
 ========

regex_match $a : U \to X$
nn candidate a : U \to
nn candidate a hypercovering
nn a hypercovering
new_annotation [(186729, 186742, 'VAR'), (186746, 186761, 'TYPE')]
===== sent \end{lemma}

\begin{proof}
This is a special case of
Lemma \ref{lemma-hypercovering-X-descent-bounded-abelian}.
\end{proof}

\begin{lemma}
\label{lemma-compare-cohomology-hypercovering-X-simple}
Let $\mathcal{C}$ be a site with fibre products and $X \in \Ob(\mathcal{C})$.
Let $a : U \to X$ be a hypercovering of $X$ in $\mathcal{C}$ as defined above.
 ========

regex_match $\mathcal{C}$
nn candidate a site
nn a site
new_annotation [(187097, 187110, 'VAR'), (187114, 187120, 'TYPE')]
===== sent \end{lemma}

\begin{proof}
This is a special case of
Lemma \ref{lemma-hypercovering-X-descent-bounded-abelian}.
\end{proof}

\begin{lemma}
\label{lemma-compare-cohomology-hypercovering-X-simple}
Let $\mathcal{C}$ be a site with fibre products and $X \in \Ob(\mathcal{C})$.
Let $a : U \to X$ be a hypercovering of $X$ in $\mathcal{C}$ as defined above.
 ========

regex_match $a : U \to X$
nn candidate a : U \to
nn candidate a hypercovering
nn a hypercovering
new_annotation [(187175, 187188, 'VAR'), (187192, 187207, 'TYPE')]
===== sent Then we have a canonical isomorphism
$$
R\Gamma(X, E) = R\Gamma((\mathcal{C}/U)_{total}, a^{-1}E)
$$
for $E \in D^+(\mathcal{C}/X)$.
\end{lemma}

\begin{proof}
This is a special case of
Lemma \ref{lemma-compare-cohomology-hypercovering-X}.
\end{proof}

\begin{lemma}
\label{lemma-hypercovering-X-simple-equivalence-bounded}
Let $\mathcal{C}$ be a site with fibre product and $X \in ========

regex_match $\mathcal{C}$
nn candidate a site
nn a site
new_annotation [(187578, 187591, 'VAR'), (187595, 187601, 'TYPE')]
===== sent Let $a : U \to X$ be a hypercovering of $X$ in $\mathcal{C}$ as defined above.
 ========

regex_match $a : U \to X$
nn candidate a : U \to
nn candidate a hypercovering
nn a hypercovering
new_annotation [(187655, 187668, 'VAR'), (187672, 187687, 'TYPE')]
===== sent Let $\mathcal{A} \subset \textit{Ab}((\mathcal{C}/U)_{total})$
denote the weak Serre subcategory of cartesian abelian sheaves.
 ========

new_annotation []
===== sent \end{proof}

\begin{lemma}
\label{lemma-sr-when-fibre-products}
Let $U$ be a simplicial object of a site $\mathcal{C}$
with fibre products.
 ========

regex_match $U$
nn candidate a simplicial object
nn a simplicial object
new_annotation [(188190, 188193, 'VAR'), (188197, 188216, 'TYPE')]
===== sent \end{proof}







\section{Hypercovering by a simplicial object of the site: modules}
\label{section-hypercovering-modules}

\noindent
Let $\mathcal{C}$ be a site with fibre products and $X \in \Ob(\mathcal{C})$.
Let $\mathcal{O}_\mathcal{C}$ be a sheaf of rings on $\mathcal{C}$.
Let $U \to X$ be a hypercovering of $X$ in $\mathcal{C}$ as defined
in Section \ref{section-hypercovering}. In this section we study the
augmentation
$$
a :
(\Sh((\mathcal{C}/U)_{total}), \mathcal{O})
\longrightarrow
(\Sh(\mathcal{C}/X), \mathcal{O}_X)
$$
we obtain by thinking of $U$ as a simiplical semi-representable
object of $\mathcal{C}/X$ whose degree $n$ part is the singleton
element $\{U_n/X\}$ and applying the constructions in
Remark \ref{remark-augmentation-ringed-over-object}.
 ========

regex_match $\mathcal{C}$
nn candidate a site
nn a site
new_annotation [(189906, 189919, 'VAR'), (189923, 189929, 'TYPE')]
===== sent \end{proof}







\section{Hypercovering by a simplicial object of the site: modules}
\label{section-hypercovering-modules}

\noindent
Let $\mathcal{C}$ be a site with fibre products and $X \in \Ob(\mathcal{C})$.
Let $\mathcal{O}_\mathcal{C}$ be a sheaf of rings on $\mathcal{C}$.
Let $U \to X$ be a hypercovering of $X$ in $\mathcal{C}$ as defined
in Section \ref{section-hypercovering}. In this section we study the
augmentation
$$
a :
(\Sh((\mathcal{C}/U)_{total}), \mathcal{O})
\longrightarrow
(\Sh(\mathcal{C}/X), \mathcal{O}_X)
$$
we obtain by thinking of $U$ as a simiplical semi-representable
object of $\mathcal{C}/X$ whose degree $n$ part is the singleton
element $\{U_n/X\}$ and applying the constructions in
Remark \ref{remark-augmentation-ringed-over-object}.
 ========

regex_match $\mathcal{O}_\mathcal{C}$
nn candidate a sheaf
nn a sheaf
new_annotation [(189984, 190009, 'VAR'), (190013, 190020, 'TYPE')]
===== sent \end{proof}







\section{Hypercovering by a simplicial object of the site: modules}
\label{section-hypercovering-modules}

\noindent
Let $\mathcal{C}$ be a site with fibre products and $X \in \Ob(\mathcal{C})$.
Let $\mathcal{O}_\mathcal{C}$ be a sheaf of rings on $\mathcal{C}$.
Let $U \to X$ be a hypercovering of $X$ in $\mathcal{C}$ as defined
in Section \ref{section-hypercovering}. In this section we study the
augmentation
$$
a :
(\Sh((\mathcal{C}/U)_{total}), \mathcal{O})
\longrightarrow
(\Sh(\mathcal{C}/X), \mathcal{O}_X)
$$
we obtain by thinking of $U$ as a simiplical semi-representable
object of $\mathcal{C}/X$ whose degree $n$ part is the singleton
element $\{U_n/X\}$ and applying the constructions in
Remark \ref{remark-augmentation-ringed-over-object}.
 ========

regex_match $U \to X$
nn candidate $U
nn candidate a hypercovering
nn a hypercovering
new_annotation [(190052, 190061, 'VAR'), (190065, 190080, 'TYPE')]
===== sent Thus all the results in this section are immediate consequences
of the corresponding results in
Section \ref{section-cohomological-descent-hypercoverings-X-modules}.

\begin{lemma}
\label{lemma-hypercovering-X-simple-descent-modules}
Let $\mathcal{C}$ be a site with fibre products and $X \in \Ob(\mathcal{C})$.
Let $\mathcal{O}_\mathcal{C}$ be a sheaf of rings.
 ========

regex_match $\mathcal{C}$
nn candidate a site
nn a site
new_annotation [(190778, 190791, 'VAR'), (190795, 190801, 'TYPE')]
===== sent Thus all the results in this section are immediate consequences
of the corresponding results in
Section \ref{section-cohomological-descent-hypercoverings-X-modules}.

\begin{lemma}
\label{lemma-hypercovering-X-simple-descent-modules}
Let $\mathcal{C}$ be a site with fibre products and $X \in \Ob(\mathcal{C})$.
Let $\mathcal{O}_\mathcal{C}$ be a sheaf of rings.
 ========

regex_match $\mathcal{O}_\mathcal{C}$
nn candidate a sheaf
nn a sheaf
new_annotation [(190856, 190881, 'VAR'), (190885, 190892, 'TYPE')]
===== sent Let $U$ be a hypercovering of $X$ in $\mathcal{C}$. ========

regex_match $U$
nn candidate a hypercovering
nn a hypercovering
new_annotation [(190907, 190910, 'VAR'), (190914, 190929, 'TYPE')]
===== sent \end{lemma}

\begin{proof}
This is a special case of
Lemma \ref{lemma-hypercovering-X-descent-modules}.
\end{proof}

\begin{lemma}
\label{lemma-hypercovering-X-simple-descent-bounded-modules}
Let $\mathcal{C}$ be a site with fibre products and $X \in \Ob(\mathcal{C})$.
Let $\mathcal{O}_\mathcal{C}$ be a sheaf of rings.
 ========

regex_match $\mathcal{C}$
nn candidate a site
nn a site
new_annotation [(191366, 191379, 'VAR'), (191383, 191389, 'TYPE')]
===== sent \end{lemma}

\begin{proof}
This is a special case of
Lemma \ref{lemma-hypercovering-X-descent-modules}.
\end{proof}

\begin{lemma}
\label{lemma-hypercovering-X-simple-descent-bounded-modules}
Let $\mathcal{C}$ be a site with fibre products and $X \in \Ob(\mathcal{C})$.
Let $\mathcal{O}_\mathcal{C}$ be a sheaf of rings.
 ========

regex_match $\mathcal{O}_\mathcal{C}$
nn candidate a sheaf
nn a sheaf
new_annotation [(191444, 191469, 'VAR'), (191473, 191480, 'TYPE')]
===== sent Let $U$ be a hypercovering of $X$ in $\mathcal{C}$. For
$E \in D^+(\mathcal{O}_X)$ the map
$$
E \longrightarrow Ra_*La^*E
$$
is an isomorphism.
 ========

regex_match $U$
nn candidate a hypercovering
nn a hypercovering
new_annotation [(191495, 191498, 'VAR'), (191502, 191517, 'TYPE')]
===== sent \end{lemma}

\begin{proof}
This is a special case of
Lemma \ref{lemma-hypercovering-X-descent-bounded-modules}.
\end{proof}

\begin{lemma}
\label{lemma-compare-cohomology-hypercovering-X-simple-modules}
Let $\mathcal{C}$ be a site with fibre products and $X \in \Ob(\mathcal{C})$.
Let $\mathcal{O}_\mathcal{C}$ be a sheaf of rings.
 ========

regex_match $\mathcal{C}$
nn candidate a site
nn a site
new_annotation [(191842, 191855, 'VAR'), (191859, 191865, 'TYPE')]
===== sent \end{lemma}

\begin{proof}
This is a special case of
Lemma \ref{lemma-hypercovering-X-descent-bounded-modules}.
\end{proof}

\begin{lemma}
\label{lemma-compare-cohomology-hypercovering-X-simple-modules}
Let $\mathcal{C}$ be a site with fibre products and $X \in \Ob(\mathcal{C})$.
Let $\mathcal{O}_\mathcal{C}$ be a sheaf of rings.
 ========

regex_match $\mathcal{O}_\mathcal{C}$
nn candidate a sheaf
nn a sheaf
new_annotation [(191920, 191945, 'VAR'), (191949, 191956, 'TYPE')]
===== sent Let $U$ be a hypercovering of $X$ in $\mathcal{C}$.
Then we have a canonical isomorphism
$$
R\Gamma(X, E) = R\Gamma((\mathcal{C}/U)_{total}, La^*E)
$$
for $E \in D^+(\mathcal{O}_\mathcal{C})$.
\end{lemma}

\begin{proof}
 ========

regex_match $U$
nn candidate a hypercovering
nn a hypercovering
new_annotation [(191971, 191974, 'VAR'), (191978, 191993, 'TYPE')]
===== sent This is a special case of
Lemma \ref{lemma-compare-cohomology-hypercovering-X-modules}.
\end{proof}

\begin{lemma}
\label{lemma-hypercovering-X-simple-equivalence-bounded-modules}
Let $\mathcal{C}$ be a site with fibre products and $X \in \Ob(\mathcal{C})$.
Let $\mathcal{O}_\mathcal{C}$ be a sheaf of rings.
 ========

regex_match $\mathcal{C}$
nn candidate a site
nn a site
new_annotation [(192371, 192384, 'VAR'), (192388, 192394, 'TYPE')]
===== sent This is a special case of
Lemma \ref{lemma-compare-cohomology-hypercovering-X-modules}.
\end{proof}

\begin{lemma}
\label{lemma-hypercovering-X-simple-equivalence-bounded-modules}
Let $\mathcal{C}$ be a site with fibre products and $X \in \Ob(\mathcal{C})$.
Let $\mathcal{O}_\mathcal{C}$ be a sheaf of rings.
 ========

regex_match $\mathcal{O}_\mathcal{C}$
nn candidate a sheaf
nn a sheaf
new_annotation [(192449, 192474, 'VAR'), (192478, 192485, 'TYPE')]
===== sent Let $U$ be a hypercovering of $X$ in $\mathcal{C}$.
Let $\mathcal{A} \subset \textit{Mod}(\mathcal{O})$
denote the weak Serre subcategory of cartesian $\mathcal{O}$-modules.
 ========

regex_match $U$
nn candidate a hypercovering
nn a hypercovering
new_annotation [(192500, 192503, 'VAR'), (192507, 192522, 'TYPE')]
===== sent Let $U$ be a hypercovering of $X$ in $\mathcal{C}$.
Let $\mathcal{A} \subset \textit{Mod}(\mathcal{O})$
denote the weak Serre subcategory of cartesian $\mathcal{O}$-modules.
 ========

new_annotation []
===== sent \medskip\noindent
Let $(\mathcal{C}, \mathcal{O}_\mathcal{C})$ be a ringed site.
 ========

regex_match $(\mathcal{C}, \mathcal{O}_\mathcal{C})$
nn candidate a ringed site
nn a ringed site
new_annotation [(193563, 193603, 'VAR'), (193607, 193620, 'TYPE')]
===== sent \begin{example}
\label{example-quasi-coherent-spaces-etale}
Let $S$ be a scheme and let $X$ be an algebraic space over $S$.
Let $\mathcal{C} = X_{spaces, \etale}$ be the \'etale site
on the category of algebraic spaces \'etale over $X$, see
Properties of Spaces, Definition
\ref{spaces-properties-definition-spaces-etale-site}.
Denote $\mathcal{O}_\mathcal{C}$ the structure sheaf, i.e., the
sheaf given by the rule $U \mapsto \Gamma(U, \mathcal{O}_U)$.
Denote $\mathcal{A}_U$ the category of quasi-coherent $\mathcal{O}_U$-modules.
 ========

regex_match $S$
nn candidate a scheme
nn a scheme
new_annotation [(195517, 195520, 'VAR'), (195524, 195532, 'TYPE')]
===== sent \begin{example}
\label{example-quasi-coherent-spaces-etale}
Let $S$ be a scheme and let $X$ be an algebraic space over $S$.
Let $\mathcal{C} = X_{spaces, \etale}$ be the \'etale site
on the category of algebraic spaces \'etale over $X$, see
Properties of Spaces, Definition
\ref{spaces-properties-definition-spaces-etale-site}.
Denote $\mathcal{O}_\mathcal{C}$ the structure sheaf, i.e., the
sheaf given by the rule $U \mapsto \Gamma(U, \mathcal{O}_U)$.
Denote $\mathcal{A}_U$ the category of quasi-coherent $\mathcal{O}_U$-modules.
 ========

regex_match $X$
nn candidate $X$
nn candidate an algebraic space
nn an algebraic space
new_annotation [(195541, 195544, 'VAR'), (195548, 195566, 'TYPE')]
===== sent \begin{example}
\label{example-quasi-coherent-spaces-etale}
Let $S$ be a scheme and let $X$ be an algebraic space over $S$.
Let $\mathcal{C} = X_{spaces, \etale}$ be the \'etale site
on the category of algebraic spaces \'etale over $X$, see
Properties of Spaces, Definition
\ref{spaces-properties-definition-spaces-etale-site}.
Denote $\mathcal{O}_\mathcal{C}$ the structure sheaf, i.e., the
sheaf given by the rule $U \mapsto \Gamma(U, \mathcal{O}_U)$.
Denote $\mathcal{A}_U$ the category of quasi-coherent $\mathcal{O}_U$-modules.
 ========

regex_match $\mathcal{C} = X_{spaces, \etale}$
nn candidate $\mathcal{C
nn candidate the \'etale site
nn the \'etale site
new_annotation [(195581, 195615, 'VAR'), (195619, 195635, 'TYPE')]
===== sent Let $\mathcal{B} = \Ob(\mathcal{C})$ and for $V \in \mathcal{B}$
set $d_V = 0$ and let $\text{Cov}_V$ denote
the coverings $\{V_i \to V\}$ with $V_i$ affine for all $i$.
Then the assumptions (1), (2), (3) are satisfied.
 ========

new_annotation []
===== sent Let $\mathcal{B} = \Ob(\mathcal{C})$ and for $V \in \mathcal{B}$
set $d_V = 0$ and let $\text{Cov}_V$ denote
the coverings $\{V_i \to V\}$ with $V_i$ affine for all $i$.
Then the assumptions (1), (2), (3) are satisfied.
 ========

new_annotation []
===== sent See Properties of Spaces, Lemmas
\ref{spaces-properties-lemma-pullback-quasi-coherent} and
\ref{spaces-properties-lemma-properties-quasi-coherent}
for properties (1) and (2) and the vanishing in (3) follows from
Cohomology of Schemes, Lemma
\ref{coherent-lemma-quasi-coherent-affine-cohomology-zero}
and the discussion in Cohomology of Spaces, Section
\ref{spaces-cohomology-section-higher-direct-image}.
\end{example}

\begin{example}
\label{example-etale}
Let $S$ be one of the following types of schemes
\begin{enumerate}
\item the spectrum of a finite field,
\item the spectrum of a separably closed field,
\item the spectrum of a strictly henselian Noetherian local ring,
\item the spectrum of a henselian Noetherian local ring with
finite residue field,
\item add more here.
 ========

new_annotation []
===== sent \end{enumerate}
Let $\Lambda$ be a finite ring whose order is invertible on $S$.
Let $\mathcal{C} \subset (\Sch/S)_\etale$
be the full subcategory consisting of schemes locally of finite
type over $S$ endowed with the \'etale topology.
 ========

regex_match $\Lambda$
nn candidate a finite ring
nn a finite ring
new_annotation [(197007, 197016, 'VAR'), (197020, 197033, 'TYPE')]
===== sent \end{enumerate}
Let $\Lambda$ be a finite ring whose order is invertible on $S$.
Let $\mathcal{C} \subset (\Sch/S)_\etale$
be the full subcategory consisting of schemes locally of finite
type over $S$ endowed with the \'etale topology.
 ========

new_annotation []
===== sent Let $\mathcal{O}_\mathcal{C} = \underline{\Lambda}$ be the
constant sheaf. ========

new_annotation []
===== sent Let $\mathcal{B} \subset \Ob(\mathcal{C})$
be the set of quasi-compact objects. ========

new_annotation []
===== sent For $V \in \mathcal{B}$ set
$$
d_V = 1 + 2\dim(S) +
\sup\nolimits_{v \in V}(\text{trdeg}_{\kappa(s)}(\kappa(v)) +
2 \dim \mathcal{O}_{V, v})
$$
and let $\text{Cov}_V$ denote the \'etale coverings $\{V_i \to V\}$
with $V_i$ quasi-compact for all $i$.
Our choice of bound $d_V$ comes from Gabber's theorem
on cohomological dimension. ========

new_annotation []
===== sent \end{example}

\noindent
Let $(\mathcal{C}, \mathcal{O}_\mathcal{C})$ be a ringed site.
 ========

regex_match $(\mathcal{C}, \mathcal{O}_\mathcal{C})$
nn candidate a ringed site
nn a ringed site
new_annotation [(198216, 198256, 'VAR'), (198260, 198273, 'TYPE')]
===== sent In particular, given a simplicial semi-representable object $K$
it is unambiguous to say what it means for an object $\mathcal{F}$ of
$\textit{Mod}(\mathcal{O})$ as in Remark \ref{remark-augmentation-ringed}
to have restrictions $\mathcal{F}_n$ in
$\mathcal{A}_{K_n}$ for all $n$.

\begin{lemma}
\label{lemma-hypercovering-equivalence-modules}
Let $(\mathcal{C}, \mathcal{O}_\mathcal{C})$ be a ringed site.
 ========

regex_match $(\mathcal{C}, \mathcal{O}_\mathcal{C})$
nn candidate a ringed site
nn a ringed site
new_annotation [(199341, 199381, 'VAR'), (199385, 199398, 'TYPE')]
===== sent Assume $\mathcal{C}$ has equalizers and fibre products and
let $K$ be a hypercovering.
 ========

regex_match $K$
nn candidate a hypercovering
nn a hypercovering
new_annotation [(199663, 199666, 'VAR'), (199670, 199685, 'TYPE')]
===== sent Let $((\mathcal{C}/K)_{total}, \mathcal{O})$ be as in
Remark \ref{remark-augmentation-ringed}.
Let $\mathcal{A}_{total} \subset \textit{Mod}(\mathcal{O})$
denote the weak Serre subcategory of cartesian $\mathcal{O}$-modules
$\mathcal{F}$ whose restriction $\mathcal{F}_n$ is in
$\mathcal{A}_{K_n}$ for all $n$ (as defined above).
 ========

new_annotation []
===== sent Let $((\mathcal{C}/K)_{total}, \mathcal{O})$ be as in
Remark \ref{remark-augmentation-ringed}.
Let $\mathcal{A}_{total} \subset \textit{Mod}(\mathcal{O})$
denote the weak Serre subcategory of cartesian $\mathcal{O}$-modules
$\mathcal{F}$ whose restriction $\mathcal{F}_n$ is in
$\mathcal{A}_{K_n}$ for all $n$ (as defined above).
 ========

new_annotation []
===== sent For assumption (5) let $\mathcal{B}_{total}$ be the set of
objects $U/U_{n, i}$ of the site $(\mathcal{C}/K)_{total}$
such that $U \in \mathcal{B}$ where $\mathcal{B}$ is as in
(\ref{item-bounded-dimension}). ========

regex_match $\mathcal{B}_{total}$
nn candidate the set
nn the set
new_annotation [(202580, 202601, 'VAR'), (202605, 202612, 'TYPE')]
===== sent \begin{situation}
\label{situation-locally-given}
Let $(\mathcal{C}, \mathcal{O}_\mathcal{C})$ be a ringed site. ========

regex_match $(\mathcal{C}, \mathcal{O}_\mathcal{C})$
nn candidate a ringed site
nn a ringed site
new_annotation [(204215, 204255, 'VAR'), (204259, 204272, 'TYPE')]
===== sent Let $L$ be a simplicial object of $\text{SR}(\mathcal{B})$.
Consider the simplicial object $K = u(L)$ of $\text{SR}(\mathcal{C})$
and let $((\mathcal{C}/K)_{total}, \mathcal{O})$ be as in
Remark \ref{remark-augmentation-ringed}.
There exists a cartesian object $E$ of $D(\mathcal{O})$
such that writing $L_n = \{U_{n, i}\}_{i \in I_n}$
the restriction of $E$ to $D(\mathcal{O}_{\mathcal{C}/u(U_{n, i})})$
is $E_{U_{n, i}}$ compatibly (see proof for details).
 ========

regex_match $L$
nn candidate a simplicial object
nn a simplicial object
new_annotation [(205929, 205932, 'VAR'), (205936, 205955, 'TYPE')]
===== sent Let $L$ be a simplicial object of $\text{SR}(\mathcal{B})$.
Consider the simplicial object $K = u(L)$ of $\text{SR}(\mathcal{C})$
and let $((\mathcal{C}/K)_{total}, \mathcal{O})$ be as in
Remark \ref{remark-augmentation-ringed}.
There exists a cartesian object $E$ of $D(\mathcal{O})$
such that writing $L_n = \{U_{n, i}\}_{i \in I_n}$
the restriction of $E$ to $D(\mathcal{O}_{\mathcal{C}/u(U_{n, i})})$
is $E_{U_{n, i}}$ compatibly (see proof for details).
 ========

new_annotation []
===== sent Let's work in the category $\textit{LC}$ of Hausdorff and locally
quasi-compact topological spaces and continuous maps, see
Cohomology on Sites, Section \ref{sites-cohomology-section-cohomology-LC}.
Let $X$ be an object of $\textit{LC}$ and let $U$ be a simplicial
object of $\textit{LC}$. Assume we have an augmentation
$$
a : U \to X
$$
We say that $U$ is a {\it proper hypercovering} of $X$ if
\begin{enumerate}
\item $U_0 \to X$ is a proper surjective map,
\item $U_1 \to U_0 \times_X U_0$ is a proper surjective map,
\item $U_{n + 1} \to (\text{cosk}_n\text{sk}_n U)_{n + 1}$
is a proper surjective map for $n \geq 1$.
\end{enumerate}
The category $\textit{LC}$ has all finite limits, hence the
coskeleta used in the formulation above exist.
 ========

regex_match $X$
nn candidate an object
nn an object
new_annotation [(212749, 212752, 'VAR'), (212756, 212765, 'TYPE')]
===== sent Let's work in the category $\textit{LC}$ of Hausdorff and locally
quasi-compact topological spaces and continuous maps, see
Cohomology on Sites, Section \ref{sites-cohomology-section-cohomology-LC}.
Let $X$ be an object of $\textit{LC}$ and let $U$ be a simplicial
object of $\textit{LC}$. Assume we have an augmentation
$$
a : U \to X
$$
We say that $U$ is a {\it proper hypercovering} of $X$ if
\begin{enumerate}
\item $U_0 \to X$ is a proper surjective map,
\item $U_1 \to U_0 \times_X U_0$ is a proper surjective map,
\item $U_{n + 1} \to (\text{cosk}_n\text{sk}_n U)_{n + 1}$
is a proper surjective map for $n \geq 1$.
\end{enumerate}
The category $\textit{LC}$ has all finite limits, hence the
coskeleta used in the formulation above exist.
 ========

regex_match $U$
nn candidate a simplicial
object
nn a simplicial
object
new_annotation [(212791, 212794, 'VAR'), (212798, 212817, 'TYPE')]
===== sent \begin{lemma}
\label{lemma-compare-simplicial-objects}
Let $U$ be a simplicial object of $\textit{LC}$ and let $a : U \to X$
be an augmentation. ========

regex_match $U$
nn candidate a simplicial object
nn a simplicial object
new_annotation [(213942, 213945, 'VAR'), (213949, 213968, 'TYPE')]
===== sent \begin{lemma}
\label{lemma-compare-simplicial-objects}
Let $U$ be a simplicial object of $\textit{LC}$ and let $a : U \to X$
be an augmentation. ========

new_annotation []
===== sent \end{proof}

\begin{lemma}
\label{lemma-descent-sheaves-for-proper-hypercovering}
Let $U$ be a simplicial object of $\textit{LC}$ and let $a : U \to X$
be an augmentation. ========

regex_match $U$
nn candidate a simplicial object
nn a simplicial object
new_annotation [(215326, 215329, 'VAR'), (215333, 215352, 'TYPE')]
===== sent \end{proof}

\begin{lemma}
\label{lemma-descent-sheaves-for-proper-hypercovering}
Let $U$ be a simplicial object of $\textit{LC}$ and let $a : U \to X$
be an augmentation. ========

new_annotation []
===== sent Since $a_{qc}^{-1}\mathcal{H} = h^{-1}\mathcal{G}$
we deduce that $h^{-1}\mathcal{G} \cong h^{-1}a^{-1}a_*\mathcal{G}$
which ends the proof by fully faithfulness of $h^{-1}$.
\end{proof}

\begin{lemma}
\label{lemma-cohomological-descent-for-proper-hypercovering}
Let $U$ be a simplicial object of $\textit{LC}$ and let $a : U \to X$
be an augmentation. ========

regex_match $U$
nn candidate a simplicial object
nn a simplicial object
new_annotation [(218508, 218511, 'VAR'), (218515, 218534, 'TYPE')]
===== sent Since $a_{qc}^{-1}\mathcal{H} = h^{-1}\mathcal{G}$
we deduce that $h^{-1}\mathcal{G} \cong h^{-1}a^{-1}a_*\mathcal{G}$
which ends the proof by fully faithfulness of $h^{-1}$.
\end{proof}

\begin{lemma}
\label{lemma-cohomological-descent-for-proper-hypercovering}
Let $U$ be a simplicial object of $\textit{LC}$ and let $a : U \to X$
be an augmentation. ========

new_annotation []
===== sent Hence $Rh_*h^{-1}$ is the identity functor on
$D^+(U_{Zar})$ by
Lemma \ref{lemma-direct-image-morphism-simplicial-sites}.
We have
\begin{align*}
Ra_*(a^{-1}K)
& =
Ra_*Rh_*h^{-1}a^{-1}K \\
& =
Rh_{-1, *}Ra_{qc, *}a_{qc}^{-1}(h_{-1})^{-1}K \\
& =
Rh_{-1, *}(h_{-1})^{-1}K \\
& =
K
\end{align*}
The first equality by the discussion above, the second equality
because of the commutativity of the diagram in
Lemma \ref{lemma-compare-simplicial-objects}, the third equality by
Lemma \ref{lemma-hypercovering-X-simple-descent-bounded-abelian}
($U$ is a hypercovering of $X$ in $\textit{LC}_{qc}$ by
Cohomology on Sites, Lemma
\ref{sites-cohomology-lemma-proper-surjective-is-qc-covering}),
and the last equality by the already used Cohomology on Sites, Lemma
\ref{sites-cohomology-lemma-cohomological-descent-LC}.
\end{proof}

\begin{lemma}
\label{lemma-compute-via-proper-hypercovering}
Let $U$ be a simplicial object of $\textit{LC}$ and let $a : U \to X$
be an augmentation. ========

regex_match $U$
nn candidate a simplicial object
nn a simplicial object
new_annotation [(219931, 219934, 'VAR'), (219938, 219957, 'TYPE')]
===== sent Hence $Rh_*h^{-1}$ is the identity functor on
$D^+(U_{Zar})$ by
Lemma \ref{lemma-direct-image-morphism-simplicial-sites}.
We have
\begin{align*}
Ra_*(a^{-1}K)
& =
Ra_*Rh_*h^{-1}a^{-1}K \\
& =
Rh_{-1, *}Ra_{qc, *}a_{qc}^{-1}(h_{-1})^{-1}K \\
& =
Rh_{-1, *}(h_{-1})^{-1}K \\
& =
K
\end{align*}
The first equality by the discussion above, the second equality
because of the commutativity of the diagram in
Lemma \ref{lemma-compare-simplicial-objects}, the third equality by
Lemma \ref{lemma-hypercovering-X-simple-descent-bounded-abelian}
($U$ is a hypercovering of $X$ in $\textit{LC}_{qc}$ by
Cohomology on Sites, Lemma
\ref{sites-cohomology-lemma-proper-surjective-is-qc-covering}),
and the last equality by the already used Cohomology on Sites, Lemma
\ref{sites-cohomology-lemma-cohomological-descent-LC}.
\end{proof}

\begin{lemma}
\label{lemma-compute-via-proper-hypercovering}
Let $U$ be a simplicial object of $\textit{LC}$ and let $a : U \to X$
be an augmentation. ========

new_annotation []
===== sent This follows from
Lemma \ref{lemma-cohomological-descent-for-proper-hypercovering}
because $R\Gamma(U_{Zar}, -) = R\Gamma(X, -) \circ Ra_*$ by
Cohomology on Sites, Remark \ref{sites-cohomology-remark-before-Leray}.
\end{proof}

\begin{lemma}
\label{lemma-proper-hypercovering-equivalence-bounded}
Let $U$ be a simplicial object of $\textit{LC}$ and let $a : U \to X$
be an augmentation.
 ========

regex_match $U$
nn candidate a simplicial object
nn a simplicial object
new_annotation [(220535, 220538, 'VAR'), (220542, 220561, 'TYPE')]
===== sent This follows from
Lemma \ref{lemma-cohomological-descent-for-proper-hypercovering}
because $R\Gamma(U_{Zar}, -) = R\Gamma(X, -) \circ Ra_*$ by
Cohomology on Sites, Remark \ref{sites-cohomology-remark-before-Leray}.
\end{proof}

\begin{lemma}
\label{lemma-proper-hypercovering-equivalence-bounded}
Let $U$ be a simplicial object of $\textit{LC}$ and let $a : U \to X$
be an augmentation.
 ========

new_annotation []
===== sent Let $\mathcal{A} \subset \textit{Ab}(U_{Zar})$
denote the weak Serre subcategory of cartesian abelian sheaves.
 ========

new_annotation []
===== sent Use
Lemmas \ref{lemma-equivalence-bounded},
\ref{lemma-descent-sheaves-for-proper-hypercovering}, and
\ref{lemma-cohomological-descent-for-proper-hypercovering}.
\end{proof}

\begin{lemma}
\label{lemma-spectral-sequence-proper-hypercovering}
Let $U$ be a simplicial object of $\textit{LC}$ and let
$a : U \to X$ be an augmentation. ========

regex_match $U$
nn candidate a simplicial object
nn a simplicial object
new_annotation [(221432, 221435, 'VAR'), (221439, 221458, 'TYPE')]
===== sent Let $\mathcal{F}$ be an abelian sheaf
on $X$. Let $\mathcal{F}_n$ be the pullback to $U_n$.
If $U$ is a proper hypercovering of $X$, then
there exists a canonical spectral sequence
$$
E_1^{p, q} = H^q(U_p, \mathcal{F}_p)
$$
converging to $H^{p + q}(X, \mathcal{F})$.
\end{lemma}

\begin{proof}
Immediate consequence of Lemmas \ref{lemma-compute-via-proper-hypercovering}
and \ref{lemma-simplicial-sheaf-cohomology}.
\end{proof}




\section{Simplicial schemes}
\label{section-simplicial}

\noindent
A {\it simplicial scheme} is a simplicial object in the category of schemes,
see Simplicial, Definition \ref{simplicial-definition-simplicial-object}.
Recall that a simplicial scheme looks like
$$
\xymatrix{
X_2
\ar@<2ex>[r]
\ar@<0ex>[r]
\ar@<-2ex>[r]
&
X_1
\ar@<1ex>[r]
\ar@<-1ex>[r]
\ar@<1ex>[l]
\ar@<-1ex>[l]
&
X_0
\ar@<0ex>[l]
}
$$
 ========

regex_match $\mathcal{F}$
nn candidate an abelian sheaf
nn an abelian sheaf
new_annotation [(221522, 221535, 'VAR'), (221539, 221555, 'TYPE')]
===== sent Let $\mathcal{F}$ be an abelian sheaf
on $X$. Let $\mathcal{F}_n$ be the pullback to $U_n$.
If $U$ is a proper hypercovering of $X$, then
there exists a canonical spectral sequence
$$
E_1^{p, q} = H^q(U_p, \mathcal{F}_p)
$$
converging to $H^{p + q}(X, \mathcal{F})$.
\end{lemma}

\begin{proof}
Immediate consequence of Lemmas \ref{lemma-compute-via-proper-hypercovering}
and \ref{lemma-simplicial-sheaf-cohomology}.
\end{proof}




\section{Simplicial schemes}
\label{section-simplicial}

\noindent
A {\it simplicial scheme} is a simplicial object in the category of schemes,
see Simplicial, Definition \ref{simplicial-definition-simplicial-object}.
Recall that a simplicial scheme looks like
$$
\xymatrix{
X_2
\ar@<2ex>[r]
\ar@<0ex>[r]
\ar@<-2ex>[r]
&
X_1
\ar@<1ex>[r]
\ar@<-1ex>[r]
\ar@<1ex>[l]
\ar@<-1ex>[l]
&
X_0
\ar@<0ex>[l]
}
$$
 ========

regex_match $\mathcal{F}_n$
nn candidate the pullback
nn the pullback
new_annotation [(221568, 221583, 'VAR'), (221587, 221599, 'TYPE')]
===== sent An {\it augmentation} of a simplicial scheme $X$ is a morphism
of schemes $a_0 : X_0 \to S$ such that $a_0 \circ d^1_0 = a_0 \circ d^1_1$.
See Simplicial, Section \ref{simplicial-section-augmentation}.

\medskip\noindent
Let $X$ be a simplicial scheme. ========

regex_match $X$
nn candidate a simplicial scheme
nn a simplicial scheme
new_annotation [(223519, 223522, 'VAR'), (223526, 223545, 'TYPE')]
===== sent The difference between $\mathcal{C}_{total}$ and $X_{Zar}$
is similar to the difference between the small Zariski site $S_{Zar}$
and the underlying topological space of $S$.
We will silently identify these sites in what follows.

\medskip\noindent
Let $X_{Zar}$ be the site associated to a simplicial scheme $X$.
There is a sheaf of rings $\mathcal{O}$ on $X_{Zar}$ whose restriction
to $X_n$ is the structure sheaf ========

regex_match $X_{Zar}$
nn candidate the site
nn the site
new_annotation [(224691, 224700, 'VAR'), (224704, 224712, 'TYPE')]
===== sent $\mathcal{O}_{X_n}$. This follows
from Lemma \ref{lemma-describe-sheaves-simplicial-site} or from
Lemma \ref{lemma-describe-sheaves-simplicial-site-site}. We will say
{\it $\mathcal{O}$ is the structure sheaf of the simplicial scheme $X$}.
At this point all the material developed for simplicial (ringed) sites
applies, see Sections \ref{section-simplicial-sites},
\ref{section-augmentation-simplicial-sites},
\ref{section-morphism-simplicial-sites},
\ref{section-simplicial-sites-modules},
\ref{section-cohomology-simplicial-sites},
\ref{section-cohomology-augmentation-simplicial-sites},
\ref{section-cohomology-simplicial-sites-modules},
\ref{section-cohomology-augmentation-ringed-simplicial-sites},
\ref{section-cartesian},
\ref{section-glueing}, and
\ref{section-glueing-modules}.

\medskip\noindent
Let $X$ be a simplicial scheme with structure sheaf $\mathcal{O}$.
As on any ringed topos, there is a notion
of a {\it quasi-coherent $\mathcal{O}$-module on $X_{Zar}$}, see
Modules on Sites, Definition \ref{sites-modules-definition-site-local}.
 ========

regex_match $X$
nn candidate a simplicial scheme
nn a simplicial scheme
new_annotation [(225665, 225668, 'VAR'), (225672, 225691, 'TYPE')]
===== sent However, a quasi-coherent $\mathcal{O}$-module on $X_{Zar}$ is
just a cartesian $\mathcal{O}$-module $\mathcal{F}$ whose restrictions
$\mathcal{F}_n$ are quasi-coherent on $X_n$, see
Lemma \ref{lemma-quasi-coherent-sheaf}.

\medskip\noindent
Let $h : X \to Y$ be a morphism of simplicial schemes. ========

regex_match $h : X \to Y$
nn candidate a morphism
nn a morphism
new_annotation [(226153, 226166, 'VAR'), (226170, 226180, 'TYPE')]
===== sent Either by
Lemma \ref{lemma-simplicial-space-site-functorial} or by
(the proof of) Lemma \ref{lemma-morphism-simplicial-sites}
we obtain a morphism of sites $h_{Zar} : X_{Zar} \to Y_{Zar}$.
Recall that $h_{Zar}^{-1}$ and $h_{Zar, *}$ have a simple
description in terms of the components, see
Lemma \ref{lemma-describe-functoriality} or
Lemma \ref{lemma-morphism-simplicial-sites}.
Let $\mathcal{O}_X$, resp.\ $\mathcal{O}_Y$ denote the structure
sheaf of $X$, resp.\ $Y$. We define
$h_{Zar}^\sharp : h_{Zar, *}\mathcal{O}_X \to \mathcal{O}_Y$
to be the map of sheaves of rings on $Y_{Zar}$ given by
$h_n^\sharp : h_{n, *}\mathcal{O}_{X_n} \to \mathcal{O}_{Y_n}$ on $Y_n$.
We obtain a morphism of ringed sites
$$
h_{Zar} : (X_{Zar}, \mathcal{O}_X) \longrightarrow (Y_{Zar}, \mathcal{O}_Y)
$$

\medskip\noindent
Let $X$ be a simplicial scheme with structure sheaf $\mathcal{O}$.
Let $S$ be a scheme and let $a_0 : X_0 \to S$ be an augmentation of $X$.
Either by
Lemma \ref{lemma-augmentation} or by
Lemma \ref{lemma-augmentation-site}
we obtain a corresponding morphism of topoi $a : \Sh(X_{Zar}) \to \Sh(S)$.
Observe that $a^{-1}\mathcal{G}$ is the sheaf on $X_{Zar}$ with components
$a_n^{-1}\mathcal{G}$. Hence we can use the maps
$a_n^\sharp : a_n^{-1}\mathcal{O}_S \to \mathcal{O}_{X_n}$ to define
a map $a^\sharp : a^{-1}\mathcal{O}_S \to \mathcal{O}$, or equivalently
by adjunction a map $a^\sharp : \mathcal{O}_S \to a_*\mathcal{O}$
(which as usual has the same name). ========

new_annotation []
===== sent Either by
Lemma \ref{lemma-simplicial-space-site-functorial} or by
(the proof of) Lemma \ref{lemma-morphism-simplicial-sites}
we obtain a morphism of sites $h_{Zar} : X_{Zar} \to Y_{Zar}$.
Recall that $h_{Zar}^{-1}$ and $h_{Zar, *}$ have a simple
description in terms of the components, see
Lemma \ref{lemma-describe-functoriality} or
Lemma \ref{lemma-morphism-simplicial-sites}.
Let $\mathcal{O}_X$, resp.\ $\mathcal{O}_Y$ denote the structure
sheaf of $X$, resp.\ $Y$. We define
$h_{Zar}^\sharp : h_{Zar, *}\mathcal{O}_X \to \mathcal{O}_Y$
to be the map of sheaves of rings on $Y_{Zar}$ given by
$h_n^\sharp : h_{n, *}\mathcal{O}_{X_n} \to \mathcal{O}_{Y_n}$ on $Y_n$.
We obtain a morphism of ringed sites
$$
h_{Zar} : (X_{Zar}, \mathcal{O}_X) \longrightarrow (Y_{Zar}, \mathcal{O}_Y)
$$

\medskip\noindent
Let $X$ be a simplicial scheme with structure sheaf $\mathcal{O}$.
Let $S$ be a scheme and let $a_0 : X_0 \to S$ be an augmentation of $X$.
Either by
Lemma \ref{lemma-augmentation} or by
Lemma \ref{lemma-augmentation-site}
we obtain a corresponding morphism of topoi $a : \Sh(X_{Zar}) \to \Sh(S)$.
Observe that $a^{-1}\mathcal{G}$ is the sheaf on $X_{Zar}$ with components
$a_n^{-1}\mathcal{G}$. Hence we can use the maps
$a_n^\sharp : a_n^{-1}\mathcal{O}_S \to \mathcal{O}_{X_n}$ to define
a map $a^\sharp : a^{-1}\mathcal{O}_S \to \mathcal{O}$, or equivalently
by adjunction a map $a^\sharp : \mathcal{O}_S \to a_*\mathcal{O}$
(which as usual has the same name). ========

regex_match $X$
nn candidate a simplicial scheme
nn a simplicial scheme
new_annotation [(227017, 227020, 'VAR'), (227024, 227043, 'TYPE')]
===== sent Either by
Lemma \ref{lemma-simplicial-space-site-functorial} or by
(the proof of) Lemma \ref{lemma-morphism-simplicial-sites}
we obtain a morphism of sites $h_{Zar} : X_{Zar} \to Y_{Zar}$.
Recall that $h_{Zar}^{-1}$ and $h_{Zar, *}$ have a simple
description in terms of the components, see
Lemma \ref{lemma-describe-functoriality} or
Lemma \ref{lemma-morphism-simplicial-sites}.
Let $\mathcal{O}_X$, resp.\ $\mathcal{O}_Y$ denote the structure
sheaf of $X$, resp.\ $Y$. We define
$h_{Zar}^\sharp : h_{Zar, *}\mathcal{O}_X \to \mathcal{O}_Y$
to be the map of sheaves of rings on $Y_{Zar}$ given by
$h_n^\sharp : h_{n, *}\mathcal{O}_{X_n} \to \mathcal{O}_{Y_n}$ on $Y_n$.
We obtain a morphism of ringed sites
$$
h_{Zar} : (X_{Zar}, \mathcal{O}_X) \longrightarrow (Y_{Zar}, \mathcal{O}_Y)
$$

\medskip\noindent
Let $X$ be a simplicial scheme with structure sheaf $\mathcal{O}$.
Let $S$ be a scheme and let $a_0 : X_0 \to S$ be an augmentation of $X$.
Either by
Lemma \ref{lemma-augmentation} or by
Lemma \ref{lemma-augmentation-site}
we obtain a corresponding morphism of topoi $a : \Sh(X_{Zar}) \to \Sh(S)$.
Observe that $a^{-1}\mathcal{G}$ is the sheaf on $X_{Zar}$ with components
$a_n^{-1}\mathcal{G}$. Hence we can use the maps
$a_n^\sharp : a_n^{-1}\mathcal{O}_S \to \mathcal{O}_{X_n}$ to define
a map $a^\sharp : a^{-1}\mathcal{O}_S \to \mathcal{O}$, or equivalently
by adjunction a map $a^\sharp : \mathcal{O}_S \to a_*\mathcal{O}$
(which as usual has the same name). ========

regex_match $S$
nn candidate a scheme
nn a scheme
new_annotation [(227084, 227087, 'VAR'), (227091, 227099, 'TYPE')]
===== sent Either by
Lemma \ref{lemma-simplicial-space-site-functorial} or by
(the proof of) Lemma \ref{lemma-morphism-simplicial-sites}
we obtain a morphism of sites $h_{Zar} : X_{Zar} \to Y_{Zar}$.
Recall that $h_{Zar}^{-1}$ and $h_{Zar, *}$ have a simple
description in terms of the components, see
Lemma \ref{lemma-describe-functoriality} or
Lemma \ref{lemma-morphism-simplicial-sites}.
Let $\mathcal{O}_X$, resp.\ $\mathcal{O}_Y$ denote the structure
sheaf of $X$, resp.\ $Y$. We define
$h_{Zar}^\sharp : h_{Zar, *}\mathcal{O}_X \to \mathcal{O}_Y$
to be the map of sheaves of rings on $Y_{Zar}$ given by
$h_n^\sharp : h_{n, *}\mathcal{O}_{X_n} \to \mathcal{O}_{Y_n}$ on $Y_n$.
We obtain a morphism of ringed sites
$$
h_{Zar} : (X_{Zar}, \mathcal{O}_X) \longrightarrow (Y_{Zar}, \mathcal{O}_Y)
$$

\medskip\noindent
Let $X$ be a simplicial scheme with structure sheaf $\mathcal{O}$.
Let $S$ be a scheme and let $a_0 : X_0 \to S$ be an augmentation of $X$.
Either by
Lemma \ref{lemma-augmentation} or by
Lemma \ref{lemma-augmentation-site}
we obtain a corresponding morphism of topoi $a : \Sh(X_{Zar}) \to \Sh(S)$.
Observe that $a^{-1}\mathcal{G}$ is the sheaf on $X_{Zar}$ with components
$a_n^{-1}\mathcal{G}$. Hence we can use the maps
$a_n^\sharp : a_n^{-1}\mathcal{O}_S \to \mathcal{O}_{X_n}$ to define
a map $a^\sharp : a^{-1}\mathcal{O}_S \to \mathcal{O}$, or equivalently
by adjunction a map $a^\sharp : \mathcal{O}_S \to a_*\mathcal{O}$
(which as usual has the same name). ========

regex_match $a_0 : X_0 \to S$
nn candidate $a_0
nn candidate an augmentation
nn an augmentation
new_annotation [(227108, 227125, 'VAR'), (227129, 227144, 'TYPE')]
===== sent \begin{definition}
\label{definition-cartesian-morphism}
Let $a : Y \to X$ be a morphism of simplicial schemes.
 ========

regex_match $a : Y \to X$
nn candidate a : Y \to X$
nn candidate a morphism
nn a morphism
new_annotation [(228874, 228887, 'VAR'), (228891, 228901, 'TYPE')]
===== sent In this lemma we denote $f^* : \Sch/X \to \Sch/Y$
the base change functor associated to a morphism of schemes $f :Y \to X$.

\begin{lemma}
\label{lemma-characterize-cartesian-schemes}
Let $X$ be a simplicial scheme. ========

regex_match $X$
nn candidate a simplicial scheme
nn a simplicial scheme
new_annotation [(229612, 229615, 'VAR'), (229619, 229638, 'TYPE')]
===== sent \end{proof}

\begin{definition}
\label{definition-fibre-products-simplicial-scheme}
Let $f : X \to S$ be a morphism of schemes. ========

regex_match $f : X \to S$
nn candidate X \to
nn candidate a morphism
nn a morphism
new_annotation [(232372, 232385, 'VAR'), (232389, 232399, 'TYPE')]
===== sent The morphism $s^0_0$ is the diagonal morphism
$X \to X \times_S X$.

\begin{lemma}
\label{lemma-cartesian-over}
Let $f : X \to S$ be a morphism of schemes.
 ========

regex_match $f : X \to S$
nn candidate X \to
nn candidate a morphism
nn a morphism
new_annotation [(232999, 233012, 'VAR'), (233016, 233026, 'TYPE')]
===== sent Let $\pi : Y \to (X/S)_\bullet$ be a cartesian morphism
of simplicial schemes.
 ========

regex_match $\pi : Y \to (X/S)_\bullet$
nn candidate a cartesian morphism
nn a cartesian morphism
new_annotation [(233043, 233070, 'VAR'), (233074, 233094, 'TYPE')]
===== sent \label{lemma-cartesian-equivalent-descent-datum}
Let $f : X \to S$ be a morphism of schemes. ========

regex_match $f : X \to S$
nn candidate X \to
nn candidate a morphism
nn a morphism
new_annotation [(233903, 233916, 'VAR'), (233920, 233930, 'TYPE')]
===== sent $$
We claim that the ``pullback'' $f_\bullet^*$ along the morphism
$f_\bullet : (X'/S')_\bullet \to (X/S)_\bullet$ corresponds via
Lemma \ref{lemma-cartesian-equivalent-descent-datum}
with the pullback defined in terms of descent data in
the aforementioned
Descent, Lemma \ref{descent-lemma-pullback}.







\section{Quasi-coherent modules on simplicial schemes}
\label{section-modules-simplicial}

\begin{lemma}
\label{lemma-pullback-cartesian-module}
Let $f : V \to U$ be a morphism of simplicial schemes. ========

regex_match $f : V \to U$
nn candidate V \to
nn candidate a morphism
nn a morphism
new_annotation [(235498, 235511, 'VAR'), (235515, 235525, 'TYPE')]
===== sent \end{proof}

\begin{lemma}
\label{lemma-pushforward-cartesian-module}
Let $f : V \to U$ be a cartesian morphism of simplicial schemes.
 ========

regex_match $f : V \to U$
nn candidate V \to
nn candidate a cartesian morphism
nn a cartesian morphism
new_annotation [(236262, 236275, 'VAR'), (236279, 236299, 'TYPE')]
===== sent Thus $\mathcal{F}$ is quasi-coherent by
Lemma \ref{lemma-quasi-coherent-sheaf}.
\end{proof}

\begin{lemma}
\label{lemma-adjoint-functors-cartesian-modules}
Let $f : V \to U$ be a cartesian morphism of
simplicial schemes. ========

regex_match $f : V \to U$
nn candidate V \to
nn candidate a cartesian morphism
nn a cartesian morphism
new_annotation [(237557, 237570, 'VAR'), (237574, 237594, 'TYPE')]
===== sent \end{proof}

\begin{lemma}
\label{lemma-cartesian-modules-with-section}
Let $f : X \to S$ be a morphism of schemes which has a
section\footnote{In fact, it would be enough to assume that $f$
has fpqc locally on $S$ a section, since we have descent of
quasi-coherent modules by Descent,
Section \ref{descent-section-fpqc-descent-quasi-coherent}.}.
 ========

regex_match $f : X \to S$
nn candidate X \to
nn candidate a morphism
nn a morphism
new_annotation [(238222, 238235, 'VAR'), (238239, 238249, 'TYPE')]
===== sent Let $(X/S)_\bullet$ be the simplicial
scheme associated to $X \to S$, see
Definition \ref{definition-fibre-products-simplicial-scheme}.
Then pullback defines an equivalence between the category of
quasi-coherent $\mathcal{O}_S$-modules and the category of
quasi-coherent modules on $((X/S)_\bullet)_{Zar}$.
\end{lemma}

\begin{proof}
Let $\sigma : S \to X$ be a section of $f$. Let $(\mathcal{F}, \alpha)$
be a pair as in Lemma \ref{lemma-characterize-cartesian-modules}.
Set $\mathcal{G} = \sigma^*\mathcal{F}$. Consider the diagram
$$
\xymatrix{
X \ar[r]_-{(\sigma \circ f, 1)} \ar[d]_f &
X \times_S X \ar[d]^{\text{pr}_0} \ar[r]_-{\text{pr}_1} & X \\
S \ar[r]^\sigma & X
}
$$
Note that $\text{pr}_0 = d^1_1$ and $\text{pr}_1 = d^1_0$. ========

regex_match $(X/S)_\bullet$
nn candidate X
nn candidate the simplicial
scheme
nn the simplicial
scheme
new_annotation [(238497, 238512, 'VAR'), (238516, 238537, 'TYPE')]
===== sent Let $(X/S)_\bullet$ be the simplicial
scheme associated to $X \to S$, see
Definition \ref{definition-fibre-products-simplicial-scheme}.
Then pullback defines an equivalence between the category of
quasi-coherent $\mathcal{O}_S$-modules and the category of
quasi-coherent modules on $((X/S)_\bullet)_{Zar}$.
\end{lemma}

\begin{proof}
Let $\sigma : S \to X$ be a section of $f$. Let $(\mathcal{F}, \alpha)$
be a pair as in Lemma \ref{lemma-characterize-cartesian-modules}.
Set $\mathcal{G} = \sigma^*\mathcal{F}$. Consider the diagram
$$
\xymatrix{
X \ar[r]_-{(\sigma \circ f, 1)} \ar[d]_f &
X \times_S X \ar[d]^{\text{pr}_0} \ar[r]_-{\text{pr}_1} & X \\
S \ar[r]^\sigma & X
}
$$
Note that $\text{pr}_0 = d^1_1$ and $\text{pr}_1 = d^1_0$. ========

regex_match $\sigma : S \to X$
nn candidate a section
nn a section
new_annotation [(238831, 238849, 'VAR'), (238853, 238862, 'TYPE')]
===== sent Let $(X/S)_\bullet$ be the simplicial
scheme associated to $X \to S$, see
Definition \ref{definition-fibre-products-simplicial-scheme}.
Then pullback defines an equivalence between the category of
quasi-coherent $\mathcal{O}_S$-modules and the category of
quasi-coherent modules on $((X/S)_\bullet)_{Zar}$.
\end{lemma}

\begin{proof}
Let $\sigma : S \to X$ be a section of $f$. Let $(\mathcal{F}, \alpha)$
be a pair as in Lemma \ref{lemma-characterize-cartesian-modules}.
Set $\mathcal{G} = \sigma^*\mathcal{F}$. Consider the diagram
$$
\xymatrix{
X \ar[r]_-{(\sigma \circ f, 1)} \ar[d]_f &
X \times_S X \ar[d]^{\text{pr}_0} \ar[r]_-{\text{pr}_1} & X \\
S \ar[r]^\sigma & X
}
$$
Note that $\text{pr}_0 = d^1_1$ and $\text{pr}_1 = d^1_0$. ========

new_annotation []
===== sent \begin{lemma}
\label{lemma-groupoid-simplicial}
Let $(U, R, s, t, c, e, i)$ be a groupoid scheme over $S$.
There exists a simplicial scheme $X$ over $S$
with the following properties
\begin{enumerate}
\item $X_0 = U$, $X_1 = R$, $X_2 = R \times_{s, U, t} R$,
\item $s_0^0 = e : X_0 \to X_1$,
\item $d^1_0 = s : X_1 \to X_0$, $d^1_1 = t : X_1 \to X_0$,
\item $s_0^1 = (e \circ t, 1) : X_1 \to X_2$,
$s_1^1 = (1, e \circ t) : X_1 \to X_2$,
\item $d^2_0 = \text{pr}_1 : X_2 \to X_1$,
$d^2_1 = c : X_2 \to X_1$,
$d^2_2 = \text{pr}_0$, and
\item $X = \text{cosk}_2 \text{sk}_2 X$.
\end{enumerate}
For all $n$ we have $X_n = R \times_{s, U, t} \ldots \times_{s, U, t} R$
with $n$ factors. ========

regex_match $(U, R, s, t, c, e, i)$
nn candidate $(U
nn candidate R
nn candidate c
nn candidate a groupoid scheme
nn a groupoid scheme
new_annotation [(240017, 240040, 'VAR'), (240044, 240061, 'TYPE')]
===== sent \end{proof}

\begin{lemma}
\label{lemma-quasi-coherent-groupoid-simplicial}
Let $S$ be a scheme. ========

regex_match $S$
nn candidate a scheme
nn a scheme
new_annotation [(242308, 242311, 'VAR'), (242315, 242323, 'TYPE')]
===== sent Let $(U, R, s, t, c)$ be a groupoid scheme
over $S$. Let $X$ be the simplicial scheme over $S$ constructed
in Lemma \ref{lemma-groupoid-simplicial}.
Then the category of quasi-coherent modules on $(U, R, s, t, c)$
is equivalent to the category of quasi-coherent modules on $X_{Zar}$.
\end{lemma}

\begin{proof}
This is clear from Lemmas
\ref{lemma-quasi-coherent-sheaf} and
\ref{lemma-characterize-cartesian-modules}
and Groupoids, Definition \ref{groupoids-definition-groupoid-module}.
\end{proof}

\noindent
In the following lemma we will use the concept of a cartesian
morphism $V \to U$ of simplicial schemes as defined in
Definition \ref{definition-cartesian-morphism}.

\begin{lemma}
\label{lemma-quasi-coherent-groupoid-R-cartesian}
Let $(U, R, s, t, c)$ be a groupoid scheme over a scheme $S$.
Let $X$ be the simplicial scheme over $S$ constructed
in Lemma \ref{lemma-groupoid-simplicial}.
Let $(R/U)_\bullet$ be the simplicial
scheme associated to $s : R \to U$, see
Definition \ref{definition-fibre-products-simplicial-scheme}.
There exists a cartesian morphism $t_\bullet : (R/U)_\bullet \to X$
of simplicial schemes with low degree morphisms given by
$$
\xymatrix{
R \times_{s, U, s} R \times_{s, U, s} R
\ar@<3ex>[r]_-{\text{pr}_{12}}
\ar@<0ex>[r]_-{\text{pr}_{02}}
\ar@<-3ex>[r]_-{\text{pr}_{01}}
\ar[dd]_{(r_0, r_1, r_2) \mapsto (r_0 \circ r_1^{-1}, r_1 \circ r_2^{-1})} &
R \times_{s, U, s} R
\ar@<1ex>[r]_-{\text{pr}_1} \ar@<-2ex>[r]_-{\text{pr}_0}
\ar[dd]_{(r_0, r_1) \mapsto r_0 \circ r_1^{-1}} &
R \ar[dd]^t
\\
\\
R \times_{s, U, t} R
\ar@<3ex>[r]_{\text{pr}_1}
\ar@<0ex>[r]_c
\ar@<-3ex>[r]_{\text{pr}_0} &
R \ar@<1ex>[r]_s \ar@<-2ex>[r]_t &
U
}
$$
\end{lemma}

\begin{proof}
For arbitrary $n$ we define $(R/U)_\bullet \to X_n$ by the rule
$$
(r_0, \ldots, r_n)
\longrightarrow
(r_0 \circ r_1^{-1}, \ldots, r_{n - 1} \circ r_n^{-1})
$$
Compatibility with degeneracy maps is clear from the description of the
degeneracies in Lemma \ref{lemma-groupoid-simplicial}.
We omit the verification that the maps respect the morphisms $s^n_j$.
Groupoids, Lemma \ref{groupoids-lemma-diagram-pull}
(with the roles of $s$ and $t$ reversed)
shows that the two right squares are cartesian. ========

regex_match $(U, R, s, t, c)$
nn candidate U
nn candidate R
nn candidate a groupoid scheme
nn a groupoid scheme
new_annotation [(242329, 242346, 'VAR'), (242350, 242367, 'TYPE')]
===== sent Let $(U, R, s, t, c)$ be a groupoid scheme
over $S$. Let $X$ be the simplicial scheme over $S$ constructed
in Lemma \ref{lemma-groupoid-simplicial}.
Then the category of quasi-coherent modules on $(U, R, s, t, c)$
is equivalent to the category of quasi-coherent modules on $X_{Zar}$.
\end{lemma}

\begin{proof}
This is clear from Lemmas
\ref{lemma-quasi-coherent-sheaf} and
\ref{lemma-characterize-cartesian-modules}
and Groupoids, Definition \ref{groupoids-definition-groupoid-module}.
\end{proof}

\noindent
In the following lemma we will use the concept of a cartesian
morphism $V \to U$ of simplicial schemes as defined in
Definition \ref{definition-cartesian-morphism}.

\begin{lemma}
\label{lemma-quasi-coherent-groupoid-R-cartesian}
Let $(U, R, s, t, c)$ be a groupoid scheme over a scheme $S$.
Let $X$ be the simplicial scheme over $S$ constructed
in Lemma \ref{lemma-groupoid-simplicial}.
Let $(R/U)_\bullet$ be the simplicial
scheme associated to $s : R \to U$, see
Definition \ref{definition-fibre-products-simplicial-scheme}.
There exists a cartesian morphism $t_\bullet : (R/U)_\bullet \to X$
of simplicial schemes with low degree morphisms given by
$$
\xymatrix{
R \times_{s, U, s} R \times_{s, U, s} R
\ar@<3ex>[r]_-{\text{pr}_{12}}
\ar@<0ex>[r]_-{\text{pr}_{02}}
\ar@<-3ex>[r]_-{\text{pr}_{01}}
\ar[dd]_{(r_0, r_1, r_2) \mapsto (r_0 \circ r_1^{-1}, r_1 \circ r_2^{-1})} &
R \times_{s, U, s} R
\ar@<1ex>[r]_-{\text{pr}_1} \ar@<-2ex>[r]_-{\text{pr}_0}
\ar[dd]_{(r_0, r_1) \mapsto r_0 \circ r_1^{-1}} &
R \ar[dd]^t
\\
\\
R \times_{s, U, t} R
\ar@<3ex>[r]_{\text{pr}_1}
\ar@<0ex>[r]_c
\ar@<-3ex>[r]_{\text{pr}_0} &
R \ar@<1ex>[r]_s \ar@<-2ex>[r]_t &
U
}
$$
\end{lemma}

\begin{proof}
For arbitrary $n$ we define $(R/U)_\bullet \to X_n$ by the rule
$$
(r_0, \ldots, r_n)
\longrightarrow
(r_0 \circ r_1^{-1}, \ldots, r_{n - 1} \circ r_n^{-1})
$$
Compatibility with degeneracy maps is clear from the description of the
degeneracies in Lemma \ref{lemma-groupoid-simplicial}.
We omit the verification that the maps respect the morphisms $s^n_j$.
Groupoids, Lemma \ref{groupoids-lemma-diagram-pull}
(with the roles of $s$ and $t$ reversed)
shows that the two right squares are cartesian. ========

regex_match $X$
nn candidate the simplicial scheme
nn the simplicial scheme
new_annotation [(242382, 242385, 'VAR'), (242389, 242410, 'TYPE')]
===== sent Let $(U, R, s, t, c)$ be a groupoid scheme
over $S$. Let $X$ be the simplicial scheme over $S$ constructed
in Lemma \ref{lemma-groupoid-simplicial}.
Then the category of quasi-coherent modules on $(U, R, s, t, c)$
is equivalent to the category of quasi-coherent modules on $X_{Zar}$.
\end{lemma}

\begin{proof}
This is clear from Lemmas
\ref{lemma-quasi-coherent-sheaf} and
\ref{lemma-characterize-cartesian-modules}
and Groupoids, Definition \ref{groupoids-definition-groupoid-module}.
\end{proof}

\noindent
In the following lemma we will use the concept of a cartesian
morphism $V \to U$ of simplicial schemes as defined in
Definition \ref{definition-cartesian-morphism}.

\begin{lemma}
\label{lemma-quasi-coherent-groupoid-R-cartesian}
Let $(U, R, s, t, c)$ be a groupoid scheme over a scheme $S$.
Let $X$ be the simplicial scheme over $S$ constructed
in Lemma \ref{lemma-groupoid-simplicial}.
Let $(R/U)_\bullet$ be the simplicial
scheme associated to $s : R \to U$, see
Definition \ref{definition-fibre-products-simplicial-scheme}.
There exists a cartesian morphism $t_\bullet : (R/U)_\bullet \to X$
of simplicial schemes with low degree morphisms given by
$$
\xymatrix{
R \times_{s, U, s} R \times_{s, U, s} R
\ar@<3ex>[r]_-{\text{pr}_{12}}
\ar@<0ex>[r]_-{\text{pr}_{02}}
\ar@<-3ex>[r]_-{\text{pr}_{01}}
\ar[dd]_{(r_0, r_1, r_2) \mapsto (r_0 \circ r_1^{-1}, r_1 \circ r_2^{-1})} &
R \times_{s, U, s} R
\ar@<1ex>[r]_-{\text{pr}_1} \ar@<-2ex>[r]_-{\text{pr}_0}
\ar[dd]_{(r_0, r_1) \mapsto r_0 \circ r_1^{-1}} &
R \ar[dd]^t
\\
\\
R \times_{s, U, t} R
\ar@<3ex>[r]_{\text{pr}_1}
\ar@<0ex>[r]_c
\ar@<-3ex>[r]_{\text{pr}_0} &
R \ar@<1ex>[r]_s \ar@<-2ex>[r]_t &
U
}
$$
\end{lemma}

\begin{proof}
For arbitrary $n$ we define $(R/U)_\bullet \to X_n$ by the rule
$$
(r_0, \ldots, r_n)
\longrightarrow
(r_0 \circ r_1^{-1}, \ldots, r_{n - 1} \circ r_n^{-1})
$$
Compatibility with degeneracy maps is clear from the description of the
degeneracies in Lemma \ref{lemma-groupoid-simplicial}.
We omit the verification that the maps respect the morphisms $s^n_j$.
Groupoids, Lemma \ref{groupoids-lemma-diagram-pull}
(with the roles of $s$ and $t$ reversed)
shows that the two right squares are cartesian. ========

regex_match $(U, R, s, t, c)$
nn candidate U
nn candidate R
nn candidate a groupoid scheme
nn a groupoid scheme
new_annotation [(243069, 243086, 'VAR'), (243090, 243107, 'TYPE')]
===== sent Let $(U, R, s, t, c)$ be a groupoid scheme
over $S$. Let $X$ be the simplicial scheme over $S$ constructed
in Lemma \ref{lemma-groupoid-simplicial}.
Then the category of quasi-coherent modules on $(U, R, s, t, c)$
is equivalent to the category of quasi-coherent modules on $X_{Zar}$.
\end{lemma}

\begin{proof}
This is clear from Lemmas
\ref{lemma-quasi-coherent-sheaf} and
\ref{lemma-characterize-cartesian-modules}
and Groupoids, Definition \ref{groupoids-definition-groupoid-module}.
\end{proof}

\noindent
In the following lemma we will use the concept of a cartesian
morphism $V \to U$ of simplicial schemes as defined in
Definition \ref{definition-cartesian-morphism}.

\begin{lemma}
\label{lemma-quasi-coherent-groupoid-R-cartesian}
Let $(U, R, s, t, c)$ be a groupoid scheme over a scheme $S$.
Let $X$ be the simplicial scheme over $S$ constructed
in Lemma \ref{lemma-groupoid-simplicial}.
Let $(R/U)_\bullet$ be the simplicial
scheme associated to $s : R \to U$, see
Definition \ref{definition-fibre-products-simplicial-scheme}.
There exists a cartesian morphism $t_\bullet : (R/U)_\bullet \to X$
of simplicial schemes with low degree morphisms given by
$$
\xymatrix{
R \times_{s, U, s} R \times_{s, U, s} R
\ar@<3ex>[r]_-{\text{pr}_{12}}
\ar@<0ex>[r]_-{\text{pr}_{02}}
\ar@<-3ex>[r]_-{\text{pr}_{01}}
\ar[dd]_{(r_0, r_1, r_2) \mapsto (r_0 \circ r_1^{-1}, r_1 \circ r_2^{-1})} &
R \times_{s, U, s} R
\ar@<1ex>[r]_-{\text{pr}_1} \ar@<-2ex>[r]_-{\text{pr}_0}
\ar[dd]_{(r_0, r_1) \mapsto r_0 \circ r_1^{-1}} &
R \ar[dd]^t
\\
\\
R \times_{s, U, t} R
\ar@<3ex>[r]_{\text{pr}_1}
\ar@<0ex>[r]_c
\ar@<-3ex>[r]_{\text{pr}_0} &
R \ar@<1ex>[r]_s \ar@<-2ex>[r]_t &
U
}
$$
\end{lemma}

\begin{proof}
For arbitrary $n$ we define $(R/U)_\bullet \to X_n$ by the rule
$$
(r_0, \ldots, r_n)
\longrightarrow
(r_0 \circ r_1^{-1}, \ldots, r_{n - 1} \circ r_n^{-1})
$$
Compatibility with degeneracy maps is clear from the description of the
degeneracies in Lemma \ref{lemma-groupoid-simplicial}.
We omit the verification that the maps respect the morphisms $s^n_j$.
Groupoids, Lemma \ref{groupoids-lemma-diagram-pull}
(with the roles of $s$ and $t$ reversed)
shows that the two right squares are cartesian. ========

regex_match $X$
nn candidate the simplicial scheme
nn the simplicial scheme
new_annotation [(243131, 243134, 'VAR'), (243138, 243159, 'TYPE')]
===== sent Let $(U, R, s, t, c)$ be a groupoid scheme
over $S$. Let $X$ be the simplicial scheme over $S$ constructed
in Lemma \ref{lemma-groupoid-simplicial}.
Then the category of quasi-coherent modules on $(U, R, s, t, c)$
is equivalent to the category of quasi-coherent modules on $X_{Zar}$.
\end{lemma}

\begin{proof}
This is clear from Lemmas
\ref{lemma-quasi-coherent-sheaf} and
\ref{lemma-characterize-cartesian-modules}
and Groupoids, Definition \ref{groupoids-definition-groupoid-module}.
\end{proof}

\noindent
In the following lemma we will use the concept of a cartesian
morphism $V \to U$ of simplicial schemes as defined in
Definition \ref{definition-cartesian-morphism}.

\begin{lemma}
\label{lemma-quasi-coherent-groupoid-R-cartesian}
Let $(U, R, s, t, c)$ be a groupoid scheme over a scheme $S$.
Let $X$ be the simplicial scheme over $S$ constructed
in Lemma \ref{lemma-groupoid-simplicial}.
Let $(R/U)_\bullet$ be the simplicial
scheme associated to $s : R \to U$, see
Definition \ref{definition-fibre-products-simplicial-scheme}.
There exists a cartesian morphism $t_\bullet : (R/U)_\bullet \to X$
of simplicial schemes with low degree morphisms given by
$$
\xymatrix{
R \times_{s, U, s} R \times_{s, U, s} R
\ar@<3ex>[r]_-{\text{pr}_{12}}
\ar@<0ex>[r]_-{\text{pr}_{02}}
\ar@<-3ex>[r]_-{\text{pr}_{01}}
\ar[dd]_{(r_0, r_1, r_2) \mapsto (r_0 \circ r_1^{-1}, r_1 \circ r_2^{-1})} &
R \times_{s, U, s} R
\ar@<1ex>[r]_-{\text{pr}_1} \ar@<-2ex>[r]_-{\text{pr}_0}
\ar[dd]_{(r_0, r_1) \mapsto r_0 \circ r_1^{-1}} &
R \ar[dd]^t
\\
\\
R \times_{s, U, t} R
\ar@<3ex>[r]_{\text{pr}_1}
\ar@<0ex>[r]_c
\ar@<-3ex>[r]_{\text{pr}_0} &
R \ar@<1ex>[r]_s \ar@<-2ex>[r]_t &
U
}
$$
\end{lemma}

\begin{proof}
For arbitrary $n$ we define $(R/U)_\bullet \to X_n$ by the rule
$$
(r_0, \ldots, r_n)
\longrightarrow
(r_0 \circ r_1^{-1}, \ldots, r_{n - 1} \circ r_n^{-1})
$$
Compatibility with degeneracy maps is clear from the description of the
degeneracies in Lemma \ref{lemma-groupoid-simplicial}.
We omit the verification that the maps respect the morphisms $s^n_j$.
Groupoids, Lemma \ref{groupoids-lemma-diagram-pull}
(with the roles of $s$ and $t$ reversed)
shows that the two right squares are cartesian. ========

regex_match $(R/U)_\bullet$
nn candidate R
nn candidate the simplicial
scheme
nn the simplicial
scheme
new_annotation [(243227, 243242, 'VAR'), (243246, 243267, 'TYPE')]
===== sent \begin{lemma}
\label{lemma-equivalence-relation}
Let $f : X \to S$ be a morphism of schemes.
 ========

regex_match $f : X \to S$
nn candidate X \to
nn candidate a morphism
nn a morphism
new_annotation [(245024, 245037, 'VAR'), (245041, 245051, 'TYPE')]
===== sent Let $\pi : Y \to (X/S)_\bullet$ be a cartesian morphism of simplicial
schemes, see Definitions \ref{definition-cartesian-morphism} and
\ref{definition-fibre-products-simplicial-scheme}.
Then the morphism
$$
j = (d^1_1, d^1_0) : Y_1 \to Y_0 \times_S Y_0
$$
defines an equivalence relation on $Y_0$ over $S$,
see Groupoids, Definition \ref{groupoids-definition-equivalence-relation}.
\end{lemma}

\begin{proof}
Note that $j$ is a monomorphism. ========

regex_match $\pi : Y \to (X/S)_\bullet$
nn candidate $\pi
nn candidate a cartesian morphism
nn a cartesian morphism
new_annotation [(245068, 245095, 'VAR'), (245099, 245119, 'TYPE')]
===== sent \medskip\noindent
Let $T/S$ be a scheme. ========

regex_match $T/S$
nn candidate a scheme
nn a scheme
new_annotation [(247336, 247341, 'VAR'), (247345, 247353, 'TYPE')]
===== sent Let $a \sim b$ for $a, b \in Y_0(T)$
be synonymous with $(a, b) \in Y_1(T)$.
 ========

new_annotation []
===== sent \begin{lemma}
\label{lemma-equivalence-classes-points}
Let $X \to S$ be a morphism of schemes. ========

regex_match $X \to S$
nn candidate $X
nn candidate a morphism
nn a morphism
new_annotation [(247898, 247907, 'VAR'), (247911, 247921, 'TYPE')]
===== sent d^1_1(y_1) = y, d^1_0(y_1) = y'\}
$$
as a subset of $Y_0$. Then $y \in T_y$ and
$T_y \cap T_{y'} \not = \emptyset \Rightarrow T_y = T_{y'}$.
\end{lemma}

\begin{proof}
Combine Lemma \ref{lemma-equivalence-relation} and
Groupoids, Lemma
\ref{groupoids-lemma-pre-equivalence-equivalence-relation-points}.
\end{proof}

\begin{lemma}
\label{lemma-quasi-compact}
Let $X \to S$ be a morphism of schemes.
 ========

regex_match $X \to S$
nn candidate $X
nn candidate a morphism
nn a morphism
new_annotation [(248453, 248462, 'VAR'), (248466, 248476, 'TYPE')]
===== sent Let $y \in Y_0$ be a point. ========

regex_match $y \in Y_0$
nn candidate $y
nn candidate a point
nn a point
new_annotation [(248570, 248581, 'VAR'), (248585, 248592, 'TYPE')]
===== sent If $X \to S$ is quasi-compact, then
$$
T_y = \{y' \in Y_0 \mid \exists\ y_1 \in Y_1:
d^1_1(y_1) = y, d^1_0(y_1) = y'\}
$$
is a quasi-compact subset of $Y_0$.
\end{lemma}

\begin{proof}
Let $F_y$ be the scheme theoretic fibre of $d^1_1 : Y_1 \to Y_0$
at $y$. Then we see that $T_y$ is the image of the morphism
$$
\xymatrix{
F_y \ar[r] \ar[d] &
Y_1 \ar[r]^{d^1_0} \ar[d]^{d^1_1} &
Y_0 \\
 ========

regex_match $F_y$
nn candidate the scheme
nn the scheme
new_annotation [(248783, 248788, 'VAR'), (248792, 248802, 'TYPE')]
===== sent \end{proof}

\begin{lemma}
\label{lemma-descent-disjoint-union-Artinian-along-fields}
Let $X \to S$ be a quasi-compact flat surjective morphism.
 ========

regex_match $X \to S$
nn candidate $X
nn candidate a quasi-compact flat surjective morphism
nn a quasi-compact flat surjective morphism
new_annotation [(249150, 249159, 'VAR'), (249163, 249203, 'TYPE')]
===== sent Let $(V, \varphi)$ be a descent datum relative
to $X \to ========

regex_match $(V, \varphi)$
nn candidate $(V
nn candidate a descent datum
nn a descent datum
new_annotation [(249209, 249223, 'VAR'), (249227, 249242, 'TYPE')]
===== sent \end{lemma}

\begin{proof}
Let $Y \to (X/S)_\bullet$ be the cartesian morphism of simplicial
schemes corresponding to $(V, \varphi)$ by
Lemma \ref{lemma-cartesian-equivalent-descent-datum}.
Observe that $Y_0 = V$.
Write $V = \coprod_{i \in I} \Spec(A_i)$ with each $A_i$ local
Artinian. ========

regex_match $Y \to (X/S)_\bullet$
nn candidate $Y
nn candidate the cartesian morphism
nn the cartesian morphism
new_annotation [(249388, 249409, 'VAR'), (249413, 249435, 'TYPE')]
===== sent Moreover, let $v_i \in V$ be the unique closed point of
$\Spec(A_i)$ for all $i \in I$. ========

regex_match $v_i \in V$
nn candidate the unique closed point
nn the unique closed point
new_annotation [(249658, 249669, 'VAR'), (249673, 249696, 'TYPE')]
===== sent Let $\overline{I} = I/\sim$. ========

new_annotation []
===== sent \section{Simplicial algebraic spaces}
\label{section-simplicial-algebraic-spaces}

\noindent
Let $S$ be a scheme. ========

regex_match $S$
nn candidate a scheme
nn a scheme
new_annotation [(251101, 251104, 'VAR'), (251108, 251116, 'TYPE')]
===== sent Let $X$ be a simplicial algebraic space. ========

regex_match $X$
nn candidate a simplicial algebraic space
nn a simplicial algebraic space
new_annotation [(252841, 252844, 'VAR'), (252848, 252876, 'TYPE')]
===== sent We will say that $X_\etale$ is the {\it small \'etale site of $X$}
and its topos is the {\it small \'etale topos of $X$}.

\medskip\noindent
Let $X_\etale$ be the small \'etale site of a simplicial algebraic space $X$.
There is a sheaf of rings $\mathcal{O}$ on $X_\etale$ whose restriction
to $X_n$ is the structure sheaf ========

regex_match $X_\etale$
nn candidate the small \'etale site
nn the small \'etale site
new_annotation [(254646, 254656, 'VAR'), (254660, 254682, 'TYPE')]
===== sent Let $X$ be a simplicial algebraic space with structure sheaf $\mathcal{O}$.
As on any ringed topos, there is a notion
of a {\it quasi-coherent $\mathcal{O}$-module on $X_\etale$}, see
Modules on Sites, Definition \ref{sites-modules-definition-site-local}.
 ========

regex_match $X$
nn candidate a simplicial algebraic space
nn a simplicial algebraic space
new_annotation [(255584, 255587, 'VAR'), (255591, 255619, 'TYPE')]
===== sent However, a quasi-coherent $\mathcal{O}$-module on $X_\etale$ is
just a cartesian $\mathcal{O}$-module $\mathcal{F}$ whose restrictions
$\mathcal{F}_n$ are quasi-coherent on $X_n$, see
Lemma \ref{lemma-quasi-coherent-sheaf}.

\medskip\noindent
Let $h : X \to Y$ be a morphism of simplicial algebraic spaces over $S$.
 ========

regex_match $h : X \to Y$
nn candidate a morphism
nn a morphism
new_annotation [(256083, 256096, 'VAR'), (256100, 256110, 'TYPE')]
===== sent \to \Sh(Y_\etale)$.
Recall that $h_\etale^{-1}$ and $h_{\etale, *}$ have a simple
description in terms of the components, see
Lemma \ref{lemma-morphism-simplicial-sites}.
Let $\mathcal{O}_X$, resp.\ $\mathcal{O}_Y$ denote the structure
sheaf of $X$, resp.\ $Y$. We define
$h_\etale^\sharp : h_{\etale, *}\mathcal{O}_X \to \mathcal{O}_Y$
to be the map of sheaves of rings on $Y_\etale$ given by
$h_n^\sharp : h_{n, *}\mathcal{O}_{X_n} \to \mathcal{O}_{Y_n}$ on $Y_n$.
We obtain a morphism of ringed topoi
$$
h_\etale :
(\Sh(X_\etale), \mathcal{O}_X)
\longrightarrow
(\Sh(Y_\etale), \mathcal{O}_Y)
$$

\medskip\noindent
Let $X$ be a simplicial algebraic space with structure sheaf $\mathcal{O}$.
Let $X_{-1}$ be an algebraic space over $S$ and let $a_0 : X_0 \to X_{-1}$
be an augmentation of $X$. By
Lemma \ref{lemma-augmentation-site}
applied to the morphism of sites
$(a_0)_{spaces, \etale} :
X_{0, spaces, \etale} \to X_{-1, spaces, \etale}$
we obtain a corresponding morphism of topoi
$a : \Sh(X_\etale) \to \Sh(X_{-1, \etale})$.
Observe that $a^{-1}\mathcal{G}$ is the sheaf on
$X_\etale$ with components $a_n^{-1}\mathcal{G}$. Hence we can use the maps
$a_n^\sharp : a_n^{-1}\mathcal{O}_{X_{-1}} \to \mathcal{O}_{X_n}$ to define
a map $a^\sharp : a^{-1}\mathcal{O}_{X_{-1}} \to \mathcal{O}$, or equivalently
by adjunction a map $a^\sharp : \mathcal{O}_{X_{-1}} \to a_*\mathcal{O}$
(which as usual has the same name). ========

new_annotation []
===== sent \to \Sh(Y_\etale)$.
Recall that $h_\etale^{-1}$ and $h_{\etale, *}$ have a simple
description in terms of the components, see
Lemma \ref{lemma-morphism-simplicial-sites}.
Let $\mathcal{O}_X$, resp.\ $\mathcal{O}_Y$ denote the structure
sheaf of $X$, resp.\ $Y$. We define
$h_\etale^\sharp : h_{\etale, *}\mathcal{O}_X \to \mathcal{O}_Y$
to be the map of sheaves of rings on $Y_\etale$ given by
$h_n^\sharp : h_{n, *}\mathcal{O}_{X_n} \to \mathcal{O}_{Y_n}$ on $Y_n$.
We obtain a morphism of ringed topoi
$$
h_\etale :
(\Sh(X_\etale), \mathcal{O}_X)
\longrightarrow
(\Sh(Y_\etale), \mathcal{O}_Y)
$$

\medskip\noindent
Let $X$ be a simplicial algebraic space with structure sheaf $\mathcal{O}$.
Let $X_{-1}$ be an algebraic space over $S$ and let $a_0 : X_0 \to X_{-1}$
be an augmentation of $X$. By
Lemma \ref{lemma-augmentation-site}
applied to the morphism of sites
$(a_0)_{spaces, \etale} :
X_{0, spaces, \etale} \to X_{-1, spaces, \etale}$
we obtain a corresponding morphism of topoi
$a : \Sh(X_\etale) \to \Sh(X_{-1, \etale})$.
Observe that $a^{-1}\mathcal{G}$ is the sheaf on
$X_\etale$ with components $a_n^{-1}\mathcal{G}$. Hence we can use the maps
$a_n^\sharp : a_n^{-1}\mathcal{O}_{X_{-1}} \to \mathcal{O}_{X_n}$ to define
a map $a^\sharp : a^{-1}\mathcal{O}_{X_{-1}} \to \mathcal{O}$, or equivalently
by adjunction a map $a^\sharp : \mathcal{O}_{X_{-1}} \to a_*\mathcal{O}$
(which as usual has the same name). ========

regex_match $X$
nn candidate a simplicial algebraic space
nn a simplicial algebraic space
new_annotation [(257079, 257082, 'VAR'), (257086, 257114, 'TYPE')]
===== sent \to \Sh(Y_\etale)$.
Recall that $h_\etale^{-1}$ and $h_{\etale, *}$ have a simple
description in terms of the components, see
Lemma \ref{lemma-morphism-simplicial-sites}.
Let $\mathcal{O}_X$, resp.\ $\mathcal{O}_Y$ denote the structure
sheaf of $X$, resp.\ $Y$. We define
$h_\etale^\sharp : h_{\etale, *}\mathcal{O}_X \to \mathcal{O}_Y$
to be the map of sheaves of rings on $Y_\etale$ given by
$h_n^\sharp : h_{n, *}\mathcal{O}_{X_n} \to \mathcal{O}_{Y_n}$ on $Y_n$.
We obtain a morphism of ringed topoi
$$
h_\etale :
(\Sh(X_\etale), \mathcal{O}_X)
\longrightarrow
(\Sh(Y_\etale), \mathcal{O}_Y)
$$

\medskip\noindent
Let $X$ be a simplicial algebraic space with structure sheaf $\mathcal{O}$.
Let $X_{-1}$ be an algebraic space over $S$ and let $a_0 : X_0 \to X_{-1}$
be an augmentation of $X$. By
Lemma \ref{lemma-augmentation-site}
applied to the morphism of sites
$(a_0)_{spaces, \etale} :
X_{0, spaces, \etale} \to X_{-1, spaces, \etale}$
we obtain a corresponding morphism of topoi
$a : \Sh(X_\etale) \to \Sh(X_{-1, \etale})$.
Observe that $a^{-1}\mathcal{G}$ is the sheaf on
$X_\etale$ with components $a_n^{-1}\mathcal{G}$. Hence we can use the maps
$a_n^\sharp : a_n^{-1}\mathcal{O}_{X_{-1}} \to \mathcal{O}_{X_n}$ to define
a map $a^\sharp : a^{-1}\mathcal{O}_{X_{-1}} \to \mathcal{O}$, or equivalently
by adjunction a map $a^\sharp : \mathcal{O}_{X_{-1}} \to a_*\mathcal{O}$
(which as usual has the same name). ========

regex_match $X_{-1}$
nn candidate an algebraic space
nn an algebraic space
new_annotation [(257155, 257163, 'VAR'), (257167, 257185, 'TYPE')]
===== sent \to \Sh(Y_\etale)$.
Recall that $h_\etale^{-1}$ and $h_{\etale, *}$ have a simple
description in terms of the components, see
Lemma \ref{lemma-morphism-simplicial-sites}.
Let $\mathcal{O}_X$, resp.\ $\mathcal{O}_Y$ denote the structure
sheaf of $X$, resp.\ $Y$. We define
$h_\etale^\sharp : h_{\etale, *}\mathcal{O}_X \to \mathcal{O}_Y$
to be the map of sheaves of rings on $Y_\etale$ given by
$h_n^\sharp : h_{n, *}\mathcal{O}_{X_n} \to \mathcal{O}_{Y_n}$ on $Y_n$.
We obtain a morphism of ringed topoi
$$
h_\etale :
(\Sh(X_\etale), \mathcal{O}_X)
\longrightarrow
(\Sh(Y_\etale), \mathcal{O}_Y)
$$

\medskip\noindent
Let $X$ be a simplicial algebraic space with structure sheaf $\mathcal{O}$.
Let $X_{-1}$ be an algebraic space over $S$ and let $a_0 : X_0 \to X_{-1}$
be an augmentation of $X$. By
Lemma \ref{lemma-augmentation-site}
applied to the morphism of sites
$(a_0)_{spaces, \etale} :
X_{0, spaces, \etale} \to X_{-1, spaces, \etale}$
we obtain a corresponding morphism of topoi
$a : \Sh(X_\etale) \to \Sh(X_{-1, \etale})$.
Observe that $a^{-1}\mathcal{G}$ is the sheaf on
$X_\etale$ with components $a_n^{-1}\mathcal{G}$. Hence we can use the maps
$a_n^\sharp : a_n^{-1}\mathcal{O}_{X_{-1}} \to \mathcal{O}_{X_n}$ to define
a map $a^\sharp : a^{-1}\mathcal{O}_{X_{-1}} \to \mathcal{O}$, or equivalently
by adjunction a map $a^\sharp : \mathcal{O}_{X_{-1}} \to a_*\mathcal{O}$
(which as usual has the same name). ========

new_annotation []
===== sent This has the advantage of replacing the references to
More on Cohomology of Spaces, Section
\ref{spaces-more-cohomology-section-fppf-etale}
with references to
\'Etale Cohomology, Section \ref{etale-cohomology-section-fppf-etale}.

\medskip\noindent
We fix a base scheme $S$.
Let $X$ be an algebraic space over $S$ and let $U$ be a simplicial
algebraic space over $S$. Assume we have an augmentation
$$
a : U \to X
$$
See Section \ref{section-simplicial-algebraic-spaces}.
We say that $U$ is an {\it fppf hypercovering} of $X$ if
\begin{enumerate}
\item $U_0 \to X$ is flat, locally of finite presentation, and surjective,
\item $U_1 \to U_0 \times_X U_0$ is flat, locally of finite presentation, and
surjective,
\item $U_{n + 1} \to (\text{cosk}_n\text{sk}_n U)_{n + 1}$
is flat, locally of finite presentation, and surjective for $n \geq 1$.
\end{enumerate}
The category of algebraic spaces over $S$ has all finite limits, hence the
coskeleta used in the formulation above exist.
 ========

regex_match $X$
nn candidate an algebraic space
nn an algebraic space
new_annotation [(259525, 259528, 'VAR'), (259532, 259550, 'TYPE')]
===== sent This has the advantage of replacing the references to
More on Cohomology of Spaces, Section
\ref{spaces-more-cohomology-section-fppf-etale}
with references to
\'Etale Cohomology, Section \ref{etale-cohomology-section-fppf-etale}.

\medskip\noindent
We fix a base scheme $S$.
Let $X$ be an algebraic space over $S$ and let $U$ be a simplicial
algebraic space over $S$. Assume we have an augmentation
$$
a : U \to X
$$
See Section \ref{section-simplicial-algebraic-spaces}.
We say that $U$ is an {\it fppf hypercovering} of $X$ if
\begin{enumerate}
\item $U_0 \to X$ is flat, locally of finite presentation, and surjective,
\item $U_1 \to U_0 \times_X U_0$ is flat, locally of finite presentation, and
surjective,
\item $U_{n + 1} \to (\text{cosk}_n\text{sk}_n U)_{n + 1}$
is flat, locally of finite presentation, and surjective for $n \geq 1$.
\end{enumerate}
The category of algebraic spaces over $S$ has all finite limits, hence the
coskeleta used in the formulation above exist.
 ========

regex_match $U$
nn candidate a simplicial
algebraic space
nn a simplicial
algebraic space
new_annotation [(259568, 259571, 'VAR'), (259575, 259603, 'TYPE')]
===== sent Namely, the fppf topology is stronger than the \'etale topology and we have
(a) a flat, locally finitely presented, surjective map defines
an fppf covering, and
(b) fppf cohomology of sheaves pulled back from the small \'etale site
agrees with \'etale cohomology as we have seen in
More on Cohomology of Spaces, Section
\ref{spaces-more-cohomology-section-fppf-etale}.

\begin{lemma}
\label{lemma-compare-simplicial-objects-fppf-etale}
Let $S$ be a scheme. ========

regex_match $S$
nn candidate a scheme
nn a scheme
new_annotation [(260884, 260887, 'VAR'), (260891, 260899, 'TYPE')]
===== sent Let $X$ be an algebraic space over $S$.
Let $U$ be a simplicial algebraic space over $S$.
Let $a : U \to X$ be an augmentation. ========

regex_match $X$
nn candidate an algebraic space
nn an algebraic space
new_annotation [(260905, 260908, 'VAR'), (260912, 260930, 'TYPE')]
===== sent Let $X$ be an algebraic space over $S$.
Let $U$ be a simplicial algebraic space over $S$.
Let $a : U \to X$ be an augmentation. ========

regex_match $U$
nn candidate a simplicial algebraic space
nn a simplicial algebraic space
new_annotation [(260945, 260948, 'VAR'), (260952, 260980, 'TYPE')]
===== sent Let $X$ be an algebraic space over $S$.
Let $U$ be a simplicial algebraic space over $S$.
Let $a : U \to X$ be an augmentation. ========

regex_match $a : U \to X$
nn candidate a : U
nn candidate an augmentation
nn an augmentation
new_annotation [(260995, 261008, 'VAR'), (261012, 261027, 'TYPE')]
===== sent \end{proof}

\begin{lemma}
\label{lemma-descent-sheaves-for-fppf-hypercovering}
Let $S$ be a scheme. ========

regex_match $S$
nn candidate a scheme
nn a scheme
new_annotation [(263071, 263074, 'VAR'), (263078, 263086, 'TYPE')]
===== sent Let $X$ be an algebraic space over $S$.
Let $U$ be a simplicial algebraic space over $S$. Let $a : U \to X$
be an augmentation. ========

regex_match $X$
nn candidate an algebraic space
nn an algebraic space
new_annotation [(263092, 263095, 'VAR'), (263099, 263117, 'TYPE')]
===== sent Let $X$ be an algebraic space over $S$.
Let $U$ be a simplicial algebraic space over $S$. Let $a : U \to X$
be an augmentation. ========

regex_match $U$
nn candidate a simplicial algebraic space
nn a simplicial algebraic space
new_annotation [(263132, 263135, 'VAR'), (263139, 263167, 'TYPE')]
===== sent Let $X$ be an algebraic space over $S$.
Let $U$ be a simplicial algebraic space over $S$. Let $a : U \to X$
be an augmentation. ========

new_annotation []
===== sent \end{proof}

\begin{lemma}
\label{lemma-cohomological-descent-for-fppf-hypercovering}
Let $S$ be a scheme. ========

regex_match $S$
nn candidate a scheme
nn a scheme
new_annotation [(266498, 266501, 'VAR'), (266505, 266513, 'TYPE')]
===== sent Let $X$ be an algebraic space over $S$.
Let $U$ be a simplicial algebraic space over $S$. Let $a : U \to X$
be an augmentation. ========

regex_match $X$
nn candidate an algebraic space
nn an algebraic space
new_annotation [(266519, 266522, 'VAR'), (266526, 266544, 'TYPE')]
===== sent Let $X$ be an algebraic space over $S$.
Let $U$ be a simplicial algebraic space over $S$. Let $a : U \to X$
be an augmentation. ========

regex_match $U$
nn candidate a simplicial algebraic space
nn a simplicial algebraic space
new_annotation [(266559, 266562, 'VAR'), (266566, 266594, 'TYPE')]
===== sent Let $X$ be an algebraic space over $S$.
Let $U$ be a simplicial algebraic space over $S$. Let $a : U \to X$
be an augmentation. ========

new_annotation []
===== sent Here $a : \Sh(U_\etale) \to \Sh(X_\etale)$
is as in Section \ref{section-simplicial-algebraic-spaces}.
\end{lemma}

\begin{proof}
Consider the diagram of Lemma \ref{lemma-compare-simplicial-objects-fppf-etale}.
Observe that $Rh_{n, *}h_n^{-1}$ is the identity functor
on $D^+(U_{n, \etale})$ by
More on Cohomology of Spaces, Lemma
\ref{spaces-more-cohomology-lemma-cohomological-descent-etale-fppf}.
Hence $Rh_*h^{-1}$ is the identity functor on
$D^+(U_\etale)$ by
Lemma \ref{lemma-direct-image-morphism-simplicial-sites}.
We have
\begin{align*}
Ra_*(a^{-1}K)
& =
Ra_*Rh_*h^{-1}a^{-1}K \\
& =
Rh_{-1, *}Ra_{fppf, *}a_{fppf}^{-1}(h_{-1})^{-1}K \\
& =
Rh_{-1, *}(h_{-1})^{-1}K \\
& =
K
\end{align*}
The first equality by the discussion above, the second equality
because of the commutativity of the diagram in
Lemma \ref{lemma-compare-simplicial-objects}, the third equality by
Lemma \ref{lemma-hypercovering-X-simple-descent-bounded-abelian}
as $U$ is a hypercovering of $X$ in $(\textit{Spaces}/S)_{fppf}$,
and the last equality by the already used
More on Cohomology of Spaces, Lemma
\ref{spaces-more-cohomology-lemma-cohomological-descent-etale-fppf}.
\end{proof}

\begin{lemma}
\label{lemma-compute-via-fppf-hypercovering}
Let $S$ be a scheme. ========

regex_match $S$
nn candidate a scheme
nn a scheme
new_annotation [(267999, 268002, 'VAR'), (268006, 268014, 'TYPE')]
===== sent Let $X$ be an algebraic space over $S$.
Let $U$ be a simplicial algebraic space over $S$. Let $a : U \to X$
be an augmentation. ========

regex_match $X$
nn candidate an algebraic space
nn an algebraic space
new_annotation [(268020, 268023, 'VAR'), (268027, 268045, 'TYPE')]
===== sent Let $X$ be an algebraic space over $S$.
Let $U$ be a simplicial algebraic space over $S$. Let $a : U \to X$
be an augmentation. ========

regex_match $U$
nn candidate a simplicial algebraic space
nn a simplicial algebraic space
new_annotation [(268060, 268063, 'VAR'), (268067, 268095, 'TYPE')]
===== sent Let $X$ be an algebraic space over $S$.
Let $U$ be a simplicial algebraic space over $S$. Let $a : U \to X$
be an augmentation. ========

new_annotation []
===== sent This follows from
Lemma \ref{lemma-cohomological-descent-for-fppf-hypercovering}
because $R\Gamma(U_\etale, -) = R\Gamma(X_\etale, -) \circ Ra_*$ by
Cohomology on Sites, Remark \ref{sites-cohomology-remark-before-Leray}.
\end{proof}

\begin{lemma}
\label{lemma-fppf-hypercovering-equivalence-bounded}
Let $S$ be a scheme. ========

regex_match $S$
nn candidate a scheme
nn a scheme
new_annotation [(268717, 268720, 'VAR'), (268724, 268732, 'TYPE')]
===== sent Let $X$ be an algebraic space over $S$.
Let $U$ be a simplicial algebraic space over $S$. Let $a : U \to X$
be an augmentation.
 ========

regex_match $X$
nn candidate an algebraic space
nn an algebraic space
new_annotation [(268738, 268741, 'VAR'), (268745, 268763, 'TYPE')]
===== sent Let $X$ be an algebraic space over $S$.
Let $U$ be a simplicial algebraic space over $S$. Let $a : U \to X$
be an augmentation.
 ========

regex_match $U$
nn candidate a simplicial algebraic space
nn a simplicial algebraic space
new_annotation [(268778, 268781, 'VAR'), (268785, 268813, 'TYPE')]
===== sent Let $X$ be an algebraic space over $S$.
Let $U$ be a simplicial algebraic space over $S$. Let $a : U \to X$
be an augmentation.
 ========

new_annotation []
===== sent Let $\mathcal{A} \subset \textit{Ab}(U_\etale)$
denote the weak Serre subcategory of cartesian abelian sheaves.
 ========

new_annotation []
===== sent Use
Lemmas \ref{lemma-equivalence-bounded},
\ref{lemma-descent-sheaves-for-fppf-hypercovering}, and
\ref{lemma-cohomological-descent-for-fppf-hypercovering}.
\end{proof}

\begin{lemma}
\label{lemma-spectral-sequence-fppf-hypercovering}
Let $S$ be a scheme. ========

regex_match $S$
nn candidate a scheme
nn a scheme
new_annotation [(269702, 269705, 'VAR'), (269709, 269717, 'TYPE')]
===== sent Let $X$ be an algebraic space over $S$.
Let $U$ be a simplicial algebraic space over $S$. Let $a : U \to X$
be an augmentation. ========

regex_match $X$
nn candidate an algebraic space
nn an algebraic space
new_annotation [(269723, 269726, 'VAR'), (269730, 269748, 'TYPE')]
===== sent Let $X$ be an algebraic space over $S$.
Let $U$ be a simplicial algebraic space over $S$. Let $a : U \to X$
be an augmentation. ========

regex_match $U$
nn candidate a simplicial algebraic space
nn a simplicial algebraic space
new_annotation [(269763, 269766, 'VAR'), (269770, 269798, 'TYPE')]
===== sent Let $X$ be an algebraic space over $S$.
Let $U$ be a simplicial algebraic space over $S$. Let $a : U \to X$
be an augmentation. ========

new_annotation []
===== sent Let $\mathcal{F}$ be an abelian sheaf
on $X_\etale$. Let $\mathcal{F}_n$ be the pullback to $U_{n, \etale}$.
If $U$ is an fppf hypercovering of $X$, then
there exists a canonical spectral sequence
$$
E_1^{p, q} = H^q_\etale(U_p, \mathcal{F}_p)
$$
converging to $H^{p + q}_\etale(X, \mathcal{F})$.
\end{lemma}

\begin{proof}
Immediate consequence of Lemmas \ref{lemma-compute-via-fppf-hypercovering}
and \ref{lemma-simplicial-sheaf-cohomology-site}.
\end{proof}








\section{Fppf hypercoverings of algebraic spaces: modules}
\label{section-fppf-hypercovering-modules}

\noindent
 ========

regex_match $\mathcal{F}$
nn candidate an abelian sheaf
nn an abelian sheaf
new_annotation [(269851, 269864, 'VAR'), (269868, 269884, 'TYPE')]
===== sent Let $\mathcal{F}$ be an abelian sheaf
on $X_\etale$. Let $\mathcal{F}_n$ be the pullback to $U_{n, \etale}$.
If $U$ is an fppf hypercovering of $X$, then
there exists a canonical spectral sequence
$$
E_1^{p, q} = H^q_\etale(U_p, \mathcal{F}_p)
$$
converging to $H^{p + q}_\etale(X, \mathcal{F})$.
\end{lemma}

\begin{proof}
Immediate consequence of Lemmas \ref{lemma-compute-via-fppf-hypercovering}
and \ref{lemma-simplicial-sheaf-cohomology-site}.
\end{proof}








\section{Fppf hypercoverings of algebraic spaces: modules}
\label{section-fppf-hypercovering-modules}

\noindent
 ========

regex_match $\mathcal{F}_n$
nn candidate the pullback
nn the pullback
new_annotation [(269904, 269919, 'VAR'), (269923, 269935, 'TYPE')]
===== sent \begin{lemma}
\label{lemma-compare-simplicial-objects-fppf-etale-modules}
Let $S$ be a scheme. ========

regex_match $S$
nn candidate a scheme
nn a scheme
new_annotation [(270822, 270825, 'VAR'), (270829, 270837, 'TYPE')]
===== sent Let $X$ be an algebraic space over $S$.
Let $U$ be a simplicial algebraic space over $S$.
Let $a : U \to X$ be an augmentation. ========

regex_match $X$
nn candidate an algebraic space
nn an algebraic space
new_annotation [(270843, 270846, 'VAR'), (270850, 270868, 'TYPE')]
===== sent Let $X$ be an algebraic space over $S$.
Let $U$ be a simplicial algebraic space over $S$.
Let $a : U \to X$ be an augmentation. ========

regex_match $U$
nn candidate a simplicial algebraic space
nn a simplicial algebraic space
new_annotation [(270883, 270886, 'VAR'), (270890, 270918, 'TYPE')]
===== sent Let $X$ be an algebraic space over $S$.
Let $U$ be a simplicial algebraic space over $S$.
Let $a : U \to X$ be an augmentation. ========

regex_match $a : U \to X$
nn candidate a : U
nn candidate an augmentation
nn an augmentation
new_annotation [(270933, 270946, 'VAR'), (270950, 270965, 'TYPE')]
===== sent \end{proof}

\begin{lemma}
\label{lemma-descent-qcoh-for-fppf-hypercovering}
Let $S$ be a scheme. ========

regex_match $S$
nn candidate a scheme
nn a scheme
new_annotation [(273280, 273283, 'VAR'), (273287, 273295, 'TYPE')]
===== sent Let $X$ be an algebraic space over $S$.
Let $U$ be a simplicial algebraic space over $S$. Let $a : U \to X$
be an augmentation. ========

regex_match $X$
nn candidate an algebraic space
nn an algebraic space
new_annotation [(273301, 273304, 'VAR'), (273308, 273326, 'TYPE')]
===== sent Let $X$ be an algebraic space over $S$.
Let $U$ be a simplicial algebraic space over $S$. Let $a : U \to X$
be an augmentation. ========

regex_match $U$
nn candidate a simplicial algebraic space
nn a simplicial algebraic space
new_annotation [(273341, 273344, 'VAR'), (273348, 273376, 'TYPE')]
===== sent Let $X$ be an algebraic space over $S$.
Let $U$ be a simplicial algebraic space over $S$. Let $a : U \to X$
be an augmentation. ========

new_annotation []
===== sent In turn we see that $\mathcal{H} = (h_{-1})^*\mathcal{F}$
with $\mathcal{F} = h_{-1, *}\mathcal{H}$ in $\QCoh(\mathcal{O}_X)$.
Going around the diagram we deduce that
$h^*\mathcal{G} \cong h^*a^*\mathcal{F}$.
By fully faithfulness of $h^*$ we conclude that
$a^*\mathcal{F} \cong \mathcal{G}$.
Since $\mathcal{F} = h_{-1, *}a_{fppf, *}h^*\mathcal{G} =
a_*h_*h^*\mathcal{G} = a_*\mathcal{G}$ we also obtain
the statement that the quasi-inverse is given by $a_*$.
\end{proof}

\begin{lemma}
\label{lemma-cohomological-descent-qcoh-for-fppf-hypercovering}
Let $S$ be a scheme. ========

regex_match $S$
nn candidate a scheme
nn a scheme
new_annotation [(276425, 276428, 'VAR'), (276432, 276440, 'TYPE')]
===== sent Let $X$ be an algebraic space over $S$.
Let $U$ be a simplicial algebraic space over $S$. Let $a : U \to X$
be an augmentation. ========

regex_match $X$
nn candidate an algebraic space
nn an algebraic space
new_annotation [(276446, 276449, 'VAR'), (276453, 276471, 'TYPE')]
===== sent Let $X$ be an algebraic space over $S$.
Let $U$ be a simplicial algebraic space over $S$. Let $a : U \to X$
be an augmentation. ========

regex_match $U$
nn candidate a simplicial algebraic space
nn a simplicial algebraic space
new_annotation [(276486, 276489, 'VAR'), (276493, 276521, 'TYPE')]
===== sent Let $X$ be an algebraic space over $S$.
Let $U$ be a simplicial algebraic space over $S$. Let $a : U \to X$
be an augmentation. ========

new_annotation []
===== sent Here $a : \Sh(U_\etale) \to \Sh(X_\etale)$
is as in Section \ref{section-simplicial-algebraic-spaces}.
\end{lemma}

\begin{proof}
Consider the diagram of Lemma \ref{lemma-compare-simplicial-objects-fppf-etale}.
Let $\mathcal{F}_n = a_n^*\mathcal{F}$ be the $n$th component of
$a^*\mathcal{F}$. This is a quasi-coherent $\mathcal{O}_{U_n}$-module.
 ========

regex_match $\mathcal{F}_n = a_n^*\mathcal{F}$
nn candidate the $n$th component
nn the $n$th component
new_annotation [(276968, 277002, 'VAR'), (277006, 277025, 'TYPE')]
===== sent Then $\mathcal{F}_n = Rh_{n, *}h_n^*\mathcal{F}_n$
by More on Cohomology of Spaces, Lemma
\ref{spaces-more-cohomology-lemma-cohomological-descent-etale-fppf-modules}.
Hence $a^*\mathcal{F} = Rh_*h^*a^*\mathcal{F}$ by
Lemma \ref{lemma-direct-image-morphism-simplicial-sites-modules}.
We have
\begin{align*}
Ra_*(a^*\mathcal{F})
& =
Ra_*Rh_*h^*a^*\mathcal{F} \\
& =
Rh_{-1, *}Ra_{fppf, *}a_{fppf}^*(h_{-1})^*\mathcal{F} \\
& =
Rh_{-1, *}(h_{-1})^*\mathcal{F} \\
& =
\mathcal{F}
\end{align*}
The first equality by the discussion above, the second equality
because of the commutativity of the diagram in
Lemma \ref{lemma-compare-simplicial-objects}, the third equality by
Lemma \ref{lemma-hypercovering-X-simple-descent-bounded-modules}
as $U$ is a hypercovering of $X$ in $(\textit{Spaces}/S)_{fppf}$
and $La_{fppf}^* = a_{fppf}^*$ as $a_{fppf}$ is flat
(namely $a_{fppf}^{-1}\mathcal{O}_{big} = \mathcal{O}_{big, total}$,
see Remark \ref{remark-augmentation-ringed}), and
the last equality by the already used
More on Cohomology of Spaces, Lemma
\ref{spaces-more-cohomology-lemma-cohomological-descent-etale-fppf-modules}.
\end{proof}

\begin{lemma}
\label{lemma-coh-descent-qcoh-unbounded-for-fppf-hypercovering}
Let $S$ be a scheme. ========

regex_match $S$
nn candidate a scheme
nn a scheme
new_annotation [(278316, 278319, 'VAR'), (278323, 278331, 'TYPE')]
===== sent Let $X$ be an algebraic space over $S$.
Let $U$ be a simplicial algebraic space over $S$. Let $a : U \to X$
be an augmentation. ========

regex_match $X$
nn candidate an algebraic space
nn an algebraic space
new_annotation [(278337, 278340, 'VAR'), (278344, 278362, 'TYPE')]
===== sent Let $X$ be an algebraic space over $S$.
Let $U$ be a simplicial algebraic space over $S$. Let $a : U \to X$
be an augmentation. ========

regex_match $U$
nn candidate a simplicial algebraic space
nn a simplicial algebraic space
new_annotation [(278377, 278380, 'VAR'), (278384, 278412, 'TYPE')]
===== sent Let $X$ be an algebraic space over $S$.
Let $U$ be a simplicial algebraic space over $S$. Let $a : U \to X$
be an augmentation. ========

new_annotation []
===== sent \end{proof}

\begin{lemma}
\label{lemma-compute-via-fppf-hypercovering-modules}
Let $S$ be a scheme. ========

regex_match $S$
nn candidate a scheme
nn a scheme
new_annotation [(281884, 281887, 'VAR'), (281891, 281899, 'TYPE')]
===== sent Let $X$ be an algebraic space over $S$.
Let $U$ be a simplicial algebraic space over $S$. Let $a : U \to X$
be an augmentation. ========

regex_match $X$
nn candidate an algebraic space
nn an algebraic space
new_annotation [(281905, 281908, 'VAR'), (281912, 281930, 'TYPE')]
===== sent Let $X$ be an algebraic space over $S$.
Let $U$ be a simplicial algebraic space over $S$. Let $a : U \to X$
be an augmentation. ========

regex_match $U$
nn candidate a simplicial algebraic space
nn a simplicial algebraic space
new_annotation [(281945, 281948, 'VAR'), (281952, 281980, 'TYPE')]
===== sent Let $X$ be an algebraic space over $S$.
Let $U$ be a simplicial algebraic space over $S$. Let $a : U \to X$
be an augmentation. ========

new_annotation []
===== sent Let $S$ be a scheme. ========

regex_match $S$
nn candidate a scheme
nn a scheme
new_annotation [(282619, 282622, 'VAR'), (282626, 282634, 'TYPE')]
===== sent Let $X$ be an algebraic space over $S$.
Let $U$ be a simplicial algebraic space over $S$. Let $a : U \to X$
be an augmentation. ========

regex_match $X$
nn candidate an algebraic space
nn an algebraic space
new_annotation [(282640, 282643, 'VAR'), (282647, 282665, 'TYPE')]
===== sent Let $X$ be an algebraic space over $S$.
Let $U$ be a simplicial algebraic space over $S$. Let $a : U \to X$
be an augmentation. ========

regex_match $U$
nn candidate a simplicial algebraic space
nn a simplicial algebraic space
new_annotation [(282680, 282683, 'VAR'), (282687, 282715, 'TYPE')]
===== sent Let $X$ be an algebraic space over $S$.
Let $U$ be a simplicial algebraic space over $S$. Let $a : U \to X$
be an augmentation. ========

new_annotation []
===== sent Let $\mathcal{F}$ be quasi-coherent
$\mathcal{O}_X$-module. ========

new_annotation []
===== sent Let $\mathcal{F}_n$ be the pullback to
$U_{n, \etale}$. If $U$ is an fppf hypercovering of $X$, then
there exists a canonical spectral sequence
$$
E_1^{p, q} = H^q_\etale(U_p, \mathcal{F}_p)
$$
converging to $H^{p + q}_\etale(X, \mathcal{F})$.
\end{lemma}

\begin{proof}
Immediate consequence of
Lemmas \ref{lemma-compute-via-fppf-hypercovering-modules}
and \ref{lemma-simplicial-module-cohomology-site}.
\end{proof}







\section{Fppf descent of complexes}
\label{section-fppf-descent-derived}

\noindent
 ========

regex_match $\mathcal{F}_n$
nn candidate the pullback
nn the pullback
new_annotation [(282828, 282843, 'VAR'), (282847, 282859, 'TYPE')]
===== sent \begin{lemma}
\label{lemma-fppf-neg-ext-zero-hom}
Let $X$ be an algebraic space over a scheme $S$.
Let $K, E \in D_\QCoh(\mathcal{O}_X)$.
Let $a : U \to X$ be an fppf hypercovering.
 ========

regex_match $X$
nn candidate an algebraic space
nn an algebraic space
new_annotation [(283546, 283549, 'VAR'), (283553, 283571, 'TYPE')]
===== sent \begin{lemma}
\label{lemma-fppf-neg-ext-zero-hom}
Let $X$ be an algebraic space over a scheme $S$.
Let $K, E \in D_\QCoh(\mathcal{O}_X)$.
Let $a : U \to X$ be an fppf hypercovering.
 ========

new_annotation []
===== sent \begin{lemma}
\label{lemma-fppf-neg-ext-zero-hom}
Let $X$ be an algebraic space over a scheme $S$.
Let $K, E \in D_\QCoh(\mathcal{O}_X)$.
Let $a : U \to X$ be an fppf hypercovering.
 ========

regex_match $a : U \to X$
nn candidate a : U
nn candidate an fppf hypercovering
nn an fppf hypercovering
new_annotation [(283634, 283647, 'VAR'), (283651, 283672, 'TYPE')]
===== sent Part (1) follows by applying part (2) with
$K[i]$ and $E$ for $i > 0$.
\end{proof}

\begin{lemma}
\label{lemma-fppf-glue-neg-ext-zero}
Let $X$ be an algebraic space over a scheme $S$.
Let $a : U \to X$ be an fppf hypercovering.
 ========

regex_match $X$
nn candidate an algebraic space
nn an algebraic space
new_annotation [(285133, 285136, 'VAR'), (285140, 285158, 'TYPE')]
===== sent Part (1) follows by applying part (2) with
$K[i]$ and $E$ for $i > 0$.
\end{proof}

\begin{lemma}
\label{lemma-fppf-glue-neg-ext-zero}
Let $X$ be an algebraic space over a scheme $S$.
Let $a : U \to X$ be an fppf hypercovering.
 ========

regex_match $a : U \to X$
nn candidate a : U
nn candidate an fppf hypercovering
nn an fppf hypercovering
new_annotation [(285182, 285195, 'VAR'), (285199, 285220, 'TYPE')]
===== sent This has the advantage of replacing the references to
More on Cohomology of Spaces, Section
\ref{spaces-more-cohomology-section-ph-etale}
with references to
\'Etale Cohomology, Section \ref{etale-cohomology-section-ph-etale}.

\medskip\noindent
We fix a base scheme $S$.
Let $X$ be an algebraic space over $S$ and let $U$ be a simplicial
algebraic space over $S$. Assume we have an augmentation
$$
a : U \to X
$$
See Section \ref{section-simplicial-algebraic-spaces}.
We say that $U$ is a {\it proper hypercovering} of $X$ if
\begin{enumerate}
\item $U_0 \to X$ is proper and surjective,
\item $U_1 \to U_0 \times_X U_0$ is proper and surjective,
\item $U_{n + 1} \to (\text{cosk}_n\text{sk}_n U)_{n + 1}$
is proper and surjective for $n \geq 1$.
\end{enumerate}
The category of algebraic spaces over $S$ has all finite limits, hence the
coskeleta used in the formulation above exist.
 ========

regex_match $X$
nn candidate an algebraic space
nn an algebraic space
new_annotation [(288122, 288125, 'VAR'), (288129, 288147, 'TYPE')]
===== sent This has the advantage of replacing the references to
More on Cohomology of Spaces, Section
\ref{spaces-more-cohomology-section-ph-etale}
with references to
\'Etale Cohomology, Section \ref{etale-cohomology-section-ph-etale}.

\medskip\noindent
We fix a base scheme $S$.
Let $X$ be an algebraic space over $S$ and let $U$ be a simplicial
algebraic space over $S$. Assume we have an augmentation
$$
a : U \to X
$$
See Section \ref{section-simplicial-algebraic-spaces}.
We say that $U$ is a {\it proper hypercovering} of $X$ if
\begin{enumerate}
\item $U_0 \to X$ is proper and surjective,
\item $U_1 \to U_0 \times_X U_0$ is proper and surjective,
\item $U_{n + 1} \to (\text{cosk}_n\text{sk}_n U)_{n + 1}$
is proper and surjective for $n \geq 1$.
\end{enumerate}
The category of algebraic spaces over $S$ has all finite limits, hence the
coskeleta used in the formulation above exist.
 ========

regex_match $U$
nn candidate a simplicial
algebraic space
nn a simplicial
algebraic space
new_annotation [(288165, 288168, 'VAR'), (288172, 288200, 'TYPE')]
===== sent \begin{lemma}
\label{lemma-compare-simplicial-objects-ph-etale}
Let $S$ be a scheme. ========

regex_match $S$
nn candidate a scheme
nn a scheme
new_annotation [(289663, 289666, 'VAR'), (289670, 289678, 'TYPE')]
===== sent Let $X$ be an algebraic space over $S$.
Let $U$ be a simplicial algebraic space over $S$.
Let $a : U \to X$ be an augmentation. ========

regex_match $X$
nn candidate an algebraic space
nn an algebraic space
new_annotation [(289684, 289687, 'VAR'), (289691, 289709, 'TYPE')]
===== sent Let $X$ be an algebraic space over $S$.
Let $U$ be a simplicial algebraic space over $S$.
Let $a : U \to X$ be an augmentation. ========

regex_match $U$
nn candidate a simplicial algebraic space
nn a simplicial algebraic space
new_annotation [(289724, 289727, 'VAR'), (289731, 289759, 'TYPE')]
===== sent Let $X$ be an algebraic space over $S$.
Let $U$ be a simplicial algebraic space over $S$.
Let $a : U \to X$ be an augmentation. ========

regex_match $a : U \to X$
nn candidate a : U
nn candidate an augmentation
nn an augmentation
new_annotation [(289774, 289787, 'VAR'), (289791, 289806, 'TYPE')]
===== sent \end{proof}

\begin{lemma}
\label{lemma-descent-sheaves-for-ph-hypercovering}
Let $S$ be a scheme. ========

regex_match $S$
nn candidate a scheme
nn a scheme
new_annotation [(291857, 291860, 'VAR'), (291864, 291872, 'TYPE')]
===== sent Let $X$ be an algebraic space over $S$.
Let $U$ be a simplicial algebraic space over $S$. Let $a : U \to X$
be an augmentation. ========

regex_match $X$
nn candidate an algebraic space
nn an algebraic space
new_annotation [(291878, 291881, 'VAR'), (291885, 291903, 'TYPE')]
===== sent Let $X$ be an algebraic space over $S$.
Let $U$ be a simplicial algebraic space over $S$. Let $a : U \to X$
be an augmentation. ========

regex_match $U$
nn candidate a simplicial algebraic space
nn a simplicial algebraic space
new_annotation [(291918, 291921, 'VAR'), (291925, 291953, 'TYPE')]
===== sent Let $X$ be an algebraic space over $S$.
Let $U$ be a simplicial algebraic space over $S$. Let $a : U \to X$
be an augmentation. ========

new_annotation []
===== sent Since $a_{ph}^{-1}\mathcal{H} = h^{-1}\mathcal{G}$
we deduce that $h^{-1}\mathcal{G} \cong h^{-1}a^{-1}a_*\mathcal{G}$
which ends the proof by fully faithfulness of $h^{-1}$.
\end{proof}

\begin{lemma}
\label{lemma-cohomological-descent-for-ph-hypercovering}
Let $S$ be a scheme. ========

regex_match $S$
nn candidate a scheme
nn a scheme
new_annotation [(295534, 295537, 'VAR'), (295541, 295549, 'TYPE')]
===== sent Let $X$ be an algebraic space over $S$.
Let $U$ be a simplicial algebraic space over $S$. Let $a : U \to X$
be an augmentation. ========

regex_match $X$
nn candidate an algebraic space
nn an algebraic space
new_annotation [(295555, 295558, 'VAR'), (295562, 295580, 'TYPE')]
===== sent Let $X$ be an algebraic space over $S$.
Let $U$ be a simplicial algebraic space over $S$. Let $a : U \to X$
be an augmentation. ========

regex_match $U$
nn candidate a simplicial algebraic space
nn a simplicial algebraic space
new_annotation [(295595, 295598, 'VAR'), (295602, 295630, 'TYPE')]
===== sent Let $X$ be an algebraic space over $S$.
Let $U$ be a simplicial algebraic space over $S$. Let $a : U \to X$
be an augmentation. ========

new_annotation []
===== sent Here $a : \Sh(U_\etale) \to \Sh(X_\etale)$
is as in Section \ref{section-simplicial-algebraic-spaces}.
\end{lemma}

\begin{proof}
Consider the diagram of Lemma \ref{lemma-compare-simplicial-objects-ph-etale}.
Observe that $Rh_{n, *}h_n^{-1}$ is the identity functor
on $D^+(U_{n, \etale})$ by
More on Cohomology of Spaces, Lemma
\ref{spaces-more-cohomology-lemma-cohomological-descent-etale-ph}.
Hence $Rh_*h^{-1}$ is the identity functor on
$D^+(U_\etale)$ by
Lemma \ref{lemma-direct-image-morphism-simplicial-sites}.
We have
\begin{align*}
Ra_*(a^{-1}K)
& =
Ra_*Rh_*h^{-1}a^{-1}K \\
& =
Rh_{-1, *}Ra_{ph, *}a_{ph}^{-1}(h_{-1})^{-1}K \\
& =
Rh_{-1, *}(h_{-1})^{-1}K \\
& =
K
\end{align*}
The first equality by the discussion above, the second equality
because of the commutativity of the diagram in
Lemma \ref{lemma-compare-simplicial-objects}, the third equality by
Lemma \ref{lemma-hypercovering-X-simple-descent-bounded-abelian}
as $U$ is a hypercovering of $X$ in $(\textit{Spaces}/S)_{ph}$
by Topologies on Spaces, Lemma
\ref{spaces-topologies-lemma-surjective-proper-ph},
and the last equality by the already used
More on Cohomology of Spaces, Lemma
\ref{spaces-more-cohomology-lemma-cohomological-descent-etale-ph}.
\end{proof}

\begin{lemma}
\label{lemma-compute-via-ph-hypercovering}
Let $S$ be a scheme. ========

regex_match $S$
nn candidate a scheme
nn a scheme
new_annotation [(297104, 297107, 'VAR'), (297111, 297119, 'TYPE')]
===== sent Let $X$ be an algebraic space over $S$.
Let $U$ be a simplicial algebraic space over $S$. Let $a : U \to X$
be an augmentation. ========

regex_match $X$
nn candidate an algebraic space
nn an algebraic space
new_annotation [(297125, 297128, 'VAR'), (297132, 297150, 'TYPE')]
===== sent Let $X$ be an algebraic space over $S$.
Let $U$ be a simplicial algebraic space over $S$. Let $a : U \to X$
be an augmentation. ========

regex_match $U$
nn candidate a simplicial algebraic space
nn a simplicial algebraic space
new_annotation [(297165, 297168, 'VAR'), (297172, 297200, 'TYPE')]
===== sent Let $X$ be an algebraic space over $S$.
Let $U$ be a simplicial algebraic space over $S$. Let $a : U \to X$
be an augmentation. ========

new_annotation []
===== sent This follows from
Lemma \ref{lemma-cohomological-descent-for-ph-hypercovering}
because $R\Gamma(U_\etale, -) = R\Gamma(X_\etale, -) \circ Ra_*$ by
Cohomology on Sites, Remark \ref{sites-cohomology-remark-before-Leray}.
\end{proof}

\begin{lemma}
\label{lemma-ph-hypercovering-equivalence-bounded}
Let $S$ be a scheme. ========

regex_match $S$
nn candidate a scheme
nn a scheme
new_annotation [(297819, 297822, 'VAR'), (297826, 297834, 'TYPE')]
===== sent Let $X$ be an algebraic space over $S$.
Let $U$ be a simplicial algebraic space over $S$. Let $a : U \to X$
be an augmentation.
 ========

regex_match $X$
nn candidate an algebraic space
nn an algebraic space
new_annotation [(297840, 297843, 'VAR'), (297847, 297865, 'TYPE')]
===== sent Let $X$ be an algebraic space over $S$.
Let $U$ be a simplicial algebraic space over $S$. Let $a : U \to X$
be an augmentation.
 ========

regex_match $U$
nn candidate a simplicial algebraic space
nn a simplicial algebraic space
new_annotation [(297880, 297883, 'VAR'), (297887, 297915, 'TYPE')]
===== sent Let $X$ be an algebraic space over $S$.
Let $U$ be a simplicial algebraic space over $S$. Let $a : U \to X$
be an augmentation.
 ========

new_annotation []
===== sent Let $\mathcal{A} \subset \textit{Ab}(U_\etale)$
denote the weak Serre subcategory of cartesian abelian sheaves.
 ========

new_annotation []
===== sent Use
Lemmas \ref{lemma-equivalence-bounded},
\ref{lemma-descent-sheaves-for-ph-hypercovering}, and
\ref{lemma-cohomological-descent-for-ph-hypercovering}.
\end{proof}

\begin{lemma}
\label{lemma-spectral-sequence-ph-hypercovering}
Let $S$ be a scheme. ========

regex_match $S$
nn candidate a scheme
nn a scheme
new_annotation [(298799, 298802, 'VAR'), (298806, 298814, 'TYPE')]
===== sent Let $X$ be an algebraic space over $S$.
Let $U$ be a simplicial algebraic space over $S$. Let $a : U \to X$
be an augmentation. ========

regex_match $X$
nn candidate an algebraic space
nn an algebraic space
new_annotation [(298820, 298823, 'VAR'), (298827, 298845, 'TYPE')]
===== sent Let $X$ be an algebraic space over $S$.
Let $U$ be a simplicial algebraic space over $S$. Let $a : U \to X$
be an augmentation. ========

regex_match $U$
nn candidate a simplicial algebraic space
nn a simplicial algebraic space
new_annotation [(298860, 298863, 'VAR'), (298867, 298895, 'TYPE')]
===== sent Let $X$ be an algebraic space over $S$.
Let $U$ be a simplicial algebraic space over $S$. Let $a : U \to X$
be an augmentation. ========

new_annotation []
===== sent Let $\mathcal{F}$ be an abelian sheaf
on $X_\etale$. Let $\mathcal{F}_n$ be the pullback to $U_{n, \etale}$.
If $U$ is a ph hypercovering of $X$, then
there exists a canonical spectral sequence
$$
E_1^{p, q} = H^q_\etale(U_p, \mathcal{F}_p)
$$
converging to $H^{p + q}_\etale(X, \mathcal{F})$.
\end{lemma}

\begin{proof}
Immediate consequence of Lemmas \ref{lemma-compute-via-ph-hypercovering}
and \ref{lemma-simplicial-sheaf-cohomology-site}.
\end{proof}










\input{chapters}

\bibliography{my}
\bibliographystyle{amsalpha}

\end{document}
 ========

regex_match $\mathcal{F}$
nn candidate an abelian sheaf
nn an abelian sheaf
new_annotation [(298948, 298961, 'VAR'), (298965, 298981, 'TYPE')]
===== sent Let $\mathcal{F}$ be an abelian sheaf
on $X_\etale$. Let $\mathcal{F}_n$ be the pullback to $U_{n, \etale}$.
If $U$ is a ph hypercovering of $X$, then
there exists a canonical spectral sequence
$$
E_1^{p, q} = H^q_\etale(U_p, \mathcal{F}_p)
$$
converging to $H^{p + q}_\etale(X, \mathcal{F})$.
\end{lemma}

\begin{proof}
Immediate consequence of Lemmas \ref{lemma-compute-via-ph-hypercovering}
and \ref{lemma-simplicial-sheaf-cohomology-site}.
\end{proof}










\input{chapters}

\bibliography{my}
\bibliographystyle{amsalpha}

\end{document}
 ========

regex_match $\mathcal{F}_n$
nn candidate the pullback
nn the pullback
new_annotation [(299001, 299016, 'VAR'), (299020, 299032, 'TYPE')]
file:  stacks-sheaves.tex
===== sent \begin{definition}
\label{definition-presheaves}
Let $p : \mathcal{X} \to (\Sch/S)_{fppf}$ be a category fibred in
groupoids.
 ========

regex_match $p : \mathcal{X} \to (\Sch/S)_{fppf}$
nn candidate a category
nn a category
new_annotation [(3183, 3220, 'VAR'), (3224, 3234, 'TYPE')]
===== sent The category of {\it abelian presheaves}, i.e., presheaves of abelian
groups, is denoted $\textit{PAb}(\mathcal{X})$.

\medskip\noindent
Let $f : \mathcal{X} \to \mathcal{Y}$ be a $1$-morphism of categories
fibred in groupoids over $(\Sch/S)_{fppf}$. Recall that this
means just that $f$ is a functor over $(\Sch/S)_{fppf}$.
The material in
Sites, Section \ref{sites-section-more-functoriality-PSh}
provides us with a pair of adjoint functors\footnote{These functors
will be denoted $f^{-1}$ and $f_*$ after
Lemma \ref{lemma-functoriality-sheaves}
has been proved.}
 ========

regex_match $f : \mathcal{X} \to \mathcal{Y}$
nn candidate \mathcal{X
nn candidate categories
nn categories
new_annotation [(3971, 4004, 'VAR'), (4026, 4036, 'TYPE')]
===== sent \begin{lemma}
\label{lemma-1-morphisms-presheaves}
Let $f : \mathcal{X} \to \mathcal{Y}$ and $g : \mathcal{Y} \to \mathcal{Z}$
be $1$-morphisms of categories fibred in groupoids over
$(\Sch/S)_{fppf}$. ========

new_annotation []
===== sent Then $(g \circ f)^p = f^p \circ g^p$ and
there is a canonical isomorphism
${}_p(g \circ f) \to {}_pg \circ {}_pf$
compatible with with adjointness of $(f^p, {}_pf)$, $(g^p, {}_pg)$, and
$((g \circ f)^p, {}_p(g \circ f))$.
\end{lemma}

\begin{proof}
Let $\mathcal{H}$ be a presheaf on $\mathcal{Z}$. Then
$(g \circ f)^p\mathcal{H} = f^p (g^p\mathcal{H})$ is given
by the equalities
$$
(g \circ f)^p\mathcal{H}(x) = \mathcal{H}((g \circ f)(x))
= \mathcal{H}(g(f(x))) = f^p (g^p\mathcal{H})(x).
 ========

regex_match $\mathcal{H}$
nn candidate a presheaf
nn a presheaf
new_annotation [(5838, 5851, 'VAR'), (5855, 5865, 'TYPE')]
===== sent Next, we define the transformation ${}_p(g \circ f) \to {}_pg \circ {}_pf$.
Let $\mathcal{F}$ be a presheaf on $\mathcal{X}$.
 ========

regex_match $\mathcal{F}$
nn candidate a presheaf
nn a presheaf
new_annotation [(6251, 6264, 'VAR'), (6268, 6278, 'TYPE')]
===== sent \end{proof}

\begin{lemma}
\label{lemma-2-morphisms-presheaves}
Let $f, g : \mathcal{X} \to \mathcal{Y}$ be $1$-morphisms of categories
fibred in groupoids over $(\Sch/S)_{fppf}$. Let $t : f ========

new_annotation []
===== sent \end{proof}

\begin{lemma}
\label{lemma-2-morphisms-presheaves}
Let $f, g : \mathcal{X} \to \mathcal{Y}$ be $1$-morphisms of categories
fibred in groupoids over $(\Sch/S)_{fppf}$. Let $t : f ========

new_annotation []
===== sent \to g$
be a $2$-morphism of categories fibred in groupoids over
$(\Sch/S)_{fppf}$. Assigned to $t$ there are canonical
isomorphisms of functors
$$
t^p : g^p \longrightarrow f^p
\quad\text{and}\quad
{}_pt : {}_pf \longrightarrow {}_pg
$$
which compatible with adjointness of $(f^p, {}_pf)$ and
$(g^p, {}_pg)$ and with
vertical and horizontal composition of $2$-morphisms.
\end{lemma}

\begin{proof}
Let $\mathcal{G}$ be a presheaf on $\mathcal{Y}$. Then
$t^p : g^p\mathcal{G} \to f^p\mathcal{G}$ is given by the family
of maps
$$
g^p\mathcal{G}(x) = \mathcal{G}(g(x))
\xrightarrow{\mathcal{G}(t_x)}
\mathcal{G}(f(x)) = f^p\mathcal{G}(x)
$$
parametrized by $x \in \Ob(\mathcal{X})$. ========

regex_match $\mathcal{G}$
nn candidate a presheaf
nn a presheaf
new_annotation [(8595, 8608, 'VAR'), (8612, 8622, 'TYPE')]
===== sent namely $\Sch_\alpha/S$.

\medskip\noindent
It follows that if we have a category
$p : \mathcal{X} \to (\Sch/S)_{fppf}$ fibred in groupoids, then
$\mathcal{X}$ inherits a Zariski, \'etale, smooth, syntomic, and
fppf topology, see
Stacks, Definition \ref{stacks-definition-topology-inherited}.

\begin{definition}
\label{definition-inherited-topologies}
Let $\mathcal{X}$ be a category fibred in groupoids over
$(\Sch/S)_{fppf}$.
\begin{enumerate}
\item The {\it associated Zariski site}, denoted $\mathcal{X}_{Zar}$,
is the structure of site on $\mathcal{X}$ inherited from
$(\Sch/S)_{Zar}$.
\item ========

regex_match $\mathcal{X}$
nn candidate a category
nn a category
new_annotation [(13008, 13021, 'VAR'), (13025, 13035, 'TYPE')]
===== sent \begin{definition}
\label{definition-sheaves}
Let $\mathcal{X}$ be a category fibred in groupoids over
$(\Sch/S)_{fppf}$. Let $\mathcal{F}$ be a presheaf on $\mathcal{X}$.
\begin{enumerate}
\item We say $\mathcal{F}$ is a {\it Zariski sheaf}, or a
{\it sheaf for the Zariski topology} if $\mathcal{F}$
is a sheaf on the associated Zariski site $\mathcal{X}_{Zar}$.
\item We say $\mathcal{F}$ is an {\it \'etale sheaf}, or a
{\it sheaf for the \'etale topology} if $\mathcal{F}$
is a sheaf on the associated \'etale site $\mathcal{X}_\etale$.
\item We say $\mathcal{F}$ is a {\it smooth sheaf}, or a
{\it sheaf for the smooth topology} if $\mathcal{F}$
is a sheaf on the associated smooth site $\mathcal{X}_{smooth}$.
\item We say $\mathcal{F}$ is a {\it syntomic sheaf}, or a
{\it sheaf for the syntomic topology} if $\mathcal{F}$
is a sheaf on the associated syntomic site $\mathcal{X}_{syntomic}$.
\item We say $\mathcal{F}$ is an {\it fppf sheaf}, or a {\it sheaf},
or a {\it sheaf for the fppf topology} if $\mathcal{F}$
is a sheaf on the associated fppf site $\mathcal{X}_{fppf}$.
\end{enumerate}
A morphism of sheaves is just a morphism of presheaves. ========

regex_match $\mathcal{X}$
nn candidate a category
nn a category
new_annotation [(14800, 14813, 'VAR'), (14817, 14827, 'TYPE')]
===== sent \begin{definition}
\label{definition-sheaves}
Let $\mathcal{X}$ be a category fibred in groupoids over
$(\Sch/S)_{fppf}$. Let $\mathcal{F}$ be a presheaf on $\mathcal{X}$.
\begin{enumerate}
\item We say $\mathcal{F}$ is a {\it Zariski sheaf}, or a
{\it sheaf for the Zariski topology} if $\mathcal{F}$
is a sheaf on the associated Zariski site $\mathcal{X}_{Zar}$.
\item We say $\mathcal{F}$ is an {\it \'etale sheaf}, or a
{\it sheaf for the \'etale topology} if $\mathcal{F}$
is a sheaf on the associated \'etale site $\mathcal{X}_\etale$.
\item We say $\mathcal{F}$ is a {\it smooth sheaf}, or a
{\it sheaf for the smooth topology} if $\mathcal{F}$
is a sheaf on the associated smooth site $\mathcal{X}_{smooth}$.
\item We say $\mathcal{F}$ is a {\it syntomic sheaf}, or a
{\it sheaf for the syntomic topology} if $\mathcal{F}$
is a sheaf on the associated syntomic site $\mathcal{X}_{syntomic}$.
\item We say $\mathcal{F}$ is an {\it fppf sheaf}, or a {\it sheaf},
or a {\it sheaf for the fppf topology} if $\mathcal{F}$
is a sheaf on the associated fppf site $\mathcal{X}_{fppf}$.
\end{enumerate}
A morphism of sheaves is just a morphism of presheaves. ========

regex_match $\mathcal{F}$
nn candidate a presheaf
nn a presheaf
new_annotation [(14876, 14889, 'VAR'), (14893, 14903, 'TYPE')]
===== sent \begin{lemma}
\label{lemma-functoriality-sheaves}
Let $f : \mathcal{X} \to \mathcal{Y}$ be a $1$-morphism of categories
fibred in groupoids over $(\Sch/S)_{fppf}$. Let
$\tau \in \{Zar, \etale, smooth, syntomic, fppf\}$.
The functors ${}_pf$ and $f^p$ of (\ref{equation-pushforward-pullback})
transform $\tau$ sheaves into $\tau$ sheaves and define a morphism
of topoi
$f : \Sh(\mathcal{X}_\tau) \to \Sh(\mathcal{Y}_\tau)$.
\end{lemma}

\begin{proof}
This follows immediately from
Stacks, Lemma \ref{stacks-lemma-topology-inherited-functorial}.
\end{proof}

\noindent
 ========

regex_match $f : \mathcal{X} \to \mathcal{Y}$
nn candidate categories
nn categories
new_annotation [(17547, 17580, 'VAR'), (17602, 17612, 'TYPE')]
===== sent \begin{definition}
\label{definition-morphism}
Let $f : \mathcal{X} \to \mathcal{Y}$ be a morphism of categories
fibred in groupoids over $(\Sch/S)_{fppf}$. We denote
$$
f = (f^{-1}, f_*) :
\Sh(\mathcal{X}_{fppf})
\longrightarrow
\Sh(\mathcal{Y}_{fppf})
$$
the {\it associated morphism of fppf topoi} constructed above.
 ========

regex_match $f : \mathcal{X} \to \mathcal{Y}$
nn candidate \mathcal{X
nn candidate a morphism
nn a morphism
new_annotation [(18409, 18442, 'VAR'), (18446, 18456, 'TYPE')]
===== sent \section{Computing pushforward}
\label{section-pushforward}

\noindent
Let $f : \mathcal{X} \to \mathcal{Y}$ be a $1$-morphism of categories
fibred in groupoids over $(\Sch/S)_{fppf}$. Let $\mathcal{F}$
be a presheaf on $\mathcal{X}$. Let $y \in \Ob(\mathcal{Y})$.
We can compute $f_*\mathcal{F}(y)$ in the following way. ========

regex_match $f : \mathcal{X} \to \mathcal{Y}$
nn candidate categories
nn categories
new_annotation [(19194, 19227, 'VAR'), (19249, 19259, 'TYPE')]
===== sent \section{Computing pushforward}
\label{section-pushforward}

\noindent
Let $f : \mathcal{X} \to \mathcal{Y}$ be a $1$-morphism of categories
fibred in groupoids over $(\Sch/S)_{fppf}$. Let $\mathcal{F}$
be a presheaf on $\mathcal{X}$. Let $y \in \Ob(\mathcal{Y})$.
We can compute $f_*\mathcal{F}(y)$ in the following way. ========

new_annotation []
===== sent \section{Computing pushforward}
\label{section-pushforward}

\noindent
Let $f : \mathcal{X} \to \mathcal{Y}$ be a $1$-morphism of categories
fibred in groupoids over $(\Sch/S)_{fppf}$. Let $\mathcal{F}$
be a presheaf on $\mathcal{X}$. Let $y \in \Ob(\mathcal{Y})$.
We can compute $f_*\mathcal{F}(y)$ in the following way. ========

new_annotation []
===== sent \begin{lemma}
\label{lemma-base-change}
Let $S$ be a scheme. ========

regex_match $S$
nn candidate a scheme
nn a scheme
new_annotation [(20462, 20465, 'VAR'), (20469, 20477, 'TYPE')]
===== sent \begin{lemma}
\label{lemma-representable}
Let $f : \mathcal{X} \to \mathcal{Y}$ be a $1$-morphism of categories
fibred in groupoids over $(\Sch/S)_{fppf}$. The following are
equivalent
\begin{enumerate}
\item $f$ is representable, and
\item for every $y \in \Ob(\mathcal{Y})$ the functor
$\mathcal{X}^{opp} \to \textit{Sets}$,
$x \mapsto \Mor_\mathcal{Y}(f(x), y)$
is representable.
 ========

regex_match $f : \mathcal{X} \to \mathcal{Y}$
nn candidate \mathcal{X
nn candidate categories
nn categories
new_annotation [(21556, 21589, 'VAR'), (21611, 21621, 'TYPE')]
===== sent This is true because $U' \to U$ represents
the base change
$(\Sch/V')_{fppf} \times_{y', \mathcal{Y}} \mathcal{X} \to
(\Sch/V)_{fppf} \times_{y, \mathcal{Y}} \mathcal{X}$
of $V' \to V$.

\begin{lemma}
\label{lemma-representable-pushforward}
Let $f : \mathcal{X} \to \mathcal{Y}$ be a representable $1$-morphism of
categories fibred in groupoids over $(\Sch/S)_{fppf}$. Let
$\tau \in \{Zar, \etale, smooth, syntomic, fppf\}$.
Then the functor $u : \mathcal{Y}_\tau \to \mathcal{X}_\tau$ is continuous
and defines a morphism of sites $\mathcal{X}_\tau \to \mathcal{Y}_\tau$
which induces the same morphism of topoi
$\Sh(\mathcal{X}_\tau) \to \Sh(\mathcal{Y}_\tau)$
as the morphism $f$ constructed in
Lemma \ref{lemma-functoriality-sheaves}.
 ========

regex_match $f : \mathcal{X} \to \mathcal{Y}$
nn candidate \mathcal{X
nn candidate categories
nn categories
new_annotation [(24123, 24156, 'VAR'), (24192, 24202, 'TYPE')]
===== sent Moreover, $f_*\mathcal{F}(y) = \mathcal{F}(u(y))$ for any presheaf
$\mathcal{F}$ on $\mathcal{X}$.
\end{lemma}

\begin{proof}
Let $\{y_i \to y\}$ be a $\tau$-covering in $\mathcal{Y}$. ========

regex_match $\{y_i \to y\}$
new_annotation []
===== sent \end{proof}





\section{The structure sheaf}
\label{section-structure-sheaf}

\noindent
Let $\tau \in \{Zar, \etale, smooth, syntomic, fppf\}$.
Let $p : \mathcal{X} \to (\Sch/S)_{fppf}$ be a category
fibred in groupoids. ========

new_annotation []
===== sent \end{proof}





\section{The structure sheaf}
\label{section-structure-sheaf}

\noindent
Let $\tau \in \{Zar, \etale, smooth, syntomic, fppf\}$.
Let $p : \mathcal{X} \to (\Sch/S)_{fppf}$ be a category
fibred in groupoids. ========

regex_match $p : \mathcal{X} \to (\Sch/S)_{fppf}$
nn candidate a category
nn a category
new_annotation [(26056, 26093, 'VAR'), (26097, 26107, 'TYPE')]
===== sent \medskip\noindent
Recall that the site $(\Sch/S)_{fppf}$ is a ringed site
with structure sheaf $\mathcal{O}$ defined by the rule
$$
(\Sch/S)^{opp} \longrightarrow \textit{Rings},
\quad
U/S \longmapsto \Gamma(U, \mathcal{O}_U)
$$
see
Descent, Definition \ref{descent-definition-structure-sheaf}.

\begin{definition}
\label{definition-structure-sheaf}
Let $p : \mathcal{X} \to (\Sch/S)_{fppf}$ be a category
fibred in groupoids. ========

regex_match $p : \mathcal{X} \to (\Sch/S)_{fppf}$
nn candidate a category
nn a category
new_annotation [(27022, 27059, 'VAR'), (27063, 27073, 'TYPE')]
===== sent \begin{lemma}
\label{lemma-functoriality-structure-sheaf}
Let $f : \mathcal{X} \to \mathcal{Y}$ be a $1$-morphism of categories
fibred in groupoids over $(\Sch/S)_{fppf}$. Let
$\tau \in \{Zar, \etale, smooth, syntomic, fppf\}$.
There is a canonical identification
$f^{-1}\mathcal{O}_\mathcal{X} = \mathcal{O}_\mathcal{Y}$
which turns
$f : \Sh(\mathcal{X}_\tau) \to \Sh(\mathcal{Y}_\tau)$
into a morphism of ringed topoi.
 ========

regex_match $f : \mathcal{X} \to \mathcal{Y}$
nn candidate \mathcal{X
nn candidate categories
nn categories
new_annotation [(27733, 27766, 'VAR'), (27788, 27798, 'TYPE')]
===== sent \begin{definition}
\label{definition-modules}
Let $\mathcal{X}$ be a category fibred in groupoids over
$(\Sch/S)_{fppf}$.
\begin{enumerate}
\item A {\it presheaf of modules on $\mathcal{X}$} is a
presheaf of $\mathcal{O}_\mathcal{X}$-modules. ========

regex_match $\mathcal{X}$
nn candidate a category
nn a category
new_annotation [(29189, 29202, 'VAR'), (29206, 29216, 'TYPE')]
===== sent \begin{lemma}
\label{lemma-compare-with-scheme}
Let $S$ be a scheme. ========

regex_match $S$
nn candidate a scheme
nn a scheme
new_annotation [(33844, 33847, 'VAR'), (33851, 33859, 'TYPE')]
===== sent Let $\mathcal{X}$ be a category fibred
in groupoids over $(\Sch/S)$. Assume $\mathcal{X}$ is representable
by a scheme $X$. For $\tau \in \{Zar,\linebreak[0] \etale,\linebreak[0]
smooth,\linebreak[0] syntomic,\linebreak[0] fppf\}$
there is a canonical equivalence
$$
(\mathcal{X}_\tau, \mathcal{O}_\mathcal{X}) =
((\Sch/X)_\tau, \mathcal{O}_X)
$$
of ringed sites.
 ========

regex_match $\mathcal{X}$
nn candidate a category
nn a category
new_annotation [(33865, 33878, 'VAR'), (33882, 33892, 'TYPE')]
===== sent This follows by choosing an equivalence
$(\Sch/X)_\tau \to \mathcal{X}$ of categories fibred in groupoids
over $(\Sch/S)_{fppf}$ and using the functoriality of
the construction $\mathcal{X} \leadsto \mathcal{X}_\tau$.
\end{proof}

\begin{lemma}
\label{lemma-compare-with-morphism-of-schemes}
Let $S$ be a scheme. ========

regex_match $S$
nn candidate a scheme
nn a scheme
new_annotation [(34548, 34551, 'VAR'), (34555, 34563, 'TYPE')]
===== sent Let $f : \mathcal{X} \to \mathcal{Y}$ be a morphism
of categories fibred in groupoids over $S$.
Assume $\mathcal{X}$, $\mathcal{Y}$ are representable by schemes
$X$, $Y$. Let $f : X \to Y$ be the morphism of schemes corresponding
to $f$. For $\tau \in \{Zar,\linebreak[0] \etale,\linebreak[0]
smooth,\linebreak[0] syntomic,\linebreak[0] fppf\}$
the morphism of ringed topoi
$f : (\Sh(\mathcal{X}_\tau), \mathcal{O}_\mathcal{X}) \to
(\Sh(\mathcal{Y}_\tau), ========

regex_match $f : \mathcal{X} \to \mathcal{Y}$
nn candidate $
nn candidate a morphism
nn a morphism
new_annotation [(34569, 34602, 'VAR'), (34606, 34616, 'TYPE')]
===== sent Let $f : \mathcal{X} \to \mathcal{Y}$ be a morphism
of categories fibred in groupoids over $S$.
Assume $\mathcal{X}$, $\mathcal{Y}$ are representable by schemes
$X$, $Y$. Let $f : X \to Y$ be the morphism of schemes corresponding
to $f$. For $\tau \in \{Zar,\linebreak[0] \etale,\linebreak[0]
smooth,\linebreak[0] syntomic,\linebreak[0] fppf\}$
the morphism of ringed topoi
$f : (\Sh(\mathcal{X}_\tau), \mathcal{O}_\mathcal{X}) \to
(\Sh(\mathcal{Y}_\tau), ========

regex_match $f : X \to Y$
nn candidate X \to
nn candidate the morphism
nn the morphism
new_annotation [(34740, 34753, 'VAR'), (34757, 34769, 'TYPE')]
===== sent \begin{lemma}
\label{lemma-localizing}
Let $p : \mathcal{X} \to (\Sch/S)_{fppf}$ be a category fibred
in groupoids. ========

regex_match $p : \mathcal{X} \to (\Sch/S)_{fppf}$
nn candidate a category
nn a category
new_annotation [(35587, 35624, 'VAR'), (35628, 35638, 'TYPE')]
===== sent Let $\tau \in \{Zar, \etale, smooth, syntomic, fppf\}$.
Let $x \in \Ob(\mathcal{X})$ lying over $U = p(x)$.
The functor $p$ induces an equivalence of sites
$\mathcal{X}_\tau/x \to ========

new_annotation []
===== sent Let $\tau \in \{Zar, \etale, smooth, syntomic, fppf\}$.
Let $x \in \Ob(\mathcal{X})$ lying over $U = p(x)$.
The functor $p$ induces an equivalence of sites
$\mathcal{X}_\tau/x \to ========

new_annotation []
===== sent \begin{definition}
\label{definition-pullback}
Let $p : \mathcal{X} \to (\Sch/S)_{fppf}$ be a category fibred
in groupoids. ========

regex_match $p : \mathcal{X} \to (\Sch/S)_{fppf}$
nn candidate a category
nn a category
new_annotation [(36118, 36155, 'VAR'), (36159, 36169, 'TYPE')]
===== sent Let $x \in \Ob(\mathcal{X})$ lying over $U = p(x)$.
Let $\mathcal{F}$ be a presheaf on $\mathcal{X}$.
\begin{enumerate}
\item The {\it pullback $x^{-1}\mathcal{F}$ of $\mathcal{F}$} is the
restriction $\mathcal{F}|_{(\mathcal{X}/x)}$ viewed as a presheaf on
$(\Sch/U)_{fppf}$ via the equivalence
$\mathcal{X}/x \to (\Sch/U)_{fppf}$ of
Lemma \ref{lemma-localizing}.
\item ========

new_annotation []
===== sent Let $x \in \Ob(\mathcal{X})$ lying over $U = p(x)$.
Let $\mathcal{F}$ be a presheaf on $\mathcal{X}$.
\begin{enumerate}
\item The {\it pullback $x^{-1}\mathcal{F}$ of $\mathcal{F}$} is the
restriction $\mathcal{F}|_{(\mathcal{X}/x)}$ viewed as a presheaf on
$(\Sch/U)_{fppf}$ via the equivalence
$\mathcal{X}/x \to (\Sch/U)_{fppf}$ of
Lemma \ref{lemma-localizing}.
\item ========

regex_match $\mathcal{F}$
nn candidate a presheaf
nn a presheaf
new_annotation [(36247, 36260, 'VAR'), (36264, 36274, 'TYPE')]
===== sent \medskip\noindent
Let $p : \mathcal{X} \to (\Sch/S)_{fppf}$ be a category fibred
in groupoids. ========

regex_match $p : \mathcal{X} \to (\Sch/S)_{fppf}$
nn candidate a category
nn a category
new_annotation [(37382, 37419, 'VAR'), (37423, 37433, 'TYPE')]
===== sent Let $\varphi : x \to y$ be a morphism of $\mathcal{X}$
lying over the morphism of schemes $a : U \to V$.
Recall that $a$ induces a morphism of small \'etale
sites $a_{small} : U_\etale \to V_\etale$, see
\'Etale Cohomology, Section \ref{etale-cohomology-section-functoriality}.
Let $\mathcal{F}$ be a presheaf on $\mathcal{X}$.
Let $\mathcal{F}|_{U_\etale}$ and
$\mathcal{F}|_{V_\etale}$ be the restrictions of $\mathcal{F}$
via $x$ and $y$. There is a natural {\it comparison} map
\begin{equation}
\label{equation-comparison-push}
c_\varphi :
\mathcal{F}|_{V_\etale}
\longrightarrow
a_{small, *}(\mathcal{F}|_{U_\etale})
\end{equation}
of presheaves on $U_\etale$. ========

regex_match $\varphi : x \to y$
nn candidate $\varphi
nn candidate a morphism
nn a morphism
new_annotation [(37459, 37478, 'VAR'), (37482, 37492, 'TYPE')]
===== sent Let $\varphi : x \to y$ be a morphism of $\mathcal{X}$
lying over the morphism of schemes $a : U \to V$.
Recall that $a$ induces a morphism of small \'etale
sites $a_{small} : U_\etale \to V_\etale$, see
\'Etale Cohomology, Section \ref{etale-cohomology-section-functoriality}.
Let $\mathcal{F}$ be a presheaf on $\mathcal{X}$.
Let $\mathcal{F}|_{U_\etale}$ and
$\mathcal{F}|_{V_\etale}$ be the restrictions of $\mathcal{F}$
via $x$ and $y$. There is a natural {\it comparison} map
\begin{equation}
\label{equation-comparison-push}
c_\varphi :
\mathcal{F}|_{V_\etale}
\longrightarrow
a_{small, *}(\mathcal{F}|_{U_\etale})
\end{equation}
of presheaves on $U_\etale$. ========

regex_match $\mathcal{F}$
nn candidate a presheaf
nn a presheaf
new_annotation [(37737, 37750, 'VAR'), (37754, 37764, 'TYPE')]
===== sent Let $\varphi : x \to y$ be a morphism of $\mathcal{X}$
lying over the morphism of schemes $a : U \to V$.
Recall that $a$ induces a morphism of small \'etale
sites $a_{small} : U_\etale \to V_\etale$, see
\'Etale Cohomology, Section \ref{etale-cohomology-section-functoriality}.
Let $\mathcal{F}$ be a presheaf on $\mathcal{X}$.
Let $\mathcal{F}|_{U_\etale}$ and
$\mathcal{F}|_{V_\etale}$ be the restrictions of $\mathcal{F}$
via $x$ and $y$. There is a natural {\it comparison} map
\begin{equation}
\label{equation-comparison-push}
c_\varphi :
\mathcal{F}|_{V_\etale}
\longrightarrow
a_{small, *}(\mathcal{F}|_{U_\etale})
\end{equation}
of presheaves on $U_\etale$. ========

new_annotation []
===== sent \begin{lemma}
\label{lemma-comparison}
Let $\mathcal{F}$ be an \'etale sheaf on $\mathcal{X} \to (\Sch/S)_{fppf}$.
\begin{enumerate}
\item ========

regex_match $\mathcal{F}$
nn candidate an \'etale sheaf
nn an \'etale sheaf
new_annotation [(39455, 39468, 'VAR'), (39472, 39488, 'TYPE')]
===== sent \begin{lemma}
\label{lemma-localizing-structure-sheaf}
Let $p : \mathcal{X} \to (\Sch/S)_{fppf}$ be a category fibred
in groupoids. ========

regex_match $p : \mathcal{X} \to (\Sch/S)_{fppf}$
nn candidate a category
nn a category
new_annotation [(41076, 41113, 'VAR'), (41117, 41127, 'TYPE')]
===== sent Let $\tau \in \{Zar, \etale, smooth, syntomic, fppf\}$.
Let $x \in \Ob(\mathcal{X})$ lying over $U = p(x)$.
 ========

new_annotation []
===== sent Let $\tau \in \{Zar, \etale, smooth, syntomic, fppf\}$.
Let $x \in \Ob(\mathcal{X})$ lying over $U = p(x)$.
 ========

new_annotation []
===== sent \end{proof}

\noindent
Let $\mathcal{X}$ be a category fibred in groupoids over $(\Sch/S)_{fppf}$.
Let $\mathcal{F}$ be a (pre)sheaf of modules on $\mathcal{X}$ as in
Definition \ref{definition-modules}.
Let $x$ be an object of $\mathcal{X}$ lying over $U$. Then
Lemma \ref{lemma-localizing-structure-sheaf}
guarantees that the restriction
$x^{-1}\mathcal{F}$ is a (pre)sheaf of modules on $(\Sch/U)_{fppf}$.
We will sometimes write $x^*\mathcal{F} = x^{-1}\mathcal{F}$ in this case.
 ========

regex_match $\mathcal{X}$
nn candidate a category
nn a category
new_annotation [(41552, 41565, 'VAR'), (41569, 41579, 'TYPE')]
===== sent \end{proof}

\noindent
Let $\mathcal{X}$ be a category fibred in groupoids over $(\Sch/S)_{fppf}$.
Let $\mathcal{F}$ be a (pre)sheaf of modules on $\mathcal{X}$ as in
Definition \ref{definition-modules}.
Let $x$ be an object of $\mathcal{X}$ lying over $U$. Then
Lemma \ref{lemma-localizing-structure-sheaf}
guarantees that the restriction
$x^{-1}\mathcal{F}$ is a (pre)sheaf of modules on $(\Sch/U)_{fppf}$.
We will sometimes write $x^*\mathcal{F} = x^{-1}\mathcal{F}$ in this case.
 ========

regex_match $\mathcal{F}$
nn candidate a (pre)sheaf
nn a (pre)sheaf
new_annotation [(41628, 41641, 'VAR'), (41645, 41657, 'TYPE')]
===== sent \end{proof}

\noindent
Let $\mathcal{X}$ be a category fibred in groupoids over $(\Sch/S)_{fppf}$.
Let $\mathcal{F}$ be a (pre)sheaf of modules on $\mathcal{X}$ as in
Definition \ref{definition-modules}.
Let $x$ be an object of $\mathcal{X}$ lying over $U$. Then
Lemma \ref{lemma-localizing-structure-sheaf}
guarantees that the restriction
$x^{-1}\mathcal{F}$ is a (pre)sheaf of modules on $(\Sch/U)_{fppf}$.
We will sometimes write $x^*\mathcal{F} = x^{-1}\mathcal{F}$ in this case.
 ========

regex_match $x$
nn candidate an object
nn an object
new_annotation [(41733, 41736, 'VAR'), (41740, 41749, 'TYPE')]
===== sent \begin{lemma}
\label{lemma-enough-points}
Let $p : \mathcal{X} \to (\Sch/S)_{fppf}$ be a category fibred
in groupoids. ========

regex_match $p : \mathcal{X} \to (\Sch/S)_{fppf}$
nn candidate a category
nn a category
new_annotation [(43112, 43149, 'VAR'), (43153, 43163, 'TYPE')]
===== sent Let $\tau \in \{Zar, \etale, smooth, syntomic, fppf\}$.
The site $\mathcal{X}_\tau$ has enough points.
 ========

new_annotation []
===== sent \begin{lemma}
\label{lemma-compare}
Let $S$ be a scheme. ========

regex_match $S$
nn candidate a scheme
nn a scheme
new_annotation [(44131, 44134, 'VAR'), (44138, 44146, 'TYPE')]
===== sent Let $\mathcal{X} \to (\Sch/S)_{fppf}$ be a category
fibred in groupoids. ========

regex_match $\mathcal{X} \to (\Sch/S)_{fppf}$
nn candidate a category
nn a category
new_annotation [(44152, 44185, 'VAR'), (44189, 44199, 'TYPE')]
===== sent Let $U$ be a scheme and let
$\varphi : U \to F$ be surjective and \'etale. ========

regex_match $U$
nn candidate a scheme
nn a scheme
new_annotation [(46299, 46302, 'VAR'), (46306, 46314, 'TYPE')]
===== sent \end{proof}

\noindent
Assume $\mathcal{X}$ is an algebraic stack represented by the
algebraic space $F$.
Let $j : \mathcal{S}_F ========

new_annotation []
===== sent \begin{lemma}
\label{lemma-compare-morphism}
Let $S$ be a scheme. ========

regex_match $S$
nn candidate a scheme
nn a scheme
new_annotation [(47906, 47909, 'VAR'), (47913, 47921, 'TYPE')]
===== sent Let $f : \mathcal{X} \to \mathcal{Y}$ be a morphism
of categories fibred in groupoids over $(\Sch/S)_{fppf}$. Assume
$\mathcal{X}$, $\mathcal{Y}$ are representable by algebraic spaces $F$, $G$.
Denote $f : F \to G$ the induced morphism of algebraic spaces, and
$f_{small} : F_\etale \to G_\etale$
the corresponding morphism of ringed topoi. ========

regex_match $f : \mathcal{X} \to \mathcal{Y}$
nn candidate $
nn candidate a morphism
nn a morphism
new_annotation [(47927, 47960, 'VAR'), (47964, 47974, 'TYPE')]
===== sent Similarly for $G$.

\medskip\noindent
Let $\mathcal{G}$ be a sheaf on $\mathcal{X}_\etale$.
To prove (2) we compute $\pi_{G, *}f_*\mathcal{G}$ and
$f_{small, *}\pi_{F, *}\mathcal{G}$. To do this let $V \to G$ be an object
of $G_\etale$. Then
$$
\pi_{G, *}f_*\mathcal{G}(V) = f_*\mathcal{G}(V/G) =
\Gamma\Big(
(\Sch/V)_{fppf} \times_{\mathcal{Y}} \mathcal{X},
\ \text{pr}^{-1}\mathcal{G}\Big)
$$
see (\ref{equation-pushforward}). ========

regex_match $\mathcal{G}$
nn candidate a sheaf
nn a sheaf
new_annotation [(49720, 49733, 'VAR'), (49737, 49744, 'TYPE')]
===== sent Similarly for $G$.

\medskip\noindent
Let $\mathcal{G}$ be a sheaf on $\mathcal{X}_\etale$.
To prove (2) we compute $\pi_{G, *}f_*\mathcal{G}$ and
$f_{small, *}\pi_{F, *}\mathcal{G}$. To do this let $V \to G$ be an object
of $G_\etale$. Then
$$
\pi_{G, *}f_*\mathcal{G}(V) = f_*\mathcal{G}(V/G) =
\Gamma\Big(
(\Sch/V)_{fppf} \times_{\mathcal{Y}} \mathcal{X},
\ \text{pr}^{-1}\mathcal{G}\Big)
$$
see (\ref{equation-pushforward}). ========

regex_match $V \to G$
nn candidate $V
nn candidate an object
nn an object
new_annotation [(49877, 49886, 'VAR'), (49890, 49899, 'TYPE')]
===== sent Moreover, by the material in
Properties of Spaces, Section \ref{spaces-properties-section-etale-site}
(especially,
Properties of Spaces,
Remark \ref{spaces-properties-remark-explain-equivalence} and
Lemma \ref{spaces-properties-lemma-functoriality-etale-site})
we see that there is an equalizer diagram
$$
\xymatrix{
f_{small, *}\pi_{F, *}\mathcal{G}(V)
\ar[r] &
\pi_{F, *}\mathcal{G}(U/F) \ar@<1ex>[r] \ar@<-1ex>[r] &
\pi_{F, *}\mathcal{G}(R/F)
}
$$
Since we also have $\pi_{F, *}\mathcal{G}(U/F) = \mathcal{G}(U/F)$
and $\pi_{F, *}\mathcal{G}(U/F) = \mathcal{G}(U/F)$
we obtain a canonical identification
$f_{small, *}\pi_{F, *}\mathcal{G}(V) = \pi_{G, *}f_*\mathcal{G}(V)$.
We omit the proof that this is compatible with restriction mappings
and that it is functorial in $\mathcal{G}$.
\end{proof}

\noindent
Let $f : \mathcal{X} \to \mathcal{Y}$ and $f : F \to G$ be as in the
second part of the lemma above. ========

new_annotation []
===== sent \begin{definition}
\label{definition-quasi-coherent}
Let $p : \mathcal{X} \to (\Sch/S)_{fppf}$ be a category fibred
in groupoids. ========

regex_match $p : \mathcal{X} \to (\Sch/S)_{fppf}$
nn candidate a category
nn a category
new_annotation [(54851, 54888, 'VAR'), (54892, 54902, 'TYPE')]
===== sent \begin{lemma}
\label{lemma-pullback-quasi-coherent}
Let $f : \mathcal{X} \to \mathcal{Y}$ be a $1$-morphism of categories
fibred in groupoids over $(\Sch/S)_{fppf}$.
The pullback functor
$f^* = f^{-1} : \textit{Mod}(\mathcal{O}_\mathcal{Y}) \to
\textit{Mod}(\mathcal{O}_\mathcal{X})$
preserves quasi-coherent sheaves.
 ========

regex_match $f : \mathcal{X} \to \mathcal{Y}$
nn candidate \mathcal{X
nn candidate categories
nn categories
new_annotation [(56078, 56111, 'VAR'), (56133, 56143, 'TYPE')]
===== sent \begin{lemma}
\label{lemma-characterize-quasi-coherent}
Let $p : \mathcal{X} \to (\Sch/S)_{fppf}$ be a category
fibred in groupoids. ========

regex_match $p : \mathcal{X} \to (\Sch/S)_{fppf}$
nn candidate a category
nn a category
new_annotation [(56742, 56779, 'VAR'), (56783, 56793, 'TYPE')]
===== sent Let $\mathcal{F}$
be a sheaf of $\mathcal{O}_\mathcal{X}$-modules. ========

new_annotation []
===== sent \begin{definition}
\label{definition-locally-quasi-coherent}
Let $p : \mathcal{X} \to (\Sch/S)_{fppf}$ be a category
fibred in groupoids. ========

regex_match $p : \mathcal{X} \to (\Sch/S)_{fppf}$
nn candidate a category
nn a category
new_annotation [(57732, 57769, 'VAR'), (57773, 57783, 'TYPE')]
===== sent Let $\mathcal{F}$
be a presheaf of $\mathcal{O}_\mathcal{X}$-modules.
 ========

new_annotation []
===== sent \begin{lemma}
\label{lemma-quasi-coherent}
Let $p : \mathcal{X} \to (\Sch/S)_{fppf}$ be a category
fibred in groupoids. ========

regex_match $p : \mathcal{X} \to (\Sch/S)_{fppf}$
nn candidate a category
nn a category
new_annotation [(59044, 59081, 'VAR'), (59085, 59095, 'TYPE')]
===== sent Let $\mathcal{F}$
be a presheaf of $\mathcal{O}_\mathcal{X}$-modules. ========

new_annotation []
===== sent Let $y$ be an object of $\mathcal{X}$ with $V = p(y)$.
We have seen that $\mathcal{X}/y = (\Sch/V)_{fppf}$. By
Descent, Proposition \ref{descent-proposition-equivalence-quasi-coherent}
it follows that $y^*\mathcal{F}$ is the quasi-coherent module
associated to a (usual) quasi-coherent module $\mathcal{F}_V$ on
the scheme $V$. Hence certainly the comparison maps
(\ref{equation-comparison-modules}) are isomorphisms.

 ========

regex_match $y$
nn candidate an object
nn an object
new_annotation [(59963, 59966, 'VAR'), (59970, 59979, 'TYPE')]
===== sent Let $y$ be an object of $\mathcal{X}$ with $V = p(y)$. Denote
$\mathcal{F}_V$ the quasi-coherent module on
the scheme $V$ corresponding to the restriction
$y^*\mathcal{F}|_{V_\etale}$ which is quasi-coherent by
assumption (1), see
Descent, Proposition \ref{descent-proposition-equivalence-quasi-coherent}.
Condition (2) now signifies that the restrictions
$x^*\mathcal{F}|_{U_\etale}$ for $x$ over $y$ are each
isomorphic to the (\'etale sheaf associated to the) pullback of $\mathcal{F}_V$
via the corresponding morphism of schemes $U \to ========

regex_match $y$
nn candidate an object
nn an object
new_annotation [(60462, 60465, 'VAR'), (60469, 60478, 'TYPE')]
===== sent Hence $y^*\mathcal{F}$ is the sheaf on $(\Sch/V)_{fppf}$
associated to $\mathcal{F}_V$. Hence it is quasi-coherent (by
Descent, Proposition \ref{descent-proposition-equivalence-quasi-coherent}
again) and we see that $\mathcal{F}$ is quasi-coherent on $\mathcal{X}$ by
Lemma \ref{lemma-characterize-quasi-coherent}.
\end{proof}

\begin{lemma}
\label{lemma-pullback-lqc}
Let $f : \mathcal{X} \to \mathcal{Y}$ be a $1$-morphism of categories
fibred in groupoids over $(\Sch/S)_{fppf}$. The pullback functor
$f^* = f^{-1} :
\textit{Mod}(\mathcal{Y}_\etale, \mathcal{O}_\mathcal{Y})
\to
\textit{Mod}(\mathcal{X}_\etale, \mathcal{O}_\mathcal{X})$
preserves locally quasi-coherent sheaves.
 ========

regex_match $f : \mathcal{X} \to \mathcal{Y}$
nn candidate \mathcal{X
nn candidate categories
nn categories
new_annotation [(61375, 61408, 'VAR'), (61430, 61440, 'TYPE')]
===== sent \end{lemma}

\begin{proof}
Let $\mathcal{G}$ be locally quasi-coherent on $\mathcal{Y}$.
Choose an object $x$ of $\mathcal{X}$ lying over the scheme $U$.
The restriction $x^*f^*\mathcal{G}|_{U_\etale}$ equals
$(f \circ x)^*\mathcal{G}|_{U_\etale}$
hence is a quasi-coherent sheaf by assumption on $\mathcal{G}$.
\end{proof}

\begin{lemma}
\label{lemma-lqc-colimits}
Let $p : \mathcal{X} \to (\Sch/S)_{fppf}$ be a category fibred in
groupoids.
 ========

new_annotation []
===== sent \end{lemma}

\begin{proof}
Let $\mathcal{G}$ be locally quasi-coherent on $\mathcal{Y}$.
Choose an object $x$ of $\mathcal{X}$ lying over the scheme $U$.
The restriction $x^*f^*\mathcal{G}|_{U_\etale}$ equals
$(f \circ x)^*\mathcal{G}|_{U_\etale}$
hence is a quasi-coherent sheaf by assumption on $\mathcal{G}$.
\end{proof}

\begin{lemma}
\label{lemma-lqc-colimits}
Let $p : \mathcal{X} \to (\Sch/S)_{fppf}$ be a category fibred in
groupoids.
 ========

regex_match $p : \mathcal{X} \to (\Sch/S)_{fppf}$
nn candidate a category
nn a category
new_annotation [(62055, 62092, 'VAR'), (62096, 62106, 'TYPE')]
===== sent Let $\varphi : \mathcal{F} \to \mathcal{G}$ be a morphism of
$\textit{LQCoh}(\mathcal{O}_\mathcal{X})$ and let
$\mathcal{K} = \Ker(\varphi)$ computed in
$\textit{Mod}(\mathcal{X}_\etale, \mathcal{O}_\mathcal{X})$.
If we can show that $\mathcal{K}$ is a locally quasi-coherent module,
then the proof of (2) is complete. ========

regex_match $\varphi : \mathcal{F} \to \mathcal{G}$
nn candidate $\varphi
nn candidate a morphism
nn a morphism
new_annotation [(64540, 64579, 'VAR'), (64583, 64593, 'TYPE')]
===== sent \begin{lemma}
\label{lemma-qc-colimits}
Let $p : \mathcal{X} \to (\Sch/S)_{fppf}$ be a category
fibred in groupoids.
 ========

regex_match $p : \mathcal{X} \to (\Sch/S)_{fppf}$
nn candidate a category
nn a category
new_annotation [(65615, 65652, 'VAR'), (65656, 65666, 'TYPE')]
===== sent Given $\mathcal{F}, \mathcal{G}$ in
$\QCoh(\mathcal{O}_\mathcal{X})$
the tensor product $\mathcal{F} \otimes_{\mathcal{O}_\mathcal{X}} \mathcal{G}$
in $\textit{Mod}(\mathcal{O}_\mathcal{X})$
is an object of $\QCoh(\mathcal{O}_\mathcal{X})$.
\item Given $\mathcal{F}, \mathcal{G}$ in
$\QCoh(\mathcal{O}_\mathcal{X})$
with $\mathcal{F}$ locally of finite presentation on
$\mathcal{X}_{fppf}$ the sheaf
$\SheafHom_{\mathcal{O}_\mathcal{X}}(\mathcal{F}, \mathcal{G})$
in $\textit{Mod}(\mathcal{O}_\mathcal{X})$
is an object of $\QCoh(\mathcal{O}_\mathcal{X})$.
\end{enumerate}
\end{lemma}

\begin{proof}
Let $\mathcal{I} \to \QCoh(\mathcal{O}_\mathcal{X})$,
$i \mapsto \mathcal{F}_i$ be a diagram.
 ========

new_annotation []
===== sent \begin{lemma}
\label{lemma-stackification}
Let $f : \mathcal{X} \to \mathcal{Y}$ be a $1$-morphism of categories
fibred in groupoids over $(\Sch/S)_{fppf}$. If
$f$ induces an equivalence of stackifications, then the morphism
of topoi
$f : \Sh(\mathcal{X}_{fppf}) \to \Sh(\mathcal{Y}_{fppf})$
is an equivalence.
 ========

regex_match $f : \mathcal{X} \to \mathcal{Y}$
nn candidate \mathcal{X
nn candidate categories
nn categories
new_annotation [(67931, 67964, 'VAR'), (67986, 67996, 'TYPE')]
===== sent \end{proof}

\begin{lemma}
\label{lemma-stackification-quasi-coherent}
Let $f : \mathcal{X} \to \mathcal{Y}$ be a $1$-morphism of categories
fibred in groupoids over $(\Sch/S)_{fppf}$. If
$f$ induces an equivalence of stackifications, then $f^*$
induces equivalences
$\textit{Mod}(\mathcal{O}_\mathcal{X}) \to
\textit{Mod}(\mathcal{O}_\mathcal{Y})$
and
$\QCoh(\mathcal{O}_\mathcal{X}) \to
\QCoh(\mathcal{O}_\mathcal{Y})$.
\end{lemma}

\begin{proof}
We may assume $\mathcal{Y}$ is the stackification of $\mathcal{X}$.
The first assertion is clear from
Lemma \ref{lemma-stackification}
and
$\mathcal{O}_\mathcal{X} = f^{-1}\mathcal{O}_\mathcal{Y}$.
Pullback of quasi-coherent sheaves are quasi-coherent, see
Lemma \ref{lemma-pullback-quasi-coherent}.
 ========

regex_match $f : \mathcal{X} \to \mathcal{Y}$
nn candidate \mathcal{X
nn candidate categories
nn categories
new_annotation [(68802, 68835, 'VAR'), (68857, 68867, 'TYPE')]
===== sent To see this, let $y$ be an object of $\mathcal{Y}$.
Translating the condition that $\mathcal{Y}$ is the stackification
of $\mathcal{X}$ we see there exists an fppf covering $\{y_i \to y\}$
in $\mathcal{Y}$ such that $y_i \cong f(x_i)$ for some
$x_i$ object of $\mathcal{X}$. Say $x_i$ and $y_i$ lie over the scheme $U_i$.
Then $f^*\mathcal{G}$ being quasi-coherent, means that $x_i^*f^*\mathcal{G}$
is quasi-coherent. ========

regex_match $y$
nn candidate $y$
nn candidate an object
nn an object
new_annotation [(69586, 69589, 'VAR'), (69593, 69602, 'TYPE')]
===== sent \begin{proposition}
\label{proposition-quasi-coherent}
Let $(U, R, s, t, c)$ be a groupoid in algebraic spaces over $S$.
Let $\mathcal{X} = [U/R]$ be the quotient stack.
 ========

regex_match $(U, R, s, t, c)$
nn candidate U
nn candidate R
nn candidate a groupoid
nn a groupoid
new_annotation [(70894, 70911, 'VAR'), (70915, 70925, 'TYPE')]
===== sent \begin{proposition}
\label{proposition-quasi-coherent}
Let $(U, R, s, t, c)$ be a groupoid in algebraic spaces over $S$.
Let $\mathcal{X} = [U/R]$ be the quotient stack.
 ========

regex_match $\mathcal{X} = [U/R]$
nn candidate $\mathcal{X
nn candidate [U/R]$
nn candidate the quotient stack
nn the quotient stack
new_annotation [(70960, 70981, 'VAR'), (70985, 71003, 'TYPE')]
===== sent Let $\mathcal{F}$ be a quasi-coherent sheaf on $\mathcal{X}$.
Then we obtain for every $x = (T, u)$ a quasi-coherent sheaf
$\mathcal{F}_{(T, u)} = x^*\mathcal{F}|_{T_\etale}$ on $T$.
 ========

regex_match $\mathcal{F}$
nn candidate a quasi-coherent sheaf
nn a quasi-coherent sheaf
new_annotation [(72249, 72262, 'VAR'), (72266, 72288, 'TYPE')]
===== sent \begin{lemma}
\label{lemma-map-from-quasi-coherent}
Let $(U, R, s, t, c)$ be a groupoid in algebraic spaces over $S$.
Assume $s, t$ are flat and locally of finite presentation.
 ========

regex_match $(U, R, s, t, c)$
nn candidate U
nn candidate R
nn candidate a groupoid
nn a groupoid
new_annotation [(76238, 76255, 'VAR'), (76259, 76269, 'TYPE')]
===== sent Let $\mathcal{X} = [U/R]$ be the quotient stack. ========

regex_match $\mathcal{X} = [U/R]$
nn candidate $\mathcal{X
nn candidate [U/R]$
nn candidate the quotient stack
nn the quotient stack
new_annotation [(76363, 76384, 'VAR'), (76388, 76406, 'TYPE')]
===== sent Let $\mathcal{F}$ be a quasi-coherent
$\mathcal{O}_\mathcal{X}$-module, and let $\mathcal{H}$ be any object
of $\textit{Mod}(\mathcal{O}_\mathcal{X})$.
The map
$$
\Hom_{\mathcal{O}_\mathcal{X}}(\mathcal{F}, \mathcal{H})
\longrightarrow
\Hom_{\mathcal{O}_U}(x^*\mathcal{F}|_{U_\etale},
x^*\mathcal{H}|_{U_\etale}),
\quad
\phi \longmapsto x^*\phi|_{U_\etale}
$$
is injective and its image consists of exactly those
$\varphi : x^*\mathcal{F}|_{U_\etale} \to
x^*\mathcal{H}|_{U_\etale}$ which give rise to a commutative
diagram
$$
\xymatrix{
s_{small}^*(x^*\mathcal{F}|_{U_\etale})
\ar[r] \ar[d]^{s_{small}^*\varphi} &
(x \circ s)^*\mathcal{F}|_{R_\etale} =
(x \circ t)^*\mathcal{F}|_{R_\etale} &
t_{small}^*(x^*\mathcal{F}|_{U_\etale})
\ar[l] \ar[d]_{t_{small}^*\varphi} \\
 ========

regex_match $\mathcal{F}$
nn candidate $\mathcal{H}$
nn $\mathcal{H}$
new_annotation [(76475, 76488, 'VAR'), (76551, 76564, 'TYPE')]
===== sent Let $\mathcal{F}$ be a quasi-coherent
$\mathcal{O}_\mathcal{X}$-module, and let $\mathcal{H}$ be any object
of $\textit{Mod}(\mathcal{O}_\mathcal{X})$.
The map
$$
\Hom_{\mathcal{O}_\mathcal{X}}(\mathcal{F}, \mathcal{H})
\longrightarrow
\Hom_{\mathcal{O}_U}(x^*\mathcal{F}|_{U_\etale},
x^*\mathcal{H}|_{U_\etale}),
\quad
\phi \longmapsto x^*\phi|_{U_\etale}
$$
is injective and its image consists of exactly those
$\varphi : x^*\mathcal{F}|_{U_\etale} \to
x^*\mathcal{H}|_{U_\etale}$ which give rise to a commutative
diagram
$$
\xymatrix{
s_{small}^*(x^*\mathcal{F}|_{U_\etale})
\ar[r] \ar[d]^{s_{small}^*\varphi} &
(x \circ s)^*\mathcal{F}|_{R_\etale} =
(x \circ t)^*\mathcal{F}|_{R_\etale} &
t_{small}^*(x^*\mathcal{F}|_{U_\etale})
\ar[l] \ar[d]_{t_{small}^*\varphi} \\
 ========

new_annotation []
===== sent Hint: the restriction maps
of $\mathcal{F}$ are made explicit in terms of $(\mathcal{G}, \alpha)$
in the proof of
Proposition \ref{proposition-quasi-coherent}.
\end{proof}







\section{Quasi-coherent sheaves on algebraic stacks}
\label{section-quasi-coherent-algebraic-stacks}

\noindent
Let $\mathcal{X}$ be an algebraic stack over $S$. By
Algebraic Stacks, Lemma \ref{algebraic-lemma-stack-presentation}
we can find an equivalence $[U/R] \to \mathcal{X}$
where ========

regex_match $\mathcal{X}$
nn candidate an algebraic stack
nn an algebraic stack
new_annotation [(79270, 79283, 'VAR'), (79287, 79305, 'TYPE')]
===== sent \begin{lemma}
\label{lemma-quasi-coherent-algebraic-stack}
Let $\mathcal{X}$ be an algebraic stack over $S$.
\begin{enumerate}
\item If $[U/R] \to \mathcal{X}$ is a presentation of $\mathcal{X}$
then there is a canonical equivalence
$\QCoh(\mathcal{O}_\mathcal{X}) \cong
\QCoh(U, R, s, t, c)$.
\item The category $\QCoh(\mathcal{O}_\mathcal{X})$ is abelian.
 ========

regex_match $\mathcal{X}$
nn candidate an algebraic stack
nn an algebraic stack
new_annotation [(80959, 80972, 'VAR'), (80976, 80994, 'TYPE')]
===== sent \end{proof}

\begin{proposition}
\label{proposition-coherator}
Let $\mathcal{X}$ be an algebraic stack over $S$.
\begin{enumerate}
\item The category $\QCoh(\mathcal{O}_\mathcal{X})$ is a Grothendieck
abelian category. ========

regex_match $\mathcal{X}$
nn candidate an algebraic stack
nn an algebraic stack
new_annotation [(82317, 82330, 'VAR'), (82334, 82352, 'TYPE')]
===== sent \end{proof}








\section{Cohomology}
\label{section-cohomology-general}

\noindent
Let $S$ be a scheme and let $\mathcal{X}$ be a category fibred in groupoids
over $(\Sch/S)_{fppf}$. For any $\tau \in \{Zariski, \etale, smooth,
syntomic, fppf\}$ the categories $\textit{Ab}(\mathcal{X}_\tau)$ and
$\textit{Mod}(\mathcal{X}_\tau, \mathcal{O}_\mathcal{X})$ have
enough injectives, see
Injectives, Theorems \ref{injectives-theorem-sheaves-injectives} and
\ref{injectives-theorem-sheaves-modules-injectives}.
 ========

regex_match $S$
nn candidate a scheme
nn a scheme
new_annotation [(86069, 86072, 'VAR'), (86076, 86084, 'TYPE')]
===== sent \end{proof}








\section{Cohomology}
\label{section-cohomology-general}

\noindent
Let $S$ be a scheme and let $\mathcal{X}$ be a category fibred in groupoids
over $(\Sch/S)_{fppf}$. For any $\tau \in \{Zariski, \etale, smooth,
syntomic, fppf\}$ the categories $\textit{Ab}(\mathcal{X}_\tau)$ and
$\textit{Mod}(\mathcal{X}_\tau, \mathcal{O}_\mathcal{X})$ have
enough injectives, see
Injectives, Theorems \ref{injectives-theorem-sheaves-injectives} and
\ref{injectives-theorem-sheaves-modules-injectives}.
 ========

regex_match $\mathcal{X}$
nn candidate a category
nn a category
new_annotation [(86093, 86106, 'VAR'), (86110, 86120, 'TYPE')]
===== sent Moreover, if
$f : \mathcal{X} \to \mathcal{Y}$ is a $1$-morphism of categories fibred
in groupoids over $(\Sch/S)_{fppf}$, then we obtain the higher direct
images $R^if_*\mathcal{F}$ in $\textit{Ab}(\mathcal{Y}_\tau)$ or
$\textit{Mod}(\mathcal{Y}_\tau, \mathcal{O}_\mathcal{Y})$.
Of course, as explained in
Cohomology on Sites, Section \ref{sites-cohomology-section-derived-functors}
there are also derived versions of $H^p(-)$ and $R^if_*$.

\begin{lemma}
\label{lemma-cohomology-restriction}
Let $S$ be a scheme. ========

regex_match $S$
nn candidate a scheme
nn a scheme
new_annotation [(87434, 87437, 'VAR'), (87441, 87449, 'TYPE')]
===== sent Let $\mathcal{X}$ be a category fibred in groupoids
over $(\Sch/S)_{fppf}$. Let $\tau \in \{Zariski, \etale, smooth,
syntomic, fppf\}$. Let $x \in \Ob(\mathcal{X})$ be an object lying
over the scheme $U$. Let $\mathcal{F}$ be
an object of $\textit{Ab}(\mathcal{X}_\tau)$ or
$\textit{Mod}(\mathcal{X}_\tau, \mathcal{O}_\mathcal{X})$. ========

regex_match $\mathcal{X}$
nn candidate a category
nn a category
new_annotation [(87455, 87468, 'VAR'), (87472, 87482, 'TYPE')]
===== sent Let $\mathcal{X}$ be a category fibred in groupoids
over $(\Sch/S)_{fppf}$. Let $\tau \in \{Zariski, \etale, smooth,
syntomic, fppf\}$. Let $x \in \Ob(\mathcal{X})$ be an object lying
over the scheme $U$. Let $\mathcal{F}$ be
an object of $\textit{Ab}(\mathcal{X}_\tau)$ or
$\textit{Mod}(\mathcal{X}_\tau, \mathcal{O}_\mathcal{X})$. ========

new_annotation []
===== sent Let $\mathcal{X}$ be a category fibred in groupoids
over $(\Sch/S)_{fppf}$. Let $\tau \in \{Zariski, \etale, smooth,
syntomic, fppf\}$. Let $x \in \Ob(\mathcal{X})$ be an object lying
over the scheme $U$. Let $\mathcal{F}$ be
an object of $\textit{Ab}(\mathcal{X}_\tau)$ or
$\textit{Mod}(\mathcal{X}_\tau, \mathcal{O}_\mathcal{X})$. ========

regex_match $x \in \Ob(\mathcal{X})$
nn candidate an object
nn an object
new_annotation [(87591, 87615, 'VAR'), (87619, 87628, 'TYPE')]
===== sent Let $\mathcal{X}$ be a category fibred in groupoids
over $(\Sch/S)_{fppf}$. Let $\tau \in \{Zariski, \etale, smooth,
syntomic, fppf\}$. Let $x \in \Ob(\mathcal{X})$ be an object lying
over the scheme $U$. Let $\mathcal{F}$ be
an object of $\textit{Ab}(\mathcal{X}_\tau)$ or
$\textit{Mod}(\mathcal{X}_\tau, \mathcal{O}_\mathcal{X})$. ========

new_annotation []
===== sent \begin{lemma}
\label{lemma-pushforward-injective}
Let $f : \mathcal{X} \to \mathcal{Y}$ be a $1$-morphism of categories
fibred in groupoids over $(\Sch/S)_{fppf}$. Let
$\tau \in \{Zar, \etale, smooth, syntomic, fppf\}$.
\begin{enumerate}
\item $f_*\mathcal{I}$ is injective in $\textit{Ab}(\mathcal{Y}_\tau)$
for $\mathcal{I}$ injective in $\textit{Ab}(\mathcal{X}_\tau)$, and
\item $f_*\mathcal{I}$ is injective in
$\textit{Mod}(\mathcal{Y}_\tau, \mathcal{O}_\mathcal{Y})$
for $\mathcal{I}$ injective in
$\textit{Mod}(\mathcal{X}_\tau, \mathcal{O}_\mathcal{X})$.
\end{enumerate}
\end{lemma}

\begin{proof}
This follows formally from the fact that $f^{-1}$ is an exact
left adjoint of $f_*$, see
Homology, Lemma \ref{homology-lemma-adjoint-preserve-injectives}.
\end{proof}

\noindent
In the rest of this section we prove that pullback $f^{-1}$ has a left
adjoint $f_!$ on abelian sheaves and modules. ========

regex_match $f : \mathcal{X} \to \mathcal{Y}$
nn candidate \mathcal{X
nn candidate categories
nn categories
new_annotation [(88544, 88577, 'VAR'), (88599, 88609, 'TYPE')]
===== sent \begin{lemma}
\label{lemma-fibre-products}
Let $p : \mathcal{X} \to (\Sch/S)_{fppf}$
be a category fibred in groupoids.
 ========

new_annotation []
===== sent \end{enumerate}
\end{lemma}

\begin{proof}
Part (1) follows
Categories, Lemma \ref{categories-lemma-fibred-groupoids-fibre-product-goes-up}
as $(\Sch/S)_{fppf}$ has fibre products.

\medskip\noindent
Let $a, b : x \to y$ be morphisms of $\mathcal{X}$.
Set $U = p(x)$ and $V = p(y)$. The category of schemes has equalizers
hence we can let $W \to U$ be the equalizer of $p(a)$ and $p(b)$.
Denote $c : z \to x$ a morphism of $\mathcal{X}$ lying over $W \to U$.
The equalizer of $a$ and $b$, if it exists, is the equalizer of $a \circ c$
and $b \circ c$. ========

new_annotation []
===== sent \end{enumerate}
\end{lemma}

\begin{proof}
Part (1) follows
Categories, Lemma \ref{categories-lemma-fibred-groupoids-fibre-product-goes-up}
as $(\Sch/S)_{fppf}$ has fibre products.

\medskip\noindent
Let $a, b : x \to y$ be morphisms of $\mathcal{X}$.
Set $U = p(x)$ and $V = p(y)$. The category of schemes has equalizers
hence we can let $W \to U$ be the equalizer of $p(a)$ and $p(b)$.
Denote $c : z \to x$ a morphism of $\mathcal{X}$ lying over $W \to U$.
The equalizer of $a$ and $b$, if it exists, is the equalizer of $a \circ c$
and $b \circ c$. ========

regex_match $W \to U$
nn candidate $
nn candidate the equalizer
nn the equalizer
new_annotation [(90471, 90480, 'VAR'), (90484, 90497, 'TYPE')]
===== sent If $\mathcal{X} = [U/R]$ for some groupoid in algebraic spaces
$(U, R, s, t, c)$ over $S$, then the hypothesis of (2) holds by
Bootstrap, Lemma \ref{bootstrap-lemma-quotient-stack-isom}.
If $\mathcal{X}$ is an algebraic stack, then we can choose a
presentation $[U/R] \cong \mathcal{X}$ by
Algebraic Stacks, Lemma \ref{algebraic-lemma-stack-presentation}.
\end{proof}

\begin{lemma}
\label{lemma-fibre-products-morphism}
Let $f : \mathcal{X} \to \mathcal{Y}$ be a $1$-morphism of categories
fibred in groupoids over $(\Sch/S)_{fppf}$.
\begin{enumerate}
\item The functor $f$ transforms fibre products into fibre products.
 ========

regex_match $f : \mathcal{X} \to \mathcal{Y}$
nn candidate \mathcal{X
nn candidate categories
nn categories
new_annotation [(92095, 92128, 'VAR'), (92150, 92160, 'TYPE')]
===== sent \medskip\noindent
Let $x \to x'$ be the equalizer of two morphisms $a, b : x' \to x''$
in $\mathcal{X}$. We will show that $f(x) \to f(x')$ is the equalizer
of $f(a)$ and $f(b)$. Let $y \to f(x)$ be a morphism of $\mathcal{Y}$
equalizing $f(a)$ and $f(b)$. Say $x, x', x''$ lie over the schemes
$U, U', U''$ and $y$ lies over $V$. Denote $h : V \to U'$ the image
of $y \to f(x)$ in the category of schemes. ========

regex_match $x \to x'$
nn candidate the equalizer
nn the equalizer
new_annotation [(92692, 92702, 'VAR'), (92706, 92719, 'TYPE')]
===== sent \medskip\noindent
Let $x \to x'$ be the equalizer of two morphisms $a, b : x' \to x''$
in $\mathcal{X}$. We will show that $f(x) \to f(x')$ is the equalizer
of $f(a)$ and $f(b)$. Let $y \to f(x)$ be a morphism of $\mathcal{Y}$
equalizing $f(a)$ and $f(b)$. Say $x, x', x''$ lie over the schemes
$U, U', U''$ and $y$ lies over $V$. Denote $h : V \to U'$ the image
of $y \to f(x)$ in the category of schemes. ========

regex_match $y \to f(x)$
nn candidate $y
nn candidate a morphism
nn a morphism
new_annotation [(92853, 92865, 'VAR'), (92869, 92879, 'TYPE')]
===== sent Hence, as $f$ is faithful, we see that
$h^*x' \to x'$ equalizes $a$ and $b$. Thus we obtain a unique morphism
$h^*x' \to x$ whose image $y = f(h^*x') \to f(x)$ is the desired morphism
in $\mathcal{Y}$.
\end{proof}

\begin{lemma}
\label{lemma-fibre-products-preserve-properties}
Let $f : \mathcal{X} \to \mathcal{Y}$, $g : \mathcal{Z} \to \mathcal{Y}$
be faithful $1$-morphisms of categories
fibred in groupoids over $(\Sch/S)_{fppf}$.
\begin{enumerate}
\item the functor $\mathcal{X} \times_\mathcal{Y} \mathcal{Z} \to \mathcal{Y}$
is faithful, and
\item if $\mathcal{X}, \mathcal{Z}$ have equalizers, so does
$\mathcal{X} \times_\mathcal{Y} \mathcal{Z}$.
\end{enumerate}
\end{lemma}

\begin{proof}
We think of objects in $\mathcal{X} \times_\mathcal{Y} \mathcal{Z}$ as
quadruples $(U, x, z, \alpha)$ where $\alpha : f(x) \to g(z)$ is an
isomorphism over $U$, see
Categories, Lemma \ref{categories-lemma-2-product-categories-over-C}.
 ========

new_annotation []
===== sent \begin{lemma}
\label{lemma-pullback-injective}
Let $f : \mathcal{X} \to \mathcal{Y}$ be a $1$-morphism of categories
fibred in groupoids over $(\Sch/S)_{fppf}$. Let
$\tau \in \{Zar, \etale, smooth, syntomic, fppf\}$.
The functor
$f^{-1} : \textit{Ab}(\mathcal{Y}_\tau) \to \textit{Ab}(\mathcal{X}_\tau)$
has a left adjoint
$f_! : \textit{Ab}(\mathcal{X}_\tau) \to \textit{Ab}(\mathcal{Y}_\tau)$.
If $f$ is faithful and $\mathcal{X}$ has equalizers, then
\begin{enumerate}
\item $f_!$ is exact, and
\item $f^{-1}\mathcal{I}$ is injective in $\textit{Ab}(\mathcal{X}_\tau)$
for $\mathcal{I}$ injective in $\textit{Ab}(\mathcal{Y}_\tau)$.
\end{enumerate}
\end{lemma}

\begin{proof}
By
Stacks, Lemma \ref{stacks-lemma-topology-inherited-functorial}
the functor $f$ is continuous and cocontinuous. ========

regex_match $f : \mathcal{X} \to \mathcal{Y}$
nn candidate \mathcal{X
nn candidate categories
nn categories
new_annotation [(95928, 95961, 'VAR'), (95983, 95993, 'TYPE')]
===== sent Part (2) follows from this formally, see
Homology, Lemma \ref{homology-lemma-adjoint-preserve-injectives}.
\end{proof}

\begin{lemma}
\label{lemma-pullback-injective-modules}
Let $f : \mathcal{X} \to \mathcal{Y}$ be a $1$-morphism of categories
fibred in groupoids over ========

regex_match $f : \mathcal{X} \to \mathcal{Y}$
nn candidate \mathcal{X
nn candidate categories
nn categories
new_annotation [(97341, 97374, 'VAR'), (97396, 97406, 'TYPE')]
===== sent If $f$ is faithful and $\mathcal{X}$ has equalizers, then
\begin{enumerate}
\item $f_!$ is exact, and
\item $f^{-1}\mathcal{I}$ is injective in
$\textit{Mod}(\mathcal{X}_\tau, \mathcal{O}_\mathcal{X})$
for $\mathcal{I}$ injective in
$\textit{Mod}(\mathcal{Y}_\tau, \mathcal{O}_\mathcal{X})$.
\end{enumerate}
\end{lemma}

\begin{proof}
Recall that $f$ is a continuous and cocontinuous functor of sites
and that $f^{-1}\mathcal{O}_\mathcal{Y} = \mathcal{O}_\mathcal{X}$. Hence
Modules on Sites, Lemma \ref{sites-modules-lemma-lower-shriek-modules}
implies $f^*$ has a left adjoint $f_!^{Mod}$.
Let $x$ be an object of $\mathcal{X}$ lying over the scheme $U$.
 ========

regex_match $x$
nn candidate an object
nn an object
new_annotation [(98489, 98492, 'VAR'), (98496, 98505, 'TYPE')]
===== sent \medskip\noindent
Let $\mathcal{U} \to \mathcal{X}$ be as above and let $\mathcal{F}$
be an abelian presheaf on $\mathcal{X}$.
The {\it {\v C}ech complex} associated to the situation is denoted
$\check{\mathcal{C}}^\bullet(\mathcal{U} \to \mathcal{X}, \mathcal{F})$.
It is the cochain complex associated to the cosimplicial abelian group
above, see
Simplicial, Section \ref{simplicial-section-dold-kan-cosimplicial}.
It has terms
$$
\check{\mathcal{C}}^n(\mathcal{U} \to \mathcal{X}, \mathcal{F})
= \Gamma(\mathcal{U}_n, f_n^{-1}\mathcal{F}).
 ========

new_annotation []
===== sent \medskip\noindent
Let $\mathcal{U} \to \mathcal{X}$ be as above and let $\mathcal{F}$
be an abelian presheaf on $\mathcal{X}$.
The {\it {\v C}ech complex} associated to the situation is denoted
$\check{\mathcal{C}}^\bullet(\mathcal{U} \to \mathcal{X}, \mathcal{F})$.
It is the cochain complex associated to the cosimplicial abelian group
above, see
Simplicial, Section \ref{simplicial-section-dold-kan-cosimplicial}.
It has terms
$$
\check{\mathcal{C}}^n(\mathcal{U} \to \mathcal{X}, \mathcal{F})
= \Gamma(\mathcal{U}_n, f_n^{-1}\mathcal{F}).
 ========

new_annotation []
===== sent In the situation of (1) let $t : f \circ h \to e \circ g$ be a $2$-morphism.
 ========

regex_match $t : f \circ h \to e \circ g$
nn candidate let $t
new_annotation []
===== sent \end{proof}









\section{The relative {\v C}ech complex}
\label{section-sheaf-cech-complex}

\noindent
Let $f : \mathcal{U} \to \mathcal{X}$ be a $1$-morphism of categories
fibred in groupoids over $(\Sch/S)_{fppf}$ as in
(\ref{equation-covering}). ========

regex_match $f : \mathcal{U} \to \mathcal{X}$
nn candidate \mathcal{U
nn candidate categories
nn categories
new_annotation [(106459, 106492, 'VAR'), (106514, 106524, 'TYPE')]
===== sent \mathcal{Y} \ar[r]^e & \mathcal{X}
}
$$
be a $2$-fibre product of categories fibred in groupoids over
$(\Sch/S)_{fppf}$ and let $\mathcal{F}$ be an abelian presheaf
on $\mathcal{X}$. ========

regex_match $\mathcal{F}$
nn candidate $\mathcal{F}$
nn candidate an abelian presheaf
nn an abelian presheaf
new_annotation [(111826, 111839, 'VAR'), (111843, 111862, 'TYPE')]
===== sent \end{lemma}

\begin{proof}
Let $y$ be an object of $\mathcal{Y}$ lying over the scheme $T$.
Set $x = e(y)$. We are going to show that the map induces an isomorphism
on sections over $y$. Note that
$$
\Gamma(y, e^{-1}\mathcal{K}^\bullet(f, \mathcal{F})) =
\Gamma(x, \mathcal{K}^\bullet(f, \mathcal{F})) =
\check{\mathcal{C}}^\bullet(
(\Sch/T)_{fppf} \times_{x, \mathcal{X}} \mathcal{U} \to
(\Sch/T)_{fppf}, x^{-1}\mathcal{F})
$$
by
Remark \ref{remark-cech-complex-sections}. On the other hand,
$$
\Gamma(y, \mathcal{K}^\bullet(g, e^{-1}\mathcal{F})) =
\check{\mathcal{C}}^\bullet(
(\Sch/T)_{fppf} \times_{y, \mathcal{Y}} \mathcal{V} \to
(\Sch/T)_{fppf}, y^{-1}e^{-1}\mathcal{F})
$$
also by
Remark \ref{remark-cech-complex-sections}.
Note that $y^{-1}e^{-1}\mathcal{F} = x^{-1}\mathcal{F}$
and since the diagram is $2$-cartesian the $1$-morphism
$$
(\Sch/T)_{fppf} \times_{y, \mathcal{Y}} \mathcal{V} \to
(\Sch/T)_{fppf} \times_{x, \mathcal{X}} \mathcal{U}
$$
is an equivalence. ========

regex_match $y$
nn candidate an object
nn an object
new_annotation [(112114, 112117, 'VAR'), (112121, 112130, 'TYPE')]
===== sent \begin{lemma}
\label{lemma-check-exactness-covering}
Let $f : \mathcal{U} \to \mathcal{X}$ be a $1$-morphism of categories fibred
in groupoids over $(\Sch/S)_{fppf}$. Let
$\tau \in \{Zar, \etale, smooth, syntomic, fppf\}$.
Let
$$
\mathcal{F} \to \mathcal{G} \to \mathcal{H}
$$
be a complex in $\textit{Ab}(\mathcal{X}_\tau)$. Assume that
\begin{enumerate}
\item for every object $x$ of $\mathcal{X}$ there exists a covering
$\{x_i \to x\}$ in $\mathcal{X}_\tau$ such that each $x_i$ is isomorphic
to $f(u_i)$ for some object $u_i$ of $\mathcal{U}$, and
\item $f^{-1}\mathcal{F} \to f^{-1}\mathcal{G} ========

regex_match $f : \mathcal{U} \to \mathcal{X}$
nn candidate \mathcal{U
nn candidate categories
nn categories
new_annotation [(113273, 113306, 'VAR'), (113328, 113338, 'TYPE')]
===== sent \end{lemma}

\begin{proof}
Let $x$ be an object of $\mathcal{X}$ lying over the scheme $T$.
Consider the sequence
$x^{-1}\mathcal{F} \to x^{-1}\mathcal{G} \to x^{-1}\mathcal{H}$
of abelian sheaves on $(\Sch/T)_\tau$. It suffices to show
this sequence is exact. ========

regex_match $x$
nn candidate an object
nn an object
new_annotation [(113970, 113973, 'VAR'), (113977, 113986, 'TYPE')]
===== sent \end{proof}

\begin{proposition}
\label{proposition-exactness-cech-complex}
Let $f : \mathcal{U} \to \mathcal{X}$ be a $1$-morphism of categories fibred
in groupoids over $(\Sch/S)_{fppf}$. Let
$\tau \in \{Zar, \etale, smooth, syntomic, fppf\}$.
If
\begin{enumerate}
\item $\mathcal{F}$ is an abelian sheaf on $\mathcal{X}_\tau$, and
\item for every object $x$ of $\mathcal{X}$ there exists a covering
$\{x_i \to x\}$ in $\mathcal{X}_\tau$ such that each $x_i$ is isomorphic
to $f(u_i)$ for some object $u_i$ of $\mathcal{U}$,
\end{enumerate}
then the extended relative {\v C}ech complex
$$
\ldots \to 0 \to
\mathcal{F} \to
f_{0, *}f_0^{-1}\mathcal{F} \to
f_{1, *}f_1^{-1}\mathcal{F} \to
f_{2, *}f_2^{-1}\mathcal{F} \to \ldots
$$
is exact in $\textit{Ab}(\mathcal{X}_\tau)$.
\end{proposition}

\begin{proof}
By
Lemma \ref{lemma-check-exactness-covering}
it suffices to check exactness after pulling back to $\mathcal{U}$.
By
Lemma \ref{lemma-base-change-cech-complex}
the pullback of the extended relative {\v C}ech complex is isomorphic
to the extend relative {\v C}ech complex for the morphism
$\mathcal{U} \times_\mathcal{X} \mathcal{U} \to \mathcal{U}$
and an abelian sheaf on $\mathcal{U}_\tau$. Since there is a section
$\Delta_{\mathcal{U}/\mathcal{X}} : \mathcal{U} \to
\mathcal{U} \times_\mathcal{X} \mathcal{U}$ exactness follows from
Lemma \ref{lemma-homotopy-sheafified}.
\end{proof}

\noindent
Using this we can construct the {\v C}ech-to-cohomology spectral sequence
as follows. ========

regex_match $f : \mathcal{U} \to \mathcal{X}$
nn candidate \mathcal{U
nn candidate categories
nn categories
new_annotation [(114866, 114899, 'VAR'), (114921, 114931, 'TYPE')]
===== sent \begin{lemma}
\label{lemma-cech-to-cohomology}
Let $f : \mathcal{U} \to \mathcal{X}$ be a $1$-morphism of categories fibred
in groupoids over $(\Sch/S)_{fppf}$. Let
$\tau \in \{Zar, \etale, smooth, syntomic, fppf\}$.
Assume
\begin{enumerate}
\item $\mathcal{F}$ is an abelian sheaf on $\mathcal{X}_\tau$,
\item for every object $x$ of $\mathcal{X}$ there exists a covering
$\{x_i \to x\}$ in $\mathcal{X}_\tau$ such that each $x_i$ is isomorphic
to $f(u_i)$ for some object $u_i$ of $\mathcal{U}$,
\item the category $\mathcal{U}$ has equalizers, and
\item the functor $f$ is faithful.
 ========

regex_match $f : \mathcal{U} \to \mathcal{X}$
nn candidate \mathcal{U
nn candidate categories
nn categories
new_annotation [(116452, 116485, 'VAR'), (116507, 116517, 'TYPE')]
===== sent Let $\mathcal{I}$ be an injective object
of $\textit{Ab}(\mathcal{X}_\tau)$. By
Lemma \ref{lemma-pullback-injective}
we see $f_p^{-1}\mathcal{I}$ is an injective object of
$\textit{Ab}((\mathcal{U}_p)_\tau)$.
Hence $f_{p, *}f_p^{-1}\mathcal{I}$ is an injective object of
$\textit{Ab}(\mathcal{X}_\tau)$ by
Lemma \ref{lemma-pushforward-injective}.
Hence
Proposition \ref{proposition-exactness-cech-complex}
shows that the extended relative {\v C}ech complex
$$
\ldots \to 0 \to
\mathcal{I} \to
f_{0, *}f_0^{-1}\mathcal{I} \to
f_{1, *}f_1^{-1}\mathcal{I} \to
f_{2, *}f_2^{-1}\mathcal{I} \to \ldots
$$
is an exact complex in $\textit{Ab}(\mathcal{X}_\tau)$ all of whose
terms are injective. ========

regex_match $\mathcal{I}$
nn candidate an injective object
nn an injective object
new_annotation [(117561, 117574, 'VAR'), (117578, 117597, 'TYPE')]
===== sent \begin{lemma}
\label{lemma-cech-to-cohomology-modules}
Let $f : \mathcal{U} \to \mathcal{X}$ be a $1$-morphism of categories fibred
in groupoids over $(\Sch/S)_{fppf}$. Let
$\tau \in \{Zar, \etale, smooth, syntomic, fppf\}$.
Assume
\begin{enumerate}
\item $\mathcal{F}$ is an object of
$\textit{Mod}(\mathcal{X}_\tau, \mathcal{O}_\mathcal{X})$,
\item for every object $x$ of $\mathcal{X}$ there exists a covering
$\{x_i \to x\}$ in $\mathcal{X}_\tau$ such that each $x_i$ is isomorphic
to $f(u_i)$ for some object $u_i$ of $\mathcal{U}$,
\item the category $\mathcal{U}$ has equalizers, and
\item the functor $f$ is faithful.
 ========

regex_match $f : \mathcal{U} \to \mathcal{X}$
nn candidate \mathcal{U
nn candidate categories
nn categories
new_annotation [(119414, 119447, 'VAR'), (119469, 119479, 'TYPE')]
===== sent \begin{lemma}
\label{lemma-surjective-flat-locally-finite-presentation}
Let $f : \mathcal{X} \to \mathcal{Y}$ be a $1$-morphism of
categories fibred in groupoids over $(\Sch/S)_{fppf}$.
\begin{enumerate}
\item Assume that $f$ is representable by algebraic spaces, surjective,
flat, and locally of finite presentation. ========

regex_match $f : \mathcal{X} \to \mathcal{Y}$
nn candidate \mathcal{X
nn candidate categories
nn categories
new_annotation [(120820, 120853, 'VAR'), (120875, 120885, 'TYPE')]
===== sent By
More on Morphisms, Lemma \ref{more-morphisms-lemma-etale-dominates-smooth}
there exists an \'etale covering $\{V_i \to V\}$ such that
each $V_i \to V$ factors through $U$. Denote $x_i$ the object of
$\mathcal{X}$ over $V_i$ corresponding to the $1$-morphism
$$
(\Sch/V_i)_{fppf} \to (\Sch/U)_{fppf} \to \mathcal{X}.
$$
Then $\{f(x_i) \to y\}$ is
the desired \'etale covering of $\mathcal{Y}$.
\end{proof}

\begin{lemma}
\label{lemma-cech-to-cohomology-relative}
Let $f : \mathcal{U} \to \mathcal{X}$ and
$g : \mathcal{X} \to \mathcal{Y}$
be composable $1$-morphisms of categories fibred
in groupoids over $(\Sch/S)_{fppf}$. Let
$\tau \in \{Zar, \etale, smooth, syntomic, \linebreak[0] fppf\}$.
Assume
\begin{enumerate}
\item $\mathcal{F}$ is an abelian sheaf on $\mathcal{X}_\tau$,
\item for every object $x$ of $\mathcal{X}$ there exists a covering
$\{x_i \to x\}$ in $\mathcal{X}_\tau$ such that each $x_i$ is isomorphic
to $f(u_i)$ for some object $u_i$ of $\mathcal{U}$,
\item the category $\mathcal{U}$ has equalizers, and
\item the functor $f$ is faithful.
 ========

new_annotation []
===== sent \end{proof}

\begin{lemma}
\label{lemma-cech-to-cohomology-relative-modules}
Let $f : \mathcal{U} \to \mathcal{X}$ and
$g : \mathcal{X} \to \mathcal{Y}$
be composable $1$-morphisms of categories fibred
in groupoids over $(\Sch/S)_{fppf}$. Let
$\tau \in \{Zar, \etale, smooth, syntomic, \linebreak[0] fppf\}$.
Assume
\begin{enumerate}
\item $\mathcal{F}$ is an object of
$\textit{Mod}(\mathcal{X}_\tau, \mathcal{O}_\mathcal{X})$,
\item for every object $x$ of $\mathcal{X}$ there exists a covering
$\{x_i \to x\}$ in $\mathcal{X}_\tau$ such that each $x_i$ is isomorphic
to $f(u_i)$ for some object $u_i$ of $\mathcal{U}$,
\item the category $\mathcal{U}$ has equalizers, and
\item the functor $f$ is faithful.
 ========

new_annotation []
===== sent Let $\mathcal{X}$ be an algebraic stack over $S$. In the sections above
we have seen how to define sheaves for the \'etale, ..., fppf
topologies on $\mathcal{X}$. ========

regex_match $\mathcal{X}$
nn candidate an algebraic stack
nn an algebraic stack
new_annotation [(127286, 127299, 'VAR'), (127303, 127321, 'TYPE')]
===== sent \begin{proposition}
\label{proposition-smooth-covering-compute-cohomology}
Let $f : \mathcal{U} \to \mathcal{X}$ be a $1$-morphism of algebraic stacks.
 ========

regex_match $f : \mathcal{U} \to \mathcal{X}$
nn candidate \mathcal{U
nn candidate algebraic stacks
nn algebraic stacks
new_annotation [(129852, 129885, 'VAR'), (129907, 129923, 'TYPE')]
===== sent \begin{enumerate}
\item Let $\mathcal{F}$ be an abelian \'etale sheaf on $\mathcal{X}$.
Assume that $f$ is representable by algebraic spaces, surjective, and smooth.
 ========

regex_match $\mathcal{F}$
nn candidate an abelian \'etale sheaf
nn an abelian \'etale sheaf
new_annotation [(129953, 129966, 'VAR'), (129970, 129994, 'TYPE')]
===== sent Let $\mathcal{F}$ be an abelian sheaf on $\mathcal{X}$.
Assume that $f$ is representable by algebraic spaces, surjective, flat,
and locally of finite presentation. ========

regex_match $\mathcal{F}$
nn candidate an abelian sheaf
nn an abelian sheaf
new_annotation [(130256, 130269, 'VAR'), (130273, 130289, 'TYPE')]
===== sent \end{proof}










\section{Higher direct images and algebraic stacks}
\label{section-higher-direct-images}

\noindent
Let $g : \mathcal{X} \to \mathcal{Y}$ be a $1$-morphism of algebraic stacks
over $S$. ========

regex_match $g : \mathcal{X} \to \mathcal{Y}$
nn candidate $g
nn candidate algebraic stacks
nn algebraic stacks
new_annotation [(131219, 131252, 'VAR'), (131274, 131290, 'TYPE')]
===== sent \begin{proposition}
\label{proposition-smooth-covering-compute-direct-image}
Let $f : \mathcal{U} \to \mathcal{X}$ and $g : \mathcal{X} \to \mathcal{Y}$
be composable $1$-morphisms of algebraic stacks.
 ========

new_annotation []
===== sent \begin{lemma}
\label{lemma-pushforward-restriction}
Let $S$ be a scheme. ========

regex_match $S$
nn candidate a scheme
nn a scheme
new_annotation [(135068, 135071, 'VAR'), (135075, 135083, 'TYPE')]
===== sent Let $f : \mathcal{X} \to \mathcal{Y}$ be a
$1$-morphism of algebraic stacks\footnote{This result should hold
for any $1$-morphism of categories fibred in groupoids over
$(\Sch/S)_{fppf}$.} over $S$.
Let $\tau \in \{Zariski,\linebreak[0] \etale,\linebreak[0]
smooth,\linebreak[0] syntomic,\linebreak[0] fppf\}$.
Let $\mathcal{F}$ be
an object of $\textit{Ab}(\mathcal{X}_\tau)$ or
$\textit{Mod}(\mathcal{X}_\tau, \mathcal{O}_\mathcal{X})$.
 ========

new_annotation []
===== sent Let $f : \mathcal{X} \to \mathcal{Y}$ be a
$1$-morphism of algebraic stacks\footnote{This result should hold
for any $1$-morphism of categories fibred in groupoids over
$(\Sch/S)_{fppf}$.} over $S$.
Let $\tau \in \{Zariski,\linebreak[0] \etale,\linebreak[0]
smooth,\linebreak[0] syntomic,\linebreak[0] fppf\}$.
Let $\mathcal{F}$ be
an object of $\textit{Ab}(\mathcal{X}_\tau)$ or
$\textit{Mod}(\mathcal{X}_\tau, \mathcal{O}_\mathcal{X})$.
 ========

new_annotation []
===== sent Let $f : \mathcal{X} \to \mathcal{Y}$ be a
$1$-morphism of algebraic stacks\footnote{This result should hold
for any $1$-morphism of categories fibred in groupoids over
$(\Sch/S)_{fppf}$.} over $S$.
Let $\tau \in \{Zariski,\linebreak[0] \etale,\linebreak[0]
smooth,\linebreak[0] syntomic,\linebreak[0] fppf\}$.
Let $\mathcal{F}$ be
an object of $\textit{Ab}(\mathcal{X}_\tau)$ or
$\textit{Mod}(\mathcal{X}_\tau, \mathcal{O}_\mathcal{X})$.
 ========

new_annotation []
===== sent \begin{lemma}
\label{lemma-base-change-higher-direct-images}
Let $S$ be a scheme. ========

regex_match $S$
nn candidate a scheme
nn a scheme
new_annotation [(137108, 137111, 'VAR'), (137115, 137123, 'TYPE')]
===== sent \begin{lemma}
\label{lemma-compare-injectives}
Let $S$ be a scheme. ========

regex_match $S$
nn candidate a scheme
nn a scheme
new_annotation [(138745, 138748, 'VAR'), (138752, 138760, 'TYPE')]
===== sent Let $\mathcal{X}$ be an algebraic stack over $S$
representable by the algebraic space $F$.
\begin{enumerate}
\item If $\mathcal{I}$ injective in $\textit{Ab}(\mathcal{X}_\etale)$, then
$\mathcal{I}|_{F_\etale}$ is injective in $\textit{Ab}(F_\etale)$,
\item If $\mathcal{I}^\bullet$ is a K-injective complex in
$\textit{Ab}(\mathcal{X}_\etale)$, then $\mathcal{I}^\bullet|_{F_\etale}$
is a K-injective complex in $\textit{Ab}(F_\etale)$.
\end{enumerate}
The same does not hold for modules.
 ========

regex_match $\mathcal{X}$
nn candidate an algebraic stack
nn an algebraic stack
new_annotation [(138766, 138779, 'VAR'), (138783, 138801, 'TYPE')]
===== sent \end{lemma}

\begin{proof}
This follows formally from the fact that the restriction functor
$\pi_{F, *} = i_F^{-1}$ (see Lemma \ref{lemma-compare})
is right adjoint to the exact functor $\pi_F^{-1}$, see
Homology, Lemma \ref{homology-lemma-adjoint-preserve-injectives} and
Derived Categories, Lemma \ref{derived-lemma-adjoint-preserve-K-injectives}.
To see that the lemma does not hold for modules, we refer the
reader to \'Etale Cohomology, Lemma
\ref{etale-cohomology-lemma-compare-injectives}.
\end{proof}

\begin{lemma}
\label{lemma-compare-morphism-cohomology}
Let $S$ be a scheme. ========

regex_match $S$
nn candidate a scheme
nn a scheme
new_annotation [(139822, 139825, 'VAR'), (139829, 139837, 'TYPE')]
===== sent Let $f : \mathcal{X} \to \mathcal{Y}$ be a morphism
of algebraic stacks over $S$. Assume $\mathcal{X}$, $\mathcal{Y}$ ========

regex_match $f : \mathcal{X} \to \mathcal{Y}$
nn candidate $
nn candidate a morphism
nn a morphism
new_annotation [(139843, 139876, 'VAR'), (139880, 139890, 'TYPE')]
===== sent \end{proof}

\begin{lemma}
\label{lemma-compare-representable-morphism-cohomology}
Let $S$ be a scheme. ========

regex_match $S$
nn candidate a scheme
nn a scheme
new_annotation [(141185, 141188, 'VAR'), (141192, 141200, 'TYPE')]
===== sent \begin{lemma}
\label{lemma-lqc-flat-base-change-fppf-sheaf}
Let $S$ be a scheme. ========

regex_match $S$
nn candidate a scheme
nn a scheme
new_annotation [(142424, 142427, 'VAR'), (142431, 142439, 'TYPE')]
===== sent Let $\mathcal{X}$ be an algebraic stack over $S$.
Let $\mathcal{F}$ be a presheaf of $\mathcal{O}_\mathcal{X}$-modules.
 ========

regex_match $\mathcal{X}$
nn candidate an algebraic stack
nn an algebraic stack
new_annotation [(142445, 142458, 'VAR'), (142462, 142480, 'TYPE')]
===== sent Let $\mathcal{X}$ be an algebraic stack over $S$.
Let $\mathcal{F}$ be a presheaf of $\mathcal{O}_\mathcal{X}$-modules.
 ========

regex_match $\mathcal{F}$
nn candidate a presheaf
nn a presheaf
new_annotation [(142495, 142508, 'VAR'), (142512, 142522, 'TYPE')]
===== sent \end{lemma}

\begin{proof}
Let $\{x_i \to x\}$ be an fppf covering of $\mathcal{X}$ lying over the
fppf covering $\{f_i : U_i \to U\}$ of schemes over $S$.
By assumption the restriction $\mathcal{G} = \mathcal{F}|_{U_\etale}$
is quasi-coherent and the comparison maps
$f_{i, small}^*\mathcal{G} \to \mathcal{F}|_{U_{i, \etale}}$
are isomorphisms. ========

regex_match $\{x_i \to x\}$
nn candidate an fppf
nn an fppf
new_annotation [(143062, 143077, 'VAR'), (143081, 143088, 'TYPE')]
===== sent Hence the sheaf condition for $\mathcal{F}$
and the covering $\{x_i \to x\}$ is equivalent to the sheaf condition
for $\mathcal{G}^a$ on $(\Sch/U)_{fppf}$ and the covering $\{U_i \to U\}$
which holds by
Descent, Lemma \ref{descent-lemma-sheaf-condition-holds}.
\end{proof}

\begin{lemma}
\label{lemma-compare-fppf-etale}
Let $S$ be a scheme. ========

regex_match $S$
nn candidate a scheme
nn a scheme
new_annotation [(143703, 143706, 'VAR'), (143710, 143718, 'TYPE')]
===== sent Let $\mathcal{X}$ be an algebraic stack over $S$.
Let $\mathcal{F}$ be a presheaf $\mathcal{O}_\mathcal{X}$-module such that
\begin{enumerate}
\item[(a)] $\mathcal{F}$ is locally quasi-coherent, and
\item[(b)] for any morphism $\varphi : x \to y$ of $\mathcal{X}$ which lies
over a morphism of schemes $f : U \to V$ which is flat and
locally of finite presentation, the comparison map
$c_\varphi : f_{small}^*\mathcal{F}|_{V_\etale} \to
\mathcal{F}|_{U_\etale}$ of
(\ref{equation-comparison-modules}) is an isomorphism.
 ========

regex_match $\mathcal{X}$
nn candidate an algebraic stack
nn an algebraic stack
new_annotation [(143724, 143737, 'VAR'), (143741, 143759, 'TYPE')]
===== sent Let $\mathcal{X}$ be an algebraic stack over $S$.
Let $\mathcal{F}$ be a presheaf $\mathcal{O}_\mathcal{X}$-module such that
\begin{enumerate}
\item[(a)] $\mathcal{F}$ is locally quasi-coherent, and
\item[(b)] for any morphism $\varphi : x \to y$ of $\mathcal{X}$ which lies
over a morphism of schemes $f : U \to V$ which is flat and
locally of finite presentation, the comparison map
$c_\varphi : f_{small}^*\mathcal{F}|_{V_\etale} \to
\mathcal{F}|_{U_\etale}$ of
(\ref{equation-comparison-modules}) is an isomorphism.
 ========

regex_match $\mathcal{F}$
nn candidate a presheaf
nn a presheaf
new_annotation [(143774, 143787, 'VAR'), (143791, 143801, 'TYPE')]
===== sent \begin{lemma}
\label{lemma-cohomology-on-subcategory}
Let $S$ be a scheme. ========

regex_match $S$
nn candidate a scheme
nn a scheme
new_annotation [(147424, 147427, 'VAR'), (147431, 147439, 'TYPE')]
===== sent Let $\mathcal{X}$ be an algebraic stack over $S$.
Let $\tau = \etale$ (resp.\ $\tau = fppf$). Let
$\mathcal{X}' \subset \mathcal{X}$ be a full subcategory with the
following properties
\begin{enumerate}
\item if $x \to x'$ is a morphism of $\mathcal{X}$ which lies over a
smooth (resp.\ flat and locally finitely presented) morphism of
schemes and $x' \in \Ob(\mathcal{X}')$, then $x \in \Ob(\mathcal{X}')$, and
\item there exists an object $x \in \Ob(\mathcal{X}')$ lying over
a scheme $U$ such that the associated $1$-morphism
$x : (\Sch/U)_{fppf} \to \mathcal{X}$ is smooth and surjective.
 ========

regex_match $\mathcal{X}$
nn candidate an algebraic stack
nn an algebraic stack
new_annotation [(147445, 147458, 'VAR'), (147462, 147480, 'TYPE')]
===== sent Let $\mathcal{X}$ be an algebraic stack over $S$.
Let $\tau = \etale$ (resp.\ $\tau = fppf$). Let
$\mathcal{X}' \subset \mathcal{X}$ be a full subcategory with the
following properties
\begin{enumerate}
\item if $x \to x'$ is a morphism of $\mathcal{X}$ which lies over a
smooth (resp.\ flat and locally finitely presented) morphism of
schemes and $x' \in \Ob(\mathcal{X}')$, then $x \in \Ob(\mathcal{X}')$, and
\item there exists an object $x \in \Ob(\mathcal{X}')$ lying over
a scheme $U$ such that the associated $1$-morphism
$x : (\Sch/U)_{fppf} \to \mathcal{X}$ is smooth and surjective.
 ========

new_annotation []
===== sent Let $h_{x, n}$ be
the $(n + 1)$-fold product $h_x \times \ldots \times h_x$.
 ========

new_annotation []
file:  cotangent.tex
===== sent \section{The cotangent complex of a ring map}
\label{section-cotangent-ring-map}

\noindent
Let $A$ be a ring. ========

regex_match $A$
nn candidate a ring
nn a ring
new_annotation [(2845, 2848, 'VAR'), (2852, 2858, 'TYPE')]
===== sent Let $\textit{Alg}_A$ be the category of $A$-algebras.
 ========

regex_match $\textit{Alg}_A$
nn candidate the category
nn the category
new_annotation [(2864, 2880, 'VAR'), (2884, 2896, 'TYPE')]
===== sent Let $X_\bullet$ be the simplicial object of
$\text{Fun}(\textit{Alg}_A, \textit{Alg}_A)$ constructed in
Simplicial, Section \ref{simplicial-section-standard}.

\medskip\noindent
Consider an $A$-algebra $B$. Denote $P_\bullet = X_\bullet(B)$ the resulting
simplicial $A$-algebra. ========

regex_match $X_\bullet$
nn candidate the simplicial object
nn the simplicial object
new_annotation [(3146, 3157, 'VAR'), (3161, 3182, 'TYPE')]
===== sent \begin{definition}
\label{definition-standard-resolution}
Let $A \to B$ be a ring map. ========

regex_match $A \to B$
nn candidate a ring map
nn a ring map
new_annotation [(3740, 3749, 'VAR'), (3753, 3763, 'TYPE')]
===== sent A} \otimes_{P_n, \epsilon_n} B$ and $L_{B/A}^m = 0$
for $m > 0$.

\begin{remark}
\label{remark-variant-cotangent-complex}
Let $A \to B$ be a ring map. ========

regex_match $A \to B$
nn candidate a ring map
nn a ring map
new_annotation [(4808, 4817, 'VAR'), (4821, 4831, 'TYPE')]
===== sent Let $\mathcal{A}$ be the category of
arrows $\psi : C \to B$ of $A$-algebras and let $\mathcal{S}$ be
the category of maps $E \to B$ where $E$ is a set. ========

regex_match $\mathcal{A}$
nn candidate the category
nn the category
new_annotation [(4837, 4850, 'VAR'), (4854, 4866, 'TYPE')]
===== sent Let $\mathcal{A}$ be the category of
arrows $\psi : C \to B$ of $A$-algebras and let $\mathcal{S}$ be
the category of maps $E \to B$ where $E$ is a set. ========

new_annotation []
===== sent There are adjoint
functors $i : \mathcal{A} \to \mathcal{S}$ (the forgetful functor)
and $F : \mathcal{S} \to \mathcal{A}$ which sends $E \to B$ to
$A[E] \to B$. Let $X_\bullet$ be the simplicial object of
$\text{Fun}(\mathcal{A}, \mathcal{A})$ constructed in
Simplicial, Section \ref{simplicial-section-standard}.
The diagram
$$
\xymatrix{
\mathcal{A} \ar[d] \ar[r] & \mathcal{S} \ar@<1ex>[l] \ar[d] \\
 ========

regex_match $X_\bullet$
nn candidate the simplicial object
nn the simplicial object
new_annotation [(5152, 5163, 'VAR'), (5167, 5188, 'TYPE')]
===== sent \end{remark}

\begin{lemma}
\label{lemma-colimit-cotangent-complex}
Let $A_i \to B_i$ be a system of ring maps over a directed index
set $I$. ========

regex_match $A_i \to B_i$
nn candidate a system
nn a system
new_annotation [(5632, 5645, 'VAR'), (5649, 5657, 'TYPE')]
===== sent \end{proof}





\section{Simplicial resolutions and derived lower shriek}
\label{section-compute-L-pi-shriek}

\noindent
Let $A \to B$ be a ring map. ========

regex_match $A \to B$
nn candidate a ring map
nn a ring map
new_annotation [(6201, 6210, 'VAR'), (6214, 6224, 'TYPE')]
===== sent Let $\mathcal{C} = \mathcal{C}_{B/A}$ denote the {\bf opposite}
of this category. ========

new_annotation []
===== sent With notation as above let $P_\bullet$ be a simplicial $A$-algebra
endowed with an augmentation $\epsilon : P_\bullet \to B$.
Assume each $P_n$ is a polynomial algebra over $A$ and ========

regex_match $P_\bullet$
nn candidate $P_\bullet$
nn candidate an augmentation $\epsilon
nn an augmentation $\epsilon
new_annotation [(8041, 8052, 'VAR'), (8094, 8119, 'TYPE')]
===== sent We see that
$$
S_\bullet = \Mor_{\textit{Sets}}((E, \beta|_E), (P_\bullet, \epsilon))
$$
Let $*$ be the constant simplicial set on a singleton. ========

regex_match $*$
nn candidate a singleton
nn a singleton
new_annotation [(8997, 9000, 'VAR'), (9035, 9046, 'TYPE')]
===== sent For $b \in B$
let $F_{b, \bullet}$ be the simplicial set defined by the cartesian
diagram
$$
\xymatrix{
F_{b, \bullet} \ar[r] \ar[d] & P_\bullet \ar[d]_\epsilon ========

regex_match $F_{b, \bullet}$
nn candidate $F_{b
nn candidate the simplicial set
nn the simplicial set
new_annotation [(9066, 9082, 'VAR'), (9086, 9104, 'TYPE')]
===== sent \begin{lemma}
\label{lemma-pi-shriek-standard}
Let $A \to B$ be a ring map. ========

regex_match $A \to B$
nn candidate a ring map
nn a ring map
new_annotation [(9896, 9905, 'VAR'), (9909, 9919, 'TYPE')]
===== sent Let $\epsilon : P_\bullet \to B$
be the standard resolution of $B$ over $A$. ========

new_annotation []
===== sent Let $\pi$ be as in
(\ref{equation-pi}). ========

new_annotation []
===== sent \end{proof}

\begin{proof}[Second proof]
We will use the criterion of Cohomology on Sites, Lemma
\ref{sites-cohomology-lemma-compute-by-cosimplicial-resolution}.
Let $U = (Q, \beta)$ be an object of $\mathcal{C}$.
We have to show that
$$
S_\bullet = \Mor_\mathcal{C}((Q, \beta), (P_\bullet, \epsilon))
$$
is homotopy equivalent to a singleton. ========

regex_match $U = (Q, \beta)$
nn candidate $U
nn candidate an object
nn an object
new_annotation [(11112, 11128, 'VAR'), (11132, 11141, 'TYPE')]
===== sent \end{proof}

\begin{lemma}
\label{lemma-compute-cotangent-complex}
Let $A \to B$ be a ring map.   ========

regex_match $A \to B$
nn candidate a ring map
nn a ring map
new_annotation [(11875, 11884, 'VAR'), (11888, 11898, 'TYPE')]
===== sent Let $\pi$ and $i$ be as in (\ref{equation-pi}).
 ========

new_annotation []
===== sent \begin{lemma}
\label{lemma-polynomial}
Let $A$ be a Noetherian ring. ========

regex_match $A$
nn candidate a Noetherian ring
nn a Noetherian ring
new_annotation [(15912, 15915, 'VAR'), (15919, 15936, 'TYPE')]
===== sent Let $A \to B$ be a finite type ring map.
 ========

regex_match $A \to B$
nn candidate $
nn candidate a finite type ring map
nn a finite type ring map
new_annotation [(15942, 15951, 'VAR'), (15955, 15977, 'TYPE')]
===== sent Let $\mathcal{A}$ be the category of $A$-algebra maps $C \to B$. Let
$n \geq 0$ and let $P_\bullet$ be a simplicial object of $\mathcal{A}$
such that
\begin{enumerate}
\item $P_\bullet \to B$ is a trivial Kan fibration of simplicial sets,
\item $P_k$ is finite type over $A$ for $k \leq n$,
\item $P_\bullet = \text{cosk}_n \text{sk}_n P_\bullet$ as simplicial
objects of $\mathcal{A}$.
\end{enumerate}
 ========

regex_match $\mathcal{A}$
nn candidate the category
nn the category
new_annotation [(15983, 15996, 'VAR'), (16000, 16012, 'TYPE')]
===== sent Let $\mathcal{A}$ be the category of $A$-algebra maps $C \to B$. Let
$n \geq 0$ and let $P_\bullet$ be a simplicial object of $\mathcal{A}$
such that
\begin{enumerate}
\item $P_\bullet \to B$ is a trivial Kan fibration of simplicial sets,
\item $P_k$ is finite type over $A$ for $k \leq n$,
\item $P_\bullet = \text{cosk}_n \text{sk}_n P_\bullet$ as simplicial
objects of $\mathcal{A}$.
\end{enumerate}
 ========

regex_match $P_\bullet$
nn candidate $P_\bullet$
nn candidate a simplicial object
nn a simplicial object
new_annotation [(16067, 16078, 'VAR'), (16082, 16101, 'TYPE')]
===== sent \medskip\noindent
Let $1 \leq k_0 \leq n$ and assume $Q_{k, m}$ is of finite type
over $A$ for all $1 \leq k \leq k_0$ and $k ========

new_annotation []
===== sent \end{proof}

\begin{proposition}
\label{proposition-polynomial}
Let $A$ be a Noetherian ring. ========

regex_match $A$
nn candidate a Noetherian ring
nn a Noetherian ring
new_annotation [(20652, 20655, 'VAR'), (20659, 20676, 'TYPE')]
===== sent Let $A \to B$ be a finite type ring map.
 ========

regex_match $A \to B$
nn candidate $
nn candidate a finite type ring map
nn a finite type ring map
new_annotation [(20682, 20691, 'VAR'), (20695, 20717, 'TYPE')]
===== sent \end{proposition}

\begin{proof}
Let $\mathcal{A}$ be the category of $A$-algebra maps $C \to B$.
 ========

regex_match $\mathcal{A}$
nn candidate the category
nn the category
new_annotation [(20997, 21010, 'VAR'), (21014, 21026, 'TYPE')]
===== sent \end{proof}

\begin{lemma}
\label{lemma-pi-shriek-finite}
Let $A$ be a Noetherian ring. ========

regex_match $A$
nn candidate a Noetherian ring
nn a Noetherian ring
new_annotation [(24650, 24653, 'VAR'), (24657, 24674, 'TYPE')]
===== sent Let $A \to B$ be a finite type ring map.
 ========

regex_match $A \to B$
nn candidate $
nn candidate a finite type ring map
nn a finite type ring map
new_annotation [(24680, 24689, 'VAR'), (24693, 24715, 'TYPE')]
===== sent Let $\pi$, $\underline{B}$ be as in (\ref{equation-pi}).
 ========

new_annotation []
===== sent \end{proof}

\begin{lemma}
\label{lemma-cotangent-finite}
Let $A$ be a Noetherian ring. ========

regex_match $A$
nn candidate a Noetherian ring
nn a Noetherian ring
new_annotation [(25312, 25315, 'VAR'), (25319, 25336, 'TYPE')]
===== sent Let $A \to B$ be a finite type ring map.
 ========

regex_match $A \to B$
nn candidate $
nn candidate a finite type ring map
nn a finite type ring map
new_annotation [(25342, 25351, 'VAR'), (25355, 25377, 'TYPE')]
===== sent Then $H^n(L_{B/A})$ is a finite $B$-module for all $n \in \mathbf{Z}$.
\end{lemma}

\begin{proof}
Apply Lemmas \ref{lemma-compute-cotangent-complex} and
\ref{lemma-pi-shriek-finite}.
\end{proof}

\begin{remark}[Resolutions]
\label{remark-resolution}
Let $A \to B$ be any ring map. ========

new_annotation []
===== sent \end{remark}

\begin{lemma}
\label{lemma-O-homology-B-homology}
Let $A \to B$ be a ring map. ========

regex_match $A \to B$
nn candidate a ring map
nn a ring map
new_annotation [(27687, 27696, 'VAR'), (27700, 27710, 'TYPE')]
===== sent Let $\pi$, $\mathcal{O}$, $\underline{B}$
be as in (\ref{equation-pi}). ========

new_annotation []
===== sent \end{proof}

\begin{lemma}
\label{lemma-apply-O-B-comparison}
Let $A \to B$ be a ring map. ========

regex_match $A \to B$
nn candidate a ring map
nn a ring map
new_annotation [(28810, 28819, 'VAR'), (28823, 28833, 'TYPE')]
===== sent Let $\pi$, $\mathcal{O}$, $\underline{B}$
be as in (\ref{equation-pi}). ========

new_annotation []
===== sent \begin{lemma}
\label{lemma-special-case-triangle}
Let $A \to B \to C$ be ring maps. ========

new_annotation []
===== sent \end{proof}

\begin{example}
\label{example-resolution-length-two}
Let $A \to B$ be a ring map. ========

regex_match $A \to B$
nn candidate a ring map
nn a ring map
new_annotation [(30537, 30546, 'VAR'), (30550, 30560, 'TYPE')]
===== sent Let $\mathcal{C} = \mathcal{C}_{B/A}$ and $\mathcal{C}' = \mathcal{C}_{B'/A}'$
be the categories considered in Section \ref{section-compute-L-pi-shriek}.
There is a functor
$$
u : \mathcal{C} \longrightarrow \mathcal{C}',\quad
(P, \alpha) \longmapsto (P \otimes_A A', c \circ (\alpha \otimes 1))
$$
where $c : B \otimes_A A' \to B'$ is the obvious map. ========

new_annotation []
===== sent We will apply Cohomology on Sites, Lemma
\ref{sites-cohomology-lemma-get-it-now}. Let $P_\bullet \to B$ be a resolution.
 ========

regex_match $P_\bullet \to B$
nn candidate a resolution
nn a resolution
new_annotation [(37308, 37325, 'VAR'), (37329, 37341, 'TYPE')]
===== sent \end{remark}

\begin{lemma}
\label{lemma-cotangent-complex-product}
Let $A \to B$ and $A \to C$ be ring maps.
 ========

new_annotation []
===== sent \Spec(A)
}
$$
Let $\mathcal{C}_{B/A}$, $\mathcal{C}_{C/A}$, $\mathcal{C}_{C/B}$
be the categories considered in Section \ref{section-compute-L-pi-shriek}.
There are functors
$$
\begin{matrix}
u_1 : \mathcal{C}_{C/B/A} \to \mathcal{C}_{B/A}, &
(P \to B, Q \to C) \mapsto (P \to B) \\
 ========

new_annotation []
===== sent }\mathcal{F}$ by
Lemma \ref{lemma-pi-lower-shriek-polynomial-algebra}.

\medskip\noindent
Let $U = (\beta : P \to B, \gamma : Q \to C)$ be an object of
$\mathcal{C}_{C/B/A}$. We may write $P = A[S]$ and $Q = A[S \amalg T]$
by the definition of our category $\mathcal{C}_{C ========

regex_match $U = (\beta : P \to B, \gamma : Q \to C)$
nn candidate $U
nn candidate an object
nn an object
new_annotation [(47904, 47945, 'VAR'), (47949, 47958, 'TYPE')]
===== sent Moreover, observe
that the fibre category of $u_1$ over $P \to B$ is the category
$\mathcal{C}_{C/P}$. Let $\mathcal{F}$ be an abelian sheaf on
$\mathcal{C}_{B/A}$. Since we have a fibred category we may apply
Cohomology on Sites, Lemma
\ref{sites-cohomology-lemma-compute-left-derived-pi-shriek}.
 ========

regex_match $\mathcal{F}$
nn candidate an abelian sheaf
nn an abelian sheaf
new_annotation [(50622, 50635, 'VAR'), (50639, 50655, 'TYPE')]
===== sent \end{proof}

\begin{proposition}
\label{proposition-triangle}
Let $A \to B \to C$ be ring maps. ========

new_annotation []
===== sent Let $A \to B \to C$ be ring maps and assume that $B \to C$ is injective.
 ========

new_annotation []
===== sent Let $P_\bullet \to B$ be the standard resolution of $B$ over $A$ and
let $Q_\bullet \to C$ be the standard resolution of $C$ over $B$.
Picture
$$
\xymatrix{
P_\bullet : &
A[A[A[B]]] \ar[d]
\ar@<2ex>[r]
\ar@<0ex>[r]
\ar@<-2ex>[r]
&
A[A[B]] \ar[d]
\ar@<1ex>[r]
\ar@<-1ex>[r]
\ar@<1ex>[l]
\ar@<-1ex>[l]
&
A[B] \ar[d] \ar@<0ex>[l] \ar[r] &
B \\
 ========

regex_match $P_\bullet \to B$
nn candidate the standard resolution
nn the standard resolution
new_annotation [(54933, 54950, 'VAR'), (54954, 54977, 'TYPE')]
===== sent Let $P_\bullet \to B$ be the standard resolution of $B$ over $A$ and
let $Q_\bullet \to C$ be the standard resolution of $C$ over $B$.
Picture
$$
\xymatrix{
P_\bullet : &
A[A[A[B]]] \ar[d]
\ar@<2ex>[r]
\ar@<0ex>[r]
\ar@<-2ex>[r]
&
A[A[B]] \ar[d]
\ar@<1ex>[r]
\ar@<-1ex>[r]
\ar@<1ex>[l]
\ar@<-1ex>[l]
&
A[B] \ar[d] \ar@<0ex>[l] \ar[r] &
B \\
 ========

regex_match $Q_\bullet \to C$
nn candidate the standard resolution
nn the standard resolution
new_annotation [(55002, 55019, 'VAR'), (55023, 55046, 'TYPE')]
===== sent \end{remark}

\begin{remark}
\label{remark-explicit-map}
Let $A \to B \to C$ be ring maps with $B \to C$ injective.
 ========

new_annotation []
===== sent Recall the notation $P_\bullet$, $Q_\bullet$, $\overline{Q}_\bullet$ of
Remark \ref{remark-triangle}.
Let $R_\bullet$ be the standard resolution of $C$ over $B$.
In this remark we explain how to get the canonical identification
of $\Omega_{\overline{Q}_\bullet/B} \otimes_{\overline{Q}_\bullet} C$
with $L_{C/B} = \Omega_{R_\bullet/B} \otimes_{R_\bullet} C$.
 ========

regex_match $R_\bullet$
nn candidate the standard resolution
nn the standard resolution
new_annotation [(57545, 57556, 'VAR'), (57560, 57583, 'TYPE')]
===== sent Let $S_\bullet \to B$ be the standard resolution of $B$ over $B$.
Note that the functoriality map $S_\bullet \to R_\bullet$ identifies
$R_n$ as a polynomial algebra over $S_n$ because $B \to C$ is injective.
 ========

regex_match $S_\bullet \to B$
nn candidate the standard resolution
nn the standard resolution
new_annotation [(57802, 57819, 'VAR'), (57823, 57846, 'TYPE')]
===== sent Let $A \to A' \to B$ be ring maps such that $B = B \otimes_A^\mathbf{L} A'$.
This happens for example ========

new_annotation []
===== sent \begin{lemma}
\label{lemma-localize-at-bottom}
Let $A \to A' \to B$ be ring maps such that $B = B \otimes_A^\mathbf{L} A'$.
 ========

new_annotation []
===== sent \end{proof}

\begin{lemma}
\label{lemma-derived-diagonal}
Let $A \to B$ be a ring map such that $B = B \otimes_A^\mathbf{L} B$.
Then $L_{B/A} = 0$ in $D(B)$.
\end{lemma}

\begin{proof}
This is true because $L_{B/A} = L_{B/B} = 0$ by
Lemmas \ref{lemma-localize-at-bottom} and
\ref{lemma-cotangent-complex-polynomial-algebra}.
\end{proof}

\begin{lemma}
\label{lemma-bootstrap}
 ========

regex_match $A \to B$
nn candidate a ring map
nn a ring map
new_annotation [(60798, 60807, 'VAR'), (60811, 60821, 'TYPE')]
===== sent Let $A \to B$ be a ring map such that $\text{Tor}^A_i(B, B) = 0$ for $i > 0$
and such that $L_{B/B \otimes_A B} = 0$.
Then $L_{B/A} = 0$ in $D(B)$.
\end{lemma}

\begin{proof}
By Lemma \ref{lemma-flat-base-change-cotangent-complex} we see that
$L_{B/A} \otimes_B^\mathbf{L} (B \otimes_A B) = L_{B \otimes_A B/B}$.
 ========

regex_match $A \to B$
nn candidate a ring map
nn a ring map
new_annotation [(61116, 61125, 'VAR'), (61129, 61139, 'TYPE')]
===== sent Part (6) follows from Lemma \ref{lemma-colimit-cotangent-complex}.
Parts (7) and (8) follows from the fact that (strict) henselizations
are filtered colimits of \'etale ring extensions of $A$, see
Algebra, Lemmas \ref{algebra-lemma-henselization-different} and
\ref{algebra-lemma-strict-henselization-different}.
\end{proof}

\begin{lemma}
\label{lemma-localize-on-top}
Let $A \to B \to C$ be ring maps such that $L_{C/B} = 0$.
Then $L_{C/A} = L_{B/A} \otimes_B^\mathbf{L} C$.
 ========

new_annotation []
===== sent \end{proof}

\begin{lemma}
\label{lemma-localize}
Let $A \to B$ be ring maps and $S \subset A$, $T \subset B$ multiplicative
subsets such that $S$ maps into $T$.
Then $L_{T^{-1}B/S^{-1}A} = L_{B/A} \otimes_B T^{-1}B$
in $D(T^{-1}B)$.
\end{lemma}

\begin{proof}
Lemma \ref{lemma-localize-on-top} shows that
$L_{T^{-1}B/ ========

new_annotation []
===== sent A} = L_{B/A} \otimes_B T^{-1}B$
and Lemma \ref{lemma-localize-at-bottom}
shows that $L_{T^{-1}B/A} = L_{T^{-1}B/S^{-1}A}$.
\end{proof}

\begin{lemma}
\label{lemma-cotangent-complex-henselization}
Let $A \to B$ be a local ring homomorphism of local rings.
 ========

regex_match $A \to B$
nn candidate a local ring homomorphism
nn a local ring homomorphism
new_annotation [(64158, 64167, 'VAR'), (64171, 64196, 'TYPE')]
===== sent Let $A^h \to B^h$, resp.\ $A^{sh} \to B^{sh}$ be the induced
maps of henselizations, resp.\ strict henselizations.
 ========

new_annotation []
===== sent \end{proof}




\section{Smooth ring maps}
\label{section-smooth}

\noindent
Let $C \to B$ be a surjection of rings with kernel $I$. ========

regex_match $C \to B$
nn candidate a surjection
nn a surjection
new_annotation [(65033, 65042, 'VAR'), (65046, 65058, 'TYPE')]
===== sent In this case Lemmas \ref{lemma-when-zero} and
Lemma \ref{lemma-localize-on-top} show that
$L_{B/A} = L_{A[x_1, \ldots, x_n]/A} \otimes B$
whence the conclusion by
Lemma \ref{lemma-cotangent-complex-polynomial-algebra}.
\end{proof}





\section{Comparison with the naive cotangent complex}
\label{section-surjections}

\noindent
The naive cotangent complex was introduced in
Algebra, Section \ref{algebra-section-netherlander}.

\begin{remark}
\label{remark-make-map}
Let $A \to B$ be a ring map.   ========

regex_match $A \to B$
nn candidate a ring map
nn a ring map
new_annotation [(66629, 66638, 'VAR'), (66642, 66652, 'TYPE')]
===== sent \begin{lemma}
\label{lemma-surjection}
\begin{slogan}
The cohomology of the cotangent complex of a surjective ring map is trivial in
degree zero; it is the kernel modulo its square in degree $-1$.
\end{slogan}
Let $A \to B$ be a surjective ring map with kernel $I$.
 ========

regex_match $A \to B$
nn candidate a surjective ring map
nn a surjective ring map
new_annotation [(68939, 68948, 'VAR'), (68952, 68973, 'TYPE')]
===== sent \end{proof}

\begin{lemma}
\label{lemma-relation-with-naive-cotangent-complex}
Let $A \to B$ be a ring map. ========

regex_match $A \to B$
nn candidate a ring map
nn a ring map
new_annotation [(70451, 70460, 'VAR'), (70464, 70474, 'TYPE')]
===== sent Let $P_\bullet$ be the standard resolution of $B$
over $A$.
 ========

regex_match $P_\bullet$
nn candidate the standard resolution
nn the standard resolution
new_annotation [(72109, 72120, 'VAR'), (72124, 72147, 'TYPE')]
===== sent Let $I = \Ker(A[B] \to B)$.
Recall that $P_0 = A[B]$. The map of the
lemma is given by the commutative diagram
$$
\xymatrix{
L_{B/A} \ar[d] & \ldots \ar[r] &
\Omega_{P_2/A} \otimes_{P_2} B
\ar[r] \ar[d] &
\Omega_{P_1/A} \otimes_{P_1} B
\ar[r] \ar[d] &
\Omega_{P_0/A} \otimes_{P_0} B
\ar[d] \\
 ========

new_annotation []
===== sent Let $\mathcal{C}'_{B/A}$
denote the full subcategory of $\mathcal{C}_{B/A}$ consisting of surjective
maps $P \to B$. The agreement of the cotangent complex with the naive
cotangent complex (Lemma \ref{lemma-relation-with-naive-cotangent-complex})
shows that we have an exact sequence of sheaves
$$
0 \to \underline{H_1(L_{B/A})} \to
\mathcal{J}/\mathcal{J}^2 \xrightarrow{\text{d}} \Omega \to
\underline{H_2(L_{B/A})} \to 0
$$
on $\mathcal{C}'_{B/A}$. It follows that $\Ker(d)$ and
$\Coker(d)$ on the whole category $\mathcal{C}_{B/A}$ have
vanishing higher homology groups, since
these are computed by the homology groups of constant simplicial abelian
groups by Lemma \ref{lemma-identify-pi-shriek}. Hence we conclude
that
$$
H_n(\mathcal{C}_{B/A}, \mathcal{J}/\mathcal{J}^2) \to H_n(L_{B/A})
$$
is an isomorphism for all $n \geq 2$. ========

new_annotation []
===== sent \begin{lemma}
\label{lemma-vanishing-symmetric-powers}
Notation and assumptions as in
Cohomology on Sites, Example \ref{sites-cohomology-example-category-to-point}.
Assume $\mathcal{C}$ has a cosimplicial object as in
Cohomology on Sites, Lemma
\ref{sites-cohomology-lemma-compute-by-cosimplicial-resolution}.
Let $\mathcal{F}$ be a flat $\underline{B}$-module such that
$H_0(\mathcal{C}, \mathcal{F}) = 0$.
Then $H_l(\mathcal{C}, \text{Sym}_{\underline{B}}^k(\mathcal{F})) = 0$
for $l < k$.
\end{lemma}

\begin{proof}
We drop the subscript ${}_{\underline{B}}$ from tensor products, wedge powers,
and symmetric powers ========

regex_match $\mathcal{F}$
nn candidate a flat $\underline{B}$-module
nn a flat $\underline{B}$-module
new_annotation [(75322, 75335, 'VAR'), (75339, 75368, 'TYPE')]
===== sent \end{remark}

\begin{lemma}
\label{lemma-map-tors-zero}
Let $A$ be a ring. ========

regex_match $A$
nn candidate a ring
nn a ring
new_annotation [(77850, 77853, 'VAR'), (77857, 77863, 'TYPE')]
===== sent Let $P = A[E]$ be a polynomial ring.
 ========

regex_match $P = A[E]$
nn candidate $
nn candidate a polynomial ring
nn a polynomial ring
new_annotation [(77869, 77879, 'VAR'), (77883, 77900, 'TYPE')]
===== sent \end{proof}

\begin{theorem}[Quillen spectral sequence]
\label{theorem-quillen-spectral-sequence}
Let $A \to B$ be a surjective ring map. ========

regex_match $A \to B$
nn candidate a surjective ring map
nn a surjective ring map
new_annotation [(79039, 79048, 'VAR'), (79052, 79073, 'TYPE')]
===== sent < k$.
\end{theorem}

\begin{proof}
Let $I \subset A$ be the kernel of $A \to B$. Let
$\mathcal{J} \subset \mathcal{O}$
be the kernel of $\mathcal{O} \to \underline{B}$. Then
$I\mathcal{O} \subset \mathcal{J}$. Set
$\mathcal{K} = \mathcal{J}/I\mathcal{O}$ and
$\overline{\mathcal{O}} = \mathcal{O}/I\mathcal{O}$.

\medskip\noindent
For every object $U = (P \to B)$ of $\mathcal{C}_{B/A}$
we can choose an isomorphism $P = A[E]$ such that the map
$P \to B$ maps each $e \in E$ to zero. ========

regex_match $I \subset A$
nn candidate I
nn candidate the kernel
nn the kernel
new_annotation [(79600, 79613, 'VAR'), (79617, 79627, 'TYPE')]
===== sent In the situation of Theorem \ref{theorem-quillen-spectral-sequence}
let $I = \Ker(A \to B)$. ========

new_annotation []
===== sent \end{remark}

\begin{remark}
\label{remark-elucidate-degree-two}
Let $A \to B$ be a ring map. ========

regex_match $A \to B$
nn candidate a ring map
nn a ring map
new_annotation [(85838, 85847, 'VAR'), (85851, 85861, 'TYPE')]
===== sent Let $P_\bullet$ be a resolution of
$B$ over $A$ (Remark \ref{remark-resolution}).
 ========

regex_match $P_\bullet$
nn candidate a resolution
nn a resolution
new_annotation [(85867, 85878, 'VAR'), (85882, 85894, 'TYPE')]
===== sent All in all we conclude
that there is an exact sequence
$$
\wedge^2_B(J_0/J_0^2) \to \text{Tor}_2^{P_0}(B, B) \to H^{-2}(L_{B/A}) \to 0
$$
In this way we obtain a direct proof of a consequence of Quillen's spectral
sequence discussed in Remark \ref{remark-elucidate-ss}.
\end{remark}






\section{Comparison with Lichtenbaum-Schlessinger}
\label{section-compare-higher}

\noindent
Let $A \to B$ be a ring map. ========

regex_match $A \to B$
nn candidate a ring map
nn a ring map
new_annotation [(87551, 87560, 'VAR'), (87564, 87574, 'TYPE')]
===== sent Let $Rel \subset F$ be the kernel of $F \to I$, in other words
$Rel$ is the set of relations among the $f_t$. Let $TrivRel \subset ========

regex_match $Rel \subset F$
nn candidate the kernel
nn the kernel
new_annotation [(88010, 88025, 'VAR'), (88029, 88039, 'TYPE')]
===== sent Let $Rel \subset F$ be the kernel of $F \to I$, in other words
$Rel$ is the set of relations among the $f_t$. Let $TrivRel \subset ========

new_annotation []
===== sent \Omega_{P/A} \otimes_P B
\end{equation}
where the last term is placed in degree $0$. The first map is the obvious
one and the second map sends the basis element corresponding to $t \in T$
to $\text{d}f_t \otimes 1$.

\begin{definition}
\label{definition-biderivation}
Let $A \to B$ be a ring map. ========

regex_match $A \to B$
nn candidate a ring map
nn a ring map
new_annotation [(88712, 88721, 'VAR'), (88725, 88735, 'TYPE')]
===== sent Let $M$ be a $(B, B)$-bimodule
over $A$. ========

regex_match $M$
nn candidate a $
nn a $
new_annotation [(88741, 88744, 'VAR'), (88748, 88751, 'TYPE')]
===== sent \begin{lemma}
\label{lemma-polynomial-ring-unique}
Let $P = A[S]$ be a polynomial ring over $A$. ========

regex_match $P = A[S]$
nn candidate $P
nn candidate a polynomial ring
nn a polynomial ring
new_annotation [(89047, 89057, 'VAR'), (89061, 89078, 'TYPE')]
===== sent Let $M$ be a $(P, P)$-bimodule
over $A$. ========

regex_match $M$
nn candidate a $(P
nn a $(P
new_annotation [(89093, 89096, 'VAR'), (89100, 89105, 'TYPE')]
===== sent There is a canonical map $L_{B/A} \to L$ in $D(A)$ which
induces an isomorphism $\tau_{\geq -2}L_{B/A} \to L$ in $D(B)$.
\end{lemma}

\begin{proof}
Let $P_\bullet \to B$ be a resolution of $B$ over $A$
(Remark \ref{remark-resolution}). ========

regex_match $P_\bullet \to B$
nn candidate a resolution
nn a resolution
new_annotation [(90020, 90037, 'VAR'), (90041, 90053, 'TYPE')]
===== sent \to B)$.
For $t \in T$ denote $f'_t$ the image of $f_t$ in $P'$.
Choose $f'_s \in P'$ such that the elements $f'_t$ for
$t \in T' = T \amalg S$ generate the kernel
of $P' \to B'$. Set $F = \bigoplus_{t \in T} P$ and
$F' = \bigoplus_{t' \in T'} P'$. Let $Rel = \Ker(F \to P)$
and $Rel' = \Ker(F' \to P')$ where the maps are given
by multiplication by $f_t$, resp.\ $f'_t$ on the coordinates.
 ========

new_annotation []
===== sent \begin{lemma}
\label{lemma-special-case}
Let $A = \mathbf{Z}[x] \to B = \mathbf{Z}$ be the ring map which sends
$x$ to $0$. Let $I = (x) \subset A$. ========

regex_match $A = \mathbf{Z}[x] \to B = \mathbf{Z}$
nn candidate $A = \mathbf{Z}[x
nn candidate the ring map
nn the ring map
new_annotation [(97133, 97171, 'VAR'), (97175, 97187, 'TYPE')]
===== sent \begin{lemma}
\label{lemma-special-case}
Let $A = \mathbf{Z}[x] \to B = \mathbf{Z}$ be the ring map which sends
$x$ to $0$. Let $I = (x) \subset A$. ========

new_annotation []
===== sent Namely, consider the distinguished triangle
$$
L_{\mathbf{Z}[x]/\mathbf{Z}} \otimes_{\mathbf{Z}[x]} \mathbf{Z} \to
L_{\mathbf{Z}/\mathbf{Z}} \to
L_{\mathbf{Z}/\mathbf{Z}[x]}\to
L_{\mathbf{Z}[x]/\mathbf{Z}} \otimes_{\mathbf{Z}[x]} \mathbf{Z}[1]
$$
The complex $L_{\mathbf{Z}[x]/\mathbf{Z}}$ is quasi-isomorphic to
$\Omega_{\mathbf{Z}[x]/\mathbf{Z}}$ by
Lemma \ref{lemma-cotangent-complex-polynomial-algebra}.
The complex $L_{\mathbf{Z}/\mathbf{Z}}$ is zero in $D(\mathbf{Z})$ by
Lemma \ref{lemma-when-zero}.
Thus we see that $L_{B/A}$ has only one nonzero cohomology group
which is as described in the lemma by Lemma \ref{lemma-surjection}.
\end{proof}

\begin{lemma}
\label{lemma-mod-regular-sequence}
Let $A \to B$ be a surjective ring map whose kernel $I$ is generated
by a regular sequence. ========

regex_match $A \to B$
nn candidate a surjective ring map
nn a surjective ring map
new_annotation [(98374, 98383, 'VAR'), (98387, 98408, 'TYPE')]
===== sent /A} \otimes_C^\mathbf{L} B \to L_{B/A} \to L_{B/C} \to
L_{C/A} \otimes_C^\mathbf{L} B[1]
$$
which shows that $L_{B/A}$ has only one nonzero cohomology group
which is as described in the lemma by Lemma \ref{lemma-surjection}.
\end{proof}

\begin{lemma}
\label{lemma-mod-Koszul-regular-ideal}
Let $A \to B$ be a surjective ring map whose kernel $I$ is Koszul.
 ========

regex_match $A \to B$
nn candidate a surjective ring map
nn a surjective ring map
new_annotation [(99508, 99517, 'VAR'), (99521, 99542, 'TYPE')]
===== sent \end{proof}

\begin{proposition}
\label{proposition-cotangent-complex-local-complete-intersection}
Let $A \to B$ be a local complete intersection map.
 ========

regex_match $A \to B$
nn candidate a local complete intersection map
nn a local complete intersection map
new_annotation [(100012, 100021, 'VAR'), (100025, 100058, 'TYPE')]
===== sent \end{proof}








\section{Tensor products and the cotangent complex}
\label{section-tensor-product}

\noindent
Let $R$ be a ring and let $A$, $B$ be $R$-algebras. ========

regex_match $R$
nn candidate a ring
nn a ring
new_annotation [(101136, 101139, 'VAR'), (101143, 101149, 'TYPE')]
===== sent \end{proof}








\section{Tensor products and the cotangent complex}
\label{section-tensor-product}

\noindent
Let $R$ be a ring and let $A$, $B$ be $R$-algebras. ========

new_annotation []
===== sent The first two statements are immediate from
Lemma \ref{lemma-flat-base-change-cotangent-complex}.
The last statement follows as $L_{A/R}$ is a complex
of free $A$-modules, hence $L_{A/R} \otimes_A^\mathbf{L} (A \otimes_R B)$
is represented by
$L_{A/R} \otimes_A (A \otimes_R B) = L_{A/R} \otimes_R B$
\end{proof}

\noindent
In general we can say this about the object $E$.

\begin{lemma}
\label{lemma-tensor-product}
Let $R$ be a ring and let $A$, $B$ be $R$-algebras. ========

regex_match $R$
nn candidate a ring
nn a ring
new_annotation [(103596, 103599, 'VAR'), (103603, 103609, 'TYPE')]
===== sent The first two statements are immediate from
Lemma \ref{lemma-flat-base-change-cotangent-complex}.
The last statement follows as $L_{A/R}$ is a complex
of free $A$-modules, hence $L_{A/R} \otimes_A^\mathbf{L} (A \otimes_R B)$
is represented by
$L_{A/R} \otimes_A (A \otimes_R B) = L_{A/R} \otimes_R B$
\end{proof}

\noindent
In general we can say this about the object $E$.

\begin{lemma}
\label{lemma-tensor-product}
Let $R$ be a ring and let $A$, $B$ be $R$-algebras. ========

new_annotation []
===== sent Let $Rel \subset F = \bigoplus_{t \in T} P$ be the kernel of the map
$F \to P$ which maps the basis vector corresponding to $t$ to $f_t$.
Set $P_A = A \otimes_R P$ and $F_A = A \otimes_R F = P_A \otimes_P F$.
Let $Rel_A$ be the kernel of the map $F_A \to P_A$. Using the exact sequence
$$
0 \to Rel \to F \to P \to B \to 0
$$
and standard short exact sequences for Tor we obtain an exact sequence
$$
A \otimes_R Rel \to Rel_A \to \text{Tor}_1^R(A, B) \to 0
$$
Note that $P_A \to A \otimes_R B$ is a surjection whose kernel is generated
by the elements $1 \otimes f_t$ in $P_A$. Denote $TrivRel_A \subset Rel_A$
the $P_A$-submodule generated by the elements
$(\ldots, 1 \otimes f_{t'}, 0, \ldots,
0, - 1 \otimes f_t \otimes 1, 0, \ldots)$.
Since $TrivRel \otimes_R ========

regex_match $Rel \subset F = \bigoplus_{t \in T} P$
nn candidate F = \bigoplus_{t \in T
nn candidate the kernel
nn the kernel
new_annotation [(104554, 104593, 'VAR'), (104597, 104607, 'TYPE')]
===== sent Let $Rel \subset F = \bigoplus_{t \in T} P$ be the kernel of the map
$F \to P$ which maps the basis vector corresponding to $t$ to $f_t$.
Set $P_A = A \otimes_R P$ and $F_A = A \otimes_R F = P_A \otimes_P F$.
Let $Rel_A$ be the kernel of the map $F_A \to P_A$. Using the exact sequence
$$
0 \to Rel \to F \to P \to B \to 0
$$
and standard short exact sequences for Tor we obtain an exact sequence
$$
A \otimes_R Rel \to Rel_A \to \text{Tor}_1^R(A, B) \to 0
$$
Note that $P_A \to A \otimes_R B$ is a surjection whose kernel is generated
by the elements $1 \otimes f_t$ in $P_A$. Denote $TrivRel_A \subset Rel_A$
the $P_A$-submodule generated by the elements
$(\ldots, 1 \otimes f_{t'}, 0, \ldots,
0, - 1 \otimes f_t \otimes 1, 0, \ldots)$.
Since $TrivRel \otimes_R ========

regex_match $Rel_A$
nn candidate the kernel
nn the kernel
new_annotation [(104763, 104770, 'VAR'), (104774, 104784, 'TYPE')]
===== sent Let $\alpha \in \Ext^1_A(\NL_{A/\mathbf{Z}}, I)$
be the element corresponding to the isomorphism class of $A'$.
The existence of $B'$ corresponds to an element
$\beta \in \Ext_B^1(\NL_{B/\mathbf{Z}}, N)$
which maps to the image of $\alpha$ in
$\Ext^1_A(\NL_{A/\mathbf{Z}}, N)$. Note that
$$
\Ext^1_A(\NL_{A/\mathbf{Z}}, N) =
\Ext^1_A(L_{A/\mathbf{Z}}, N) =
\Ext^1_B(L_{A/\mathbf{Z}} \otimes_A^\mathbf{L} B, N)
$$
and
$$
\Ext^1_B(\NL_{B/\mathbf{Z}}, N) =
\Ext^1_B(L_{B/\mathbf{Z}}, N)
$$
by Lemma \ref{lemma-relation-with-naive-cotangent-complex}.
 ========

new_annotation []
===== sent \Ext^2_B(L_{B/A}, N) \to \ldots
$$
we obtain the result with $\xi$ the image of $\alpha$.
\end{proof}





\section{The Atiyah class of a module}
\label{section-atiyah}

\noindent
Let $A \to B$ be a ring map. ========

regex_match $A \to B$
nn candidate a ring map
nn a ring map
new_annotation [(109103, 109112, 'VAR'), (109116, 109126, 'TYPE')]
===== sent Let $M$ be a $B$-module.
 ========

regex_match $M$
new_annotation []
===== sent Let $P \to B$ be an object of $\mathcal{C}_{B/A}$
(Section \ref{section-compute-L-pi-shriek}).
 ========

regex_match $P \to B$
nn candidate an object
nn an object
new_annotation [(109157, 109166, 'VAR'), (109170, 109179, 'TYPE')]
===== sent Here we use Cohomology on Sites, Remark
\ref{sites-cohomology-remark-O-homology-B-homology-general}
to get a distinguished triangle in $D(B)$ and not just in $D(A)$.

\begin{definition}
\label{definition-atiyah-class}
Let $A \to B$ be a ring map. ========

regex_match $A \to B$
nn candidate a ring map
nn a ring map
new_annotation [(110540, 110549, 'VAR'), (110553, 110563, 'TYPE')]
===== sent Let $M$ be a $B$-module.
 ========

regex_match $M$
new_annotation []
===== sent \medskip\noindent
Let $\mathcal{C}$ be a site and let $\Sh(\mathcal{C})$ denote the
associated topos. ========

regex_match $\mathcal{C}$
nn candidate a site
nn a site
new_annotation [(111106, 111119, 'VAR'), (111123, 111129, 'TYPE')]
===== sent \medskip\noindent
Let $\mathcal{C}$ be a site and let $\Sh(\mathcal{C})$ denote the
associated topos. ========

new_annotation []
===== sent Let $\mathcal{A}$ denote a sheaf of rings
on $\mathcal{C}$. Let $\mathcal{A}\textit{-Alg}$ be the category of
$\mathcal{A}$-algebras. ========

new_annotation []
===== sent Let $\mathcal{A}$ denote a sheaf of rings
on $\mathcal{C}$. Let $\mathcal{A}\textit{-Alg}$ be the category of
$\mathcal{A}$-algebras. ========

regex_match $\mathcal{A}\textit{-Alg}$
nn candidate the category
nn the category
new_annotation [(111250, 111276, 'VAR'), (111280, 111292, 'TYPE')]
===== sent Let $X_\bullet$ be the simplicial object of
$\text{Fun}(\mathcal{A}\textit{-Alg}, \mathcal{A}\textit{-Alg})$
constructed in
Simplicial, Section \ref{simplicial-section-standard}.

\medskip\noindent
Now assume that $\mathcal{A} \to \mathcal{B}$ is a homomorphism of sheaves
of rings. ========

regex_match $X_\bullet$
nn candidate the simplicial object
nn the simplicial object
new_annotation [(111638, 111649, 'VAR'), (111653, 111674, 'TYPE')]
===== sent \begin{definition}
\label{definition-standard-resolution-sheaves-rings}
Let $\mathcal{C}$ be a site.
 ========

regex_match $\mathcal{C}$
nn candidate a site
nn a site
new_annotation [(112479, 112492, 'VAR'), (112496, 112502, 'TYPE')]
===== sent Let $\mathcal{A} \to \mathcal{B}$ be a homomorphism of sheaves of rings
on $\mathcal{C}$. The {\it standard resolution of $\mathcal{B}$ over
$\mathcal{A}$} is the augmentation
$\epsilon : \mathcal{P}_\bullet \to \mathcal{B}$
with terms
$$
\mathcal{P}_0 = \mathcal{A}[\mathcal{B}],\quad
\mathcal{P}_1 = \mathcal{A}[\mathcal{A}[\mathcal{B}]],\quad \ldots
$$
and maps as constructed above.
 ========

regex_match $\mathcal{A} \to \mathcal{B}$
nn candidate $\mathcal{A
nn candidate a homomorphism
nn a homomorphism
new_annotation [(112508, 112537, 'VAR'), (112541, 112555, 'TYPE')]
===== sent We will use the module of differentials as defined in
Modules on Sites, Section \ref{sites-modules-section-differentials}.

\begin{definition}
\label{definition-cotangent-complex-morphism-sheaves-rings}
Let $\mathcal{C}$ be a site.
 ========

regex_match $\mathcal{C}$
nn candidate a site
nn a site
new_annotation [(113229, 113242, 'VAR'), (113246, 113252, 'TYPE')]
===== sent Let $\mathcal{A} \to \mathcal{B}$ be a homomorphism of sheaves of rings
on $\mathcal{C}$.
 ========

regex_match $\mathcal{A} \to \mathcal{B}$
nn candidate $\mathcal{A
nn candidate a homomorphism
nn a homomorphism
new_annotation [(113258, 113287, 'VAR'), (113291, 113305, 'TYPE')]
===== sent \begin{lemma}
\label{lemma-pullback-cotangent-morphism-topoi}
Let $f : \Sh(\mathcal{D}) \to \Sh(\mathcal{C})$ be a morphism of topoi.
 ========

regex_match $f : \Sh(\mathcal{D}) \to \Sh(\mathcal{C})$
nn candidate a morphism
nn a morphism
new_annotation [(115353, 115396, 'VAR'), (115400, 115410, 'TYPE')]
===== sent Let $\mathcal{A} \to \mathcal{B}$ be a homomorphism of sheaves of rings
on $\mathcal{C}$. Then
$f^{-1}L_{\mathcal{B}/\mathcal{A}} = L_{f^{-1}\mathcal{B}/f^{-1}\mathcal{A}}$.
\end{lemma}

\begin{proof}
 ========

regex_match $\mathcal{A} \to \mathcal{B}$
nn candidate $\mathcal{A
nn candidate a homomorphism
nn a homomorphism
new_annotation [(115425, 115454, 'VAR'), (115458, 115472, 'TYPE')]
===== sent \end{proof}

\begin{lemma}
\label{lemma-compute-L-morphism-sheaves-rings}
Let $\mathcal{C}$ be a site. ========

regex_match $\mathcal{C}$
nn candidate a site
nn a site
new_annotation [(115910, 115923, 'VAR'), (115927, 115933, 'TYPE')]
===== sent Let $\mathcal{A} \to \mathcal{B}$ be a
homomorphism of sheaves of rings on $\mathcal{C}$. Then
$H^i(L_{\mathcal{B}/\mathcal{A}})$ is the sheaf associated to the
presheaf $U \mapsto ========

new_annotation []
===== sent H^i(L_{\mathcal{B}(U)/\mathcal{A}(U)})$.
\end{lemma}

\begin{proof}
Let $\mathcal{C}'$ be the site we get by endowing $\mathcal{C}$ with the
chaotic topology (presheaves are sheaves). ========

regex_match $\mathcal{C}'$
nn candidate the site
nn the site
new_annotation [(116188, 116202, 'VAR'), (116206, 116214, 'TYPE')]
===== sent Moreover, these maps
are compatible with restriction maps and the complex
$L_{\mathcal{B}/\mathcal{A}}$
is the sheafification of the rule $U \mapsto L_{\mathcal{B}(U)/\mathcal{A}(U)}$.
\end{remark}

\begin{lemma}
\label{lemma-H0-L-morphism-sheaves-rings}
Let $\mathcal{C}$ be a site. ========

regex_match $\mathcal{C}$
nn candidate a site
nn a site
new_annotation [(117586, 117599, 'VAR'), (117603, 117609, 'TYPE')]
===== sent Let $\mathcal{A} \to \mathcal{B}$ be a
homomorphism of sheaves of rings on $\mathcal{C}$. Then
$H^0(L_{\mathcal{B}/\mathcal{A}}) = \Omega_{\mathcal{B}/\mathcal{A}}$.
\end{lemma}

\begin{proof}
Follows from Lemmas \ref{lemma-compute-L-morphism-sheaves-rings}
and \ref{lemma-identify-H0} and
Modules on Sites, Lemma \ref{sites-modules-lemma-differentials-sheafify}.
\end{proof}

\begin{lemma}
\label{lemma-compute-L-product-sheaves-rings}
Let $\mathcal{C}$ be a site. ========

new_annotation []
===== sent Let $\mathcal{A} \to \mathcal{B}$ be a
homomorphism of sheaves of rings on $\mathcal{C}$. Then
$H^0(L_{\mathcal{B}/\mathcal{A}}) = \Omega_{\mathcal{B}/\mathcal{A}}$.
\end{lemma}

\begin{proof}
Follows from Lemmas \ref{lemma-compute-L-morphism-sheaves-rings}
and \ref{lemma-identify-H0} and
Modules on Sites, Lemma \ref{sites-modules-lemma-differentials-sheafify}.
\end{proof}

\begin{lemma}
\label{lemma-compute-L-product-sheaves-rings}
Let $\mathcal{C}$ be a site. ========

regex_match $\mathcal{C}$
nn candidate a site
nn a site
new_annotation [(118052, 118065, 'VAR'), (118069, 118075, 'TYPE')]
===== sent Let $\mathcal{A} \to \mathcal{B}$
and $\mathcal{A} \to \mathcal{B}'$ be homomorphisms of sheaves of rings
on ========

new_annotation []
===== sent \begin{lemma}
\label{lemma-triangle-sheaves-rings}
Let $\mathcal{D}$ be a site. ========

regex_match $\mathcal{D}$
nn candidate a site
nn a site
new_annotation [(118810, 118823, 'VAR'), (118827, 118833, 'TYPE')]
===== sent Let $\mathcal{A} \to \mathcal{B} \to \mathcal{C}$
be homomorphisms of sheaves of rings on $\mathcal{D}$.
There is a canonical distinguished triangle
$$
L_{\mathcal{B}/\mathcal{A}} \otimes_\mathcal{B}^\mathbf{L} \mathcal{C}
\to L_{\mathcal{C}/\mathcal{A}} \to L_{\mathcal{C}/\mathcal{B}} \to
L_{\mathcal{B}/\mathcal{A}} \otimes_\mathcal{B}^\mathbf{L} \mathcal{C}[1]
$$
in $D(\mathcal{C})$.
\end{lemma}

\begin{proof}
We will use the method described in
Remarks \ref{remark-triangle} and \ref{remark-explicit-map}
to construct the triangle; we will freely use the results mentioned there.
 ========

new_annotation []
===== sent This is possible by the argument given in
Remark \ref{remark-triangle} by
Lemma \ref{lemma-compute-L-product-sheaves-rings}.
\end{proof}

\begin{lemma}
\label{lemma-stalk-cotangent-complex}
Let $\mathcal{C}$ be a site. ========

regex_match $\mathcal{C}$
nn candidate a site
nn a site
new_annotation [(121955, 121968, 'VAR'), (121972, 121978, 'TYPE')]
===== sent Let $\mathcal{A} \to \mathcal{B}$ be a
homomorphism of sheaves of rings on $\mathcal{C}$. If $p$ is a point
of $\mathcal{C}$, then
$(L_{\mathcal{B}/\mathcal{A}})_p = L_{\mathcal{B}_p/\mathcal{A}_p}$.
\end{lemma}

\begin{proof}
This is a special case of Lemma \ref{lemma-pullback-cotangent-morphism-topoi}.
\end{proof}

\noindent
For the construction of the naive cotangent complex and its properties
we refer to
Modules on Sites, Section \ref{sites-modules-section-netherlander}.

\begin{lemma}
\label{lemma-compare-cotangent-complex-with-naive}
Let $\mathcal{C}$ be a site. ========

new_annotation []
===== sent Let $\mathcal{A} \to \mathcal{B}$ be a
homomorphism of sheaves of rings on $\mathcal{C}$. If $p$ is a point
of $\mathcal{C}$, then
$(L_{\mathcal{B}/\mathcal{A}})_p = L_{\mathcal{B}_p/\mathcal{A}_p}$.
\end{lemma}

\begin{proof}
This is a special case of Lemma \ref{lemma-pullback-cotangent-morphism-topoi}.
\end{proof}

\noindent
For the construction of the naive cotangent complex and its properties
we refer to
Modules on Sites, Section \ref{sites-modules-section-netherlander}.

\begin{lemma}
\label{lemma-compare-cotangent-complex-with-naive}
Let $\mathcal{C}$ be a site. ========

regex_match $\mathcal{C}$
nn candidate a site
nn a site
new_annotation [(122530, 122543, 'VAR'), (122547, 122553, 'TYPE')]
===== sent Let $\mathcal{A} \to \mathcal{B}$ be a
homomorphism of sheaves of rings on $\mathcal{C}$.
There is a canonical map
$L_{\mathcal{B}/\mathcal{A}} \to \NL_{\mathcal{B}/\mathcal{A}}$
which identifies the naive cotangent complex with the truncation
$\tau_{\geq -1}L_{\mathcal{B}/\mathcal{A}}$.
\end{lemma}

\begin{proof}
Let $\mathcal{P}_\bullet$ be the standard resolution of $\mathcal{B}$
over $\mathcal{A}$.
Let $\mathcal{I} = \Ker(\mathcal{A}[\mathcal{B}] \to \mathcal{B})$.
Recall that $\mathcal{P}_0 = \mathcal{A}[\mathcal{B}]$. The map of the
lemma is given by the commutative diagram
$$
\xymatrix{
L_{\mathcal{B}/\mathcal{A}} \ar[d] & \ldots \ar[r] &
\Omega_{\mathcal{P}_2/\mathcal{A}} \otimes_{\mathcal{P}_2} \mathcal{B}
\ar[r] \ar[d] &
\Omega_{\mathcal{P}_1/\mathcal{A}} \otimes_{\mathcal{P}_1} \mathcal{B}
\ar[r] \ar[d] &
\Omega_{\mathcal{P}_0/\mathcal{A}} \otimes_{\mathcal{P}_0} \mathcal{B}
\ar[d] \\
 ========

new_annotation []
===== sent Let $\mathcal{A} \to \mathcal{B}$ be a
homomorphism of sheaves of rings on $\mathcal{C}$.
There is a canonical map
$L_{\mathcal{B}/\mathcal{A}} \to \NL_{\mathcal{B}/\mathcal{A}}$
which identifies the naive cotangent complex with the truncation
$\tau_{\geq -1}L_{\mathcal{B}/\mathcal{A}}$.
\end{lemma}

\begin{proof}
Let $\mathcal{P}_\bullet$ be the standard resolution of $\mathcal{B}$
over $\mathcal{A}$.
Let $\mathcal{I} = \Ker(\mathcal{A}[\mathcal{B}] \to \mathcal{B})$.
Recall that $\mathcal{P}_0 = \mathcal{A}[\mathcal{B}]$. The map of the
lemma is given by the commutative diagram
$$
\xymatrix{
L_{\mathcal{B}/\mathcal{A}} \ar[d] & \ldots \ar[r] &
\Omega_{\mathcal{P}_2/\mathcal{A}} \otimes_{\mathcal{P}_2} \mathcal{B}
\ar[r] \ar[d] &
\Omega_{\mathcal{P}_1/\mathcal{A}} \otimes_{\mathcal{P}_1} \mathcal{B}
\ar[r] \ar[d] &
\Omega_{\mathcal{P}_0/\mathcal{A}} \otimes_{\mathcal{P}_0} \mathcal{B}
\ar[d] \\
 ========

regex_match $\mathcal{P}_\bullet$
nn candidate the standard resolution
nn the standard resolution
new_annotation [(122875, 122896, 'VAR'), (122900, 122923, 'TYPE')]
===== sent Let $\mathcal{A} \to \mathcal{B}$ be a
homomorphism of sheaves of rings on $\mathcal{C}$.
There is a canonical map
$L_{\mathcal{B}/\mathcal{A}} \to \NL_{\mathcal{B}/\mathcal{A}}$
which identifies the naive cotangent complex with the truncation
$\tau_{\geq -1}L_{\mathcal{B}/\mathcal{A}}$.
\end{lemma}

\begin{proof}
Let $\mathcal{P}_\bullet$ be the standard resolution of $\mathcal{B}$
over $\mathcal{A}$.
Let $\mathcal{I} = \Ker(\mathcal{A}[\mathcal{B}] \to \mathcal{B})$.
Recall that $\mathcal{P}_0 = \mathcal{A}[\mathcal{B}]$. The map of the
lemma is given by the commutative diagram
$$
\xymatrix{
L_{\mathcal{B}/\mathcal{A}} \ar[d] & \ldots \ar[r] &
\Omega_{\mathcal{P}_2/\mathcal{A}} \otimes_{\mathcal{P}_2} \mathcal{B}
\ar[r] \ar[d] &
\Omega_{\mathcal{P}_1/\mathcal{A}} \otimes_{\mathcal{P}_1} \mathcal{B}
\ar[r] \ar[d] &
\Omega_{\mathcal{P}_0/\mathcal{A}} \otimes_{\mathcal{P}_0} \mathcal{B}
\ar[d] \\
 ========

new_annotation []
===== sent Let $\mathcal{C}'$ be the site
with the same underlying category as $\mathcal{C}$ but endowed with the
chaotic topology. ========

regex_match $\mathcal{C}'$
nn candidate the site
nn the site
new_annotation [(124870, 124884, 'VAR'), (124888, 124896, 'TYPE')]
===== sent Let $f : \Sh(\mathcal{C}) \to \Sh(\mathcal{C}')$ be
the morphism of topoi whose pullback functor is sheafification.
 ========

new_annotation []
===== sent Let $\mathcal{A}' \to \mathcal{B}'$ be the given map, but thought of
as a map of sheaves of rings on $\mathcal{C}'$. The construction above
gives a map $L_{\mathcal{B}'/\mathcal{A}'} \to \NL_{\mathcal{B}'/\mathcal{A}'}$
on $\mathcal{C}'$ whose value over any object $U$ of $\mathcal{C}'$
is just the map
$$
L_{\mathcal{B}(U)/\mathcal{A}(U)} \to \NL_{\mathcal{B}(U)/\mathcal{A}(U)}
$$
of Remark \ref{remark-explicit-comparison-map} which induces an isomorphism
on $H^0$ and $H^{-1}$. ========

regex_match $\mathcal{A}' \to \mathcal{B}'$
nn candidate the given map
nn the given map
new_annotation [(125107, 125138, 'VAR'), (125142, 125155, 'TYPE')]
===== sent \end{proof}











\section{The Atiyah class of a sheaf of modules}
\label{section-atiyah-general}

\noindent
Let $\mathcal{C}$ be a site. ========

regex_match $\mathcal{C}$
nn candidate a site
nn a site
new_annotation [(125993, 126006, 'VAR'), (126010, 126016, 'TYPE')]
===== sent Let $\mathcal{A} \to \mathcal{B}$ be a
homomorphism of sheaves of rings. ========

new_annotation []
===== sent Let $\mathcal{F}$ be a
sheaf of $\mathcal{B}$-modules. ========

new_annotation []
===== sent Let $\mathcal{P}_\bullet \to \mathcal{B}$
be the standard resolution of $\mathcal{B}$ over $\mathcal{A}$
(Section \ref{section-cotangent-complex}).
 ========

new_annotation []
===== sent D(\mathcal{B})$
of Cohomology on Sites, Remark
\ref{sites-cohomology-remark-homology-augmentation}
(here we use that $\mathcal{P}_\bullet \to \mathcal{A}$ is a resolution)
we obtain a distinguished triangle
\begin{equation}
\label{equation-atiyah-general}
L_{\mathcal{B}/\mathcal{A}} \otimes_\mathcal{B}^\mathbf{L} \mathcal{F} \to
L\pi_!\left(\mathcal{P}^1_{\mathcal{P}_\bullet/\mathcal{A}}(\mathcal{F})\right)
\to \mathcal{F} \to
L_{\mathcal{B}/\mathcal{A}} \otimes_\mathcal{B}^\mathbf{L} \mathcal{F} [1]
\end{equation}
in $D(\mathcal{B})$.

\begin{definition}
\label{definition-atiyah-class-general}
Let $\mathcal{C}$ be a site.
 ========

regex_match $\mathcal{C}$
nn candidate a site
nn a site
new_annotation [(127666, 127679, 'VAR'), (127683, 127689, 'TYPE')]
===== sent Let $\mathcal{A} \to \mathcal{B}$ be a homomorphism of sheaves of rings.
 ========

regex_match $\mathcal{A} \to \mathcal{B}$
nn candidate $\mathcal{A
nn candidate a homomorphism
nn a homomorphism
new_annotation [(127695, 127724, 'VAR'), (127728, 127742, 'TYPE')]
===== sent Let $\mathcal{F}$ be a sheaf of $\mathcal{B}$-modules.
 ========

regex_match $\mathcal{F}$
nn candidate a sheaf
nn a sheaf
new_annotation [(127768, 127781, 'VAR'), (127785, 127792, 'TYPE')]
===== sent \begin{definition}
\label{definition-cotangent-complex-morphism-ringed-spaces}
Let $f : (X, \mathcal{O}_X) \to (S, \mathcal{O}_S)$ be a morphism of
ringed spaces. ========

regex_match $f : (X, \mathcal{O}_X) \to (S, \mathcal{O}_S)$
nn candidate (X
nn candidate a morphism
nn a morphism
new_annotation [(128354, 128401, 'VAR'), (128405, 128415, 'TYPE')]
===== sent \begin{lemma}
\label{lemma-H0-L-morphism-ringed-spaces}
Let $f : (X, \mathcal{O}_X) \to (S, \mathcal{O}_S)$ be a morphism of
ringed spaces. ========

regex_match $f : (X, \mathcal{O}_X) \to (S, \mathcal{O}_S)$
nn candidate (X
nn candidate a morphism
nn a morphism
new_annotation [(129030, 129077, 'VAR'), (129081, 129091, 'TYPE')]
===== sent Then $H^0(L_{X/S}) = \Omega_{X/S}$.
\end{lemma}

\begin{proof}
Special case of Lemma \ref{lemma-H0-L-morphism-sheaves-rings}.
\end{proof}

\begin{lemma}
\label{lemma-triangle-ringed-spaces}
Let $f : X \to Y$ and $g : Y \to Z$ be morphisms of ringed spaces.
 ========

new_annotation []
===== sent Hence the distinguished triangle above is an example of the
distinguished triangle of
Lemma \ref{lemma-triangle-sheaves-rings}
with $\mathcal{A} = h^{-1}\mathcal{O}_Z$, $\mathcal{B} = f^{-1}\mathcal{O}_Y$,
and $\mathcal{C} = \mathcal{O}_X$.
\end{proof}

\begin{lemma}
\label{lemma-compare-cotangent-complex-with-naive-ringed-spaces}
Let $f : (X, \mathcal{O}_X) \to (Y, \mathcal{O}_Y)$ be a morphism of
ringed spaces. ========

regex_match $f : (X, \mathcal{O}_X) \to (Y, \mathcal{O}_Y)$
nn candidate a morphism
nn a morphism
new_annotation [(130130, 130177, 'VAR'), (130181, 130191, 'TYPE')]
===== sent Let $\alpha \in \Ext^1_{\mathcal{O}_S}(\NL_{S/\mathbf{Z}}, \mathcal{J})$
be the element corresponding to the isomorphism class of $S'$.
The existence of $X'$ corresponds to an element
$\beta \in \Ext_{\mathcal{O}_X}^1(\NL_{X/\mathbf{Z}}, \mathcal{G})$
which maps to the image of $\alpha$ in
$\Ext^1_{\mathcal{O}_X}(Lf^*\NL_{S/\mathbf{Z}}, \mathcal{G})$.
Note that
$$
\Ext^1_{\mathcal{O}_X}(Lf^*\NL_{S/\mathbf{Z}}, \mathcal{G}) =
\Ext^1_{\mathcal{O}_X}(Lf^*L_{S/\mathbf{Z}}, \mathcal{G})
$$
and
$$
\Ext^1_{\mathcal{O}_X}(\NL_{X/\mathbf{Z}}, \mathcal{G}) =
\Ext^1_{\mathcal{O}_X}(L_{X/\mathbf{Z}}, \mathcal{G})
$$
by Lemma \ref{lemma-compare-cotangent-complex-with-naive-ringed-spaces}.
The distinguished triangle of Lemma \ref{lemma-triangle-ringed-spaces}
for $X \to S \to (*, \mathbf{Z})$ gives rise to a long exact sequence
$$
\ldots \to
\Ext^1_{\mathcal{O}_X}(L_{X/\mathbf{Z}}, \mathcal{G}) \to
\Ext^1_{\mathcal{O}_X}(Lf^*L_{S/\mathbf{Z}}, \mathcal{G}) \to
\Ext^2_{\mathcal{O}_X}(L_{X/S}, \mathcal{G}) \to \ldots
$$
We obtain the result with $\xi$ the image of $\alpha$.
\end{proof}










\section{The cotangent complex of a morphism of ringed topoi}
\label{section-cotangent-morphism-ringed-topoi}

\noindent
The cotangent complex of a morphism of ringed topoi is defined
in terms of the cotangent complex we defined above.

 ========

new_annotation []
===== sent \begin{definition}
\label{definition-cotangent-complex-morphism-ringed-topoi}
Let $(f, f^\sharp) : (\Sh(\mathcal{C}), \mathcal{O}_\mathcal{C}) \to
(\Sh(\mathcal{D}), \mathcal{O}_\mathcal{D})$ be a morphism of ringed topoi.
 ========

regex_match $(f, f^\sharp) : (\Sh(\mathcal{C}), \mathcal{O}_\mathcal{C}) \to
(\Sh(\mathcal{D}), \mathcal{O}_\mathcal{D})$
nn candidate a morphism
nn a morphism
new_annotation [(134588, 134697, 'VAR'), (134701, 134711, 'TYPE')]
===== sent On the other hand, if we take $L_f$ where we think of $f$ as a morphism
between the underlying Zariski ringed topoi, then $L_f$ does agree with
the cotangent complex $L_{X/Y}$ (as defined below)
whose zeroth cohomology sheaf is $\Omega_{X/Y}$.

\begin{lemma}
\label{lemma-H0-L-morphism-ringed-topoi}
Let $f : (\Sh(\mathcal{C}), \mathcal{O}) \to
(\Sh(\mathcal{B}), \mathcal{O}_\mathcal{B})$ be a morphism of
ringed topoi. ========

regex_match $f : (\Sh(\mathcal{C}), \mathcal{O}) \to
(\Sh(\mathcal{B}), \mathcal{O}_\mathcal{B})$
nn candidate a morphism
nn a morphism
new_annotation [(135674, 135759, 'VAR'), (135763, 135773, 'TYPE')]
===== sent Then $H^0(L_f) = \Omega_f$.
\end{lemma}

\begin{proof}
Special case of Lemma \ref{lemma-H0-L-morphism-sheaves-rings}.
\end{proof}

\begin{lemma}
\label{lemma-triangle-ringed-topoi}
Let $f : (\Sh(\mathcal{C}_1), \mathcal{O}_1) \to
(\Sh(\mathcal{C}_2), \mathcal{O}_2)$ and
$g : (\Sh(\mathcal{C}_2), \mathcal{O}_2) \to
(\Sh(\mathcal{C}_3), \mathcal{O}_3)$ be morphisms of ringed topoi.
 ========

new_annotation []
===== sent Hence the distinguished triangle above is an example of the
distinguished triangle of
Lemma \ref{lemma-triangle-sheaves-rings}
with $\mathcal{A} = h^{-1}\mathcal{O}_3$, $\mathcal{B} = f^{-1}\mathcal{O}_2$,
and $\mathcal{C} = \mathcal{O}_1$.
\end{proof}

\begin{lemma}
\label{lemma-compare-cotangent-complex-with-naive-ringed-topoi}
Let $f : (\Sh(\mathcal{C}), \mathcal{O}) \to
(\Sh(\mathcal{B}), \mathcal{O}_\mathcal{B})$ be a morphism of
ringed topoi. ========

regex_match $f : (\Sh(\mathcal{C}), \mathcal{O}) \to
(\Sh(\mathcal{B}), \mathcal{O}_\mathcal{B})$
nn candidate (\Sh(\mathcal{C
nn candidate a morphism
nn a morphism
new_annotation [(136926, 137011, 'VAR'), (137015, 137025, 'TYPE')]
===== sent $$
Let $\alpha \in \Ext^1_{\mathcal{O}_\mathcal{B}}(\NL_q, \mathcal{J})$
be the element corresponding to the isomorphism class of
$\mathcal{O}_{\mathcal{B}'}$. The existence of $\mathcal{O}'$
corresponds to an element
$\beta \in \Ext_\mathcal{O}^1(\NL_p, \mathcal{G})$
which maps to the image of $\alpha$ in
$\Ext^1_{\mathcal{O}_X}(Lf^*\NL_q, \mathcal{G})$.
Note that
$$
\Ext^1_{\mathcal{O}_X}(Lf^*\NL_q, \mathcal{G}) =
\Ext^1_{\mathcal{O}_X}(Lf^*L_q, \mathcal{G})
$$
and
$$
\Ext^1_{\mathcal{O}_X}(\NL_p, \mathcal{G}) =
\Ext^1_{\mathcal{O}_X}(L_p, \mathcal{G})
$$
by Lemma \ref{lemma-compare-cotangent-complex-with-naive-ringed-topoi}.
The distinguished triangle of Lemma \ref{lemma-triangle-ringed-topoi}
for $p = q \circ f$ gives rise to a long exact sequence
$$
\ldots \to
\Ext^1_{\mathcal{O}_X}(L_p, \mathcal{G}) \to
\Ext^1_{\mathcal{O}_X}(Lf^*L_q, \mathcal{G}) \to
\Ext^2_{\mathcal{O}_X}(L_f, \mathcal{G}) \to \ldots
$$
We obtain the result with $\xi$ the image of $\alpha$.
\end{proof}












\section{The cotangent complex of a morphism of schemes}
\label{section-cotangent-morphism-schemes}

\noindent
As promised above we define the cotangent complex of a morphism of
schemes as follows.

 ========

new_annotation []
===== sent \begin{definition}
\label{definition-cotangent-morphism-schemes}
Let $f : X \to Y$ be a morphism of schemes. ========

regex_match $f : X \to Y$
nn candidate X \to
nn candidate a morphism
nn a morphism
new_annotation [(141588, 141601, 'VAR'), (141605, 141615, 'TYPE')]
===== sent The next lemma shows this definition is compatible with the definition
for ring maps and it also implies that $L_{X/Y}$ is an
object of $D_\QCoh(\mathcal{O}_X)$.

\begin{lemma}
\label{lemma-morphism-affine-schemes}
Let $f : X \to Y$ be a morphism of schemes. ========

regex_match $f : X \to Y$
nn candidate X \to
nn candidate a morphism
nn a morphism
new_annotation [(142203, 142216, 'VAR'), (142220, 142230, 'TYPE')]
===== sent Let $U = \Spec(A) \subset X$
and ========

new_annotation []
===== sent \end{proof}

\begin{lemma}
\label{lemma-scheme-over-ring}
Let $\Lambda$ be a ring. ========

regex_match $\Lambda$
nn candidate a ring
nn a ring
new_annotation [(143748, 143757, 'VAR'), (143761, 143767, 'TYPE')]
===== sent Let $X$ be a scheme over $\Lambda$.
Then
$$
L_{X/\Spec(\Lambda)} = L_{\mathcal{O}_X/\underline{\Lambda}}
$$
where $\underline{\Lambda}$ is the constant sheaf with value
$\Lambda$ on $X$.
\end{lemma}

\begin{proof}
Let $p : X \to \Spec(\Lambda)$ be the structure morphism.
 ========

regex_match $X$
nn candidate a scheme
nn a scheme
new_annotation [(143773, 143776, 'VAR'), (143780, 143788, 'TYPE')]
===== sent Let $X$ be a scheme over $\Lambda$.
Then
$$
L_{X/\Spec(\Lambda)} = L_{\mathcal{O}_X/\underline{\Lambda}}
$$
where $\underline{\Lambda}$ is the constant sheaf with value
$\Lambda$ on $X$.
\end{lemma}

\begin{proof}
Let $p : X \to \Spec(\Lambda)$ be the structure morphism.
 ========

regex_match $p : X \to \Spec(\Lambda)$
nn candidate the structure morphism
nn the structure morphism
new_annotation [(143987, 144013, 'VAR'), (144017, 144039, 'TYPE')]
===== sent Let $q : \Spec(\Lambda) \to (*, \Lambda)$ be the obvious morphism.
 ========

regex_match $q : \Spec(\Lambda) \to (*, \Lambda)$
nn candidate $q : \Spec(\Lambda) \to
nn candidate the obvious morphism
nn the obvious morphism
new_annotation [(144045, 144082, 'VAR'), (144086, 144106, 'TYPE')]
===== sent To see this it suffices to
show for $\mathfrak p \in \Spec(\Lambda)$ that
$$
(L_q)_\mathfrak p =
L_{\mathcal{O}_{\Spec(\Lambda), \mathfrak p}/\Lambda} =
L_{\Lambda_\mathfrak p/\Lambda}
$$
(Lemma \ref{lemma-stalk-cotangent-complex})
is zero which follows from Lemma \ref{lemma-when-zero}.
\end{proof}






\section{The cotangent complex of a scheme over a ring}
\label{section-cotangent-schemes-variant}

\noindent
Let $\Lambda$ be a ring and let $X$ be a scheme over $\Lambda$.
Write $L_{X/\Spec(\Lambda)} = L_{X/\Lambda}$ ========

regex_match $\Lambda$
nn candidate a ring
nn a ring
new_annotation [(144637, 144646, 'VAR'), (144650, 144656, 'TYPE')]
===== sent To see this it suffices to
show for $\mathfrak p \in \Spec(\Lambda)$ that
$$
(L_q)_\mathfrak p =
L_{\mathcal{O}_{\Spec(\Lambda), \mathfrak p}/\Lambda} =
L_{\Lambda_\mathfrak p/\Lambda}
$$
(Lemma \ref{lemma-stalk-cotangent-complex})
is zero which follows from Lemma \ref{lemma-when-zero}.
\end{proof}






\section{The cotangent complex of a scheme over a ring}
\label{section-cotangent-schemes-variant}

\noindent
Let $\Lambda$ be a ring and let $X$ be a scheme over $\Lambda$.
Write $L_{X/\Spec(\Lambda)} = L_{X/\Lambda}$ ========

regex_match $X$
nn candidate a scheme
nn a scheme
new_annotation [(144665, 144668, 'VAR'), (144672, 144680, 'TYPE')]
===== sent \medskip\noindent
Let $\Lambda$ be a ring. ========

regex_match $\Lambda$
nn candidate a ring
nn a ring
new_annotation [(145351, 145360, 'VAR'), (145364, 145370, 'TYPE')]
===== sent Let $X$ be a scheme over $\Lambda$.
Let $\mathcal{C}_{X/\Lambda}$ be the category whose objects are
commutative diagrams
\begin{equation}
\label{equation-object}
\vcenter{
\xymatrix{
X \ar[d] & U \ar[l] \ar[d] \\
 ========

regex_match $X$
nn candidate a scheme
nn a scheme
new_annotation [(145376, 145379, 'VAR'), (145383, 145391, 'TYPE')]
===== sent Let $X$ be a scheme over $\Lambda$.
Let $\mathcal{C}_{X/\Lambda}$ be the category whose objects are
commutative diagrams
\begin{equation}
\label{equation-object}
\vcenter{
\xymatrix{
X \ar[d] & U \ar[l] \ar[d] \\
 ========

regex_match $\mathcal{C}_{X/\Lambda}$
nn candidate the category
nn the category
new_annotation [(145412, 145437, 'VAR'), (145441, 145453, 'TYPE')]
===== sent \begin{definition}
\label{definition-cotangent-morphism-spaces}
Let $S$ be a scheme. ========

regex_match $S$
nn candidate a scheme
nn a scheme
new_annotation [(151833, 151836, 'VAR'), (151840, 151848, 'TYPE')]
===== sent Let $f : X \to Y$ be a morphism of algebraic spaces
over $S$. ========

regex_match $f : X \to Y$
nn candidate $
nn candidate X \to
nn candidate a morphism
nn a morphism
new_annotation [(151854, 151867, 'VAR'), (151871, 151881, 'TYPE')]
===== sent The next lemmas show this definition is compatible with the definition
for ring maps and for schemes and that $L_{X/Y}$ is an
object of $D_\QCoh(\mathcal{O}_X)$.

\begin{lemma}
\label{lemma-etale-localization}
Let $S$ be a scheme. ========

regex_match $S$
nn candidate a scheme
nn a scheme
new_annotation [(152630, 152633, 'VAR'), (152637, 152645, 'TYPE')]
===== sent Tracing through the definitions we conclude that
$L_{X/Y}|_{U_\etale} = L_{U/V}$.
\end{proof}

\begin{lemma}
\label{lemma-compare-spaces-schemes}
Let $S$ be a scheme. ========

regex_match $S$
nn candidate a scheme
nn a scheme
new_annotation [(153496, 153499, 'VAR'), (153503, 153511, 'TYPE')]
===== sent Let $f : X \to Y$ be a morphism of algebraic spaces
over $S$. Assume $X$ and $Y$ representable by schemes $X_0$ and $Y_0$.
 ========

regex_match $f : X \to Y$
nn candidate $
nn candidate X \to
nn candidate a morphism
nn a morphism
new_annotation [(153517, 153530, 'VAR'), (153534, 153544, 'TYPE')]
===== sent Then there is a canonical identification
$L_{X/Y} = \epsilon^*L_{X_0/Y_0}$ in $D(\mathcal{O}_X)$
where $\epsilon$ is as in Derived Categories of Spaces, Section
\ref{spaces-perfect-section-derived-quasi-coherent-etale}
and $L_{X_0/Y_0}$ is as in
 Definition \ref{definition-cotangent-morphism-schemes}.
\end{lemma}

\begin{proof}
Let $f_0 : X_0 \to Y_0$ be the morphism of schemes corresponding to $f$.
There is a canonical map
$\epsilon^{-1}f_0^{-1}\mathcal{O}_{Y_0} \to f_{small}^{-1}\mathcal{O}_Y$
compatible with
$\epsilon^\sharp : \epsilon^{-1}\mathcal{O}_{X_0} \to \mathcal{O}_X$
because there is a commutative diagram
$$
\xymatrix{
X_{0, Zar} \ar[d]_{f_0} & X_\etale \ar[l]^\epsilon \ar[d]^f \\
 ========

regex_match $f_0 : X_0 \to Y_0$
nn candidate $f_0
nn candidate the morphism
nn the morphism
new_annotation [(153970, 153989, 'VAR'), (153993, 154005, 'TYPE')]
===== sent Let $\overline{x} : \Spec(k) \to X_0$
be a geometric point lying over $x \in X_0$, with
$\overline{y} = f \circ \overline{x}$ lying over $y \in ========

new_annotation []
===== sent Thus the result follows from
Lemma \ref{lemma-cotangent-complex-henselization}.
\end{proof}

\begin{lemma}
\label{lemma-space-over-ring}
Let $\Lambda$ be a ring. ========

regex_match $\Lambda$
nn candidate a ring
nn a ring
new_annotation [(156298, 156307, 'VAR'), (156311, 156317, 'TYPE')]
===== sent Let $X$ be an algebraic space over $\Lambda$.
Then
$$
L_{X/\Spec(\Lambda)} = L_{\mathcal{O}_X/\underline{\Lambda}}
$$
where $\underline{\Lambda}$ is the constant sheaf with value
$\Lambda$ on $X_\etale$.
\end{lemma}

\begin{proof}
Let $p : X \to \Spec(\Lambda)$ be the structure morphism.
 ========

regex_match $X$
nn candidate an algebraic space
nn an algebraic space
new_annotation [(156323, 156326, 'VAR'), (156330, 156348, 'TYPE')]
===== sent Let $X$ be an algebraic space over $\Lambda$.
Then
$$
L_{X/\Spec(\Lambda)} = L_{\mathcal{O}_X/\underline{\Lambda}}
$$
where $\underline{\Lambda}$ is the constant sheaf with value
$\Lambda$ on $X_\etale$.
\end{lemma}

\begin{proof}
Let $p : X \to \Spec(\Lambda)$ be the structure morphism.
 ========

regex_match $p : X \to \Spec(\Lambda)$
nn candidate the structure morphism
nn the structure morphism
new_annotation [(156554, 156580, 'VAR'), (156584, 156606, 'TYPE')]
===== sent Let $q : \Spec(\Lambda)_\etale \to (*, \Lambda)$
be the obvious morphism. ========

new_annotation []
===== sent Since $\mathcal{O}_{\Spec(\Lambda)_\etale, \overline{t}}$
is a strict henselization of a local ring of $\Lambda$
(Properties of Spaces, Lemma
\ref{spaces-properties-lemma-describe-etale-local-ring})
this follows from Lemma \ref{lemma-when-zero}.
\end{proof}










\section{The cotangent complex of an algebraic space over a ring}
\label{section-cotangent-spaces-variant}

\noindent
Let $\Lambda$ be a ring and let $X$ be an algebraic space over $\Lambda$.
Write $L_{X/\Spec(\Lambda)} = L_{X/\Lambda}$ ========

regex_match $\Lambda$
nn candidate a ring
nn a ring
new_annotation [(157510, 157519, 'VAR'), (157523, 157529, 'TYPE')]
===== sent Since $\mathcal{O}_{\Spec(\Lambda)_\etale, \overline{t}}$
is a strict henselization of a local ring of $\Lambda$
(Properties of Spaces, Lemma
\ref{spaces-properties-lemma-describe-etale-local-ring})
this follows from Lemma \ref{lemma-when-zero}.
\end{proof}










\section{The cotangent complex of an algebraic space over a ring}
\label{section-cotangent-spaces-variant}

\noindent
Let $\Lambda$ be a ring and let $X$ be an algebraic space over $\Lambda$.
Write $L_{X/\Spec(\Lambda)} = L_{X/\Lambda}$ ========

regex_match $X$
nn candidate $X$
nn candidate an algebraic space
nn an algebraic space
new_annotation [(157538, 157541, 'VAR'), (157545, 157563, 'TYPE')]
===== sent \medskip\noindent
Let $\Lambda$ be a ring. ========

regex_match $\Lambda$
nn candidate a ring
nn a ring
new_annotation [(158234, 158243, 'VAR'), (158247, 158253, 'TYPE')]
===== sent Let $X$ be an algebraic space over $\Lambda$.
Let $\mathcal{C}_{X/\Lambda}$ be the category whose objects are
commutative diagrams
\begin{equation}
\label{equation-object-space}
\vcenter{
\xymatrix{
X \ar[d] & U \ar[l] \ar[d] \\
 ========

regex_match $X$
nn candidate an algebraic space
nn an algebraic space
new_annotation [(158259, 158262, 'VAR'), (158266, 158284, 'TYPE')]
===== sent Let $X$ be an algebraic space over $\Lambda$.
Let $\mathcal{C}_{X/\Lambda}$ be the category whose objects are
commutative diagrams
\begin{equation}
\label{equation-object-space}
\vcenter{
\xymatrix{
X \ar[d] & U \ar[l] \ar[d] \\
 ========

regex_match $\mathcal{C}_{X/\Lambda}$
nn candidate the category
nn the category
new_annotation [(158305, 158330, 'VAR'), (158334, 158346, 'TYPE')]
===== sent $U \to X$ of $X_\etale$
let $P_{\bullet, U}$ be the standard resolution of $\mathcal{O}_X(U)$
over $\Lambda$. ========

regex_match $P_{\bullet, U}$
nn candidate the standard resolution
nn the standard resolution
new_annotation [(161620, 161636, 'VAR'), (161640, 161663, 'TYPE')]
===== sent \left(\Omega_{\mathcal{O}/\underline{\Lambda}}
\otimes_\mathcal{O} \underline{\mathcal{O}}_X\right)(\mathbf{A}_{\bullet, U})
=
\Omega_{P_{\bullet, U}/\Lambda} \otimes_{P_{\bullet, U}} \mathcal{O}_X(U) =
L_{\mathcal{O}_X(U)/\Lambda}
$$
using notation as in Remark \ref{remark-compute-L-pi-shriek-spaces}.
Now Remark \ref{remark-map-sections-over-U} shows that
$L\pi_!(\Omega_{\mathcal{O}/\underline{\Lambda}}
\otimes_\mathcal{O} \underline{\mathcal{O}}_X)$
computes the cotangent complex of the map of rings
$\underline{\Lambda} \to \mathcal{O}_X$ on $X_\etale$.
This is what we want by Lemma \ref{lemma-space-over-ring}.
\end{proof}






\section{Fibre products of algebraic spaces and the cotangent complex}
\label{section-fibre-product}

\noindent
Let $S$ be a scheme. ========

regex_match $S$
nn candidate a scheme
nn a scheme
new_annotation [(164643, 164646, 'VAR'), (164650, 164658, 'TYPE')]
===== sent Let $X \to B$ and $Y \to B$ be morphisms of algebraic
spaces over $S$. Consider the fibre product $X \times_B Y$ with projection
morphisms $p : X \times_B Y \to X$ and $q : X \times_B Y \to Y$.
 ========

new_annotation []
===== sent In general we can say the following about the object $E$.

\begin{lemma}
\label{lemma-fibre-product}
Let $S$ be a scheme. ========

regex_match $S$
nn candidate a scheme
nn a scheme
new_annotation [(167368, 167371, 'VAR'), (167375, 167383, 'TYPE')]
===== sent Let $X \to B$ and $Y \to B$ be morphisms of algebraic
spaces over $S$. The object $E$ in (\ref{equation-fibre-product}) satisfies
$H^i(E) = 0$ for $i = 0, -1$ and for a geometric point
$(\overline{x}, \overline{y}) : \Spec(k) \to X \times_B Y$ we have
$$
H^{-2}(E)_{(\overline{x}, \overline{y})} =
\text{Tor}_1^R(A, B) \otimes_{A \otimes_R B} C
$$
where $R = \mathcal{O}_{B, \overline{b}}$, $A = \mathcal{O}_{X, \overline{x}}$,
$B = \mathcal{O}_{Y, \overline{y}}$, and
$C = \mathcal{O}_{X \times_B Y, (\overline{x}, \overline{y})}$.
\end{lemma}

\begin{proof}
The formation of the cotangent complex commutes with taking stalks
and pullbacks, see
Lemmas \ref{lemma-stalk-cotangent-complex} and
\ref{lemma-pullback-cotangent-morphism-topoi}.
Note that $C$ is a henselization of $A \otimes_R B$.
$L_{C/R} = L_{A \otimes_R B/R} \otimes_{A \otimes_R B} C$
by the results of Section \ref{section-localization}.
 ========

new_annotation []
file:  stacks-more-morphisms.tex
===== sent \item Let $\mathcal{Z}$ be an algebraic stack. ========

regex_match $\mathcal{Z}$
nn candidate an algebraic stack
nn an algebraic stack
new_annotation [(1737, 1750, 'VAR'), (1754, 1772, 'TYPE')]
===== sent \end{enumerate}
\end{definition}

\noindent
Let $\mathcal{X} \subset \mathcal{X}'$ be a thickening of algebraic stacks.
 ========

regex_match $\mathcal{X} \subset \mathcal{X}'$
nn candidate $\mathcal{X
nn candidate a thickening
nn a thickening
new_annotation [(2156, 2190, 'VAR'), (2194, 2206, 'TYPE')]
===== sent Let $U'$ be a scheme and let $U' \to \mathcal{X}'$ be a surjective smooth
morphism. ========

regex_match $U'$
nn candidate a scheme
nn a scheme
new_annotation [(2232, 2236, 'VAR'), (2240, 2248, 'TYPE')]
===== sent Let $U'$ be a scheme and let $U' \to \mathcal{X}'$ be a surjective smooth
morphism. ========

regex_match $U' \to \mathcal{X}'$
nn candidate $U
nn candidate a surjective smooth
morphism
nn a surjective smooth
morphism
new_annotation [(2257, 2278, 'VAR'), (2282, 2310, 'TYPE')]
===== sent Sometimes, by abuse of language, we say that a morphism
$\mathcal{X} \to \mathcal{X}'$ is a thickening if it is a closed
immersion inducing a bijection $|\mathcal{X}| \to |\mathcal{X}'|$.

\begin{lemma}
\label{lemma-thickening}
Let $i : \mathcal{X} \to \mathcal{X}'$ be a morphism of algebraic stacks.
 ========

regex_match $i : \mathcal{X} \to \mathcal{X}'$
nn candidate $i
nn candidate a morphism
nn a morphism
new_annotation [(2912, 2946, 'VAR'), (2950, 2960, 'TYPE')]
===== sent \begin{lemma}
\label{lemma-base-change-thickening}
Let $\mathcal{Y} \subset \mathcal{Y}'$ be a thickening of algebraic stacks.
 ========

regex_match $\mathcal{Y} \subset \mathcal{Y}'$
nn candidate $\mathcal{Y
nn candidate a thickening
nn a thickening
new_annotation [(5532, 5566, 'VAR'), (5570, 5582, 'TYPE')]
===== sent Let $\mathcal{X}' \to \mathcal{Y}'$ be a morphism of algebraic stacks
and set $\mathcal{X} = \mathcal{Y} \times_{\mathcal{Y}'} \mathcal{X}'$.
Then
$(\mathcal{X} \subset \mathcal{X}') \to (\mathcal{Y} \subset \mathcal{Y}')$
is a morphism of thickenings. ========

regex_match $\mathcal{X}' \to \mathcal{Y}'$
nn candidate a morphism
nn a morphism
new_annotation [(5608, 5639, 'VAR'), (5643, 5653, 'TYPE')]
===== sent \end{lemma}

\begin{proof}
See discussion above, Properties of Stacks, Section
\ref{stacks-properties-section-properties-morphisms}, and
More on Morphisms of Spaces, Lemma
\ref{spaces-more-morphisms-lemma-base-change-thickening}.
\end{proof}

\begin{lemma}
\label{lemma-composition-thickening}
If $\mathcal{X} \subset \mathcal{X}'$ and $\mathcal{X}' \subset \mathcal{X}''$
are thickenings of algebraic stacks, then so is
$\mathcal{X} \subset \mathcal{X}''$.
\end{lemma}

\begin{proof}
See discussion above, Properties of Stacks, Section
\ref{stacks-properties-section-properties-morphisms}, and
More on Morphisms of Spaces, Lemma
\ref{spaces-more-morphisms-lemma-composition-thickening}
\end{proof}

\begin{example}
\label{example-reduction-thickening}
Let $\mathcal{X}'$ be an algebraic stack. ========

regex_match $\mathcal{X}'$
nn candidate an algebraic stack
nn an algebraic stack
new_annotation [(6750, 6764, 'VAR'), (6768, 6786, 'TYPE')]
===== sent In other words, $\mathcal{X} = \mathcal{X}'_{red}$ if and only
if $\mathcal{X}$ is a reduced algebraic stack.
\end{example}

\begin{lemma}
\label{lemma-reduced-diagonal}
Let $(f, f') : (\mathcal{X} \subset \mathcal{X}') \to
(\mathcal{Y} \subset \mathcal{Y}')$ be a morphism of thickenings
of algebraic stacks. ========

regex_match $(f, f') : (\mathcal{X} \subset \mathcal{X}') \to
(\mathcal{Y} \subset \mathcal{Y}')$
nn candidate (\mathcal{Y} \subset
nn candidate a morphism
nn a morphism
new_annotation [(7284, 7369, 'VAR'), (7373, 7383, 'TYPE')]
===== sent Since $\mathcal{X} \to \mathcal{X}'$ is a monomorphism,
the final statement of the lemma follows from
Properties of Stacks, Lemma
\ref{stacks-properties-lemma-monomorphism-diagonal}
applied to $\mathcal{X} \to \mathcal{X}' \to \mathcal{Y}'$.
\end{proof}

\begin{lemma}
\label{lemma-thickening-diagonals}
Let $(f, f') : (\mathcal{X} \subset \mathcal{X}') \to
(\mathcal{Y} \subset \mathcal{Y}')$ be a morphism of thickenings
of algebraic stacks.
 ========

regex_match $(f, f') : (\mathcal{X} \subset \mathcal{X}') \to
(\mathcal{Y} \subset \mathcal{Y}')$
nn candidate (\mathcal{Y} \subset
nn candidate a morphism
nn a morphism
new_annotation [(8763, 8848, 'VAR'), (8852, 8862, 'TYPE')]
===== sent Let $\Delta : \mathcal{X} \to \mathcal{X} \times_\mathcal{Y} \mathcal{X}$ and
$\Delta' : \mathcal{X}' \to \mathcal{X}' \times_{\mathcal{Y}'} \mathcal{X}'$
be the corresponding diagonal morphisms.
 ========

new_annotation []
===== sent \begin{lemma}
\label{lemma-thickening-properties}
\begin{reference}
\cite[Theorem 2.2.5]{Conrad-moduli}
\end{reference}
Let $\mathcal{X} \subset \mathcal{X}'$ be a thickening of algebraic
stacks. ========

regex_match $\mathcal{X} \subset \mathcal{X}'$
nn candidate $\mathcal{X
nn candidate a thickening
nn a thickening
new_annotation [(11018, 11052, 'VAR'), (11056, 11068, 'TYPE')]
===== sent \begin{lemma}
\label{lemma-thicken-property-morphisms}
Let $(f, f') : (\mathcal{X} \subset \mathcal{X}') \to
(\mathcal{Y} \subset \mathcal{Y}')$
be a morphism of thickenings of algebraic stacks. ========

new_annotation []
===== sent Conversely, assume $f$ is representable, and let $T' \to \mathcal{Y}'$
be a morphism where $T'$ is a scheme. ========

new_annotation []
===== sent Choose an algebraic space $W'$ and a surjective smooth morphism
$W' \to V' \times_{\mathcal{Y}'} \mathcal{X}'$. Let $U, V, W$
be the base change of $U', V', W'$ by $\mathcal{B} \to \mathcal{B}'$.
 ========

new_annotation []
===== sent Choose an algebraic space $W'$ and a surjective smooth morphism
$W' \to V' \times_{\mathcal{Y}'} \mathcal{X}'$. Let $U, V, W$
be the base change of $U', V', W'$ by $\mathcal{B} \to \mathcal{B}'$.
Then the property for $f$, resp.\ $f'$
is equivalent to the property for of $W' \to V'$, resp.\ $W \to V$.
 ========

new_annotation []
===== sent $W' \to V' \times_{\mathcal{Y}'} \mathcal{X}'$. Let $U, V, W$
be the base change of $U', V', W'$ by $\mathcal{B} \to \mathcal{B}'$.
Then $W \to V \times_\mathcal{Y} \mathcal{X}$ is surjective
\'etale as well. ========

new_annotation []
===== sent Let $U, V, W$ be the base change of
$U', V', W'$ by $\mathcal{B} \to \mathcal{B}'$.
Then $W = V \times_\mathcal{Y} \mathcal{X}$ as well.
 ========

regex_match $U, V, W$
nn candidate $U
nn candidate the base change
nn the base change
new_annotation [(23259, 23268, 'VAR'), (23272, 23287, 'TYPE')]
===== sent Namely, set $b = a \circ f$. Let $f' : V' \to U'$ be the \'etale morphism
whose restriction to $V$ is $f$ (More on Morphisms of Spaces,
Lemma \ref{spaces-more-morphisms-lemma-topological-invariance}).
 ========

regex_match $f' : V' \to U'$
nn candidate V
nn candidate the \'etale morphism
nn the \'etale morphism
new_annotation [(28845, 28861, 'VAR'), (28865, 28885, 'TYPE')]
===== sent Let $y' = x' \circ f'$ and $\gamma = \text{id} : x' \circ f' \to y'$.
Set
$$
\beta = \alpha \star \text{id}_f :
x \circ b = x \circ a \circ f \to
x' \circ i \circ f = x' \circ f' \circ j = y' ========

new_annotation []
===== sent Finally, if we temporarily view the $1$-morphisms
$x'_k : U'_k \to \mathcal{X}'$ as objects of the stack
$\mathcal{X}'$ over $U'_k$ then we see that these come endowed with a
descent datum with respect to the \'etale covering
$\{U'_k \to U'\}$ given by the third component $\gamma_{kk'}$
of the morphisms $\varphi_{kk'}$ in $\mathcal{C}$.
Since $\mathcal{X}'$ is a stack
this descent datum is effective and translating back we obtain
a $1$-morphism $x' : U' \to \mathcal{X}'$ such that the compositions
$U'_k \to U' \to \mathcal{X}'$ come equipped with isomorphisms to $x'_k$
compatible with $\gamma_{kk'}$. This means that the morphisms
$\alpha_k : x \circ a_k \to x'_k \circ i_k$ glue to a morphism
$\alpha : x \circ a \to x' \circ i$. Then $\xi = (U, U', a, i, x', \alpha)$
is the desired object over $U$.
\end{proof}

\begin{lemma}
\label{lemma-etale-local-lifts}
Let $\mathcal{X} \subset \mathcal{X}'$ be a thickening of algebraic stacks.
 ========

regex_match $\mathcal{X} \subset \mathcal{X}'$
nn candidate $\mathcal{X
nn candidate a thickening
nn a thickening
new_annotation [(34284, 34318, 'VAR'), (34322, 34334, 'TYPE')]
===== sent Let $W$ be an algebraic space and let $W \to \mathcal{X}$ be a smooth morphism.
 ========

regex_match $W$
nn candidate an algebraic space
nn an algebraic space
new_annotation [(34360, 34363, 'VAR'), (34367, 34385, 'TYPE')]
===== sent Let $W$ be an algebraic space and let $W \to \mathcal{X}$ be a smooth morphism.
 ========

regex_match $W \to \mathcal{X}$
nn candidate a smooth morphism
nn a smooth morphism
new_annotation [(34394, 34413, 'VAR'), (34417, 34434, 'TYPE')]
===== sent Let $W'_w \subset U'_w$ be the closed subscheme cut out by
$f'_1, \ldots, f'_r$.
We claim that $W'_w \to \mathcal{X}'$ is smooth.
 ========

regex_match $W'_w \subset U'_w$
nn candidate the closed subscheme
nn the closed subscheme
new_annotation [(36987, 37006, 'VAR'), (37010, 37030, 'TYPE')]
===== sent \end{proof}

\begin{lemma}
\label{lemma-etale-local-lifts-isomorphic}
Let $\mathcal{X} \subset \mathcal{X}'$ be a thickening of algebraic stacks.
 ========

regex_match $\mathcal{X} \subset \mathcal{X}'$
nn candidate $\mathcal{X
nn candidate a thickening
nn a thickening
new_annotation [(38959, 38993, 'VAR'), (38997, 39009, 'TYPE')]
===== sent \end{reference}
Let $\mathcal{X} \subset \mathcal{X}'$ be a first order thickening
of algebraic stacks. ========

regex_match $\mathcal{X} \subset \mathcal{X}'$
nn candidate $\mathcal{X
nn candidate a first order thickening
nn a first order thickening
new_annotation [(50536, 50570, 'VAR'), (50574, 50598, 'TYPE')]
===== sent Let $W$ be an affine scheme and let
$W \to \mathcal{X}$ be a smooth morphism. ========

regex_match $W$
nn candidate an affine scheme
nn an affine scheme
new_annotation [(50624, 50627, 'VAR'), (50631, 50647, 'TYPE')]
===== sent \end{proof}






\section{Infinitesimal deformations}
\label{section-inf}

\noindent
We continue the discussion from
Artin's Axioms, Section \ref{artin-section-inf}.

\begin{lemma}
\label{lemma-inf-quasi-coherent}
Let $\mathcal{X}$ be an algebraic stack over a scheme $S$.
Assume $\mathcal{I}_\mathcal{X} \to \mathcal{X}$ is locally
of finite presentation. ========

regex_match $\mathcal{X}$
nn candidate an algebraic stack
nn an algebraic stack
new_annotation [(52633, 52646, 'VAR'), (52650, 52668, 'TYPE')]
===== sent Let $A \to B$ be a flat $S$-algebra homomorphism.
 ========

regex_match $A \to B$
nn candidate $
nn candidate a flat $S$-algebra homomorphism
nn a flat $S$-algebra homomorphism
new_annotation [(52776, 52785, 'VAR'), (52789, 52820, 'TYPE')]
===== sent Let $x$ be an object of $\mathcal{X}$ over $A$ and set $y = x|_B$.
Then $\text{Inf}_x(M) \otimes_A B = \text{Inf}_y(M \otimes_A B)$.
\end{lemma}

\begin{proof}
Recall that $\text{Inf}_x(M)$ is the set of automorphisms of the
trivial deformation of $x$ to $A[M]$ which induce the identity
automorphism of $x$ over $A$. ========

regex_match $x$
nn candidate an object
nn an object
new_annotation [(52826, 52829, 'VAR'), (52833, 52842, 'TYPE')]
===== sent The trivial deformation is
the pullback of $x$ to $\Spec(A[M])$ via $\Spec(A[M]) \to \Spec(A)$.
Let $G \to \Spec(A)$ be the automorphism group algebraic space of $x$
(this exists because $\mathcal{X}$ is an algebraic space).
 ========

regex_match $G \to \Spec(A)$
nn candidate the automorphism group algebraic space
nn the automorphism group algebraic space
new_annotation [(53240, 53256, 'VAR'), (53260, 53298, 'TYPE')]
===== sent Let $e : \Spec(A) \to G$ be the neutral element.
 ========

regex_match $e : \Spec(A) \to G$
nn candidate the neutral element
nn the neutral element
new_annotation [(53369, 53389, 'VAR'), (53393, 53412, 'TYPE')]
===== sent Therefore $e_B^*\Omega_{G_B/B} = e^*\Omega_{G/A} \otimes_A B$.
we conclude by More on Algebra, Remark
\ref{more-algebra-remark-pseudo-coherence-and-base-change-ext}.
\end{proof}

\begin{lemma}
\label{lemma-sheaf-of-infinitesimal-lifts}
Let $\mathcal{X}$ be an algebraic stack over a base scheme $S$.
Assume $\mathcal{I}_\mathcal{X} \to \mathcal{X}$ is locally
of finite presentation. ========

regex_match $\mathcal{X}$
nn candidate an algebraic stack
nn an algebraic stack
new_annotation [(54383, 54396, 'VAR'), (54400, 54418, 'TYPE')]
===== sent Let $(A' \to A, x)$ be a deformation situation.
 ========

regex_match $(A' \to A, x)$
nn candidate A' \to A
nn candidate a deformation situation
nn a deformation situation
new_annotation [(54531, 54546, 'VAR'), (54550, 54573, 'TYPE')]
===== sent Then the functor
$$
F : B' \longmapsto
\{\text{lifts of }x|_{B' \otimes_{A'} A}\text{ to } B'\}/\text{isomorphisms}
$$
is a sheaf on the site $(\textit{Aff}/\Spec(A'))_{fppf}$ of
Topologies, Definition \ref{topologies-definition-big-small-fppf}.
\end{lemma}

\begin{proof}
Let $\{T'_i \to T'\}_{i = 1, \ldots n}$ be a standard fppf covering
of affine schemes over $A'$. Write $T' = \Spec(B')$. ========

regex_match $\{T'_i \to T'\}_{i = 1, \ldots n}$
nn candidate \to T'\}_{i
nn candidate \ldots
nn candidate a standard fppf
nn a standard fppf
new_annotation [(54852, 54887, 'VAR'), (54891, 54906, 'TYPE')]
===== sent Denote $y = x|_B$ and $y_{i_0 \ldots i_p} = x|_{B_{i_0 \ldots i_p}}$.
Let $\gamma_i \in F(B'_i)$ such that $\gamma_{i_0}$ and $\gamma_{i_1}$
map to the same element of $F(B'_{i_0i_1})$.
We have to find a unique $\gamma \in F(B')$ mapping to
$\gamma_i$ in $F(B'_i)$.

\medskip\noindent
Choose an actual object $y'_i$ of $\textit{Lift}(y_i, ========

new_annotation []
===== sent \end{proof}

\begin{lemma}
\label{lemma-T-quasi-coherent}
Let $\mathcal{X}$ be an algebraic stack over a scheme $S$ whose
structure morphism $\mathcal{X} \to S$ is locally of finite presentation.
 ========

regex_match $\mathcal{X}$
nn candidate an algebraic stack
nn an algebraic stack
new_annotation [(58070, 58083, 'VAR'), (58087, 58105, 'TYPE')]
===== sent Let $A \to B$ be a flat $S$-algebra homomorphism.
 ========

regex_match $A \to B$
nn candidate a flat $S$-algebra homomorphism
nn a flat $S$-algebra homomorphism
new_annotation [(58208, 58217, 'VAR'), (58221, 58252, 'TYPE')]
===== sent Let $x$ be an object of $\mathcal{X}$ over $A$.
 ========

regex_match $x$
nn candidate an object
nn an object
new_annotation [(58258, 58261, 'VAR'), (58265, 58274, 'TYPE')]
===== sent Clearly, we have $y^*\Omega_{X/S} = x^*\Omega_{X/S} \otimes_A B$.
we conclude by More on Algebra, Remark
\ref{more-algebra-remark-pseudo-coherence-and-base-change-ext}.
\end{proof}

\begin{lemma}
\label{lemma-local-lift-enough}
Let $\mathcal{X}$ be an algebraic stack over a scheme $S$ whose
structure morphism $\mathcal{X} \to S$ is locally of finite presentation.
Let $(A' \to A, x)$ be a deformation situation. ========

regex_match $\mathcal{X}$
nn candidate an algebraic stack
nn an algebraic stack
new_annotation [(62018, 62031, 'VAR'), (62035, 62053, 'TYPE')]
===== sent Clearly, we have $y^*\Omega_{X/S} = x^*\Omega_{X/S} \otimes_A B$.
we conclude by More on Algebra, Remark
\ref{more-algebra-remark-pseudo-coherence-and-base-change-ext}.
\end{proof}

\begin{lemma}
\label{lemma-local-lift-enough}
Let $\mathcal{X}$ be an algebraic stack over a scheme $S$ whose
structure morphism $\mathcal{X} \to S$ is locally of finite presentation.
Let $(A' \to A, x)$ be a deformation situation. ========

regex_match $(A' \to A, x)$
nn candidate A' \to A
nn candidate a deformation situation
nn a deformation situation
new_annotation [(62156, 62171, 'VAR'), (62175, 62198, 'TYPE')]
===== sent If there exists a
faithfully flat finitely presented $A'$-algebra $B'$ and an
object $y'$ of $\mathcal{X}$ over $B'$ lifting $x|_{B' \otimes_{A'} A}$,
then there exists an object $x'$ over $A'$ lifting $x$.
\end{lemma}

\begin{proof}
Let $I = \Ker(A' \to A)$. Set $B'_1 = B' \otimes_{A'} B'$ and
$B'_2 ========

new_annotation []
===== sent = B' \otimes_{A'} B' \otimes_{A'} B'$. Let
$J = IB'$, $J_1 = IB'_1$, $J_2 = IB'_2$ and
$B = B'/J$, $B_1 = B'_1/J_1$, $B_2 = B'_2/J_2$.
Set $y = x|_B$, $y_1 = x|_{B_1}$, $y_2 = x|_{B_2}$.
Let $F$ be the fppf sheaf of
Lemma \ref{lemma-sheaf-of-infinitesimal-lifts}
(which applies, see footnote in the proof of
Lemma \ref{lemma-T-quasi-coherent}).
 ========

regex_match $F$
nn candidate the fppf sheaf
nn the fppf sheaf
new_annotation [(62693, 62696, 'VAR'), (62700, 62714, 'TYPE')]
===== sent Let $f : \mathcal{X} \to \mathcal{Y}$ be a morphism of algebraic stacks.
 ========

regex_match $f : \mathcal{X} \to \mathcal{Y}$
nn candidate \mathcal{X
nn candidate a morphism
nn a morphism
new_annotation [(65548, 65581, 'VAR'), (65585, 65595, 'TYPE')]
===== sent \end{proof}

\begin{lemma}
\label{lemma-lift-to-smooth}
Let $T \to T'$ be a first order thickening of affine schemes.
 ========

regex_match $T \to T'$
nn candidate $T
nn candidate a first order thickening
nn a first order thickening
new_annotation [(67327, 67337, 'VAR'), (67341, 67365, 'TYPE')]
===== sent Let $\mathcal{X}'$ be an algebraic stack over $T'$
whose structure morphism $\mathcal{X}' \to T'$ is smooth.
 ========

regex_match $\mathcal{X}'$
nn candidate an algebraic stack
nn an algebraic stack
new_annotation [(67389, 67403, 'VAR'), (67407, 67425, 'TYPE')]
===== sent Let $x : T \to \mathcal{X}'$ be a morphism
over $T'$. Then there exists a morphsm $x' : T' \to \mathcal{X}'$
over $T'$ with $x'|_T = x$.
\end{lemma}

\begin{proof}
We may apply the result of Lemma \ref{lemma-local-lift-enough}.
 ========

regex_match $x : T \to \mathcal{X}'$
nn candidate T \to
nn candidate a morphism
nn a morphism
new_annotation [(67498, 67522, 'VAR'), (67526, 67536, 'TYPE')]
===== sent It implies, combined with
Limits of Stacks, Proposition
\ref{stacks-limits-proposition-characterize-locally-finite-presentation},
that we can recognize whether a morphism of algebraic stacks
$f : \mathcal{X} \to \mathcal{Y}$ is smooth in terms of
``simple'' properties of the $1$-morphism of stacks in groupoids
$\mathcal{X} \to \mathcal{Y}$.

\begin{lemma}[Infinitesimal lifting criterion]
\label{lemma-smooth-formally-smooth}
Let $f : \mathcal{X} \to \mathcal{Y}$ be a morphism of algebraic stacks.
 ========

regex_match $f : \mathcal{X} \to \mathcal{Y}$
nn candidate \mathcal{X
nn candidate a morphism
nn a morphism
new_annotation [(69296, 69329, 'VAR'), (69333, 69343, 'TYPE')]
===== sent \begin{lemma}
\label{lemma-flatten-stack}
Let $f : \mathcal{X} \to Y$ be a morphism from an algebraic stack
to an algebraic space. ========

regex_match $f : \mathcal{X} \to Y$
nn candidate \mathcal{X
nn candidate a morphism
nn a morphism
new_annotation [(72216, 72239, 'VAR'), (72243, 72253, 'TYPE')]
===== sent Let $V \subset Y$ be an open subspace. ========

regex_match $V \subset Y$
nn candidate $
nn candidate an open subspace
nn an open subspace
new_annotation [(72305, 72318, 'VAR'), (72322, 72338, 'TYPE')]
===== sent \mathcal{X}$.
Let $R = U \times_\mathcal{X} U$ so that we obtain
a groupoid $(U, R, s, t, c)$ in algebraic spaces over $Y$ with
$\mathcal{X} = [U/R]$
(Algebraic Stacks, Lemma \ref{algebraic-lemma-stack-presentation}) ========

new_annotation []
===== sent Thus we obtain a $V$-admissible blowup $Y' \to Y$
such that the strict transform $U' \subset U_{Y'}$
is flat and of finite presentation over $Y'$.
Let $R' \subset R_{Y'}$ be the strict transform of $R$.
 ========

regex_match $R' \subset R_{Y'}$
nn candidate \subset
nn candidate the strict transform
nn the strict transform
new_annotation [(73460, 73479, 'VAR'), (73483, 73503, 'TYPE')]
===== sent \begin{lemma}
\label{lemma-finite-cover-factor}
Let $Y$ be a quasi-compact and quasi-separated algebraic space.
Let $V \subset Y$ be a quasi-compact open. ========

regex_match $Y$
nn candidate a quasi-compact
nn a quasi-compact
new_annotation [(74660, 74663, 'VAR'), (74667, 74682, 'TYPE')]
===== sent \begin{lemma}
\label{lemma-finite-cover-factor}
Let $Y$ be a quasi-compact and quasi-separated algebraic space.
Let $V \subset Y$ be a quasi-compact open. ========

regex_match $V \subset Y$
nn candidate $V
nn candidate a quasi-compact open
nn a quasi-compact open
new_annotation [(74724, 74737, 'VAR'), (74741, 74761, 'TYPE')]
===== sent Let $f : \mathcal{X} \to V$
be surjective, flat, and locally of finite presentation.
 ========

new_annotation []
===== sent See
Limits of Spaces, Proposition
\ref{spaces-limits-proposition-there-is-a-scheme-finite-over}.
\end{proof}

\begin{lemma}
\label{lemma-make-section}
Let $f : \mathcal{X} \to Y$ be a morphism from an algebraic stack
to an algebraic space. ========

regex_match $f : \mathcal{X} \to Y$
nn candidate \mathcal{X
nn candidate a morphism
nn a morphism
new_annotation [(75898, 75921, 'VAR'), (75925, 75935, 'TYPE')]
===== sent Let $V \subset Y$ be an open subspace.
 ========

regex_match $V \subset Y$
nn candidate $
nn candidate an open subspace
nn an open subspace
new_annotation [(75987, 76000, 'VAR'), (76004, 76020, 'TYPE')]
===== sent (Let $Y' \to Y$ be a finite surjective morphism where $Y'$ is
a scheme. ========

regex_match $Y' \to Y$
nn candidate $Y
nn candidate a finite surjective morphism
nn a finite surjective morphism
new_annotation [(77485, 77495, 'VAR'), (77499, 77527, 'TYPE')]
===== sent Let $y \in V$ be arbitrary.
 ========

new_annotation []
===== sent This is a result due to Ofer Gabber, see
\cite[Theorem 1.1]{olsson_proper}
\end{reference}
Let $f : \mathcal{X} \to Y$ be a morphism from an algebraic stack
to an algebraic space. ========

regex_match $f : \mathcal{X} \to Y$
nn candidate \mathcal{X
nn candidate a morphism
nn a morphism
new_annotation [(84834, 84857, 'VAR'), (84861, 84871, 'TYPE')]
===== sent Let $\mathcal{V} \to V$ be the morphism
which expresses $\mathcal{V}$ as a gerbe over the algebraic space $V$.
See
Morphisms of Stacks, Lemma \ref{stacks-morphisms-lemma-gerbe-over-iso-classes}
for a construction of $\mathcal{V} \to V$.
This construction in particular shows that the morphism
$\mathcal{V} \to Y$ factors as $\mathcal{V} \to V \to Y$.
Picture
$$
\xymatrix{
\mathcal{V} \ar[r] \ar[d] & \mathcal{X} \ar[d] \\
 ========

regex_match $\mathcal{V} \to V$
nn candidate $\mathcal{V
nn candidate the morphism
nn the morphism
new_annotation [(87674, 87693, 'VAR'), (87697, 87709, 'TYPE')]
===== sent Let $\mathcal{X}'$ be the scheme theoretic image of the morphism
$$
j : \mathcal{V} \longrightarrow \mathbf{P}^n_Y \times_Y \mathcal{X}
$$
and let $Y'$ be the scheme theoretic image of the morphism
$V \to \mathbf{P}^n_Y$. We obtain a commutative diagram
$$
\xymatrix{
\mathcal{V} \ar[r] \ar[d] &
\mathcal{X}' \ar[r] \ar[d] &
\mathbf{P}^n_Y \times_Y \mathcal{X} \ar[d] \ar[r] &
\mathcal{X} \ar[d] \\
 ========

regex_match $\mathcal{X}'$
nn candidate the scheme theoretic image
nn the scheme theoretic image
new_annotation [(89696, 89710, 'VAR'), (89714, 89740, 'TYPE')]
===== sent Let $\mathcal{X}'$ be the scheme theoretic image of the morphism
$$
j : \mathcal{V} \longrightarrow \mathbf{P}^n_Y \times_Y \mathcal{X}
$$
and let $Y'$ be the scheme theoretic image of the morphism
$V \to \mathbf{P}^n_Y$. We obtain a commutative diagram
$$
\xymatrix{
\mathcal{V} \ar[r] \ar[d] &
\mathcal{X}' \ar[r] \ar[d] &
\mathbf{P}^n_Y \times_Y \mathcal{X} \ar[d] \ar[r] &
\mathcal{X} \ar[d] \\
 ========

regex_match $Y'$
nn candidate the scheme theoretic image
nn the scheme theoretic image
new_annotation [(89839, 89843, 'VAR'), (89847, 89873, 'TYPE')]
===== sent \medskip\noindent
Let $f : \mathcal{X} \to \mathcal{Y}$ be a morphism of algebraic stacks
(or algebraic spaces or schemes). ========

regex_match $f : \mathcal{X} \to \mathcal{Y}$
nn candidate \mathcal{X
nn candidate a morphism
nn a morphism
new_annotation [(92858, 92891, 'VAR'), (92895, 92905, 'TYPE')]
===== sent \begin{lemma}
\label{lemma-check-separated-dvr}
Let $f : \mathcal{X} \to \mathcal{Y}$ be a morphism of algebraic stacks. ========

regex_match $f : \mathcal{X} \to \mathcal{Y}$
nn candidate \mathcal{X
nn candidate a morphism
nn a morphism
new_annotation [(97664, 97697, 'VAR'), (97701, 97711, 'TYPE')]
===== sent \end{proof}

\begin{lemma}
\label{lemma-refined-valuative-criterion-proper}
Let $f : \mathcal{X} \to \mathcal{Y}$ and $h : \mathcal{U} \to \mathcal{X}$
be morphisms of algebraic stacks. ========

new_annotation []
===== sent Let $V \to \mathcal{Y}$ be a smooth morphism where $V$ is an affine scheme.
 ========

regex_match $V \to \mathcal{Y}$
nn candidate a smooth morphism
nn a smooth morphism
new_annotation [(101477, 101496, 'VAR'), (101500, 101517, 'TYPE')]
===== sent \end{proof}

\begin{lemma}
\label{lemma-refined-valuative-criterion-separated}
Let $f : \mathcal{X} \to \mathcal{Y}$ and $h : \mathcal{U} \to \mathcal{X}$
be morphisms of algebraic stacks. ========

new_annotation []
===== sent Under suitable hypotheses
$Y$ is called a {\it moduli space} for $\mathcal{X}$. If
$\mathcal{X} = [U/R]$ is a presentation, then we obtain an
$R$-invariant morphism $U \to Y$ and (under suitable hypotheses)
$Y$ is a {\it quotient} of the groupoid $(U, R, s, t, c)$.
A discussion of the different types of quotients can be found
starting with
Quotients of Groupoids, Section \ref{groupoids-quotients-section-introduction}.

\begin{definition}
\label{definition-categorical-quotient}
Let $\mathcal{X}$ be an algebraic stack. ========

regex_match $\mathcal{X}$
nn candidate an algebraic stack
nn an algebraic stack
new_annotation [(108761, 108774, 'VAR'), (108778, 108796, 'TYPE')]
===== sent \end{enumerate}
Let $\mathcal{C}$ be a full subcategory of the category of algebraic
spaces.
 ========

regex_match $\mathcal{C}$
nn candidate a full subcategory
nn a full subcategory
new_annotation [(109252, 109265, 'VAR'), (109269, 109287, 'TYPE')]
===== sent \begin{lemma}
\label{lemma-quotient-compare}
Let $(U, R, s, t, c)$ be a groupoid in algebraic spaces with
$s, t : R \to U$ flat and locally of finite presentation.
 ========

regex_match $(U, R, s, t, c)$
nn candidate U
nn candidate R
nn candidate a groupoid
nn a groupoid
new_annotation [(110037, 110054, 'VAR'), (110058, 110068, 'TYPE')]
===== sent Given a morphism $f : \mathcal{X} \to Y$ we let $\phi : U \to Y$ be
the composition $U \to \mathcal{X} \to Y$. Since $R = U \times_\mathcal{X} U$
(Groupoids in Spaces, Lemma
\ref{spaces-groupoids-lemma-quotient-stack-2-cartesian})
it is immediate that $\phi$ is $R$-invariant.
 ========

new_annotation []
===== sent \end{proof}

\begin{lemma}
\label{lemma-check-uniform-categorical-quotient-on-affines}
Let $f : \mathcal{X} \to Y$ be a morphism from an algebraic stack
to an algebraic space. ========

regex_match $f : \mathcal{X} \to Y$
nn candidate \mathcal{X
nn candidate a morphism
nn a morphism
new_annotation [(112634, 112657, 'VAR'), (112661, 112671, 'TYPE')]
===== sent For each $i$ and $j$ choose a affine open covering
$Y_i \times_Y Y_j = \bigcup Y_{ijk}$.
Set $\mathcal{X}_i = Y_i \times_Y \mathcal{X}$ and
$\mathcal{X}_{ijk} = Y_{ijk} \times_Y \mathcal{X}$.
Let $g : \mathcal{X} \to W$ be a morphism towards
an algebraic space. ========

regex_match $g : \mathcal{X} \to W$
nn candidate $g
nn candidate a morphism
nn a morphism
new_annotation [(113212, 113235, 'VAR'), (113239, 113249, 'TYPE')]
===== sent For a discussion of the
literature, please see Guide to Literature, Subsection
\ref{guide-subsection-coarse-moduli-spaces}.

\begin{definition}
\label{definition-well-nigh-affine}
Let $\mathcal{X}$ be an algebraic stack. ========

regex_match $\mathcal{X}$
nn candidate an algebraic stack
nn an algebraic stack
new_annotation [(114449, 114462, 'VAR'), (114466, 114484, 'TYPE')]
===== sent \begin{lemma}
\label{lemma-well-nigh-affine}
Let $\mathcal{X}$ be an algebraic stack. ========

regex_match $\mathcal{X}$
nn candidate an algebraic stack
nn an algebraic stack
new_annotation [(114827, 114840, 'VAR'), (114844, 114862, 'TYPE')]
===== sent \end{lemma}

\begin{proof}
Write $\mathcal{X} = [U/R]$ with $(U, R, s, t, c)$ as in
Lemma \ref{lemma-well-nigh-affine}. Let $C$ be the ring of
$R$-invariant functions on $U$, see
Groupoids, Section \ref{groupoids-section-finite-flat}.
We set $M = \Spec(C)$. The $R$-invariant morphism
$U \to M$ corresponds to a morphism $f : \mathcal{X} \to M$ by
Lemma \ref{lemma-quotient-compare}.
 ========

regex_match $C$
nn candidate the ring
nn the ring
new_annotation [(118858, 118861, 'VAR'), (118865, 118873, 'TYPE')]
===== sent Thus $|f|$ is surjective and even bijective
by Groupoids, Lemma \ref{groupoids-lemma-points}.

\medskip\noindent
Let $C \to C'$ be a ring map. ========

regex_match $C \to C'$
nn candidate a ring map
nn a ring map
new_annotation [(120773, 120783, 'VAR'), (120787, 120797, 'TYPE')]
===== sent Let $(U', R', s', t', c')$ be
the base change of $(U, R, s, t, c)$ by $M' = \Spec(C') \to M$.
Setting $\mathcal{X}' = [U'/R']$, we observe that
$M' \times_M \mathcal{X} = \mathcal{X}'$ by
Quotients of Groupoids, Lemma
\ref{groupoids-quotients-lemma-base-change-quotient-stack}.
Let $C^1$ be the the ring of $R'$-invariant functions on $U'$.
Set $M^1 = \Spec(C^1)$ and consider the diagram
$$
\xymatrix{
\mathcal{X}' \ar[d]^{f'} \ar[r] & \mathcal{X} \ar[dd]^f \\
 ========

new_annotation []
===== sent Let $(U', R', s', t', c')$ be
the base change of $(U, R, s, t, c)$ by $M' = \Spec(C') \to M$.
Setting $\mathcal{X}' = [U'/R']$, we observe that
$M' \times_M \mathcal{X} = \mathcal{X}'$ by
Quotients of Groupoids, Lemma
\ref{groupoids-quotients-lemma-base-change-quotient-stack}.
Let $C^1$ be the the ring of $R'$-invariant functions on $U'$.
Set $M^1 = \Spec(C^1)$ and consider the diagram
$$
\xymatrix{
\mathcal{X}' \ar[d]^{f'} \ar[r] & \mathcal{X} \ar[dd]^f \\
 ========

regex_match $C^1$
nn candidate the the ring
nn the the ring
new_annotation [(121081, 121086, 'VAR'), (121090, 121102, 'TYPE')]
===== sent This follows from the discussion
above and the fact that if the
ring map $C \to C'$ is flat, then $C' \to C^1$ is an isomorphism
by Groupoids, Lemma \ref{groupoids-lemma-invariants-base-change}.
\end{proof}

\begin{lemma}
\label{lemma-well-nigh-affine-moduli-space-etale}
Let $h : \mathcal{X}' \to \mathcal{X}$ be a morphism of algebraic stacks.
 ========

regex_match $h : \mathcal{X}' \to \mathcal{X}$
nn candidate a morphism
nn a morphism
new_annotation [(122186, 122220, 'VAR'), (122224, 122234, 'TYPE')]
===== sent \medskip\noindent
Let $g : \mathcal{X} \to Y$ be a morphism where $Y$ is an algebraic space.
 ========

regex_match $g : \mathcal{X} \to Y$
nn candidate $g
nn candidate a morphism
nn a morphism
new_annotation [(125694, 125717, 'VAR'), (125721, 125731, 'TYPE')]
===== sent Let $M' \subset M$ be the equalizer
of $h_1$ and $h_2$. ========

regex_match $M' \subset M$
nn candidate M
nn candidate the equalizer
nn the equalizer
new_annotation [(125958, 125972, 'VAR'), (125976, 125989, 'TYPE')]
===== sent \medskip\noindent
Let $y \in |Y|$ be the image of $p$.
Let $(V, v) \to (Y, y)$ be an \'etale morphism with $V$ affine.
 ========

regex_match $y \in |Y|$
nn candidate $y
nn candidate the image
nn the image
new_annotation [(126914, 126925, 'VAR'), (126929, 126938, 'TYPE')]
===== sent \medskip\noindent
Let $y \in |Y|$ be the image of $p$.
Let $(V, v) \to (Y, y)$ be an \'etale morphism with $V$ affine.
 ========

regex_match $(V, v) \to (Y, y)$
nn candidate $(V
nn candidate an \'etale morphism
nn an \'etale morphism
new_annotation [(126951, 126970, 'VAR'), (126974, 126993, 'TYPE')]
===== sent By Groupoids, Lemma \ref{groupoids-lemma-find-invariant-affine}
there exists an $R'$-invariant affine open subscheme $U'' \subset U'$
containing $u'$.
Let $\mathcal{X}'' \subset \mathcal{X}'$ be
the corresponding open substack. ========

new_annotation []
===== sent \end{proof}

\begin{lemma}
\label{lemma-etale-separated-over-well-nigh-affine}
Let $h : \mathcal{X}' \to \mathcal{X}$ be a morphism of algebraic stacks.
 ========

regex_match $h : \mathcal{X}' \to \mathcal{X}$
nn candidate a morphism
nn a morphism
new_annotation [(129034, 129068, 'VAR'), (129072, 129082, 'TYPE')]
===== sent Let $\mathcal{X}'_i \subset \mathcal{X}'$ be the corresponding
open substacks. ========

regex_match $\mathcal{X}'_i \subset \mathcal{X}'$
nn candidate the corresponding
open substacks
nn the corresponding
open substacks
new_annotation [(130648, 130685, 'VAR'), (130689, 130721, 'TYPE')]
===== sent Finally, $M' \to M$ is separated because the composition
$U' \to \mathcal{X}' \to M'$ is surjective and universally closed
and we can apply Morphisms, Lemma
\ref{morphisms-lemma-image-universally-closed-separated}.
\end{proof}

\begin{lemma}
\label{lemma-etale-local-finite-inertia}
Let $\mathcal{X}$ be an algebraic stack. ========

regex_match $\mathcal{X}$
nn candidate an algebraic stack
nn an algebraic stack
new_annotation [(133224, 133237, 'VAR'), (133241, 133259, 'TYPE')]
===== sent Let $U' \subset U$ be the $R$-invariant open corresponding to
the open substack $\mathcal{U}'$.
Let $u' \in U'$ be a point of $U'$ mapping to $u$.
Observe that $t(s^{-1}(\{u'\}))$ is finite as $s : R \to U$ is finite.
 ========

regex_match $U' \subset U$
nn candidate U
nn candidate the open substack
nn the open substack
new_annotation [(134377, 134391, 'VAR'), (134435, 134452, 'TYPE')]
===== sent Let $U' \subset U$ be the $R$-invariant open corresponding to
the open substack $\mathcal{U}'$.
Let $u' \in U'$ be a point of $U'$ mapping to $u$.
Observe that $t(s^{-1}(\{u'\}))$ is finite as $s : R \to U$ is finite.
 ========

regex_match $u' \in U'$
nn candidate $u
nn candidate a point
nn a point
new_annotation [(134473, 134484, 'VAR'), (134488, 134495, 'TYPE')]
===== sent Let $R''$ be the restriction of $R$ to $U''$.
Then $\mathcal{U}'' = [U''/R'']$ is an open substack of
$\mathcal{U}'$ containing $u$, is well-nigh affine,
$\mathcal{I}_{\mathcal{U}''} \to
\mathcal{U}'' \times_\mathcal{X} \mathcal{I}_\mathcal{X}$
is an isomorphism, and $\mathcal{U}'' \to \mathcal{X}$
and is representable by algebraic spaces and \'etale.
 ========

regex_match $R''$
nn candidate the restriction
nn the restriction
new_annotation [(134806, 134811, 'VAR'), (134815, 134830, 'TYPE')]
===== sent \end{proof}

\begin{theorem}[Keel-Mori]
\label{theorem-keel-mori}
Let $\mathcal{X}$ be an algebraic stack. ========

regex_match $\mathcal{X}$
nn candidate an algebraic stack
nn an algebraic stack
new_annotation [(135654, 135667, 'VAR'), (135671, 135689, 'TYPE')]
===== sent In a similar fashion we construct a morphism $e : U \to R$
compatible with $\Delta : U \to U \times U$ and
$i : R \to R$ compatible with the flip $U \times U \to U \times U$.
Let $k$ be an algebraically closed field. ========

regex_match $k$
nn candidate an algebraically closed field
nn an algebraically closed field
new_annotation [(138694, 138697, 'VAR'), (138701, 138730, 'TYPE')]
===== sent We conclude the claim holds by
Groupoids, Lemma \ref{groupoids-lemma-etale-equivalence-relation}.

\medskip\noindent
Let $M = U/R$ be the algebraic space which is the quotient of the above
\'etale equivalence relation, see
Spaces, Theorem \ref{spaces-theorem-presentation}.
There is a canonical morphism $f : \mathcal{X} \to M$
fitting into commutative diagrams
\begin{equation}
\label{equation-fundamental-diagram}
\xymatrix{
\mathcal{X}_i \ar[r]_{g_i} \ar[d]_{f_i} & \mathcal{X} \ar[d]^f \\
 ========

regex_match $M = U/R$
nn candidate $M
nn candidate U/R$
nn candidate the algebraic space
nn the algebraic space
new_annotation [(139934, 139943, 'VAR'), (139947, 139966, 'TYPE')]
===== sent Let $a : T \to \mathcal{X}$
be an object of the left hand side. ========

new_annotation []
===== sent \begin{lemma}
\label{lemma-etale-separated-over-keel-mori}
Let $h : \mathcal{X}' \to \mathcal{X}$ be a morphism of algebraic stacks.
 ========

regex_match $h : \mathcal{X}' \to \mathcal{X}$
nn candidate a morphism
nn a morphism
new_annotation [(145253, 145287, 'VAR'), (145291, 145301, 'TYPE')]
===== sent Hence $\mathcal{I}_{\mathcal{X}'} \to \mathcal{X}'$
is finite as a base change of $\mathcal{I}_\mathcal{X} \to \mathcal{X}$.
Let $f' : \mathcal{X}' \to M'$ and $f : \mathcal{X} \to M$ be as in
Theorem \ref{theorem-keel-mori}.
We obtain a commutative diagram as in the lemma because
$f'$ is categorical moduli space.
 ========

new_annotation []
===== sent Let $f'_i : \mathcal{X}'_i \to M'_i$ be as in
Lemma \ref{lemma-well-nigh-affine-moduli-space}.
In the proof of Theorem \ref{theorem-keel-mori}
we have seen that the diagrams
$$
\xymatrix{
\mathcal{X}'_i \ar[d]_{f'_i} \ar[r]_{g'_i} &
\mathcal{X}' \ar[d]^{f'} \\
 ========

new_annotation []
===== sent \begin{lemma}
\label{lemma-keel-mori-finite-type}
Let $p : \mathcal{X} \to Y$ be a morphism of an algebraic stack to an
algebraic space. ========

regex_match $p : \mathcal{X} \to Y$
nn candidate $p
nn candidate a morphism
nn a morphism
new_annotation [(148380, 148403, 'VAR'), (148407, 148417, 'TYPE')]
===== sent \end{enumerate}
Let $f : \mathcal{X} \to M$ be the moduli space constructed in
Theorem \ref{theorem-keel-mori}.
 ========

regex_match $f : \mathcal{X} \to M$
nn candidate \mathcal{X
nn candidate the moduli space
nn the moduli space
new_annotation [(148641, 148664, 'VAR'), (148668, 148684, 'TYPE')]
===== sent \end{proof}

\begin{lemma}
\label{lemma-keel-mori-diagonal}
Let $\mathcal{X}$ be an algebraic stack. ========

regex_match $\mathcal{X}$
nn candidate an algebraic stack
nn an algebraic stack
new_annotation [(150010, 150023, 'VAR'), (150027, 150045, 'TYPE')]
===== sent Let $f : \mathcal{X} \to M$ be the moduli space constructed in
Theorem \ref{theorem-keel-mori}.
\begin{enumerate}
\item If $\mathcal{X}$ is quasi-separated, then $M$ is quasi-separated.
 ========

regex_match $f : \mathcal{X} \to M$
nn candidate \mathcal{X
nn candidate the moduli space
nn the moduli space
new_annotation [(150111, 150134, 'VAR'), (150138, 150154, 'TYPE')]
===== sent \end{proof}

\begin{lemma}
\label{lemma-keel-mori-proper}
Let $p : \mathcal{X} \to Y$ be a morphism from an algebraic stack
to an algebraic space. ========

regex_match $p : \mathcal{X} \to Y$
nn candidate $p
nn candidate a morphism
nn a morphism
new_annotation [(151271, 151294, 'VAR'), (151298, 151308, 'TYPE')]
===== sent \end{enumerate}
Let $f : \mathcal{X} \to M$ be the moduli space constructed in
Theorem \ref{theorem-keel-mori}. ========

regex_match $f : \mathcal{X} \to M$
nn candidate \mathcal{X
nn candidate the moduli space
nn the moduli space
new_annotation [(151518, 151541, 'VAR'), (151545, 151561, 'TYPE')]
file:  formal-defos.tex
===== sent \medskip\noindent
Let $\Lambda$ be a complete Noetherian local ring with residue field $k$,
and let $\mathcal{C}_\Lambda$ denote the category of Artinian local
$\Lambda$-algebras with residue field $k$. Given a functor
$F : \mathcal{C}_\Lambda \to \textit{Sets}$ such that $F(k)$
is a one element set, Schlessinger's paper introduced conditions
(H1)-(H4) such that:
\begin{enumerate}
\item $F$ has a ``hull'' if and only if (H1)-(H3) hold.
\item $F$ is prorepresentable if and only (H1)-(H4) hold.
 ========

regex_match $\Lambda$
nn candidate a complete Noetherian local ring
nn a complete Noetherian local ring
new_annotation [(679, 688, 'VAR'), (692, 724, 'TYPE')]
===== sent \medskip\noindent
Let $\Lambda$ be a complete Noetherian local ring with residue field $k$,
and let $\mathcal{C}_\Lambda$ denote the category of Artinian local
$\Lambda$-algebras with residue field $k$. Given a functor
$F : \mathcal{C}_\Lambda \to \textit{Sets}$ such that $F(k)$
is a one element set, Schlessinger's paper introduced conditions
(H1)-(H4) such that:
\begin{enumerate}
\item $F$ has a ``hull'' if and only if (H1)-(H3) hold.
\item $F$ is prorepresentable if and only (H1)-(H4) hold.
 ========

new_annotation []
===== sent \end{enumerate}
The purpose of this chapter is to generalize these results in two ways
exactly as is done in Rim's paper:
\begin{enumerate}
\item[(A)] The functor $F$ is replaced by a category $\mathcal{F}$ cofibered
in groupoids over $\mathcal{C}_\Lambda$, see
Section \ref{section-CLambda}.
\item[(B)] We let $\Lambda$ be a Noetherian ring and $\Lambda \to k$
a finite ring map to a field. ========

regex_match $\Lambda$
nn candidate a Noetherian ring
nn a Noetherian ring
new_annotation [(1466, 1475, 'VAR'), (1479, 1496, 'TYPE')]
===== sent Let $k$ be a finite type field over $S$, i.e., we are given a
finite type morphism $\Spec(k) \to S$.
 ========

regex_match $k$
nn candidate a finite type field
nn a finite type field
new_annotation [(10034, 10037, 'VAR'), (10041, 10060, 'TYPE')]
===== sent \begin{definition}
\label{definition-CLambda}
Let $\Lambda$ be a Noetherian ring and let $\Lambda \to k$ be a finite
ring map where $k$ is a field. ========

regex_match $\Lambda$
nn candidate a Noetherian ring
nn a Noetherian ring
new_annotation [(10647, 10656, 'VAR'), (10660, 10677, 'TYPE')]
===== sent \begin{definition}
\label{definition-CLambda}
Let $\Lambda$ be a Noetherian ring and let $\Lambda \to k$ be a finite
ring map where $k$ is a field. ========

regex_match $\Lambda \to k$
nn candidate a finite
ring map
nn a finite
ring map
new_annotation [(10686, 10701, 'VAR'), (10705, 10722, 'TYPE')]
===== sent \begin{definition}
\label{definition-small-extension}
Let $f: B \to A$ be a ring map in $\mathcal{C}_\Lambda$.  We say $f$
is a {\it small extension} if it is surjective and $\Ker(f)$ is a nonzero
principal ideal which is annihilated by $\mathfrak{m}_B$.
\end{definition}

\noindent
By the following lemma we can often reduce arguments involving surjective ring
maps in $\mathcal{C}_\Lambda$ to the case of small extensions.

 ========

regex_match $f: B \to A$
nn candidate a ring map
nn a ring map
new_annotation [(12324, 12336, 'VAR'), (12340, 12350, 'TYPE')]
===== sent Let $f: B \to A$ be a surjective ring map in $\mathcal{C}_\Lambda$.
Then $f$ can be factored as a composition of small extensions.
 ========

regex_match $f: B \to A$
nn candidate a surjective ring map
nn a surjective ring map
new_annotation [(12747, 12759, 'VAR'), (12763, 12784, 'TYPE')]
===== sent \end{lemma}

\begin{proof}
Let $I$ be the kernel of $f$.  The maximal ideal $\mathfrak{m}_B$ is
nilpotent since $B$ is Artinian, say $\mathfrak{m}_B^n = 0$. ========

regex_match $I$
nn candidate the kernel
nn the kernel
new_annotation [(12905, 12908, 'VAR'), (12912, 12922, 'TYPE')]
===== sent \end{proof}

\noindent
The next lemma says that we can compute the length of a module over a local
$\Lambda$-algebra with residue field $k$ in terms of the length over
$\Lambda$. To explain the notation in the statement, let $k' \subset k$
be the image of our fixed finite ring map $\Lambda \to k$. Note
that $k/k'$ is a finite extension of rings. ========

new_annotation []
===== sent \begin{lemma}
\label{lemma-length}
Let $A$ be a local $\Lambda$-algebra with residue field $k$.
Let $M$ be an $A$-module. ========

regex_match $A$
nn candidate a local $\Lambda$-algebra
nn a local $\Lambda$-algebra
new_annotation [(14554, 14557, 'VAR'), (14561, 14586, 'TYPE')]
===== sent \begin{lemma}
\label{lemma-length}
Let $A$ be a local $\Lambda$-algebra with residue field $k$.
Let $M$ be an $A$-module. ========

regex_match $M$
new_annotation []
===== sent If $\text{length}_A(M)$ is finite, then the result follows on
choosing a filtration of $M$ by $A$-submodules with simple quotients
using additivity, see
Algebra, Lemma \ref{algebra-lemma-length-additive}.
If $\text{length}_A(M)$ is infinite, the result follows from the obvious
inequality $\text{length}_A(M) \leq \text{length}_\Lambda(M)$.
\end{proof}

\begin{lemma}
\label{lemma-surjective}
Let $A \to B$ be a ring map in $\mathcal{C}_\Lambda$.
The following are equivalent
\begin{enumerate}
\item $f$ is surjective,
\item $\mathfrak m_A/\mathfrak m_A^2 \to \mathfrak m_B/\mathfrak m_B^2$
is surjective, and
\item $\mathfrak m_A/(\mathfrak m_\Lambda A + \mathfrak m_A^2)
\to \mathfrak m_B/(\mathfrak m_\Lambda B + \mathfrak m_B^2)$ is surjective.
 ========

regex_match $A \to B$
nn candidate a ring map
nn a ring map
new_annotation [(15471, 15480, 'VAR'), (15484, 15494, 'TYPE')]
===== sent Let $y \in \mathfrak m_B$.
Choose $x \in A$ with $f(x) = y$. Since $f$ induces an isomorphism
$A/\mathfrak m_A \to B/\mathfrak m_B$ we see that $x \in \mathfrak m_A$.
Hence the induced map
$\mathfrak m_A/\mathfrak m_A^2 \to \mathfrak ========

new_annotation []
===== sent \begin{definition}
\label{definition-tangent-space-ring}
Let $R \to S$ be a local homomorphism of local rings. ========

regex_match $R \to S$
nn candidate a local homomorphism
nn a local homomorphism
new_annotation [(17872, 17881, 'VAR'), (17885, 17905, 'TYPE')]
===== sent It isn't always the case that the
fibre product is an object of $\mathcal{C}_\Lambda$.

\begin{example}
\label{example-fibre-product}
Let $p$ be a prime number and let $n \in \mathbf{N}$.
Let $\Lambda = \mathbf{F}_p(t_1, t_2, \ldots, t_n)$ and let
$k = \mathbf{F}_p(x_1, \ldots, x_n)$ with map $\Lambda \to k$ given
by $t_i ========

regex_match $p$
nn candidate a prime number
nn a prime number
new_annotation [(18903, 18906, 'VAR'), (18910, 18924, 'TYPE')]
===== sent It isn't always the case that the
fibre product is an object of $\mathcal{C}_\Lambda$.

\begin{example}
\label{example-fibre-product}
Let $p$ be a prime number and let $n \in \mathbf{N}$.
Let $\Lambda = \mathbf{F}_p(t_1, t_2, \ldots, t_n)$ and let
$k = \mathbf{F}_p(x_1, \ldots, x_n)$ with map $\Lambda \to k$ given
by $t_i ========

new_annotation []
===== sent It isn't always the case that the
fibre product is an object of $\mathcal{C}_\Lambda$.

\begin{example}
\label{example-fibre-product}
Let $p$ be a prime number and let $n \in \mathbf{N}$.
Let $\Lambda = \mathbf{F}_p(t_1, t_2, \ldots, t_n)$ and let
$k = \mathbf{F}_p(x_1, \ldots, x_n)$ with map $\Lambda \to k$ given
by $t_i ========

new_annotation []
===== sent \mapsto x_i^p$. Let $A = k[\epsilon] = k[x]/(x^2)$.
Then $A$ is an object of $\mathcal{C}_\Lambda$. Suppose that
$D : k \to k$ is a derivation of $k$ over $\Lambda$, for example
$D = \partial/\partial x_i$. ========

new_annotation []
===== sent Let $f_1 : A_1 \to A$ and $f_2 : A_2 \to A$ be ring maps in
$\mathcal{C}_\Lambda$. Then:
\begin{enumerate}
\item If $f_1$ or $f_2$ is surjective, then
$A_1 \times_A A_2$ is in $\mathcal{C}_\Lambda$.
\item If $f_2$ is a small extension, then so is
$A_1 \times_A A_2 \to A_1$.
\item If the field extension $k' \subset k$ is separable, then
$A_1 \times_A A_2$ is in $\mathcal{C}_\Lambda$.
\end{enumerate}
\end{lemma}

\begin{proof}
The ring $A_1 \times_A A_2$ is a $\Lambda$-algebra via  the map
$\Lambda \to A_1 \times_A A_2$ induced by the maps
$\Lambda \to A_1$ and $\Lambda \to A_2$. ========

new_annotation []
===== sent Let $P'(T) \in k'[T]$ be the minimal polynomial of $\overline{x}$ over $k'$.
 ========

regex_match $P'(T) \in k'[T]$
nn candidate the minimal polynomial
nn the minimal polynomial
new_annotation [(22233, 22250, 'VAR'), (22254, 22276, 'TYPE')]
===== sent Next we define essential surjections in $\mathcal{C}_\Lambda$. A necessary
and sufficient condition for a surjection in $\mathcal{C}_\Lambda$ to be
essential is given in Lemma \ref{lemma-essential-surjection}.

\begin{definition}
\label{definition-essential-surjection}
Let $f: B \to A$ be a ring map in $\mathcal{C}_\Lambda$.  We say $f$
is an {\it essential surjection} if it has the following properties:
\begin{enumerate}
\item $f$ is surjective.
 ========

regex_match $f: B \to A$
nn candidate : B \to A$
nn candidate a ring map
nn a ring map
new_annotation [(23283, 23295, 'VAR'), (23299, 23309, 'TYPE')]
===== sent \begin{lemma}
\label{lemma-essential-surjection-mod-squares}
Let $f: B \to A$ be a ring map in $\mathcal{C}_\Lambda$.
The following are equivalent
\begin{enumerate}
\item $f$ is an essential surjection,
\item the map $B/\mathfrak m_B^2 \to A/\mathfrak m_A^2$ is an essential
surjection, and
\item the map
$B/(\mathfrak m_\Lambda B + \mathfrak m_B^2) \to
 ========

regex_match $f: B \to A$
nn candidate : B \to A$
nn candidate a ring map
nn a ring map
new_annotation [(23807, 23819, 'VAR'), (23823, 23833, 'TYPE')]
===== sent Let $C \to B$ be a ring map in $\mathcal{C}_\Lambda$ such
that $C \to A$ is surjective. ========

regex_match $C \to B$
nn candidate a ring map
nn a ring map
new_annotation [(24228, 24237, 'VAR'), (24241, 24251, 'TYPE')]
===== sent Let $C \to B/(\mathfrak m_\Lambda B + \mathfrak m_B^2)$
be a morphism of $\mathcal{C}_\Lambda$ such that
$C \to A/(\mathfrak m_\Lambda A + \mathfrak m_A^2)$ is surjective. ========

new_annotation []
===== sent Let $H_1(L_{k/\Lambda})$
be the first homology module of the naive cotangent complex of $k$
over $\Lambda$, see
Algebra, Definition \ref{algebra-definition-naive-cotangent-complex}.
Then we can extend (\ref{equation-sequence})
to the exact sequence
\begin{equation}
\label{equation-sequence-extended}
H_1(L_{k/\Lambda}) \to
\mathfrak m_A/\mathfrak m_A^2 \xrightarrow{\text{d}_A}
\Omega_{A/\Lambda} \otimes_A k \to
\Omega_{k/\Lambda} \to 0,
\end{equation}
see
Algebra, Lemma \ref{algebra-lemma-exact-sequence-NL}.
If $B \to A$ is a ring map in $\mathcal{C}_\Lambda$
or more generally a map of $\Lambda$-algebras equipped
with $\Lambda$-algebra surjections onto $k$, then we obtain a
commutative diagram
\begin{equation}
\label{equation-sequence-functorial}
\vcenter{
\xymatrix{
H_1(L_{k/\Lambda}) \ar[r] \ar@{=}[d] &
\mathfrak m_B/\mathfrak m_B^2 \ar[r]_{\text{d}_B} \ar[d] &
\Omega_{B/\Lambda} \otimes_B k \ar[r] \ar[d] &
\Omega_{k/\Lambda} \ar[r] \ar@{=}[d] & 0 \\
H_1(L_{k/\Lambda}) \ar[r] &
\mathfrak m_A/\mathfrak m_A^2 \ar[r]^{\text{d}_A} &
\Omega_{A/\Lambda} \otimes_A k \ar[r] &
\Omega_{k/\Lambda} \ar[r] & 0
}
}
\end{equation}
with exact rows.

\begin{lemma}
\label{lemma-H1-separable-case}
There is a canonical map
$$
\mathfrak m_\Lambda/\mathfrak m_\Lambda^2 \longrightarrow H_1(L_{k/\Lambda}).
 ========

new_annotation []
===== sent \end{proof}

\begin{lemma}
\label{lemma-essential-surjection}
Let $f: B \to A$ be a ring map in $\mathcal{C}_\Lambda$.
Notation as in (\ref{equation-sequence-functorial}).
 ========

regex_match $f: B \to A$
nn candidate B \to A$
nn candidate a ring map
nn a ring map
new_annotation [(28829, 28841, 'VAR'), (28845, 28855, 'TYPE')]
===== sent Let $g : C \to B$ be a ring map in $\mathcal{C}_\Lambda$
such that $f \circ g$ is surjective. ========

regex_match $g : C \to B$
nn candidate $
nn candidate C \to
nn candidate a ring map
nn a ring map
new_annotation [(30672, 30685, 'VAR'), (30689, 30699, 'TYPE')]
===== sent Let $K$ be the kernel. ========

regex_match $K$
nn candidate the kernel
nn the kernel
new_annotation [(31245, 31248, 'VAR'), (31252, 31262, 'TYPE')]
===== sent The map $\text{d} : B \to \Omega_{B/\Lambda}$
induces via the projection onto $K$ a map $D : B \to K$. Set
$C = \{b \in B \mid D(b) = 0\}$. The Leibniz rule shows that this is
a $\Lambda$-subalgebra of $B$. Let $\overline{x} \in k$. Choose $x \in B$
mapping to $\overline{x}$. If $D(x) \not = 0$, then we can find an element
$y \in \mathfrak m_B$ such that $D(y) = D(x)$. Hence $x - y \in C$ is
an element which maps to $\overline{x}$. Thus $C \to k$ is surjective
and $C$ is an object of $\mathcal{C}_\Lambda$. ========

new_annotation []
===== sent \end{proof}

\begin{example}
\label{example-essential-surjection}
Let $\Lambda = k[[x]]$ be the power series ring in $1$ variable over $k$.
Set $A = k$ and $B = \Lambda/(x^2)$. Then $B \to A$ is an essential
surjection by
Lemma \ref{lemma-essential-surjection}
because it is a small extension and the map $B \to A$ does not have a
right inverse (in the category $\mathcal{C}_\Lambda$). ========

regex_match $\Lambda = k[[x]]$
nn candidate the power series ring
nn the power series ring
new_annotation [(33851, 33869, 'VAR'), (33873, 33894, 'TYPE')]
===== sent \begin{definition}
\label{definition-completion-CLambda}
Let $\Lambda$ be a Noetherian ring and let $\Lambda \to k$ be a finite
ring map where $k$ is a field. ========

regex_match $\Lambda$
nn candidate a Noetherian ring
nn a Noetherian ring
new_annotation [(34928, 34937, 'VAR'), (34941, 34958, 'TYPE')]
===== sent \begin{definition}
\label{definition-completion-CLambda}
Let $\Lambda$ be a Noetherian ring and let $\Lambda \to k$ be a finite
ring map where $k$ is a field. ========

regex_match $\Lambda \to k$
nn candidate a finite
ring map
nn a finite
ring map
new_annotation [(34967, 34982, 'VAR'), (34986, 35003, 'TYPE')]
===== sent \begin{lemma}
\label{lemma-surjective-cotangent-space}
Let $f: R \to S$ be a ring map in $\widehat{\mathcal{C}}_\Lambda$.
The following are equivalent
\begin{enumerate}
\item $f$ is surjective,
\item the map
$\mathfrak m_R/\mathfrak m_R^2 \to \mathfrak ========

regex_match $f: R \to S$
nn candidate R \to
nn candidate a ring map
nn a ring map
new_annotation [(37161, 37173, 'VAR'), (37177, 37187, 'TYPE')]
===== sent Hence by
Lemma \ref{lemma-surjective}
we see that $R_n \to S_n$ is surjective for all $n$.
Now let $K_n$ be the kernel of $R_n \to S_n$.  Then the sequences
$$
0 \to K_n \to R_n \to S_n \to 0
$$
form an exact sequence of directed inverse systems. ========

regex_match $K_n$
nn candidate the kernel
nn the kernel
new_annotation [(37912, 37917, 'VAR'), (37921, 37931, 'TYPE')]
===== sent \end{lemma}

\begin{proof}
Let $R \to S_1$ and $R \to S_2$ be morphisms of
$\widehat{\mathcal{C}}_\Lambda$. Consider the ring
$C = S_1 \otimes_R S_2$.
This ring has a finitely generated maximal ideal
$\mathfrak m = \mathfrak m_{S_1} \otimes S_2 +
S_1 \otimes \mathfrak m_{S_2}$ with residue field $k$.
 ========

new_annotation []
===== sent \end{lemma}

\begin{proof}
Let $R$ and $S$ be objects of $\widehat{\mathcal{C}}_\Lambda$.
Consider the ring $C = R \otimes_\Lambda S$. There is a canonical
surjective map $C \to R \otimes_\Lambda S \to k \otimes_\Lambda k \to k$
where the last map is the multiplication map. ========

new_annotation []
===== sent \begin{lemma}
\label{lemma-derivations-finite}
Let $S$ be an object of $\widehat{\mathcal{C}}_\Lambda$.
Then $\dim_k \text{Der}_\Lambda(S, ========

regex_match $S$
nn candidate an object
nn an object
new_annotation [(41550, 41553, 'VAR'), (41557, 41566, 'TYPE')]
===== sent k) < \infty$.
\end{lemma}

\begin{proof}
Let $x_1, \ldots, x_n \in \mathfrak m_S$ map to a $k$-basis
for the relative cotangent space
$\mathfrak m_S/(\mathfrak m_\Lambda S + \mathfrak m_S^2)$.
Choose $y_1, \ldots, y_m \in S$ whose images in $k$ generate $k$
over $k'$. We claim that $\dim_k \text{Der}_\Lambda(S, k) \leq n + m$.
To see this it suffices to prove that if $D(x_i) = 0$ and
$D(y_j) = 0$, then $D ========

new_annotation []
===== sent = 0$. Let $a \in S$. We can find a
polynomial $P = \sum \lambda_J y^J$ with $\lambda_J \in \Lambda$
whose image in $k$ is the same as the image of $a$ in $k$.
Then we see that $D(a - P) = D(a) - D(P) = D(a)$ by our assumption
that $D(y_j) = 0$ for all $j$. Thus we may assume $a \in \mathfrak m_S$.
 ========

new_annotation []
===== sent $$
D(a) = \sum x_iD(a_i) + \sum a_iD(x_i) = \sum x_iD(a_i)
$$
as we assumed $D(x_i) = 0$. We have $\sum x_iD(a_i) = 0$
as multiplication by $x_i$ is zero on $k$.
\end{proof}

\begin{lemma}
\label{lemma-derivations-surjective}
Let $f : R \to S$ be a morphism of $\widehat{\mathcal{C}}_\Lambda$.
 ========

regex_match $f : R \to S$
nn candidate R \to
nn candidate a morphism
nn a morphism
new_annotation [(42639, 42652, 'VAR'), (42656, 42666, 'TYPE')]
===== sent \end{proof}

\begin{lemma}
\label{lemma-m-adic-topology}
Let $R$ be an object of $\widehat{\mathcal{C}}_\Lambda$. Let $(J_n)$ be a
decreasing sequence of ideals such that $\mathfrak m_R^n \subset J_n$.
Set ========

regex_match $R$
nn candidate an object
nn an object
new_annotation [(43973, 43976, 'VAR'), (43980, 43989, 'TYPE')]
===== sent \end{proof}

\begin{lemma}
\label{lemma-m-adic-topology}
Let $R$ be an object of $\widehat{\mathcal{C}}_\Lambda$. Let $(J_n)$ be a
decreasing sequence of ideals such that $\mathfrak m_R^n \subset J_n$.
Set ========

new_annotation []
===== sent \end{proof}

\begin{lemma}
\label{lemma-limit-artinian}
Let $\ldots \to A_3 \to A_2 \to A_1$ be a sequence of surjective
ring maps in $\mathcal{C}_\Lambda$. If
$\dim_k ========

regex_match $\ldots \to A_3 \to A_2 \to A_1$
nn candidate $\ldots
nn candidate a sequence
nn a sequence
new_annotation [(45156, 45188, 'VAR'), (45192, 45202, 'TYPE')]
===== sent Then we get a ring map
$P = \Lambda[z_1, \ldots, z_{r + t}] \to S$, $z_i \mapsto x_i$ and
$z_{r + j} \mapsto y_j$ such that the composition
$P \to S \to A_n$ is surjective for all $n$. Let $\mathfrak m \subset P$
be the kernel of $P \to k$. Let $R = P^\wedge$ be the $\mathfrak m$-adic
completion of $P$; this is an object of $\widehat{\mathcal{C}}_\Lambda$.
Since we still have the compatible system of (surjective) maps $R \to A_n$
we get a map $R \to S$. Set $J_n = \Ker(R \to A_n)$.
Set $J = \bigcap J_n$. ========

new_annotation []
===== sent Then we get a ring map
$P = \Lambda[z_1, \ldots, z_{r + t}] \to S$, $z_i \mapsto x_i$ and
$z_{r + j} \mapsto y_j$ such that the composition
$P \to S \to A_n$ is surjective for all $n$. Let $\mathfrak m \subset P$
be the kernel of $P \to k$. Let $R = P^\wedge$ be the $\mathfrak m$-adic
completion of $P$; this is an object of $\widehat{\mathcal{C}}_\Lambda$.
Since we still have the compatible system of (surjective) maps $R \to A_n$
we get a map $R \to S$. Set $J_n = \Ker(R \to A_n)$.
Set $J = \bigcap J_n$. ========

regex_match $R = P^\wedge$
nn candidate $R
nn candidate the $\mathfrak m$-adic
completion
nn the $\mathfrak m$-adic
completion
new_annotation [(46323, 46337, 'VAR'), (46341, 46374, 'TYPE')]
===== sent \end{proof}

\begin{lemma}
\label{lemma-power-series}
Let $R', R \in \Ob(\widehat{\mathcal{C}}_\Lambda)$. Suppose that
$R = R' \oplus I$ for some ideal $I$ of $R$. Let $x_1, \ldots, x_r \in I$
map to a basis of $I/\mathfrak m_R I$. Set $S = R'[[X_1, \ldots, X_r]]$
and consider the $R'$-algebra map $S \to R$ mapping $X_i$ to $x_i$.
 ========

new_annotation []
===== sent \end{proof}

\begin{lemma}
\label{lemma-power-series}
Let $R', R \in \Ob(\widehat{\mathcal{C}}_\Lambda)$. Suppose that
$R = R' \oplus I$ for some ideal $I$ of $R$. Let $x_1, \ldots, x_r \in I$
map to a basis of $I/\mathfrak m_R I$. Set $S = R'[[X_1, \ldots, X_r]]$
and consider the $R'$-algebra map $S \to R$ mapping $X_i$ to $x_i$.
 ========

new_annotation []
===== sent We assume as known the definition and basic properties of categories
{\it fibered} in groupoids, see
Categories, Section \ref{categories-section-fibred-groupoids}.

\begin{definition}
\label{definition-category-cofibred-groupoids}
Let $\mathcal{C}$ be a category.   ========

regex_match $\mathcal{C}$
nn candidate a category
nn a category
new_annotation [(48525, 48538, 'VAR'), (48542, 48552, 'TYPE')]
===== sent Let $\mathcal{C}$ be a category.
 ========

regex_match $\mathcal{C}$
nn candidate a category
nn a category
new_annotation [(49624, 49637, 'VAR'), (49641, 49651, 'TYPE')]
===== sent We sometimes write $\text{Aut}_U(x)$ for $\Mor_{\mathcal{F}(U)}(x, x)$.
\item
\label{item-pushforward}
Let $\mathcal{F}$ be a category cofibered in groupoids over
$\mathcal{C}$, let $f: U \to V$ be a morphism in $\mathcal{C}$, and
let $x \in \Ob(\mathcal{F}(U))$.
 ========

regex_match $\mathcal{F}$
nn candidate a category
nn a category
new_annotation [(50377, 50390, 'VAR'), (50394, 50404, 'TYPE')]
===== sent We sometimes write $\text{Aut}_U(x)$ for $\Mor_{\mathcal{F}(U)}(x, x)$.
\item
\label{item-pushforward}
Let $\mathcal{F}$ be a category cofibered in groupoids over
$\mathcal{C}$, let $f: U \to V$ be a morphism in $\mathcal{C}$, and
let $x \in \Ob(\mathcal{F}(U))$.
 ========

regex_match $f: U \to V$
nn candidate U \to
nn candidate a morphism
nn a morphism
new_annotation [(50452, 50464, 'VAR'), (50468, 50478, 'TYPE')]
===== sent We sometimes write $\text{Aut}_U(x)$ for $\Mor_{\mathcal{F}(U)}(x, x)$.
\item
\label{item-pushforward}
Let $\mathcal{F}$ be a category cofibered in groupoids over
$\mathcal{C}$, let $f: U \to V$ be a morphism in $\mathcal{C}$, and
let $x \in \Ob(\mathcal{F}(U))$.
 ========

new_annotation []
===== sent \item Let $\mathcal{F}$ be a category cofibered in groupoids over
$\mathcal{C}$. Given a choice of pushforwards for $\mathcal{F}$, there
is an associated pseudo-functor $\mathcal{C} \to \textit{Groupoids}$.
We will never use this construction so we give no details.
\item
\label{item-cofibered-morphism}
A morphism of categories cofibered in groupoids over $\mathcal{C}$ is a
functor commuting with the projections to $\mathcal{C}$. If $\mathcal{F}$
and $\mathcal{F}'$ are categories cofibered in groupoids over
$\mathcal{C}$, we denote the morphisms from $\mathcal{F}$ to $\mathcal{F}'$
by $\Mor_\mathcal{C}(\mathcal{F}, \mathcal{F}')$.
\item
\label{item-definition-cofibered-groupoids-2-category}
Categories cofibered in groupoids form a $(2, 1)$-category
$\text{Cof}(\mathcal{C})$. Its 1-morphisms are the morphisms described in
(\ref{item-cofibered-morphism}). If $p : \mathcal{F} \to C$ and
$p': \mathcal{F}' \to \mathcal{C}$ are categories cofibered in groupoids
and $\varphi, \psi : \mathcal{F} \to \mathcal{F}'$ are $1$-morphisms, then
a 2-morphism $t : \varphi \to \psi$ is a morphism of functors such that
$p'(t_x) = \text{id}_{p(x)}$ for all $x \in \Ob(\mathcal{F})$.
\item
\label{item-construction-associated-cofibered-groupoid}
 ========

regex_match $\mathcal{F}$
nn candidate a category
nn a category
new_annotation [(51082, 51095, 'VAR'), (51099, 51109, 'TYPE')]
===== sent Let $F : \mathcal{C} \to \textit{Groupoids}$ be a functor. ========

regex_match $F : \mathcal{C} \to \textit{Groupoids}$
nn candidate $F
nn candidate a functor
nn a functor
new_annotation [(52317, 52357, 'VAR'), (52361, 52370, 'TYPE')]
===== sent An object of $\mathcal{F}$ is a pair $(U, x)$
where $U \in \Ob(\mathcal{C})$ and $x \in \Ob(F(U))$. A
morphism $(U, x) \to (V, y)$ is a pair $(f, a)$ where
$f \in \Mor_\mathcal{C}(U, V)$ and
$a \in \Mor_{F(V)}(F(f)(x), y)$.
The functor $\mathcal{F} \to \mathcal{C}$ sends $(U, x)$ to $U$. See
Categories, Section \ref{categories-section-presheaves-groupoids}.
\item
\label{item-associated-functor-isomorphism-classes}
Let $\mathcal{F}$ be cofibered in groupoids over $\mathcal{C}$.
 ========

new_annotation []
===== sent \item
\label{item-convention-cofibered-sets}
Let $F: \mathcal{C} \to \textit{Sets}$ be a functor ========

regex_match $F: \mathcal{C} \to \textit{Sets}$
nn candidate $F
nn candidate a functor
nn a functor
new_annotation [(53696, 53730, 'VAR'), (53734, 53743, 'TYPE')]
===== sent See Categories, Section \ref{categories-section-fibred-in-sets}.
\item
\label{item-definition-yoneda}
Let $U$ be an object of $\mathcal{C}$.  We write $\underline{U}$ for the
functor
$\Mor_\mathcal{C}(U, -): \mathcal{C} \to
\textit{Sets}$.   ========

regex_match $U$
nn candidate an object
nn an object
new_annotation [(54603, 54606, 'VAR'), (54610, 54619, 'TYPE')]
===== sent If $\mathcal{F}
\to \mathcal{H}$ and $\mathcal{G} \to \mathcal{H}$ are morphisms
of categories cofibered in groupoids over $\mathcal{C}_\Lambda$, then a
construction of their 2-fiber product is given by the construction for their
2-fiber product as categories over $\mathcal{C}_\Lambda$, as described in
Categories, Lemma \ref{categories-lemma-2-product-categories-over-C}.
\item
\label{item-product}
Products of categories cofibered in groupoids: If $\mathcal{F}$ and
$\mathcal{G}$ are categories cofibered in groupoids over
$\mathcal{C}_\Lambda$ then their product is defined to be the $2$-fiber product
$\mathcal{F} \times_{\mathcal{C}_\Lambda} \mathcal{G}$ as described in
Categories, Lemma \ref{categories-lemma-2-product-categories-over-C}.
\item
\label{item-definition-restricting-base-category}
Restricting the base category: Let $p : \mathcal{F} \to \mathcal{C}$ be a
category cofibered in groupoids, and let $\mathcal{C}'$ be a full
subcategory of $\mathcal{C}$. The restriction $\mathcal{F}|_{\mathcal{C}'}$
is the full subcategory of $\mathcal{F}$ whose objects lie over
objects of ========

new_annotation []
===== sent If $\mathcal{F}
\to \mathcal{H}$ and $\mathcal{G} \to \mathcal{H}$ are morphisms
of categories cofibered in groupoids over $\mathcal{C}_\Lambda$, then a
construction of their 2-fiber product is given by the construction for their
2-fiber product as categories over $\mathcal{C}_\Lambda$, as described in
Categories, Lemma \ref{categories-lemma-2-product-categories-over-C}.
\item
\label{item-product}
Products of categories cofibered in groupoids: If $\mathcal{F}$ and
$\mathcal{G}$ are categories cofibered in groupoids over
$\mathcal{C}_\Lambda$ then their product is defined to be the $2$-fiber product
$\mathcal{F} \times_{\mathcal{C}_\Lambda} \mathcal{G}$ as described in
Categories, Lemma \ref{categories-lemma-2-product-categories-over-C}.
\item
\label{item-definition-restricting-base-category}
Restricting the base category: Let $p : \mathcal{F} \to \mathcal{C}$ be a
category cofibered in groupoids, and let $\mathcal{C}'$ be a full
subcategory of $\mathcal{C}$. The restriction $\mathcal{F}|_{\mathcal{C}'}$
is the full subcategory of $\mathcal{F}$ whose objects lie over
objects of ========

regex_match $\mathcal{C}'$
nn candidate a full
subcategory
nn a full
subcategory
new_annotation [(56020, 56034, 'VAR'), (56038, 56056, 'TYPE')]
===== sent \begin{definition}
\label{definition-prorepresentable}
Let $F : \mathcal{C}_\Lambda \to \textit{Sets}$ be a functor.
 ========

regex_match $F : \mathcal{C}_\Lambda \to \textit{Sets}$
nn candidate \mathcal{C}_\Lambda \to
nn candidate a functor
nn a functor
new_annotation [(57158, 57201, 'VAR'), (57205, 57214, 'TYPE')]
===== sent Let $x_0 \in \Ob(\mathcal{F}(k))$. Then every object of
$\mathcal{F}$ comes equipped with a unique morphism to $x_0$.
Namely, if $x$ is an object of $\mathcal{F}$ over $A$, then we
can choose a pushforward $x \to q_*x$ where $q : A \to k$ is the quotient
map. ========

new_annotation []
===== sent \end{remark}

\begin{remark}
\label{remark-localize-cofibered-groupoid}
Let $p : \mathcal{F} \to \mathcal{C}_\Lambda$ be a category cofibered in
groupoids, and let $x \in \Ob(\mathcal{F}(k))$.  We denote by
$\mathcal{F}_x$ the category of objects over $x$.
An object of $\mathcal{F}_x$ is an arrow $y \to x$.
A morphism $(y \to x) \to (z \to x)$ in $\mathcal{F}_x$ is a commutative
diagram
$$
\xymatrix{
y \ar[rr] \ar[dr] & & z \ar[dl] \\
 ========

regex_match $p : \mathcal{F} \to \mathcal{C}_\Lambda$
nn candidate $p
nn candidate a category
nn a category
new_annotation [(59180, 59221, 'VAR'), (59225, 59235, 'TYPE')]
===== sent \end{remark}

\begin{remark}
\label{remark-localize-cofibered-groupoid}
Let $p : \mathcal{F} \to \mathcal{C}_\Lambda$ be a category cofibered in
groupoids, and let $x \in \Ob(\mathcal{F}(k))$.  We denote by
$\mathcal{F}_x$ the category of objects over $x$.
An object of $\mathcal{F}_x$ is an arrow $y \to x$.
A morphism $(y \to x) \to (z \to x)$ in $\mathcal{F}_x$ is a commutative
diagram
$$
\xymatrix{
y \ar[rr] \ar[dr] & & z \ar[dl] \\
 ========

new_annotation []
===== sent \begin{definition}
\label{definition-formal-objects}
Let $\mathcal{F}$ be a category cofibered in groupoids over
$\mathcal{C}_\Lambda$. The {\it category $\widehat{\mathcal{F}}$ of formal
objects of  $\mathcal{F}$} is the category with the following objects and
morphisms.
 ========

regex_match $\mathcal{F}$
nn candidate a category
nn a category
new_annotation [(60399, 60412, 'VAR'), (60416, 60426, 'TYPE')]
===== sent \begin{enumerate}
\item A {\it formal object $\xi = (R, \xi_n, f_n)$ of $\mathcal{F}$}
consists of an object $R$ of $\widehat{\mathcal{C}}_\Lambda$, and a collection
indexed by $n \in \mathbf{N}$ of objects $\xi_n$ of
$\mathcal{F}(R/\mathfrak m_R^n)$ and morphisms
$f_n : \xi_{n + 1} \to \xi_n$ lying over the projection
$R/\mathfrak m_R^{n + 1} \to R/\mathfrak m_R^n$.
\item Let $\xi = (R, \xi_n, f_n)$ and $\eta = (S, \eta_n, g_n)$ be
formal objects of $\mathcal{F}$.  A {\it morphism $a : \xi \to \eta$ of
formal objects} consists of a map $a_0 : R \to S$ in
$\widehat{\mathcal{C}}_\Lambda$ and a collection $a_n : \xi_n \to \eta_n$
of morphisms of $\mathcal{F}$ lying over
$R/\mathfrak m_R^n \to S/\mathfrak m_S^n$,
such that for every $n$ the diagram
$$
\xymatrix{
\xi_{n + 1} \ar[r]_{f_n} \ar[d]_{a_{n + 1}} & \xi_n \ar[d]^{a_n} \\
\eta_{n + 1} \ar[r]^{g_n} & \eta_n
}
$$
commutes.
 ========

new_annotation []
===== sent \end{enumerate}
\end{definition}

\noindent
The category of formal objects comes with a functor $\widehat{p}:
\widehat{\mathcal{F}} \to \widehat{\mathcal{C}}_\Lambda$ which sends an
object $(R, \xi_n, f_n)$ to $R$ and a morphism
$(R, \xi_n, f_n) \to (S, \eta_n, g_n)$ to the map $R \to S$.

\begin{lemma}
\label{lemma-completion-cofibred}
Let $p : \mathcal{F} \to \mathcal{C}_\Lambda$ be a category cofibered in
groupoids. ========

regex_match $p : \mathcal{F} \to \mathcal{C}_\Lambda$
nn candidate $p
nn candidate a category
nn a category
new_annotation [(61846, 61887, 'VAR'), (61891, 61901, 'TYPE')]
===== sent \end{lemma}

\begin{proof}
Let $R \to S$ be a ring map in $\widehat{\mathcal{C}}_\Lambda$.
Let $(R, \xi_n, f_n)$ be an object of $\widehat{\mathcal{F}}$.
For each $n$ choose a pushforward $\xi_n \to \eta_n$ of $\xi_n$
along $R/\mathfrak m_R^n \to S/\mathfrak m_S^n$. ========

regex_match $R \to S$
nn candidate a ring map
nn a ring map
new_annotation [(62072, 62081, 'VAR'), (62085, 62095, 'TYPE')]
===== sent \end{lemma}

\begin{proof}
Let $R \to S$ be a ring map in $\widehat{\mathcal{C}}_\Lambda$.
Let $(R, \xi_n, f_n)$ be an object of $\widehat{\mathcal{F}}$.
For each $n$ choose a pushforward $\xi_n \to \eta_n$ of $\xi_n$
along $R/\mathfrak m_R^n \to S/\mathfrak m_S^n$. ========

regex_match $(R, \xi_n, f_n)$
nn candidate an object
nn an object
new_annotation [(62136, 62153, 'VAR'), (62157, 62166, 'TYPE')]
===== sent \end{proof}

\begin{definition}
\label{definition-completion}
Let $p : \mathcal{F} \to \mathcal{C}_\Lambda$ be a category cofibered in
groupoids. ========

regex_match $p : \mathcal{F} \to \mathcal{C}_\Lambda$
nn candidate $p
nn candidate a category
nn a category
new_annotation [(63573, 63614, 'VAR'), (63618, 63628, 'TYPE')]
===== sent For each $n$, let $m(n)$ be the least $m$
that $I_m \subset \mathfrak m_R^n$.   ========

regex_match $m(n)$
new_annotation []
===== sent Given an object
$(\xi_n, f_n)$ of $\widehat{\mathcal{F}}_\mathcal{I}(R)$, let
$\eta_n$ be the pushforward of $\xi_{m(n)}$ along
$R/I_{m(n)} \to R/\mathfrak m_R^n$. Let $g_n : \eta_{n + 1} \to \eta_n$
be the unique morphism of $\mathcal{F}$ lying over
$R/\mathfrak m_R^{n + 1} \to R/\mathfrak m_R^n$ such that
$$
\xymatrix{
\xi_{m(n + 1)} \ar[rrr]_{f_{m(n)} \circ \ldots \circ f_{m(n + 1) - 1}} \ar[d]
& & & \xi_{m(n)} \ar[d] \\
\eta_{n + 1} \ar[rrr]^{g_n} & & & \eta_n
}
$$
commutes (existence and uniqueness is guaranteed by the axioms of a
cofibred category). ========

new_annotation []
===== sent \end{proof}

\begin{remark}
\label{remark-different-sequence-ideals}
Let $p : \mathcal{F} \to \mathcal{C}_\Lambda$ be a category cofibered in
groupoids. ========

regex_match $p : \mathcal{F} \to \mathcal{C}_\Lambda$
nn candidate $p
nn candidate a category
nn a category
new_annotation [(66085, 66126, 'VAR'), (66130, 66140, 'TYPE')]
===== sent The categories
cofibered in groupoids $\widehat{\mathcal{F}}_\mathcal{I}$ and
$\widehat{\mathcal{F}}$ are equivalent, by using over an object
$R \in \Ob(\widehat{\mathcal{C}}_\Lambda)$
the equivalence of
Lemma \ref{lemma-formal-objects-different-filtration}.
\end{remark}

\begin{remark}
\label{remark-completion-functor}
Let $F: \mathcal{C}_\Lambda \to \textit{Sets}$ be a functor.
 ========

regex_match $F: \mathcal{C}_\Lambda \to \textit{Sets}$
nn candidate \mathcal{C}_\Lambda \to
nn candidate a functor
nn a functor
new_annotation [(67137, 67179, 'VAR'), (67183, 67192, 'TYPE')]
===== sent This agrees
with the definition in Schlessinger's paper \cite{Sch}.
\end{remark}

\begin{remark}
\label{remark-restrict-completion}
Let $\mathcal{F}$ be a category cofibred in groupoids over
$\mathcal{C}_\Lambda$. We claim that there is a canonical
equivalence
$$
can :
\widehat{\mathcal{F}}|_{\mathcal{C}_\Lambda}
\longrightarrow
\mathcal{F}.
$$
Namely, let $A \in \Ob(\mathcal{C}_\Lambda)$ and let
$(A, \xi_n, f_n)$ be an object of
$\widehat{\mathcal{F}}|_{\mathcal{C}_\Lambda}(A)$.
Since $A$ is Artinian there is a minimal $m \in \mathbf{N}$
such that $\mathfrak m_A^m = 0$. Then $can$ sends $(A, \xi_n, f_n)$ to $\xi_m$.
This functor is an equivalence of categories cofibered in groupoids by
Categories, Lemma \ref{categories-lemma-equivalence-fibred-categories}
because it is an equivalence on all fibre categories by
Lemma \ref{lemma-formal-objects-different-filtration}
and the fact that the $\mathfrak m_A$-adic topology on a local
Artinian ring $A$ comes from the zero ideal. ========

regex_match $\mathcal{F}$
nn candidate a category
nn a category
new_annotation [(67532, 67545, 'VAR'), (67549, 67559, 'TYPE')]
===== sent This agrees
with the definition in Schlessinger's paper \cite{Sch}.
\end{remark}

\begin{remark}
\label{remark-restrict-completion}
Let $\mathcal{F}$ be a category cofibred in groupoids over
$\mathcal{C}_\Lambda$. We claim that there is a canonical
equivalence
$$
can :
\widehat{\mathcal{F}}|_{\mathcal{C}_\Lambda}
\longrightarrow
\mathcal{F}.
$$
Namely, let $A \in \Ob(\mathcal{C}_\Lambda)$ and let
$(A, \xi_n, f_n)$ be an object of
$\widehat{\mathcal{F}}|_{\mathcal{C}_\Lambda}(A)$.
Since $A$ is Artinian there is a minimal $m \in \mathbf{N}$
such that $\mathfrak m_A^m = 0$. Then $can$ sends $(A, \xi_n, f_n)$ to $\xi_m$.
This functor is an equivalence of categories cofibered in groupoids by
Categories, Lemma \ref{categories-lemma-equivalence-fibred-categories}
because it is an equivalence on all fibre categories by
Lemma \ref{lemma-formal-objects-different-filtration}
and the fact that the $\mathfrak m_A$-adic topology on a local
Artinian ring $A$ comes from the zero ideal. ========

new_annotation []
===== sent We will frequently identify
$\mathcal{F}$ with a full subcategory of $\widehat{\mathcal{F}}$ via a
quasi-inverse to the functor $can$.
\end{remark}

\begin{remark}
\label{remark-completion-morphism}
Let $\varphi : \mathcal{F} \to \mathcal{G}$ be a morphism of categories
cofibered in groupoids over $\mathcal{C}_\Lambda$. Then there is an induced
morphism
$\widehat{\varphi}: \widehat{\mathcal{F}} \to \widehat{\mathcal{G}}$
of categories cofibered in groupoids over ========

regex_match $\varphi : \mathcal{F} \to \mathcal{G}$
nn candidate $\varphi
nn candidate a morphism
nn a morphism
new_annotation [(68584, 68623, 'VAR'), (68627, 68637, 'TYPE')]
===== sent Let
$\mathcal{F} \in \Ob(\text{Cof}(\mathcal{C}_\Lambda))$
and let
$\mathcal{G} \in \Ob(\text{Cof}(\widehat{\mathcal{C}}_\Lambda))$.
Then there is an equivalence of categories
$$
\Phi :
\Mor_{\mathcal{C}_\Lambda}(
\mathcal{G}|_{\mathcal{C}_\Lambda}, \mathcal{F})
\longrightarrow
\Mor_{\widehat{\mathcal{C}}_\Lambda}(\mathcal{G}, \widehat{\mathcal{F}})
$$
To describe this equivalence, we
define canonical morphisms
$\mathcal{G} \to \widehat{\mathcal{G}|_{\mathcal{C}_\Lambda}}$ and
$\widehat{\mathcal{F}}|_{\mathcal{C}_\Lambda} \to \mathcal{F}$ as follows
\begin{enumerate}
\item Let $R \in \Ob(\widehat{\mathcal{C}}_\Lambda))$ and let $\xi$
be an object of the fiber category $\mathcal{G}(R)$.
Choose a pushforward $\xi \to \xi_n$ of $\xi$ to
$R/\mathfrak m_R^n$ for each $n \in \mathbf{N}$, and let
$f_n : \xi_{n + 1} \to \xi_n$ be the induced morphism.
 ========

new_annotation []
===== sent Let
$\mathcal{F} \in \Ob(\text{Cof}(\mathcal{C}_\Lambda))$
and let
$\mathcal{G} \in \Ob(\text{Cof}(\widehat{\mathcal{C}}_\Lambda))$.
Then there is an equivalence of categories
$$
\Phi :
\Mor_{\mathcal{C}_\Lambda}(
\mathcal{G}|_{\mathcal{C}_\Lambda}, \mathcal{F})
\longrightarrow
\Mor_{\widehat{\mathcal{C}}_\Lambda}(\mathcal{G}, \widehat{\mathcal{F}})
$$
To describe this equivalence, we
define canonical morphisms
$\mathcal{G} \to \widehat{\mathcal{G}|_{\mathcal{C}_\Lambda}}$ and
$\widehat{\mathcal{F}}|_{\mathcal{C}_\Lambda} \to \mathcal{F}$ as follows
\begin{enumerate}
\item Let $R \in \Ob(\widehat{\mathcal{C}}_\Lambda))$ and let $\xi$
be an object of the fiber category $\mathcal{G}(R)$.
Choose a pushforward $\xi \to \xi_n$ of $\xi$ to
$R/\mathfrak m_R^n$ for each $n \in \mathbf{N}$, and let
$f_n : \xi_{n + 1} \to \xi_n$ be the induced morphism.
 ========

new_annotation []
===== sent \end{remark}

\begin{remark}
\label{remark-restrict-complete-continuous-functor}
Let $G : \widehat{\mathcal{C}}_\Lambda \to \textit{Sets}$
be a functor that commutes with limits.
 ========

new_annotation []
===== sent \end{remark}

\begin{remark}
\label{remark-formal-objects-yoneda}
Let $R$ be an object of $\widehat{\mathcal{C}}_\Lambda$.  It defines a functor
$\underline{R}: \widehat{\mathcal{C}}_\Lambda \to \textit{Sets}$
as described in
Remarks \ref{remarks-cofibered-groupoids} (\ref{item-definition-yoneda}).
 ========

regex_match $R$
nn candidate an object
nn an object
new_annotation [(74524, 74527, 'VAR'), (74531, 74540, 'TYPE')]
===== sent Let $m \in \mathbf{N}$ be minimal such that $\mathfrak m_A^m = 0$.
Then $\alpha$ factors through a unique $\alpha_m : R/\mathfrak m_R^m \to A$
and we can set $\underline{\xi}(\alpha) = \alpha_{m, *}\xi_m$.
We omit the description of $\underline{\xi}$ on morphisms and we
omit the proof that $\underline{\xi}$ maps to $\xi$
via (\ref{equation-formal-objects-maps}).

 ========

new_annotation []
===== sent Using $\underline{R} = \widehat{\underline{R}|_{\mathcal{C}_\Lambda}}$
(see Remark \ref{remark-restrict-complete-continuous-functor})
and unwinding the definitions of $\Phi$ and $\Psi$ we conclude that
$\iota(\xi)$ is isomorphic to the completion of $\underline{\xi}$.
\end{remark}

\begin{remark}
\label{remark-formal-objects-yoneda-map}
Let $\mathcal{F}$ be a category cofibred in groupoids over
$\mathcal{C}_\Lambda$. Let $\xi = (R, \xi_n, f_n)$ and
$\eta = (S, \eta_n, g_n)$ be formal objects of $\mathcal{F}$.
Let $a = (a_n) : \xi \to \eta$ be a morphism of formal objects, i.e.,
a morphism of $\widehat{\mathcal{F}}$. Let
$f = \widehat{p}(a) = a_0 : R \to S$ be the projection of $a$ in
$\widehat{\mathcal{C}}_\Lambda$. Then we obtain a $2$-commutative
diagram
$$
\xymatrix{
\underline{R}|_{\mathcal{C}_\Lambda} \ar[rd]_{\underline{\xi}} & &
\underline{S}|_{\mathcal{C}_\Lambda} \ar[ll]^f \ar[ld]^{\underline{\eta}} \\
& \mathcal{F}
}
$$
where $\underline{\xi}$ and $\underline{\eta}$ are the morphisms
constructed in
Remark \ref{remark-formal-objects-yoneda}.
To see this let $\alpha : S \to A$ be an object of
$\underline{S}|_{\mathcal{C}_\Lambda}$ (see loc.\ cit.).
 ========

regex_match $\mathcal{F}$
nn candidate a category
nn a category
new_annotation [(78262, 78275, 'VAR'), (78279, 78289, 'TYPE')]
===== sent Using $\underline{R} = \widehat{\underline{R}|_{\mathcal{C}_\Lambda}}$
(see Remark \ref{remark-restrict-complete-continuous-functor})
and unwinding the definitions of $\Phi$ and $\Psi$ we conclude that
$\iota(\xi)$ is isomorphic to the completion of $\underline{\xi}$.
\end{remark}

\begin{remark}
\label{remark-formal-objects-yoneda-map}
Let $\mathcal{F}$ be a category cofibred in groupoids over
$\mathcal{C}_\Lambda$. Let $\xi = (R, \xi_n, f_n)$ and
$\eta = (S, \eta_n, g_n)$ be formal objects of $\mathcal{F}$.
Let $a = (a_n) : \xi \to \eta$ be a morphism of formal objects, i.e.,
a morphism of $\widehat{\mathcal{F}}$. Let
$f = \widehat{p}(a) = a_0 : R \to S$ be the projection of $a$ in
$\widehat{\mathcal{C}}_\Lambda$. Then we obtain a $2$-commutative
diagram
$$
\xymatrix{
\underline{R}|_{\mathcal{C}_\Lambda} \ar[rd]_{\underline{\xi}} & &
\underline{S}|_{\mathcal{C}_\Lambda} \ar[ll]^f \ar[ld]^{\underline{\eta}} \\
& \mathcal{F}
}
$$
where $\underline{\xi}$ and $\underline{\eta}$ are the morphisms
constructed in
Remark \ref{remark-formal-objects-yoneda}.
To see this let $\alpha : S \to A$ be an object of
$\underline{S}|_{\mathcal{C}_\Lambda}$ (see loc.\ cit.).
 ========

new_annotation []
===== sent Using $\underline{R} = \widehat{\underline{R}|_{\mathcal{C}_\Lambda}}$
(see Remark \ref{remark-restrict-complete-continuous-functor})
and unwinding the definitions of $\Phi$ and $\Psi$ we conclude that
$\iota(\xi)$ is isomorphic to the completion of $\underline{\xi}$.
\end{remark}

\begin{remark}
\label{remark-formal-objects-yoneda-map}
Let $\mathcal{F}$ be a category cofibred in groupoids over
$\mathcal{C}_\Lambda$. Let $\xi = (R, \xi_n, f_n)$ and
$\eta = (S, \eta_n, g_n)$ be formal objects of $\mathcal{F}$.
Let $a = (a_n) : \xi \to \eta$ be a morphism of formal objects, i.e.,
a morphism of $\widehat{\mathcal{F}}$. Let
$f = \widehat{p}(a) = a_0 : R \to S$ be the projection of $a$ in
$\widehat{\mathcal{C}}_\Lambda$. Then we obtain a $2$-commutative
diagram
$$
\xymatrix{
\underline{R}|_{\mathcal{C}_\Lambda} \ar[rd]_{\underline{\xi}} & &
\underline{S}|_{\mathcal{C}_\Lambda} \ar[ll]^f \ar[ld]^{\underline{\eta}} \\
& \mathcal{F}
}
$$
where $\underline{\xi}$ and $\underline{\eta}$ are the morphisms
constructed in
Remark \ref{remark-formal-objects-yoneda}.
To see this let $\alpha : S \to A$ be an object of
$\underline{S}|_{\mathcal{C}_\Lambda}$ (see loc.\ cit.).
 ========

regex_match $a = (a_n) : \xi \to \eta$
nn candidate a morphism
nn a morphism
new_annotation [(78438, 78464, 'VAR'), (78468, 78478, 'TYPE')]
===== sent Using $\underline{R} = \widehat{\underline{R}|_{\mathcal{C}_\Lambda}}$
(see Remark \ref{remark-restrict-complete-continuous-functor})
and unwinding the definitions of $\Phi$ and $\Psi$ we conclude that
$\iota(\xi)$ is isomorphic to the completion of $\underline{\xi}$.
\end{remark}

\begin{remark}
\label{remark-formal-objects-yoneda-map}
Let $\mathcal{F}$ be a category cofibred in groupoids over
$\mathcal{C}_\Lambda$. Let $\xi = (R, \xi_n, f_n)$ and
$\eta = (S, \eta_n, g_n)$ be formal objects of $\mathcal{F}$.
Let $a = (a_n) : \xi \to \eta$ be a morphism of formal objects, i.e.,
a morphism of $\widehat{\mathcal{F}}$. Let
$f = \widehat{p}(a) = a_0 : R \to S$ be the projection of $a$ in
$\widehat{\mathcal{C}}_\Lambda$. Then we obtain a $2$-commutative
diagram
$$
\xymatrix{
\underline{R}|_{\mathcal{C}_\Lambda} \ar[rd]_{\underline{\xi}} & &
\underline{S}|_{\mathcal{C}_\Lambda} \ar[ll]^f \ar[ld]^{\underline{\eta}} \\
& \mathcal{F}
}
$$
where $\underline{\xi}$ and $\underline{\eta}$ are the morphisms
constructed in
Remark \ref{remark-formal-objects-yoneda}.
To see this let $\alpha : S \to A$ be an object of
$\underline{S}|_{\mathcal{C}_\Lambda}$ (see loc.\ cit.).
 ========

regex_match $\alpha : S \to A$
nn candidate an object
nn an object
new_annotation [(79002, 79020, 'VAR'), (79024, 79033, 'TYPE')]
===== sent Let $m \in \mathbf{N}$ be minimal such that $\mathfrak m_A^m = 0$.
We get a commutative diagram
$$
\xymatrix{
R \ar[d]^f \ar[r] & R/\mathfrak m_R^m \ar[d]_{f_m} \ar[rd]^{\beta_m} \\
S \ar[r] & S/\mathfrak m_S^m \ar[r]^{\alpha_m} & A
}
$$
such that the bottom arrows compose to give $\alpha$.
Then $\underline{\eta}(\alpha) = \alpha_{m, *}\eta_m$ and
$\underline{\xi}(\alpha \circ f) = \beta_{m, *}\xi_m$. The morphism
$a_m : \xi_m \to \eta_m$ lies over $f_m$ hence we obtain a canonical
morphism
$$
\underline{\xi}(\alpha \circ f) = \beta_{m, *}\xi_m
\longrightarrow
\underline{\eta}(\alpha) = \alpha_{m, *}\eta_m
$$
lying over $\text{id}_A$ such that
$$
\xymatrix{
\xi_m \ar[r] \ar[d]^{a_m} &  \beta_{m, *}\xi_m \ar[d] \\
\eta_m \ar[r] &  \alpha_{m, *}\eta_m
}
$$
commutes by the axioms of a category cofibred in groupoids. ========

new_annotation []
===== sent Let $\varphi : \mathcal{F} \to \mathcal{G}$ be a morphism of categories
cofibered in groupoids over $\mathcal{C}_\Lambda$.  We say  $\varphi$ is
{\it smooth} if it satisfies the following condition: Let $B \to A$ be
a surjective ring map in $\mathcal{C}_\Lambda$.  Let $y \in
\Ob(\mathcal{G}(B)), x \in \Ob(\mathcal{F}(A))$, and $y
\to \varphi(x)$ be a morphism lying over $B \to A$.   ========

regex_match $\varphi : \mathcal{F} \to \mathcal{G}$
nn candidate $\varphi
nn candidate a morphism
nn a morphism
new_annotation [(80638, 80677, 'VAR'), (80681, 80691, 'TYPE')]
===== sent Let $\varphi : \mathcal{F} \to \mathcal{G}$ be a morphism of categories
cofibered in groupoids over $\mathcal{C}_\Lambda$.  We say  $\varphi$ is
{\it smooth} if it satisfies the following condition: Let $B \to A$ be
a surjective ring map in $\mathcal{C}_\Lambda$.  Let $y \in
\Ob(\mathcal{G}(B)), x \in \Ob(\mathcal{F}(A))$, and $y
\to \varphi(x)$ be a morphism lying over $B \to A$.   ========

new_annotation []
===== sent Let $\varphi : \mathcal{F} \to \mathcal{G}$ be a morphism of categories
cofibered in groupoids over $\mathcal{C}_\Lambda$.  We say  $\varphi$ is
{\it smooth} if it satisfies the following condition: Let $B \to A$ be
a surjective ring map in $\mathcal{C}_\Lambda$.  Let $y \in
\Ob(\mathcal{G}(B)), x \in \Ob(\mathcal{F}(A))$, and $y
\to \varphi(x)$ be a morphism lying over $B \to A$.   ========

new_annotation []
===== sent \end{definition}

\begin{lemma}
\label{lemma-smoothness-small-extensions}
Let $\varphi : \mathcal{F} \to \mathcal{G}$ be a morphism of categories
cofibered in groupoids over $\mathcal{C}_\Lambda$.  Then $\varphi$ is smooth
if the condition in Definition \ref{definition-smooth-morphism} is assumed to
hold only for small extensions $B \to A$.
 ========

regex_match $\varphi : \mathcal{F} \to \mathcal{G}$
nn candidate $\varphi
nn candidate a morphism
nn a morphism
new_annotation [(81362, 81401, 'VAR'), (81405, 81415, 'TYPE')]
===== sent \end{lemma}

\begin{proof}
Let $B \to A$ be a surjective ring map in $\mathcal{C}_\Lambda$.
Let $y \in \Ob(\mathcal{G}(B))$, $x \in \Ob(\mathcal{F}(A))$,
and $y \to \varphi(x)$ be a morphism lying over $B \to A$. ========

regex_match $B \to A$
nn candidate $B
nn candidate a surjective ring map
nn a surjective ring map
new_annotation [(81658, 81667, 'VAR'), (81671, 81692, 'TYPE')]
===== sent \end{lemma}

\begin{proof}
Let $B \to A$ be a surjective ring map in $\mathcal{C}_\Lambda$.
Let $y \in \Ob(\mathcal{G}(B))$, $x \in \Ob(\mathcal{F}(A))$,
and $y \to \varphi(x)$ be a morphism lying over $B \to A$. ========

new_annotation []
===== sent Let $\varphi : \mathcal{F} \to \mathcal{G}$ be a morphism of categories
cofibered in groupoids over $\mathcal{C}_\Lambda$.   ========

regex_match $\varphi : \mathcal{F} \to \mathcal{G}$
nn candidate $\varphi
nn candidate a morphism
nn a morphism
new_annotation [(83181, 83220, 'VAR'), (83224, 83234, 'TYPE')]
===== sent Let $B \to A$ be a
ring map in $\mathcal{C}_\Lambda$.  Choices of pushforwards along $B
\to A$ for objects in the fiber categories $\mathcal{F}(B)$ and
$\mathcal{G}(B)$ determine functors $\mathcal{F}(B) \to \mathcal{F}(A)$
and $\mathcal{G}(B) \to \mathcal{G}(A)$ fitting into a $2$-commutative
diagram
$$
\xymatrix{
\mathcal{F}(B) \ar[r]^{\varphi} \ar[d] & \mathcal{G}(B) \ar[d] \\
 ========

new_annotation []
===== sent Namely, in this case
let $F, G : \mathcal{C}_\Lambda \to \textit{Sets}$ be the corresponding
functors, see
Remarks \ref{remarks-cofibered-groupoids}
(\ref{item-convention-cofibered-sets}).
 ========

regex_match $F, G : \mathcal{C}_\Lambda \to \textit{Sets}$
nn candidate \mathcal{C}_\Lambda \to
nn candidate the corresponding
functors
nn the corresponding
functors
new_annotation [(84559, 84605, 'VAR'), (84609, 84635, 'TYPE')]
===== sent \end{remark}

\begin{remark}
\label{remark-smooth-to-iso-classes}
Let $\mathcal{F}$ be a category cofibered in groupoids over
$\mathcal{C}_\Lambda$. Then the morphism
$\mathcal{F} \to \overline{\mathcal{F}}$ is smooth.
 ========

regex_match $\mathcal{F}$
nn candidate a category
nn a category
new_annotation [(84956, 84969, 'VAR'), (84973, 84983, 'TYPE')]
===== sent Namely, suppose that $f : B \to A$ is a ring map in $\mathcal{C}_\Lambda$.
Let $x \in \Ob(\mathcal{F}(A))$ and let
$\overline{y} \in \overline{\mathcal{F}}(B)$
be the isomorphism class of $y \in \Ob(\mathcal{F}(B))$ such that
$\overline{f_*y} = \overline{x}$. Then we simply take $x' = y$, the
implied morphism $x' = y \to x$ over $B \to A$, and the equality
$\overline{x'} = \overline{y}$ as the solution to
the problem posed in Definition \ref{definition-smooth-morphism}.
\end{remark}

\noindent
If $R \to S$ is a ring map $\widehat{\mathcal{C}}_\Lambda$, then there
is an induced morphism $\underline{S} \to \underline{R}$ between the
functors $\underline{S}, \underline{R}: \widehat{\mathcal{C}}_\Lambda
\to \textit{Sets}$.   ========

new_annotation []
===== sent In this situation, smoothness of the
restriction $\underline{S}|_{\mathcal{C}_\Lambda} \to
\underline{R}|_{\mathcal{C}_\Lambda}$ is a familiar notion:

\begin{lemma}
\label{lemma-smooth-morphism-power-series}
Let $R \to S$ be a ring map in $\widehat{\mathcal{C}}_\Lambda$. Then
the induced morphism
$\underline{S}|_{\mathcal{C}_\Lambda} \to \underline{R}|_{\mathcal{C}_\Lambda}$
is smooth if and only if $S$ is a power series ring over $R$.
\end{lemma}

\begin{proof}
Assume $S$ is a power series ring over $R$. Say $S = R[[x_1, \ldots, x_n]]$.
Smoothness of
$\underline{S}|_{\mathcal{C}_\Lambda} \to \underline{R}|_{\mathcal{C}_\Lambda}$
means the following (see Remark \ref{remark-compare-smooth-schlessinger}):
Given a surjective ring map $B \to A$ in
$\mathcal{C}_\Lambda$, a ring map $R \to B$, a ring map $S \to A$ such that
the solid diagram
$$
\xymatrix{
S \ar[r] \ar@{..>}[rd] & A \\
 ========

regex_match $R \to S$
nn candidate a ring map
nn a ring map
new_annotation [(86048, 86057, 'VAR'), (86061, 86071, 'TYPE')]
===== sent Let $x_1, \ldots, x_n \in S$ be elements whose images
form a basis in the relative cotangent space
$\mathfrak m_S/(\mathfrak m_R S + \mathfrak m_S^2)$ of $S$ over $R$.
Set $T = R[[X_1, \ldots, X_n]]$. Note that both
$$
S/(\mathfrak m_R S + \mathfrak m_S^2) \cong
R/\mathfrak m_R[x_1, \ldots, x_n]/(x_ix_j)
$$
and
$$
T/(\mathfrak m_R T + \mathfrak m_T^2) \cong
 ========

new_annotation []
===== sent \begin{lemma}
\label{lemma-smooth-properties}
Let $\varphi : \mathcal{F} \to \mathcal{G}$ and $\psi : \mathcal{G}
\to \mathcal{H}$ be morphisms of categories cofibered in groupoids over
$\mathcal{C}_\Lambda$.
\begin{enumerate}
\item If $\varphi$ and $\psi$ are smooth, ========

new_annotation []
===== sent Let $\varphi : \mathcal{F} \to \mathcal{G}$ be a smooth morphism of
categories cofibered in groupoids over $\mathcal{C}_\Lambda$.  Assume
$\varphi : \mathcal{F}(k) \to \mathcal{G}(k)$ is essentially surjective.
 ========

regex_match $\varphi : \mathcal{F} \to \mathcal{G}$
nn candidate $\varphi
nn candidate a smooth morphism
nn a smooth morphism
new_annotation [(90406, 90445, 'VAR'), (90449, 90466, 'TYPE')]
===== sent \end{lemma}

\begin{proof}
Let $y$ be an object of $\mathcal{G}$ lying over
$A \in \Ob(\mathcal{C}_\Lambda)$. Let $y \to y_0$ be a pushforward
of $y$ along $ ========

regex_match $y$
nn candidate an object
nn an object
new_annotation [(90791, 90794, 'VAR'), (90798, 90807, 'TYPE')]
===== sent \end{lemma}

\begin{proof}
Let $y$ be an object of $\mathcal{G}$ lying over
$A \in \Ob(\mathcal{C}_\Lambda)$. Let $y \to y_0$ be a pushforward
of $y$ along $ ========

regex_match $y \to y_0$
nn candidate a pushforward
nn a pushforward
new_annotation [(90874, 90885, 'VAR'), (90889, 90902, 'TYPE')]
===== sent Let $\eta = (R, \eta_n, g_n)$ be an object of $\widehat{\mathcal{G}}$.  We
construct an object $\xi$ of $\widehat{\mathcal{F}}$ with an isomorphism
$\eta \to \varphi(\xi)$. By the assumption on essential surjectivity of
$\varphi : \mathcal{F}(k) \to \mathcal{G}(k)$, there exists a morphism
$\eta_1 \to \varphi(\xi_1)$ in $\mathcal{G}(k)$ for some
$\xi_1 \in \Ob(\mathcal{F}(k))$. ========

regex_match $\eta = (R, \eta_n, g_n)$
nn candidate an object
nn an object
new_annotation [(91350, 91375, 'VAR'), (91379, 91388, 'TYPE')]
===== sent Continuing in this way we construct an object
$\xi = (R, \xi_n, f_n)$ of $\widehat{\mathcal{F}}$ and a morphism
$\eta \to \varphi(\xi) = (R, \varphi(\xi_n), \varphi(f_n))$
in $\widehat{\mathcal{G}}(R)$.
\end{proof}

\noindent
Later we are interested in producing smooth morphisms from
prorepresentable functors to predeformation categories $\mathcal{F}$.
By the discussion in
Remark \ref{remark-formal-objects-yoneda}
these morphisms correspond to certain formal objects of $\mathcal{F}$.
More precisely, these are the so-called versal formal objects of $\mathcal{F}$.

\begin{definition}
\label{definition-versal}
Let $\mathcal{F}$ be a category cofibered in groupoids.   ========

regex_match $\mathcal{F}$
nn candidate a category
nn a category
new_annotation [(92835, 92848, 'VAR'), (92852, 92862, 'TYPE')]
===== sent Let $\xi$ be a formal
object of $\mathcal{F}$ lying over $R \in \Ob(\widehat{\mathcal{C}}_\Lambda)$.
We say $\xi$ is {\it versal} if the corresponding morphism
$\underline{\xi}: \underline{R}|_{\mathcal{C}_\Lambda} \to \mathcal{F}$
of Remark \ref{remark-formal-objects-yoneda} is smooth.
 ========

regex_match $\xi$
nn candidate a formal
object
nn a formal
object
new_annotation [(92892, 92897, 'VAR'), (92901, 92916, 'TYPE')]
===== sent \end{definition}

\begin{remark}
\label{remark-versal-object}
Let $\mathcal{F}$ be a category cofibered in groupoids over $\mathcal
C_\Lambda$, and let $\xi$ be a formal object of $\mathcal{F}$.  It follows
from the definition of smoothness that versality of $\xi$ is equivalent to the
following condition: ========

regex_match $\mathcal{F}$
nn candidate a category
nn a category
new_annotation [(93242, 93255, 'VAR'), (93259, 93269, 'TYPE')]
===== sent \end{definition}

\begin{remark}
\label{remark-versal-object}
Let $\mathcal{F}$ be a category cofibered in groupoids over $\mathcal
C_\Lambda$, and let $\xi$ be a formal object of $\mathcal{F}$.  It follows
from the definition of smoothness that versality of $\xi$ is equivalent to the
following condition: ========

regex_match $\xi$
nn candidate a formal object
nn a formal object
new_annotation [(93328, 93333, 'VAR'), (93337, 93352, 'TYPE')]
===== sent In particular, the condition that $\xi$ be versal does not depend on
the choices of pushforwards made in the construction of
$\underline{\xi} : \underline{R}|_{\mathcal{C}_\Lambda} \to \mathcal{F}$ in
Remark \ref{remark-formal-objects-yoneda}.
\end{remark}

\begin{lemma}
\label{lemma-versal-object-quasi-initial}
Let $\mathcal{F}$ be a predeformation category.
 ========

regex_match $\mathcal{F}$
nn candidate a predeformation category
nn a predeformation category
new_annotation [(94126, 94139, 'VAR'), (94143, 94168, 'TYPE')]
===== sent Let $\xi$ be a versal formal object of $\mathcal{F}$.
For any formal object $\eta$ of $\widehat{\mathcal{F}}$,
there exists a morphism $\xi \to \eta$.
\end{lemma}

\begin{proof}
 ========

regex_match $\xi$
nn candidate a versal formal object
nn a versal formal object
new_annotation [(94174, 94179, 'VAR'), (94183, 94205, 'TYPE')]
===== sent Then
$\iota(\xi) : \underline{R} \to \widehat{\mathcal{F}}$
is the completion of $\underline{\xi}$, see
Remark \ref{remark-formal-objects-yoneda}.
By
Lemma \ref{lemma-smooth-morphism-essentially-surjective}
there exists an object $f$ of $\underline{R}$ such that
$\iota(\xi)(f) = \eta$. Then $f$ is
a ring map $f : R \to S$ in $\widehat{\mathcal{C}}_\Lambda$. And
$\iota(\xi)(f) = \eta$ means that
$f_*\xi \cong \eta$ which means exactly that there is a morphism
$\xi \to \eta$ lying over $f$.
\end{proof}





\section{Smooth or unobstructed categories}
\label{section-smooth}

\noindent
Let $p : \mathcal{F} \to \mathcal{C}_\Lambda$ be a category cofibered in
groupoids. ========

regex_match $p : \mathcal{F} \to \mathcal{C}_\Lambda$
nn candidate $p
nn candidate a category
nn a category
new_annotation [(95052, 95093, 'VAR'), (95097, 95107, 'TYPE')]
===== sent In this way
$p : \mathcal{F} \longrightarrow \mathcal{C}_\Lambda$
becomes
a morphism of categories cofibered in groupoids over $\mathcal{C}_\Lambda$.

\begin{definition}
\label{definition-cofibered-groupoid-projection-smooth}
Let $p : \mathcal{F} \to \mathcal{C}_\Lambda$ be a category cofibered in
groupoids. ========

regex_match $p : \mathcal{F} \to \mathcal{C}_\Lambda$
nn candidate $p
nn candidate a category
nn a category
new_annotation [(95492, 95533, 'VAR'), (95537, 95547, 'TYPE')]
===== sent \end{remark}

\begin{lemma}
\label{lemma-smooth}
Let $R \in \Ob(\widehat{\mathcal{C}}_\Lambda)$. The following are
equivalent
\begin{enumerate}
\item $\underline{R}|_{\mathcal{C}_\Lambda}$ is smooth,
\item $\Lambda \to R$ is formally smooth in the $\mathfrak m_R$-adic topology,
\item $\Lambda \to R$ is flat and $R \otimes_\Lambda k'$ is
geometrically regular over $k'$, and
\item $\Lambda \to R$ is flat and $k' \to R \otimes_\Lambda k'$ is
formally smooth in the $\mathfrak m_R$-adic topology.
\end{enumerate}
In the classical case, these are also equivalent to
\begin{enumerate}
\item[(5)] $R$ is isomorphic to $\Lambda[[x_1, \ldots, x_n]]$
for some $n$.
\end{enumerate}
\end{lemma}

\begin{proof}
Smoothness of
$p : \underline{R}|_{\mathcal{C}_\Lambda} \to \mathcal{C}_\Lambda$
means that given $B \to A$ surjective in $\mathcal{C}_\Lambda$
and given $R \to A$ we can find the dotted arrow in the
diagram
$$
\xymatrix{
R \ar[r] \ar@{-->}[rd] & A \\
 ========

new_annotation []
===== sent \end{proof}

\begin{lemma}
\label{lemma-smooth-power-series-classical}
Let $\mathcal{F}$ be a predeformation category.
 ========

regex_match $\mathcal{F}$
nn candidate a predeformation category
nn a predeformation category
new_annotation [(98638, 98651, 'VAR'), (98655, 98680, 'TYPE')]
===== sent Let $\xi$ be a versal formal object of $\mathcal{F}$ lying over
$R \in \Ob(\widehat{\mathcal{C}}_\Lambda)$. The following are
equivalent
\begin{enumerate}
\item $\mathcal{F}$ is unobstructed, and
\item $\Lambda \to R$ is formally smooth in the $\mathfrak m_R$-adic topology.
 ========

regex_match $\xi$
nn candidate a versal formal object
nn a versal formal object
new_annotation [(98686, 98691, 'VAR'), (98695, 98717, 'TYPE')]
===== sent Choose a smooth
$\Lambda$-algebra $P$ and a $\Lambda$-algebra surjection $P \to k$.
(For example, let $P$ be a polynomial algebra.)
 ========

regex_match $P$
nn candidate $P$
nn candidate a polynomial algebra
nn a polynomial algebra
new_annotation [(100731, 100734, 'VAR'), (100738, 100758, 'TYPE')]
===== sent Repeating finitely many times, we may assume
the map $\mathfrak m_P/\mathfrak m_P^2 \to
\Omega_{P/\Lambda} \otimes_P k$ is zero so that
the exact sequence breaks into isomorphisms
$H_1(L_{k/\Lambda}) = \mathfrak m_P/\mathfrak m_P^2$ and
$\Omega_{P/\Lambda} \otimes_P k = \Omega_{k/\Lambda}$.

\medskip\noindent
Let $R$ be the $\mathfrak m_P$-adic completion of $P$.
Then $R$ is an object of $\widehat{\mathcal{C}}_\Lambda$.
 ========

regex_match $R$
nn candidate the $\mathfrak m_P$-adic completion
nn the $\mathfrak m_P$-adic completion
new_annotation [(102035, 102038, 'VAR'), (102042, 102077, 'TYPE')]
===== sent In fact, it turns out that whenever $\underline{R}|_{\mathcal{C}_\Lambda}$
is smooth, then $R$ is isomorphic to a completion of a smooth
algebra over $\Lambda$, but we won't use this.
\end{proof}

\begin{example}
\label{example-smooth-explicit}
Here is a more explicit example of an $R$ as in
Lemma \ref{lemma-exists-smooth}.
Let $p$ be a prime number and let $n \in \mathbf{N}$.
Let $\Lambda = \mathbf{F}_p(t_1, t_2, \ldots, t_n)$ and let
$k = \mathbf{F}_p(x_1, \ldots, x_n)$ with map $\Lambda \to k$ given
by $t_i \mapsto x_i^p$. ========

regex_match $p$
nn candidate a prime number
nn a prime number
new_annotation [(103904, 103907, 'VAR'), (103911, 103925, 'TYPE')]
===== sent In fact, it turns out that whenever $\underline{R}|_{\mathcal{C}_\Lambda}$
is smooth, then $R$ is isomorphic to a completion of a smooth
algebra over $\Lambda$, but we won't use this.
\end{proof}

\begin{example}
\label{example-smooth-explicit}
Here is a more explicit example of an $R$ as in
Lemma \ref{lemma-exists-smooth}.
Let $p$ be a prime number and let $n \in \mathbf{N}$.
Let $\Lambda = \mathbf{F}_p(t_1, t_2, \ldots, t_n)$ and let
$k = \mathbf{F}_p(x_1, \ldots, x_n)$ with map $\Lambda \to k$ given
by $t_i \mapsto x_i^p$. ========

new_annotation []
===== sent In fact, it turns out that whenever $\underline{R}|_{\mathcal{C}_\Lambda}$
is smooth, then $R$ is isomorphic to a completion of a smooth
algebra over $\Lambda$, but we won't use this.
\end{proof}

\begin{example}
\label{example-smooth-explicit}
Here is a more explicit example of an $R$ as in
Lemma \ref{lemma-exists-smooth}.
Let $p$ be a prime number and let $n \in \mathbf{N}$.
Let $\Lambda = \mathbf{F}_p(t_1, t_2, \ldots, t_n)$ and let
$k = \mathbf{F}_p(x_1, \ldots, x_n)$ with map $\Lambda \to k$ given
by $t_i \mapsto x_i^p$. ========

new_annotation []
===== sent \medskip\noindent
We denote by $k[\epsilon]$ the ring of dual numbers over $k$.  More
generally, for a $k$-vector space $V$, we denote by $k[V]$ the $k$-algebra
whose underlying vector space is $k \oplus V$ and whose multiplication is given
by $(a, v) \cdot (a', v') = (aa', av' + a'v)$.  When $V = k$, $k[V]$ is the ring
of dual numbers over $k$.  For any finite dimensional $k$-vector space $V$
the ring $k[V]$ is in $\mathcal{C}_\Lambda$.

\begin{definition}
\label{definition-S1-S2}
Let $\mathcal{F}$ be a category cofibered in groupoids over $\mathcal
C_\Lambda$. We define {\it conditions (S1) and (S2)}
on $\mathcal{F}$ as follows:
\begin{enumerate}
\item[(S1)] Every diagram in $\mathcal{F}$
$$
\vcenter{
\xymatrix{
           & x_2 \ar[d] ========

regex_match $\mathcal{F}$
nn candidate a category
nn a category
new_annotation [(105652, 105665, 'VAR'), (105669, 105679, 'TYPE')]
===== sent \begin{lemma}
\label{lemma-S1-small-extensions}
Let $\mathcal{F}$ be a category cofibered in groupoids over $\mathcal
C_\Lambda$. Then $\mathcal{F}$ satisfies (S1) if the condition of (S1)
is assumed to hold only when $A_2 \to A$ is a small extension.
 ========

regex_match $\mathcal{F}$
nn candidate a category
nn a category
new_annotation [(108467, 108480, 'VAR'), (108484, 108494, 'TYPE')]
===== sent \end{remark}

\begin{lemma}
\label{lemma-S2-extensions}
Let $\mathcal{F}$ be a category cofibred in groupoids over
$\mathcal{C}_\Lambda$. If $\mathcal{F}$ satisfies (S2), then the
condition of (S2) also holds when $k[\epsilon]$ is replaced by $k[V]$
for any finite dimensional $k$-vector space $V$.
\end{lemma}

\begin{proof}
 ========

regex_match $\mathcal{F}$
nn candidate a category
nn a category
new_annotation [(109797, 109810, 'VAR'), (109814, 109824, 'TYPE')]
===== sent To see this we let $y \to y_V$ be a morphism lying over
$A \times_k k[V] \to k[V]$. We can factor the morphisms
$y \to y' \to x_{V'}$ and $y \to x_\epsilon$ through the morphism
$y \to y_V$ (by the axioms of categories cofibred in groupoids).
 ========

regex_match $y \to y_V$
nn candidate $y
nn candidate a morphism
nn a morphism
new_annotation [(112007, 112018, 'VAR'), (112022, 112032, 'TYPE')]
===== sent \end{proof}

\begin{lemma}
\label{lemma-S1-S2-associated-functor}
Let $\mathcal{F}$ be a category cofibered in groupoids over
$\mathcal{C}_\Lambda$.
\begin{enumerate}
\item If $\mathcal{F}$ satisfies (S1), then so does
$\overline{\mathcal{F}}$.
\item If $\mathcal{F}$ satisfies (S2), then so does
$\overline{\mathcal{F}}$ provided at least one of the following conditions is
satisfied
\begin{enumerate}
\item $\mathcal{F}$ is a predeformation category,
\item the category $\mathcal{F}(k)$ is a set or a setoid, or
\item for any morphism $x_\epsilon \to x_0$ of $\mathcal{F}$
lying over $k[\epsilon] \to k$ the pushforward map
$\text{Aut}_{k[\epsilon]}(x_\epsilon) \to \text{Aut}_k(x_0)$
is surjective.
 ========

regex_match $\mathcal{F}$
nn candidate a category
nn a category
new_annotation [(115163, 115176, 'VAR'), (115180, 115190, 'TYPE')]
===== sent Let $x_i \in \mathcal{F}(A_i)$ such that
the pushforwards $f_{1, *}(x_1)$ and $f_{2, *}(x_2)$ are isomorphic.
 ========

new_annotation []
===== sent \end{proof}

\begin{lemma}
\label{lemma-S1-S2-localize}
Let $\mathcal{F}$ be a category cofibered in groupoids over
$\mathcal{C}_\Lambda$. Let $x_0 \in \Ob(\mathcal{F}(k))$.
 ========

regex_match $\mathcal{F}$
nn candidate a category
nn a category
new_annotation [(118732, 118745, 'VAR'), (118749, 118759, 'TYPE')]
===== sent \end{proof}

\begin{lemma}
\label{lemma-S1-S2-localize}
Let $\mathcal{F}$ be a category cofibered in groupoids over
$\mathcal{C}_\Lambda$. Let $x_0 \in \Ob(\mathcal{F}(k))$.
 ========

new_annotation []
===== sent Let $\mathcal{F}_{x_0}$ be the category cofibred in groupoids over
$\mathcal{C}_\Lambda$ constructed in
Remark \ref{remark-localize-cofibered-groupoid}.
\begin{enumerate}
\item If $\mathcal{F}$ satisfies (S1), then so does $\mathcal{F}_{x_0}$.
\item If $\mathcal{F}$ satisfies (S2), then so does $\mathcal{F}_{x_0}$.
\end{enumerate}
\end{lemma}

\begin{proof}
Any diagram as in
Definition \ref{definition-S1-S2}
in $\mathcal{F}_{x_0}$ gives rise to a diagram in $\mathcal{F}$
and the output of condition (S1) or (S2) for this diagram in $\mathcal{F}$
can be viewed as an output for $\mathcal{F}_{x_0}$ as well.
 ========

regex_match $\mathcal{F}_{x_0}$
nn candidate the category
nn the category
new_annotation [(118850, 118869, 'VAR'), (118873, 118885, 'TYPE')]
===== sent \end{proof}

\begin{lemma}
\label{lemma-lifting-section}
Let $p: \mathcal{F} \to \mathcal{C}_\Lambda$ be a category cofibered in
groupoids. ========

regex_match $p: \mathcal{F} \to \mathcal{C}_\Lambda$
nn candidate $p
nn candidate a category
nn a category
new_annotation [(119518, 119558, 'VAR'), (119562, 119572, 'TYPE')]
===== sent Assume there is a map $h : A_1 \to A_2$ such that $f_2 = f_1 \circ h$.
Let $I = \Ker(f_2)$. Consider the ring map
$$
g : A_1 \times_A A_2 \longrightarrow k[I] = k \oplus I, \quad
(u, v) \longmapsto \overline{u} \oplus (v - h(u))
$$
Choose a pushforward $y \to g_*y$. Assume $\mathcal{F}$ satisfies (S2).
 ========

new_annotation []
===== sent \end{proof}





\section{Tangent spaces of functors}
\label{section-tangent-spaces-functors}

\noindent
Let $R$ be a ring. ========

regex_match $R$
nn candidate a ring
nn a ring
new_annotation [(123031, 123034, 'VAR'), (123038, 123044, 'TYPE')]
===== sent \begin{definition}
\label{definition-linear}
Let $L: \text{Mod}^{fg}_R \to \text{Mod}_R$,
resp.\ $L: \text{Mod}_R \to \text{Mod}_R$
be a functor.   ========

new_annotation []
===== sent \end{remark}

\begin{remark}
\label{remark-linear-functor}
If $L: \text{Mod}^{fg}_R \to \text{Mod}_R$ is an $R$-linear functor,
then $L$ preserves finite products and sends the zero module to the zero
module, see
Homology, Lemma \ref{homology-lemma-additive-additive}.
On the other hand, if a functor $\text{Mod}^{fg}_R \to \textit{Sets}$
preserves finite products and sends the zero module to a one element set,
then it has a unique lift to a $R$-linear functor, see
Lemma \ref{lemma-linear-functor}.
\end{remark}

\begin{lemma}
\label{lemma-linear-functor}
Let $L: \text{Mod}^{fg}_R \to \textit{Sets}$,
resp.\ $L: \text{Mod}_R \to \textit{Sets}$ be a
functor.   ========

new_annotation []
===== sent Let $M$ be a finitely generated $R$-module. ========

regex_match $M$
new_annotation []
===== sent By assumption $L(M \times M)$
is canonically isomorphic to $L(M) \times L(M)$.  Addition on $L(M)$ is defined
by the map $L(M) \times L(M) \cong L(M \times M) \to L(M)$.

\medskip\noindent
Zero: There is a unique map $0 \to M$. The zero element of $L(M)$ is
the image of $L(0) \to L(M)$.

\medskip\noindent
We omit the verification that this defines an $R$-module $\widetilde{L}(M)$,
the unique such that is $R$-linearly functorial in $M$.
\end{proof}

\begin{lemma}
\label{lemma-morphism-linear-functors}
Let $L_1, L_2: \text{Mod}^{fg}_R \to \textit{Sets}$ be
functors that take $0$ to a one element set and preserve finite products.
Let $t : L_1 \to L_2$ be a morphism of functors. ========

new_annotation []
===== sent By assumption $L(M \times M)$
is canonically isomorphic to $L(M) \times L(M)$.  Addition on $L(M)$ is defined
by the map $L(M) \times L(M) \cong L(M \times M) \to L(M)$.

\medskip\noindent
Zero: There is a unique map $0 \to M$. The zero element of $L(M)$ is
the image of $L(0) \to L(M)$.

\medskip\noindent
We omit the verification that this defines an $R$-module $\widetilde{L}(M)$,
the unique such that is $R$-linearly functorial in $M$.
\end{proof}

\begin{lemma}
\label{lemma-morphism-linear-functors}
Let $L_1, L_2: \text{Mod}^{fg}_R \to \textit{Sets}$ be
functors that take $0$ to a one element set and preserve finite products.
Let $t : L_1 \to L_2$ be a morphism of functors. ========

regex_match $t : L_1 \to L_2$
nn candidate $t
nn candidate a morphism
nn a morphism
new_annotation [(126315, 126332, 'VAR'), (126336, 126346, 'TYPE')]
===== sent In the case $R = K$ is a field, a $K$-linear functor
$L : \text{Mod}^{fg}_K \to \text{Mod}_K$ is determined by its value $L(K)$.

\begin{lemma}
\label{lemma-linear-functor-over-field}
Let $K$ be a field. ========

regex_match $K$
nn candidate a field
nn a field
new_annotation [(126981, 126984, 'VAR'), (126988, 126995, 'TYPE')]
===== sent Let $L: \text{Mod}^{fg}_K \to
\text{Mod}_K$ be a $K$-linear functor.   ========

regex_match $L: \text{Mod}^{fg}_K \to
\text{Mod}_K$
nn candidate $
nn candidate \text{Mod}^{fg}_K \to
nn candidate a $K$-linear functor
nn a $K$-linear functor
new_annotation [(127001, 127040, 'VAR'), (127044, 127064, 'TYPE')]
===== sent \end{proof}

\noindent
For a ring $R$ and an $R$-module $M$, let $R[M]$ be the $R$-algebra whose
underlying $R$-module is $R \oplus M$ and whose multiplication is given by
$(r, m) \cdot (r', m') = (rr', rm' + r'm)$.  When $M = R$ this is the ring of
dual numbers over $R$, which we denote by $R[\epsilon]$.

\medskip\noindent
 ========

regex_match $R[M]$
nn candidate $R$-module
nn $R$-module
new_annotation [(127708, 127714, 'VAR'), (127751, 127761, 'TYPE')]
===== sent Now let $S$ be a ring and assume $R$ is an $S$-algebra.
 ========

regex_match $S$
nn candidate $S$
nn candidate a ring
nn a ring
new_annotation [(127977, 127980, 'VAR'), (127984, 127990, 'TYPE')]
===== sent \begin{lemma}
\label{lemma-preserves-products}
Let $R$ be an $S$-algebra. ========

regex_match $R$
nn candidate an $S$-algebra
nn an $S$-algebra
new_annotation [(128377, 128380, 'VAR'), (128384, 128398, 'TYPE')]
===== sent This is merely the statement that if $M$ and $N$ are $R$-modules, then the map
$R[M \times N] \to R[M] \times_R R[N]$ is an isomorphism in
$S\text{-Alg}/R$.
\end{proof}

\begin{lemma}
\label{lemma-tangent-space-functor}
Let $R$ be an $S$-algebra, and let $\mathcal{C}$ be a strictly full
subcategory of $S\text{-Alg}/R$ containing $R[M]$ for all
$M \in \Ob(\text{Mod}^{fg}_R)$.
Let $F: \mathcal{C} \to \textit{Sets}$ be a functor. ========

regex_match $R$
nn candidate an $S$-algebra
nn an $S$-algebra
new_annotation [(128745, 128748, 'VAR'), (128752, 128766, 'TYPE')]
===== sent This is merely the statement that if $M$ and $N$ are $R$-modules, then the map
$R[M \times N] \to R[M] \times_R R[N]$ is an isomorphism in
$S\text{-Alg}/R$.
\end{proof}

\begin{lemma}
\label{lemma-tangent-space-functor}
Let $R$ be an $S$-algebra, and let $\mathcal{C}$ be a strictly full
subcategory of $S\text{-Alg}/R$ containing $R[M]$ for all
$M \in \Ob(\text{Mod}^{fg}_R)$.
Let $F: \mathcal{C} \to \textit{Sets}$ be a functor. ========

regex_match $\mathcal{C}$
nn candidate a strictly full
subcategory
nn a strictly full
subcategory
new_annotation [(128776, 128789, 'VAR'), (128793, 128820, 'TYPE')]
===== sent This is merely the statement that if $M$ and $N$ are $R$-modules, then the map
$R[M \times N] \to R[M] \times_R R[N]$ is an isomorphism in
$S\text{-Alg}/R$.
\end{proof}

\begin{lemma}
\label{lemma-tangent-space-functor}
Let $R$ be an $S$-algebra, and let $\mathcal{C}$ be a strictly full
subcategory of $S\text{-Alg}/R$ containing $R[M]$ for all
$M \in \Ob(\text{Mod}^{fg}_R)$.
Let $F: \mathcal{C} \to \textit{Sets}$ be a functor. ========

regex_match $F: \mathcal{C} \to \textit{Sets}$
nn candidate $F
nn candidate a functor
nn a functor
new_annotation [(128903, 128937, 'VAR'), (128941, 128950, 'TYPE')]
===== sent The functor $\text{Mod}_R \to S\text{-Alg}/R$ of
Lemma \ref{lemma-preserves-products}
restricts to a functor $\text{Mod}^{fg}_R \to \mathcal{C}$
by the assumption on $\mathcal{C}$. Let $L$ be ========

new_annotation []
===== sent Hence
Lemma \ref{lemma-linear-functor}
shows that $L(M) = F(R[M])$ has a natural $R$-module structure for any
$M \in \Ob(\text{Mod}^{fg}_R)$.
\end{proof}

\begin{definition}
\label{definition-tangent-space-over-R}
Let $\mathcal{C}$ be a category as in
Lemma \ref{lemma-tangent-space-functor}.
Let $F : \mathcal{C} \to \textit{Sets}$ be a functor such that
$F(R)$ is a one element set. ========

regex_match $\mathcal{C}$
nn candidate a category
nn a category
new_annotation [(130066, 130079, 'VAR'), (130083, 130093, 'TYPE')]
===== sent Hence
Lemma \ref{lemma-linear-functor}
shows that $L(M) = F(R[M])$ has a natural $R$-module structure for any
$M \in \Ob(\text{Mod}^{fg}_R)$.
\end{proof}

\begin{definition}
\label{definition-tangent-space-over-R}
Let $\mathcal{C}$ be a category as in
Lemma \ref{lemma-tangent-space-functor}.
Let $F : \mathcal{C} \to \textit{Sets}$ be a functor such that
$F(R)$ is a one element set. ========

regex_match $F : \mathcal{C} \to \textit{Sets}$
nn candidate $F
nn candidate a functor
nn a functor
new_annotation [(130145, 130180, 'VAR'), (130184, 130193, 'TYPE')]
===== sent See
Exercises, Exercise \ref{exercises-exercise-tangent-space-Zariski}.
\end{example}

\begin{lemma}
\label{lemma-morphism-tangent-spaces}
Let $F, G: \mathcal{C} \to \textit{Sets}$ be functors satisfying
the hypotheses of
Lemma \ref{lemma-tangent-space-functor}.
Let $t : F \to G$ be a morphism of functors. ========

new_annotation []
===== sent See
Exercises, Exercise \ref{exercises-exercise-tangent-space-Zariski}.
\end{example}

\begin{lemma}
\label{lemma-morphism-tangent-spaces}
Let $F, G: \mathcal{C} \to \textit{Sets}$ be functors satisfying
the hypotheses of
Lemma \ref{lemma-tangent-space-functor}.
Let $t : F \to G$ be a morphism of functors. ========

regex_match $t : F \to G$
nn candidate a morphism
nn a morphism
new_annotation [(132969, 132982, 'VAR'), (132986, 132996, 'TYPE')]
===== sent \end{example}

\begin{lemma}
\label{lemma-tangent-space-tensor}
Let $F: \mathcal{C} \to \textit{Sets}$ be a functor satisfying the
hypotheses of
Lemma \ref{lemma-tangent-space-functor}.
Assume $R = K$ is a field. ========

regex_match $F: \mathcal{C} \to \textit{Sets}$
nn candidate $F
nn candidate a functor
nn a functor
new_annotation [(134195, 134229, 'VAR'), (134233, 134242, 'TYPE')]
===== sent \begin{definition}
\label{definition-tangent-space}
Let $\mathcal{F}$ be a predeformation category.
 ========

regex_match $\mathcal{F}$
nn candidate a predeformation category
nn a predeformation category
new_annotation [(134980, 134993, 'VAR'), (134997, 135022, 'TYPE')]
===== sent \begin{lemma}
\label{lemma-tangent-space-vector-space}
Let $\mathcal{F}$ be a predeformation category such that
$\overline{\mathcal{F}}$ satisfies (S2)\footnote{For example
if $\mathcal{F}$ satisfies (S2), see
Lemma \ref{lemma-S1-S2-associated-functor}.}. ========

regex_match $\mathcal{F}$
nn candidate a predeformation category
nn a predeformation category
new_annotation [(135576, 135589, 'VAR'), (135593, 135618, 'TYPE')]
===== sent \begin{definition}
\label{definition-differential}
Let $\varphi : \mathcal{F} \to \mathcal{G}$ be a morphism predeformation
categories. ========

regex_match $\varphi : \mathcal{F} \to \mathcal{G}$
nn candidate $\varphi
nn candidate a morphism predeformation
categories
nn a morphism predeformation
categories
new_annotation [(136898, 136937, 'VAR'), (136941, 136977, 'TYPE')]
===== sent The
{\it differential $d \varphi : T \mathcal{F} \to T \mathcal{G}$ of $\varphi$}
is the map obtained by evaluating the morphism of functors
$\overline{\varphi}: \overline{\mathcal{F}} \to \overline{\mathcal{G}}$
at $A = k[\epsilon]$.
\end{definition}

\begin{lemma}
\label{lemma-k-linear-differential}
Let $\varphi : \mathcal{F} \to \mathcal{G}$ be a morphism of predeformation
categories. ========

regex_match $\varphi : \mathcal{F} \to \mathcal{G}$
nn candidate $\varphi
nn candidate a morphism
nn a morphism
new_annotation [(137286, 137325, 'VAR'), (137329, 137339, 'TYPE')]
===== sent Let $\mathcal{F}$ be a category
cofibered in groupoids over $\mathcal{C}_\Lambda$, and let
$x \in \Ob(\mathcal{F}(k))$. As in
Remark \ref{remark-localize-cofibered-groupoid},
we get a predeformation category $\mathcal{F}_x$. We define
$$
T_x\mathcal{F} = T\mathcal{F}_x
$$
to be the {\it tangent space of $\mathcal{F}$ at $x$}. If
$\varphi : \mathcal{F} \to \mathcal{G}$ is a morphism of categories cofibered
in groupoids over $\mathcal{C}_\Lambda$ and $x \in \Ob(\mathcal{F}(k))$,
then there is an induced morphism
$\varphi_x: \mathcal{F}_x \to \mathcal{G}_{\varphi(x)}$. We define the
{\it differential
$d_x \varphi : T_x \mathcal{F} \to T_{\varphi(x)} \mathcal{G}$
of $\varphi$ at $x$} to be the map
$d \varphi_x: T \mathcal{F}_x \to T \mathcal{G}_{\varphi(x)}$.
If both $\mathcal{F}$ and $\mathcal{G}$ satisfy (S2) then
all of these tangent spaces have a natural $k$-vector space structure
and all the differentials
$d_x \varphi : T_x \mathcal{F} \to T_{\varphi(x)} \mathcal{G}$
are $k$-linear (use
Lemmas \ref{lemma-S1-S2-localize} and \ref{lemma-k-linear-differential}).
 ========

regex_match $\mathcal{F}$
nn candidate a category
nn a category
new_annotation [(138017, 138030, 'VAR'), (138034, 138044, 'TYPE')]
===== sent There is a canonical identification
$$
\Mor_{\mathcal{C}_\Lambda}(k, k[\epsilon]) =
\text{Der}_\Lambda(k, k).
$$
Namely, for $D \in \text{Der}_\Lambda(k, k)$ let $f_D : k \to k[\epsilon]$
be the map $a \mapsto a + D(a)\epsilon$. More generally, given a finite
dimensional vector space $V$ over $k$ we have
$$
\Mor_{\mathcal{C}_\Lambda}(k, k[V]) =
\text{Der}_\Lambda(k, V)
$$
and we will use the same notation $f_D$ for the map associated to the
derivation $D$. We also have
$$
\Mor_{\mathcal{C}_\Lambda}(k[W], k[V]) =
\Hom_k(V, W) \oplus \text{Der}_\Lambda(k, V)
$$
where $(\varphi, D)$ corresponds to the map
$f_{\varphi, D} : a + w \mapsto a + \varphi(w) + D(a)$. We will sometimes write
$f_{1, D} : a + v \to a + v + D(a)$ for the automorphism
of $k[V]$ determined by the derivation $D : k \to V$. Note that
$f_{1, D} \circ f_{1, D'} = f_{1, D + D'}$.

\medskip\noindent
Let $\mathcal{F}$ be a predeformation category over $\mathcal{C}_\Lambda$.
Let $x_0 \in \Ob(\mathcal{F}(k))$. By the above there is a canonical
map
$$
\gamma_V :
\text{Der}_\Lambda(k, V)
\longrightarrow
\overline{\mathcal{F}}(k[V])
$$
defined by $D \mapsto f_{D, *}(x_0)$. ========

new_annotation []
===== sent There is a canonical identification
$$
\Mor_{\mathcal{C}_\Lambda}(k, k[\epsilon]) =
\text{Der}_\Lambda(k, k).
$$
Namely, for $D \in \text{Der}_\Lambda(k, k)$ let $f_D : k \to k[\epsilon]$
be the map $a \mapsto a + D(a)\epsilon$. More generally, given a finite
dimensional vector space $V$ over $k$ we have
$$
\Mor_{\mathcal{C}_\Lambda}(k, k[V]) =
\text{Der}_\Lambda(k, V)
$$
and we will use the same notation $f_D$ for the map associated to the
derivation $D$. We also have
$$
\Mor_{\mathcal{C}_\Lambda}(k[W], k[V]) =
\Hom_k(V, W) \oplus \text{Der}_\Lambda(k, V)
$$
where $(\varphi, D)$ corresponds to the map
$f_{\varphi, D} : a + w \mapsto a + \varphi(w) + D(a)$. We will sometimes write
$f_{1, D} : a + v \to a + v + D(a)$ for the automorphism
of $k[V]$ determined by the derivation $D : k \to V$. Note that
$f_{1, D} \circ f_{1, D'} = f_{1, D + D'}$.

\medskip\noindent
Let $\mathcal{F}$ be a predeformation category over $\mathcal{C}_\Lambda$.
Let $x_0 \in \Ob(\mathcal{F}(k))$. By the above there is a canonical
map
$$
\gamma_V :
\text{Der}_\Lambda(k, V)
\longrightarrow
\overline{\mathcal{F}}(k[V])
$$
defined by $D \mapsto f_{D, *}(x_0)$. ========

regex_match $\mathcal{F}$
nn candidate a predeformation category
nn a predeformation category
new_annotation [(140195, 140208, 'VAR'), (140212, 140237, 'TYPE')]
===== sent There is a canonical identification
$$
\Mor_{\mathcal{C}_\Lambda}(k, k[\epsilon]) =
\text{Der}_\Lambda(k, k).
$$
Namely, for $D \in \text{Der}_\Lambda(k, k)$ let $f_D : k \to k[\epsilon]$
be the map $a \mapsto a + D(a)\epsilon$. More generally, given a finite
dimensional vector space $V$ over $k$ we have
$$
\Mor_{\mathcal{C}_\Lambda}(k, k[V]) =
\text{Der}_\Lambda(k, V)
$$
and we will use the same notation $f_D$ for the map associated to the
derivation $D$. We also have
$$
\Mor_{\mathcal{C}_\Lambda}(k[W], k[V]) =
\Hom_k(V, W) \oplus \text{Der}_\Lambda(k, V)
$$
where $(\varphi, D)$ corresponds to the map
$f_{\varphi, D} : a + w \mapsto a + \varphi(w) + D(a)$. We will sometimes write
$f_{1, D} : a + v \to a + v + D(a)$ for the automorphism
of $k[V]$ determined by the derivation $D : k \to V$. Note that
$f_{1, D} \circ f_{1, D'} = f_{1, D + D'}$.

\medskip\noindent
Let $\mathcal{F}$ be a predeformation category over $\mathcal{C}_\Lambda$.
Let $x_0 \in \Ob(\mathcal{F}(k))$. By the above there is a canonical
map
$$
\gamma_V :
\text{Der}_\Lambda(k, V)
\longrightarrow
\overline{\mathcal{F}}(k[V])
$$
defined by $D \mapsto f_{D, *}(x_0)$. ========

new_annotation []
===== sent \begin{lemma}
\label{lemma-action-linear}
Let $\mathcal{F}$ be a predeformation category over $\mathcal{C}_\Lambda$.
If $\overline{\mathcal{F}}$ has (S2) then the maps $\gamma_V$ are
$k$-linear and we have $a_V(D, x) = x + \gamma_V(D)$.
\end{lemma}

\begin{proof}
In the proof of
Lemma \ref{lemma-tangent-space-vector-space}
we have seen that the functor $V \mapsto \overline{\mathcal{F}}(k[V])$
transforms $0$ to a singleton and products to products. ========

regex_match $\mathcal{F}$
nn candidate a predeformation category
nn a predeformation category
new_annotation [(140974, 140987, 'VAR'), (140991, 141016, 'TYPE')]
===== sent The same is
true of the functor $V \mapsto \text{Der}_\Lambda(k, V)$.
Hence $\gamma_V$ is linear by
Lemma \ref{lemma-morphism-linear-functors}.
Let $D : k \to V$ be a $\Lambda$-derivation.
 ========

regex_match $D : k \to V$
nn candidate $D
nn candidate a $\Lambda$-derivation
nn a $\Lambda$-derivation
new_annotation [(141528, 141541, 'VAR'), (141545, 141567, 'TYPE')]
===== sent \begin{lemma}
\label{lemma-versal-object-S1}
Let $\mathcal{F}$ be a predeformation category.
 ========

regex_match $\mathcal{F}$
nn candidate a predeformation category
nn a predeformation category
new_annotation [(143308, 143321, 'VAR'), (143325, 143350, 'TYPE')]
===== sent \end{lemma}

\begin{proof}
Let $\xi$ be a versal formal object of $\mathcal{F}$. Let
$$
\xymatrix{
           & x_2 \ar[d] \\
x_1 \ar[r] & x
}
$$
be a diagram in $\mathcal{F}$ such that $x_2 \to x$ lies over a
surjective ring map. ========

regex_match $\xi$
nn candidate a versal formal object
nn a versal formal object
new_annotation [(143467, 143472, 'VAR'), (143476, 143498, 'TYPE')]
===== sent \begin{lemma}
\label{lemma-versal-criterion}
Let $\mathcal{F}$ be a predeformation category satisfying (S1) and
(S2). ========

regex_match $\mathcal{F}$
nn candidate a predeformation category
nn a predeformation category
new_annotation [(144555, 144568, 'VAR'), (144572, 144597, 'TYPE')]
===== sent Let $\xi$ be a formal object of $\mathcal{F}$ corresponding to
$\underline{\xi} : \underline{R}|_{\mathcal{C}_\Lambda} \to \mathcal{F}$, see
Remark \ref{remark-formal-objects-yoneda}.
 ========

regex_match $\xi$
nn candidate a formal object
nn a formal object
new_annotation [(144628, 144633, 'VAR'), (144637, 144652, 'TYPE')]
===== sent Let $b : R \to B$ be the map guaranteed by (2). ========

regex_match $b : R \to B$
nn candidate R \to
nn candidate the map
nn the map
new_annotation [(145907, 145920, 'VAR'), (145924, 145931, 'TYPE')]
===== sent By (S1) we obtain a commutative diagram
$$
\vcenter{
\xymatrix{
z  \ar[r] \ar[d]          &  y \ar[d] \\
y' \ar[r]  &  x
}
}
\quad\text{lying over}\quad
\vcenter{
\xymatrix{
B \times_A B \ar[d] \ar[r] &   B  \ar[d]^{f} \\
B \ar[r]^{f} &   A .
}
}
$$
Set $I = \Ker(f)$. Let $\overline{g} : B \times_A B \to k[I]$
be the ring map $(u, v) \mapsto \overline{u} \oplus (v - u)$,
cf.\ Lemma \ref{lemma-lifting-along-small-extension}.
 ========

new_annotation []
===== sent \begin{lemma}
\label{lemma-largest-closed-where-lift}
Let $\mathcal{F}$ be a category cofibred in groupoids over
$\mathcal{C}_\Lambda$ which has (S1). ========

regex_match $\mathcal{F}$
nn candidate a category
nn a category
new_annotation [(148070, 148083, 'VAR'), (148087, 148097, 'TYPE')]
===== sent Let $B \to A$ be a surjection
in $\mathcal{C}_\Lambda$ with kernel $I$ annihilated by $\mathfrak m_B$.
Let $x \in \mathcal{F}(A)$. The set of ideals
$$
\mathcal{J} = \{ J \subset I \mid
\text{there exists an }y \to x\text{ lying over }B/J \to A\}
$$
has a smallest element.
 ========

regex_match $B \to A$
nn candidate $B
nn candidate a surjection
nn a surjection
new_annotation [(148167, 148176, 'VAR'), (148180, 148192, 'TYPE')]
===== sent Let $B \to A$ be a surjection
in $\mathcal{C}_\Lambda$ with kernel $I$ annihilated by $\mathfrak m_B$.
Let $x \in \mathcal{F}(A)$. The set of ideals
$$
\mathcal{J} = \{ J \subset I \mid
\text{there exists an }y \to x\text{ lying over }B/J \to A\}
$$
has a smallest element.
 ========

new_annotation []
===== sent Let $J$ and $K$ be elements of the displayed set.
 ========

new_annotation []
===== sent \begin{lemma}
\label{lemma-versal-object-existence}
Let $\mathcal{F}$ be a category cofibred in groupoids over
$\mathcal{C}_\Lambda$. Assume the following conditions hold:
\begin{enumerate}
\item ========

regex_match $\mathcal{F}$
nn candidate a category
nn a category
new_annotation [(149372, 149385, 'VAR'), (149389, 149399, 'TYPE')]
===== sent See Lemma \ref{lemma-exists-smooth}.
Let $r = \dim_k T\mathcal{F}$ and put $S = R[[X_1, \ldots, X_r]]$.

\medskip\noindent
We are going to inductively construct for $n \geq 2$ pairs
$(J_n, f_{n - 1} : \xi_n \to \xi_{n - 1})$
where $J_n \subset S$ is an decreasing sequence of ideals and
$f_{n - 1} : \xi_n \to ========

new_annotation []
===== sent Let $J_1 = \mathfrak m_S$. Let $\xi_1$ be the unique
(up to unique isomorphism) object of $\mathcal{F}$ over
$k = S/J_1 = S/\mathfrak m_S$

\medskip\noindent
Step 2. ========

new_annotation []
===== sent Let $J_1 = \mathfrak m_S$. Let $\xi_1$ be the unique
(up to unique isomorphism) object of $\mathcal{F}$ over
$k = S/J_1 = S/\mathfrak m_S$

\medskip\noindent
Step 2. ========

regex_match $\xi_1$
nn candidate unique isomorphism
nn unique isomorphism
new_annotation [(150376, 150383, 'VAR'), (150405, 150423, 'TYPE')]
===== sent Let $f_1 : \xi_2 ========

new_annotation []
===== sent \xi_{n - 1})$ has been
constructed for some $n \geq 2$. There is a minimal element $J_{n + 1}$ of
the set of ideals $J \subset S$ satisfying:
(a) $\mathfrak m_S J_n \subset J \subset J_n$ and
(b) there exists a morphism $\xi_{n + 1} \to \xi_n$ lying over
$S/J \to S/J_n$, see
Lemma \ref{lemma-largest-closed-where-lift}.
Let $f_n : \xi_{n + 1} \to \xi_n$ be any morphism of $\mathcal{F}$
lying over $S/J_{n + 1} \to S/J_n$.

\medskip\noindent
Set $J = \bigcap J_n$. Set $\overline{S} = S/J$. Set $\overline{J}_n = J_n/J$.
 ========

new_annotation []
===== sent \end{proof}

\begin{remark}
\label{remark-compare-schlessinger-H3-pre}
Let $F : \mathcal{C}_\Lambda \to \textit{Sets}$ be a predeformation functor
satisfying (S1) and (S2). ========

regex_match $F : \mathcal{C}_\Lambda \to \textit{Sets}$
nn candidate \mathcal{C}_\Lambda \to
nn candidate a predeformation functor
nn a predeformation functor
new_annotation [(155244, 155287, 'VAR'), (155291, 155315, 'TYPE')]
===== sent \medskip\noindent
Let $\mathcal{F}$ be a category cofibred in groupoids over
$\mathcal{C}_\Lambda$. For every object $x$ of $\mathcal{F}$
lying over $A \in \Ob(\mathcal{C}_\Lambda)$ consider the
category $\mathcal{S}_x$ with objects
 ========

regex_match $\mathcal{F}$
nn candidate a category
nn a category
new_annotation [(156287, 156300, 'VAR'), (156304, 156314, 'TYPE')]
===== sent $$
\Ob(\mathcal{S}_x) =
\{x' \to x \mid x' \to x\text{ lies over }A' \subset A\}
$$
and morphisms are morphisms over $x$. For every $y \to x$ in
$\mathcal{F}$ lying over $f : B \to A$
in $\mathcal{C}_\Lambda$ there is a functor
$f_* : \mathcal{S}_y \to \mathcal{S}_x$ defined as follows:
Given $y' \to y$ lying over $B' \subset B$ set $A' = f(B')$
and let $y' \to x'$ be over $B' \to f(B')$ be the pushforward of $y'$.
By the axioms of a category cofibred in groupoids we obtain a
unique morphism $x' \to x$ lying over $f(B') \to A$ such that
$$
\xymatrix{
y' \ar[d] \ar[r] & x' \ar[d] \\
y \ar[r] & x
}
$$
commutes. ========

new_annotation []
===== sent \begin{lemma}
\label{lemma-smallest-where-descends}
Let $\mathcal{F}$ be a category cofibred in groupoids over
$\mathcal{C}_\Lambda$ which has (S1).
 ========

regex_match $\mathcal{F}$
nn candidate a category
nn a category
new_annotation [(157525, 157538, 'VAR'), (157542, 157552, 'TYPE')]
===== sent Let $y' ========

new_annotation []
===== sent Suppose $f : B \to A$ is surjective and $y \to x$ lies over $f$.
Let $x' \to x$ be a minimal object of $\mathcal{S}_x$ lying over $A' \subset A$.
 ========

regex_match $x' \to x$
nn candidate a minimal object
nn a minimal object
new_annotation [(158809, 158819, 'VAR'), (158823, 158839, 'TYPE')]
===== sent \label{lemma-smallest-where-descends-versal}
Let $\mathcal{F}$ be a category cofibred in groupoids over
$\mathcal{C}_\Lambda$ which has (S1). ========

regex_match $\mathcal{F}$
nn candidate a category
nn a category
new_annotation [(159453, 159466, 'VAR'), (159470, 159480, 'TYPE')]
===== sent Let $\xi$ be a versal formal object
of $\mathcal{F}$ lying over $R$. ========

regex_match $\xi$
nn candidate a versal formal object
nn a versal formal object
new_annotation [(159550, 159555, 'VAR'), (159559, 159581, 'TYPE')]
===== sent \end{proof}

\begin{lemma}
\label{lemma-descends-versal}
Let $\mathcal{F}$ be a category cofibred in groupoids over
$\mathcal{C}_\Lambda$ which has (S1). ========

regex_match $\mathcal{F}$
nn candidate a category
nn a category
new_annotation [(162474, 162487, 'VAR'), (162491, 162501, 'TYPE')]
===== sent Let $\xi$ be a versal formal object
of $\mathcal{F}$ lying over $R$. Let $\xi' \to \xi$ be a morphism
of formal objects lying over $R' \subset R$ as constructed in
Lemma \ref{lemma-smallest-where-descends-versal}.
Then
$$
R \cong R'[[x_1, \ldots, x_r]]
$$
is a power series ring over $R'$.
 ========

regex_match $\xi$
nn candidate a versal formal object
nn a versal formal object
new_annotation [(162571, 162576, 'VAR'), (162580, 162602, 'TYPE')]
===== sent Let $\xi$ be a versal formal object
of $\mathcal{F}$ lying over $R$. Let $\xi' \to \xi$ be a morphism
of formal objects lying over $R' \subset R$ as constructed in
Lemma \ref{lemma-smallest-where-descends-versal}.
Then
$$
R \cong R'[[x_1, \ldots, x_r]]
$$
is a power series ring over $R'$.
 ========

regex_match $\xi' \to \xi$
nn candidate $\xi' \to
nn candidate a morphism
nn a morphism
new_annotation [(162640, 162654, 'VAR'), (162658, 162668, 'TYPE')]
===== sent This is an isomorphism by
Algebra, Lemma \ref{algebra-lemma-surjective-endo-noetherian-ring-is-iso}.
Hence $I = \Ker(f)$ is an ideal of $R$ such that $R = R' \oplus I$.
Let $x_1, \ldots, x_n \in I$ be elements which form a basis for
$I/\mathfrak ========

new_annotation []
===== sent \begin{definition}
\label{definition-minimal-versal}
Let $\mathcal{F}$ be a predeformation category.
 ========

regex_match $\mathcal{F}$
nn candidate a predeformation category
nn a predeformation category
new_annotation [(165215, 165228, 'VAR'), (165232, 165257, 'TYPE')]
===== sent \begin{lemma}
\label{lemma-minimal-versal}
Let $\mathcal{F}$ be a predeformation category which
has a versal formal object. ========

regex_match $\mathcal{F}$
nn candidate a predeformation category
nn a predeformation category
new_annotation [(166110, 166123, 'VAR'), (166127, 166152, 'TYPE')]
===== sent \end{enumerate}
\end{lemma}

\begin{proof}
Suppose $\mathcal{F}$ has a versal formal object $\xi$ over $R$.
Then it satisfies (S1), see
Lemma \ref{lemma-versal-object-S1}.
Let $\xi' \to \xi$ over $R' \subset R$ be any of the morphisms constructed in
Lemma \ref{lemma-smallest-where-descends-versal}.
By
Lemma \ref{lemma-descends-versal}
we see that $\xi'$ is versal, hence it is a minimal versal formal
object (by construction). ========

new_annotation []
===== sent Let $\xi$ be a formal object of the predeformation category
$\mathcal{F}$ and let
$\underline{\xi} : \underline{R}|_{\mathcal{C}_\Lambda} \to \mathcal{F}$
be the corresponding morphism. ========

regex_match $\xi$
nn candidate a formal object
nn a formal object
new_annotation [(167955, 167960, 'VAR'), (167964, 167979, 'TYPE')]
===== sent \begin{lemma}
\label{lemma-miniversal-object-unique}
Let $\mathcal{F}$ be a predeformation category.
 ========

regex_match $\mathcal{F}$
nn candidate a predeformation category
nn a predeformation category
new_annotation [(169217, 169230, 'VAR'), (169234, 169259, 'TYPE')]
===== sent Let $\xi$ be a versal formal object of $\mathcal{F}$ such that
(\ref{equation-bijective-orbits}) holds.
 ========

regex_match $\xi$
nn candidate a versal formal object
nn a versal formal object
new_annotation [(169265, 169270, 'VAR'), (169274, 169296, 'TYPE')]
===== sent Thus $d\underline{\xi}$ factors as
$$
\text{Der}_\Lambda(R, k) \to
\text{Der}_\Lambda(R', k) \to T\mathcal{F}
$$
and we see that (\ref{equation-bijective-orbits}) cannot hold
because $D : f \mapsto \partial/\partial x_1(f) \bmod \mathfrak m_R$
is an element of the kernel of the first arrow which is not in the image of
$\text{Der}_\Lambda(k, k) \to \text{Der}_\Lambda(R, k)$.
\end{proof}

\begin{lemma}
\label{lemma-miniversal-object-existence-1}
Let $\mathcal{F}$ be a predeformation category.
 ========

regex_match $\mathcal{F}$
nn candidate a predeformation category
nn a predeformation category
new_annotation [(170128, 170141, 'VAR'), (170145, 170170, 'TYPE')]
===== sent Let $\xi$ be a versal formal object of $\mathcal{F}$ such that
(\ref{equation-bijective}) holds. ========

regex_match $\xi$
nn candidate a versal formal object
nn a versal formal object
new_annotation [(170176, 170181, 'VAR'), (170185, 170207, 'TYPE')]
===== sent Let $p : \mathcal{F} \to \mathcal{C}_\Lambda$ denote the
structure morphism. ========

new_annotation []
===== sent \end{example}

\begin{lemma}
\label{lemma-construct-bijective-orbits}
Let $\mathcal{F}$ be a predeformation category satisfying
(S2) which has a versal formal object. ========

regex_match $\mathcal{F}$
nn candidate a predeformation category
nn a predeformation category
new_annotation [(172673, 172686, 'VAR'), (172690, 172715, 'TYPE')]
===== sent \end{lemma}

\begin{proof}
Let $\xi$ be a minimal versal formal object for $\mathcal{F}$, see
Lemma \ref{lemma-minimal-versal}.
Say $\xi$ lies over $R \in \Ob(\widehat{\mathcal{C}}_\Lambda)$.
In order to parse (\ref{equation-bijective-orbits}) we point out
that $T\mathcal{F}$ has a natural $k$-vector space structure
(see
Lemma \ref{lemma-tangent-space-vector-space}),
that $d\underline{\xi} : \text{Der}_\Lambda(R, k) \to T\mathcal{F}$
is linear (see
Lemma \ref{lemma-k-linear-differential}),
and that the action of $\text{Der}_\Lambda(k, k)$ is
given by addition (see
Lemma \ref{lemma-action-linear}).
 ========

regex_match $\xi$
nn candidate a minimal versal formal object
nn a minimal versal formal object
new_annotation [(172880, 172885, 'VAR'), (172889, 172919, 'TYPE')]
===== sent Let $\xi \to y$ be a morphism lying over $R \to R/\mathfrak m_R^2$.
Let $y \to x$ be a morphism lying over
$R/\mathfrak m_R^2 \to R'/\mathfrak m_R^2$.
 ========

regex_match $\xi \to y$
nn candidate a morphism
nn a morphism
new_annotation [(174746, 174757, 'VAR'), (174761, 174771, 'TYPE')]
===== sent Let $\xi \to y$ be a morphism lying over $R \to R/\mathfrak m_R^2$.
Let $y \to x$ be a morphism lying over
$R/\mathfrak m_R^2 \to R'/\mathfrak m_R^2$.
 ========

regex_match $y \to x$
nn candidate $y
nn candidate a morphism
nn a morphism
new_annotation [(174814, 174823, 'VAR'), (174827, 174837, 'TYPE')]
===== sent Let $y \to x_\epsilon$ be a morphism lying over
$R/\mathfrak m_R^2 \to k[\epsilon]$. Let $x_0$ be the unique (up to unique
isomorphism) object of $\mathcal{F}$ over $k$.
 ========

regex_match $y \to x_\epsilon$
nn candidate $y
nn candidate a morphism
nn a morphism
new_annotation [(174897, 174915, 'VAR'), (174919, 174929, 'TYPE')]
===== sent Let $y \to x_\epsilon$ be a morphism lying over
$R/\mathfrak m_R^2 \to k[\epsilon]$. Let $x_0$ be the unique (up to unique
isomorphism) object of $\mathcal{F}$ over $k$.
 ========

regex_match $x_0$
nn candidate isomorphism
nn isomorphism
new_annotation [(174982, 174987, 'VAR'), (175016, 175027, 'TYPE')]
===== sent \end{proof}

\begin{theorem}
\label{theorem-miniversal-object-existence}
Let $\mathcal{F}$ be a predeformation category.
 ========

regex_match $\mathcal{F}$
nn candidate a predeformation category
nn a predeformation category
new_annotation [(176101, 176114, 'VAR'), (176118, 176143, 'TYPE')]
===== sent \end{proof}

\begin{remark}
\label{remark-compare-schlessinger-H3}
Let $F : \mathcal{C}_\Lambda \to \textit{Sets}$ be a predeformation functor
satisfying (S1) and (S2) and $\dim_k TF < \infty$.
Recall that these conditions correspond to the conditions
(H1), (H2), and (H3) from Schlessinger's paper, see
Remark \ref{remark-compare-schlessinger-H3-pre}.
 ========

regex_match $F : \mathcal{C}_\Lambda \to \textit{Sets}$
nn candidate \mathcal{C}_\Lambda \to
nn candidate a predeformation functor
nn a predeformation functor
new_annotation [(177577, 177620, 'VAR'), (177624, 177648, 'TYPE')]
===== sent \end{remark}

\begin{remark}
\label{remark-compose-minimal-into-iso-classes}
Let $\mathcal{F}$ be a predeformation category. ========

regex_match $\mathcal{F}$
nn candidate a predeformation category
nn a predeformation category
new_annotation [(178499, 178512, 'VAR'), (178516, 178541, 'TYPE')]
===== sent \begin{definition}
\label{definition-RS}
Let $\mathcal{F}$ be a category cofibered in groupoids over $\mathcal
C_\Lambda$.  We say that $\mathcal{F}$ satisfies {\it condition (RS)}
if for every diagram in $\mathcal{F}$
$$
\vcenter{
\xymatrix{
           & x_2 \ar[d] ========

regex_match $\mathcal{F}$
nn candidate a category
nn a category
new_annotation [(182169, 182182, 'VAR'), (182186, 182196, 'TYPE')]
===== sent A_1 \ar[r]      & A.
}
}
$$
\end{definition}

\begin{lemma}
\label{lemma-RS-fiber-square}
Let $\mathcal{F}$ be a category cofibered in groupoids over
$\mathcal{C}_\Lambda$ satisfying (RS). ========

regex_match $\mathcal{F}$
nn candidate a category
nn a category
new_annotation [(182936, 182949, 'VAR'), (182953, 182963, 'TYPE')]
===== sent \end{proof}

\begin{lemma}
\label{lemma-RS-small-extension}
Let $\mathcal{F}$ be a category cofibered in groupoids over $\mathcal
C_\Lambda$. Then $\mathcal{F}$ satisfies (RS) if the condition in
Definition \ref{definition-RS} is assumed to hold only when $A_2 \to A$
is a small extension.
 ========

regex_match $\mathcal{F}$
nn candidate a category
nn a category
new_annotation [(183853, 183866, 'VAR'), (183870, 183880, 'TYPE')]
===== sent \end{lemma}

\begin{proof}
Apply Lemma \ref{lemma-factor-small-extension}.  The proof is similar to that
of Lemma \ref{lemma-smoothness-small-extensions}.
\end{proof}

\begin{lemma}
\label{lemma-RS-2-categorical}
Let $\mathcal{F}$ be a category cofibered in groupoids over
$\mathcal{C}_\Lambda$. The following are equivalent
\begin{enumerate}
\item $\mathcal{F}$ satisfies (RS),
\item the functor
$\mathcal{F}(A_1 \times_A A_2) \to
 ========

regex_match $\mathcal{F}$
nn candidate a category
nn a category
new_annotation [(184296, 184309, 'VAR'), (184313, 184323, 'TYPE')]
===== sent Let $q_1 : A_1 \to A$ and $q_2 : A_2 \to A$ be given with
$q_2$ a small extension. ========

new_annotation []
===== sent We will use the description of the $2$-fibre product
$\mathcal{F}(A_1) \times_{\mathcal{F}(A)} \mathcal{F}(A_2)$ from
Categories, Remark \ref{categories-remark-other-description-2-fibre-product}.
Hence let $y \in \mathcal{F}(A_1 \times_A A_2)$ correspond to
$(x_1, x_2, x, a_1 : x_1 \to x, a_2 : x_2 \to x)$.
Let $z$ be an object of $\mathcal{F}$ lying over $C$. ========

new_annotation []
===== sent We will use the description of the $2$-fibre product
$\mathcal{F}(A_1) \times_{\mathcal{F}(A)} \mathcal{F}(A_2)$ from
Categories, Remark \ref{categories-remark-other-description-2-fibre-product}.
Hence let $y \in \mathcal{F}(A_1 \times_A A_2)$ correspond to
$(x_1, x_2, x, a_1 : x_1 \to x, a_2 : x_2 \to x)$.
Let $z$ be an object of $\mathcal{F}$ lying over $C$. ========

regex_match $z$
nn candidate an object
nn an object
new_annotation [(185823, 185826, 'VAR'), (185830, 185839, 'TYPE')]
===== sent \end{remark}

\begin{lemma}
\label{lemma-RS-implies-S1-S2}
Let $\mathcal{F}$ be a category cofibered in groupoids over
$\mathcal{C}_\Lambda$. The condition (RS) for $\mathcal{F}$
implies both (S1) and (S2) for $\mathcal{F}$.
\end{lemma}

\begin{proof}
Using the reformulation of
Lemma \ref{lemma-RS-2-categorical}
and the explanation of (S1) following
Definition \ref{definition-S1-S2}
it is immediate that (RS) implies (S1).
 ========

regex_match $\mathcal{F}$
nn candidate a category
nn a category
new_annotation [(187025, 187038, 'VAR'), (187042, 187052, 'TYPE')]
===== sent \to \text{Aut}_A(x)$.

\begin{lemma}
\label{lemma-RS-associated-functor}
Let $\mathcal{F}$ be a category cofibered in groupoids over
$\mathcal{C}_\Lambda$ satisfying (RS).
 ========

regex_match $\mathcal{F}$
nn candidate a category
nn a category
new_annotation [(188269, 188282, 'VAR'), (188286, 188296, 'TYPE')]
===== sent Let $f_1: A_1 \to A$ and $f_2: A_2 \to A$ be ring maps in
$\mathcal{C}_\Lambda$ with $f_2$ surjective. ========

new_annotation []
===== sent Let $f_1: A_1 \to A$ and $f_2: A_2 \to A$ be ring maps in
$\mathcal{C}_\Lambda$ with $f_2$ surjective. ========

new_annotation []
===== sent Let $x' \to x$ be a morphism in $\mathcal{F}$ lying
over a surjective ring map $f: A' \to A$. ========

regex_match $x' \to x$
nn candidate x
nn candidate a morphism
nn a morphism
new_annotation [(190481, 190491, 'VAR'), (190495, 190505, 'TYPE')]
===== sent \begin{lemma}
\label{lemma-localize-RS}
Let $\mathcal{F}$ be a category cofibered in groupoids over
$\mathcal{C}_\Lambda$. Let $x_0 \in \Ob(\mathcal{F}(k))$.
 ========

regex_match $\mathcal{F}$
nn candidate a category
nn a category
new_annotation [(193528, 193541, 'VAR'), (193545, 193555, 'TYPE')]
===== sent \begin{lemma}
\label{lemma-localize-RS}
Let $\mathcal{F}$ be a category cofibered in groupoids over
$\mathcal{C}_\Lambda$. Let $x_0 \in \Ob(\mathcal{F}(k))$.
 ========

new_annotation []
===== sent Let $\mathcal{F}_{x_0}$ be the category cofibred in groupoids over
$\mathcal{C}_\Lambda$ constructed in
Remark \ref{remark-localize-cofibered-groupoid}.
If $\mathcal{F}$ satisfies (RS), then so does $\mathcal{F}_{x_0}$.
 ========

regex_match $\mathcal{F}_{x_0}$
nn candidate the category
nn the category
new_annotation [(193646, 193665, 'VAR'), (193669, 193681, 'TYPE')]
===== sent \begin{definition}
\label{definition-lifts}
Let $\mathcal{F}$ be a category cofibered in groupoids over
$\mathcal{C}_\Lambda$. Let $f: A' \to A$ be a map in $\mathcal{C}_\Lambda$.
Let $x \in \mathcal{F}(A)$. ========

regex_match $\mathcal{F}$
nn candidate a category
nn a category
new_annotation [(196531, 196544, 'VAR'), (196548, 196558, 'TYPE')]
===== sent \begin{definition}
\label{definition-lifts}
Let $\mathcal{F}$ be a category cofibered in groupoids over
$\mathcal{C}_\Lambda$. Let $f: A' \to A$ be a map in $\mathcal{C}_\Lambda$.
Let $x \in \mathcal{F}(A)$. ========

regex_match $f: A' \to A$
nn candidate a map
nn a map
new_annotation [(196614, 196627, 'VAR'), (196631, 196636, 'TYPE')]
===== sent \begin{definition}
\label{definition-lifts}
Let $\mathcal{F}$ be a category cofibered in groupoids over
$\mathcal{C}_\Lambda$. Let $f: A' \to A$ be a map in $\mathcal{C}_\Lambda$.
Let $x \in \mathcal{F}(A)$. ========

new_annotation []
===== sent The set $\text{Lift}(x, f)$ of lifts of $x$ along $f$ is the set of
isomorphism classes of $\textit{Lift}(x, f)$.
\end{definition}

\begin{remark}
\label{remark-omit-arrow}
When the map $f: A' \to A$ is clear from the context, we may write
$\textit{Lift}(x, A')$ and $\text{Lift}(x, A')$ in place of
$\textit{Lift}(x, f)$ and $\text{Lift}(x, f)$.
\end{remark}

\begin{remark}
\label{remark-tangent-space-lifting}
Let $\mathcal{F}$ be a category cofibred in groupoids over
$\mathcal{C}_\Lambda$. Let $x_0 \in \Ob(\mathcal{F}(k))$.
 ========

regex_match $\mathcal{F}$
nn candidate a category
nn a category
new_annotation [(197522, 197535, 'VAR'), (197539, 197549, 'TYPE')]
===== sent The set $\text{Lift}(x, f)$ of lifts of $x$ along $f$ is the set of
isomorphism classes of $\textit{Lift}(x, f)$.
\end{definition}

\begin{remark}
\label{remark-omit-arrow}
When the map $f: A' \to A$ is clear from the context, we may write
$\textit{Lift}(x, A')$ and $\text{Lift}(x, A')$ in place of
$\textit{Lift}(x, f)$ and $\text{Lift}(x, f)$.
\end{remark}

\begin{remark}
\label{remark-tangent-space-lifting}
Let $\mathcal{F}$ be a category cofibred in groupoids over
$\mathcal{C}_\Lambda$. Let $x_0 \in \Ob(\mathcal{F}(k))$.
 ========

new_annotation []
===== sent Let $V$ be a finite dimensional vector space.
 ========

regex_match $V$
nn candidate a finite dimensional vector space
nn a finite dimensional vector space
new_annotation [(197639, 197642, 'VAR'), (197646, 197679, 'TYPE')]
===== sent \end{remark}

\begin{remark}
\label{remark-lift-bijections}
Let $\mathcal{F}$ be a category cofibered in groupoids over $\mathcal
C_\Lambda$ satisfying (RS).   ========

regex_match $\mathcal{F}$
nn candidate a category
nn a category
new_annotation [(198253, 198266, 'VAR'), (198270, 198280, 'TYPE')]
===== sent Let $x_1 \to x$ be a morphism
lying over $A_1 \to A$. ========

regex_match $x_1 \to x$
nn candidate $
new_annotation []
===== sent \end{align*}
\end{remark}

\begin{lemma}
\label{lemma-free-transitive-action}
Let $\mathcal{F}$ be a deformation category.
 ========

regex_match $\mathcal{F}$
nn candidate a deformation category
nn a deformation category
new_annotation [(199703, 199716, 'VAR'), (199720, 199742, 'TYPE')]
===== sent Let $A' \to A$ be a surjective ring map in
$\mathcal{C}_\Lambda$ whose kernel $I$ is annihilated
by $\mathfrak m_{A'}$. Let $x \in \Ob(\mathcal{F}(A))$.
If $\text{Lift}(x, A')$ is nonempty,
then there is a free and transitive action of
$T\mathcal{F} \otimes_k I$ on $\text{Lift}(x, A')$.
\end{lemma}

\begin{proof}
Consider the ring map $g : A' \times_A A' \to k[I]$ defined by the
rule $g(a_1, a_2) = \overline{a_1} \oplus a_2 - a_1$ (compare with
Lemma \ref{lemma-lifting-along-small-extension}).
 ========

regex_match $A' \to A$
nn candidate a surjective ring map
nn a surjective ring map
new_annotation [(199748, 199758, 'VAR'), (199762, 199783, 'TYPE')]
===== sent Let $A' \to A$ be a surjective ring map in
$\mathcal{C}_\Lambda$ whose kernel $I$ is annihilated
by $\mathfrak m_{A'}$. Let $x \in \Ob(\mathcal{F}(A))$.
If $\text{Lift}(x, A')$ is nonempty,
then there is a free and transitive action of
$T\mathcal{F} \otimes_k I$ on $\text{Lift}(x, A')$.
\end{lemma}

\begin{proof}
Consider the ring map $g : A' \times_A A' \to k[I]$ defined by the
rule $g(a_1, a_2) = \overline{a_1} \oplus a_2 - a_1$ (compare with
Lemma \ref{lemma-lifting-along-small-extension}).
 ========

new_annotation []
===== sent Let $\varphi : \mathcal{F} \to \mathcal{G}$ be a morphism of deformation
categories. ========

regex_match $\varphi : \mathcal{F} \to \mathcal{G}$
nn candidate $\varphi
nn candidate a morphism
nn a morphism
new_annotation [(202900, 202939, 'VAR'), (202943, 202953, 'TYPE')]
===== sent Let $A' \to A$ be a surjective ring map whose kernel $I$
is annihilated by $\mathfrak m_{A'}$. Let
$x \in \Ob(\mathcal{F}(A))$.
 ========

regex_match $A' \to A$
nn candidate a surjective ring map
nn a surjective ring map
new_annotation [(202985, 202995, 'VAR'), (202999, 203020, 'TYPE')]
===== sent \end{remark}






\section{Schlessinger's theorem on prorepresentable functors}
\label{section-schlessingers-theorem}

\noindent
We deduce Schlessinger's theorem characterizing prorepresentable functors on
$\mathcal{C}_\Lambda$.

\begin{lemma}
\label{lemma-minimal-smooth-morphism-functors}
Let $F, G: \mathcal{C}_\Lambda \to \textit{Sets}$ be deformation
functors. ========

new_annotation []
===== sent Let $\varphi : F \to G$ be a smooth morphism which induces
an isomorphism $d\varphi : TF \to TG$ of tangent
spaces. ========

regex_match $\varphi : F \to G$
nn candidate a smooth morphism
nn a smooth morphism
new_annotation [(204058, 204077, 'VAR'), (204081, 204098, 'TYPE')]
===== sent Suppose that the
statement holds for rings of length less than $n$ and let $A'$ be a ring of
length $n$. Choose a small extension $f : A' \to A$.   ========

regex_match $A'$
nn candidate a ring
nn a ring
new_annotation [(204523, 204527, 'VAR'), (204531, 204537, 'TYPE')]
===== sent \begin{theorem}
\label{theorem-Schlessinger-prorepresentability}
Let $F: \mathcal{C}_\Lambda \to \textit{Sets}$ be a functor.
 ========

regex_match $F: \mathcal{C}_\Lambda \to \textit{Sets}$
nn candidate \mathcal{C}_\Lambda \to
nn candidate a functor
nn a functor
new_annotation [(205745, 205787, 'VAR'), (205791, 205800, 'TYPE')]
===== sent Thus we see that $\underline{\xi}$
is an isomorphism by
Lemma \ref{lemma-minimal-smooth-morphism-functors}.
\end{proof}




\section{Infinitesimal automorphisms}
\label{section-infinitesimal-automorphisms}

\noindent
Let $\mathcal{F}$ be a category cofibered in groupoids over
$\mathcal{C}_\Lambda$. Given a morphism $x' \to x$ in $\mathcal{F}$ lying over
$A' \to A$, there is an induced homomorphism
$$
\text{Aut}_{A'}(x') \to \text{Aut}_A(x).
 ========

regex_match $\mathcal{F}$
nn candidate a category
nn a category
new_annotation [(207330, 207343, 'VAR'), (207347, 207357, 'TYPE')]
===== sent We will see
that it also gives a measure of how far $\mathcal{F}$
is from $\overline{\mathcal{F}}$.

\begin{definition}
\label{definition-relative-infinitesimal-auts}
Let $\mathcal{F}$ be a category cofibered in groupoids over $\mathcal
C_\Lambda$. Let $x' \to x$ be a morphism in $\mathcal{F}$ lying over
$A' \to A$. ========

regex_match $\mathcal{F}$
nn candidate a category
nn a category
new_annotation [(207957, 207970, 'VAR'), (207974, 207984, 'TYPE')]
===== sent We will see
that it also gives a measure of how far $\mathcal{F}$
is from $\overline{\mathcal{F}}$.

\begin{definition}
\label{definition-relative-infinitesimal-auts}
Let $\mathcal{F}$ be a category cofibered in groupoids over $\mathcal
C_\Lambda$. Let $x' \to x$ be a morphism in $\mathcal{F}$ lying over
$A' \to A$. ========

regex_match $x' \to x$
nn candidate x
nn candidate a morphism
nn a morphism
new_annotation [(208039, 208049, 'VAR'), (208053, 208063, 'TYPE')]
===== sent The kernel
$$
\text{Inf}(x'/x) = \Ker(\text{Aut}_{A'}(x') \to \text{Aut}_A(x))
$$
is the {\it group of infinitesimal automorphisms of $x'$ over $x$}.
\end{definition}

\begin{definition}
\label{definition-infinitesimal-auts}
Let $\mathcal{F}$ be a category cofibered in groupoids over $\mathcal
C_\Lambda$. Let $x_0 \in \Ob(\mathcal{F}(k))$. Assume a choice of
pushforward ========

regex_match $\mathcal{F}$
nn candidate a category
nn a category
new_annotation [(208333, 208346, 'VAR'), (208350, 208360, 'TYPE')]
===== sent The kernel
$$
\text{Inf}(x'/x) = \Ker(\text{Aut}_{A'}(x') \to \text{Aut}_A(x))
$$
is the {\it group of infinitesimal automorphisms of $x'$ over $x$}.
\end{definition}

\begin{definition}
\label{definition-infinitesimal-auts}
Let $\mathcal{F}$ be a category cofibered in groupoids over $\mathcal
C_\Lambda$. Let $x_0 \in \Ob(\mathcal{F}(k))$. Assume a choice of
pushforward ========

new_annotation []
===== sent \begin{definition}
\label{definition-automorphism-functor}
Let $p : \mathcal{F} \to \mathcal{C}$ be a category cofibered in groupoids
over an arbitrary base category $\mathcal{C}$. Assume a choice of pushforwards
has been made. ========

regex_match $p : \mathcal{F} \to \mathcal{C}$
nn candidate $p
nn candidate a category
nn a category
new_annotation [(210952, 210985, 'VAR'), (210989, 210999, 'TYPE')]
===== sent Let $x \in \Ob(\mathcal{F})$ and let ========

new_annotation []
===== sent Let $x \in \Ob(\mathcal{F})$ and let ========

new_annotation []
===== sent $U = p(x)$.
Let $U/\mathcal{C}$ denote the category of objects under $U$. The
{\it automorphism functor of $x$} is the functor
$\mathit{Aut}(x) : U/\mathcal{C} \to \textit{Sets}$ sending an object
$f : U \to V$ to $\text{Aut}_V(f_*x)$ and sending a morphism
$$
\xymatrix{
V' \ar[rr] &                    & V\\
          & U \ar[ul]^{f'}  \ar[ur]_f &
}
$$
to the homomorphism
$\text{Aut}_{V'}(f'_*x) \to \text{Aut}_V(f_*x)$
coming from the unique morphism $f'_*x \to f_*x$ lying over
$V' \to V$ and compatible with $x \to f'_*x$ and $x \to f_*x$.
\end{definition}

\noindent
We will be concerned with the automorphism functors of objects in a category
cofibered in groupoids $\mathcal{F}$ over $\mathcal{C}_\Lambda$. ========

new_annotation []
===== sent \begin{lemma}
\label{lemma-Aut-functor-RS}
Let $\mathcal{F}$ be a category cofibered in groupoids over
$\mathcal{C}_\Lambda$ satisfying (RS). ========

regex_match $\mathcal{F}$
nn candidate a category
nn a category
new_annotation [(212643, 212656, 'VAR'), (212660, 212670, 'TYPE')]
===== sent \mathcal{F}(A_1) \times_{\mathcal{F}(A)} \mathcal{F}(A_2)$ in
Lemma \ref{lemma-RS-2-categorical}.
\end{proof}

\begin{lemma}
\label{lemma-Aut-functor-tangent-space}
Let $\mathcal{F}$ be a category cofibered in groupoids over
$\mathcal{C}_\Lambda$ satisfying (RS). ========

regex_match $\mathcal{F}$
nn candidate a category
nn a category
new_annotation [(213166, 213179, 'VAR'), (213183, 213193, 'TYPE')]
===== sent Let
$x \in \Ob(\mathcal{F}(A))$. Let $x_0$ be a pushforward of $x$ to
$\mathcal{F}(k)$.
\begin{enumerate}
\item $T_{\text{id}_{x_0}} \mathit{Aut}(x)$ has a natural $k$-vector
space structure such that addition agrees with composition in
$T_{\text{id}_{x_0}} \mathit{Aut}(x)$. In particular, composition in
$T_{\text{id}_{x_0}} \mathit{Aut}(x)$ is commutative.
 ========

regex_match $x_0$
nn candidate a pushforward
nn a pushforward
new_annotation [(213298, 213303, 'VAR'), (213307, 213320, 'TYPE')]
===== sent Let $x' \to x$ be a morphism lying over a ring map $A' \to A$.
 ========

regex_match $x' \to x$
nn candidate x
nn candidate a morphism
nn a morphism
new_annotation [(216511, 216521, 'VAR'), (216525, 216535, 'TYPE')]
===== sent \end{remark}

\begin{lemma}
\label{lemma-infaut-vector-space}
Let $\mathcal{F}$ be a category cofibered in groupoids over
$\mathcal{C}_\Lambda$ satisfying (RS). ========

regex_match $\mathcal{F}$
nn candidate a category
nn a category
new_annotation [(217131, 217144, 'VAR'), (217148, 217158, 'TYPE')]
===== sent Let $x_0 \in \Ob(\mathcal{F}(k))$.
Then $\text{Inf}_{x_0}(\mathcal{F})$ is equal as a set to
$T_{\text{id}_{x_0}} \mathit{Aut}(x_0)$, and so has a natural $k$-vector
space structure such that addition agrees with composition of automorphisms.
 ========

new_annotation []
===== sent \end{lemma}

\begin{proof}
The equality of sets is as in the end of
Remark \ref{remark-infaut-lifting-equalities}
and the statement about the vector space structure follows from
Lemma \ref{lemma-Aut-functor-tangent-space}.
\end{proof}

\begin{lemma}
\label{lemma-k-linear-infaut}
Let $\varphi : \mathcal{F} \to \mathcal{G}$ be a morphism of categories
cofibred in groupoids over $\mathcal{C}_\Lambda$ satisfying (RS).
 ========

regex_match $\varphi : \mathcal{F} \to \mathcal{G}$
nn candidate $\varphi
nn candidate a morphism
nn a morphism
new_annotation [(217753, 217792, 'VAR'), (217796, 217806, 'TYPE')]
===== sent Let $x_0 \in \Ob(\mathcal{F}(k))$. Then $\varphi$ induces a $k$-linear
map $\text{Inf}_{x_0}(\mathcal{F}) \to \text{Inf}_{\varphi(x_0)}(\mathcal{G})$.
\end{lemma}

\begin{proof}
It is clear that $\varphi$ induces a morphism from
$\mathit{Aut}(x_0) \to \mathit{Aut}(\varphi(x_0))$
which maps the identity to the identity. ========

new_annotation []
===== sent Hence this follows from
the result for tangent spaces, see
Lemma \ref{lemma-k-linear-differential}.
\end{proof}

\begin{lemma}
\label{lemma-lifted-automorphisms-torsor}
Let $\mathcal{F}$ be a category cofibered in groupoids over
$\mathcal{C}_\Lambda$ satisfying (RS). ========

regex_match $\mathcal{F}$
nn candidate a category
nn a category
new_annotation [(218381, 218394, 'VAR'), (218398, 218408, 'TYPE')]
===== sent Let $x' \to x$ be a
morphism lying over a surjective ring map $A' \to A$ with kernel $I$
annihilated by $\mathfrak m_{A'}$. Let $x_0$ be a pushforward of $x$ to
$\mathcal{F}(k)$. Then $\text{Inf}(x'/x)$ has a free and transitive action by
$T_{\text{id}_{x_0}} \mathit{Aut}(x') \otimes_k I
= \text{Inf}_{x_0}(\mathcal{F}) \otimes_k I$.
\end{lemma}

\begin{proof}
This is just the analogue of
Lemma \ref{lemma-free-transitive-action}
in the setting of automorphism sheaves.
 ========

new_annotation []
===== sent Let $x' \to x$ be a
morphism lying over a surjective ring map $A' \to A$ with kernel $I$
annihilated by $\mathfrak m_{A'}$. Let $x_0$ be a pushforward of $x$ to
$\mathcal{F}(k)$. Then $\text{Inf}(x'/x)$ has a free and transitive action by
$T_{\text{id}_{x_0}} \mathit{Aut}(x') \otimes_k I
= \text{Inf}_{x_0}(\mathcal{F}) \otimes_k I$.
\end{lemma}

\begin{proof}
This is just the analogue of
Lemma \ref{lemma-free-transitive-action}
in the setting of automorphism sheaves.
 ========

regex_match $x_0$
nn candidate a pushforward
nn a pushforward
new_annotation [(218604, 218609, 'VAR'), (218613, 218626, 'TYPE')]
===== sent \end{proof}

\begin{lemma}
\label{lemma-infaut-trivial}
Let $\mathcal{F}$ be a category cofibered in groupoids over
$\mathcal{C}_\Lambda$ satisfying (RS). ========

regex_match $\mathcal{F}$
nn candidate a category
nn a category
new_annotation [(219920, 219933, 'VAR'), (219937, 219947, 'TYPE')]
===== sent Let $x' \to x$ be a morphism
in $\mathcal{F}$ lying over a surjective ring map. ========

regex_match $x' \to x$
nn candidate x
nn candidate a morphism
nn a morphism
new_annotation [(220019, 220029, 'VAR'), (220033, 220043, 'TYPE')]
===== sent Let $x_0$ be a pushforward
of $x$ to $\mathcal{F}(k)$. ========

regex_match $x_0$
nn candidate $
nn candidate a pushforward
nn a pushforward
new_annotation [(220099, 220104, 'VAR'), (220108, 220121, 'TYPE')]
===== sent If $\text{Inf}_{x_0}(\mathcal{F}) = 0$ then
$\text{Inf}(x'/x) = 0$.
\end{lemma}

\begin{proof}
Follows from
Lemmas \ref{lemma-factor-small-extension} and
\ref{lemma-lifted-automorphisms-torsor}.
\end{proof}

\begin{lemma}
\label{lemma-infdef-trivial}
Let $\mathcal{F}$ be a category cofibered in groupoids over
$\mathcal{C}_\Lambda$ satisfying (RS). ========

regex_match $\mathcal{F}$
nn candidate a category
nn a category
new_annotation [(220405, 220418, 'VAR'), (220422, 220432, 'TYPE')]
===== sent Obviously if $\mathcal{F}_{x_0}$ is fibered in
setoids then $\text{Inf}_{x_0}(\mathcal{F}) = 0$.  Conversely assume
$\text{Inf}_{x_0}(\mathcal{F}) = 0$.  Let $A$ be an object of
$\mathcal{C}_\Lambda$. Then by
Lemma \ref{lemma-infaut-trivial},
$\text{Inf}(x/x_0) = 0$ for any object $x \to x_0$ of
$\mathcal{F}_{x_0}(A)$. ========

regex_match $A$
nn candidate an object
nn an object
new_annotation [(221329, 221332, 'VAR'), (221336, 221345, 'TYPE')]
===== sent \begin{lemma}
\label{lemma-deformation-categories-fiber-product-morphisms}
Let $f : \mathcal{H} \to \mathcal{F}$ and $g : \mathcal{G} \to \mathcal{F}$
be $1$-morphisms of deformation categories. ========

new_annotation []
===== sent Let $w_0 \in \Ob(\mathcal{W}(k))$ and let $x_0, y_0, z_0$ be
the image of $w_0$ in $\mathcal{F}, \mathcal{H}, \mathcal{G}$.
Then $\text{Inf}(\mathcal{W}) = \text{Inf}_{w_0}(\mathcal{W})$
and simlarly for $\mathcal{H}$, $\mathcal{G}$, and $\mathcal{F}$, see
Remark \ref{remark-trivial-aut-point}.
We apply Lemmas \ref{lemma-k-linear-differential} and
\ref{lemma-k-linear-infaut} to get all the linear maps
except for the ``boundary map''
$\delta : \text{Inf}_{x_0}(\mathcal{F}) \to T\mathcal{W}$.
We will insert suitable signs later.

 ========

new_annotation []
===== sent Let $w_0 \in \Ob(\mathcal{W}(k))$ and let $x_0, y_0, z_0$ be
the image of $w_0$ in $\mathcal{F}, \mathcal{H}, \mathcal{G}$.
Then $\text{Inf}(\mathcal{W}) = \text{Inf}_{w_0}(\mathcal{W})$
and simlarly for $\mathcal{H}$, $\mathcal{G}$, and $\mathcal{F}$, see
Remark \ref{remark-trivial-aut-point}.
We apply Lemmas \ref{lemma-k-linear-differential} and
\ref{lemma-k-linear-infaut} to get all the linear maps
except for the ``boundary map''
$\delta : \text{Inf}_{x_0}(\mathcal{F}) \to T\mathcal{W}$.
We will insert suitable signs later.

 ========

new_annotation []
===== sent \end{proof}

\begin{lemma}
\label{lemma-map-fibre-products-smooth}
Let $\mathcal{H}_1 \to \mathcal{G}$, $\mathcal{H}_2 \to \mathcal{G}$, and
$\mathcal{G} \to \mathcal{F}$ be maps of categories cofibred in groupoids
over $\mathcal{C}_\Lambda$. Assume
\begin{enumerate}
\item ========

new_annotation []
===== sent Let $A' \to A$ be a small extension in $\mathcal{C}_\Lambda$.
An object of $\mathcal{H}_1 \times_\mathcal{F} \mathcal{H}_2$
over $A'$ is a triple $(x'_1, x'_2, a')$ where
$x'_i$ is an object of $\mathcal{H}_i$ over $A'$ and
$a' : q(p_1(x'_1)) \to q(p_2(x'_2))$ is a morphism
of the fibre category of $\mathcal{F}$ over $A'$.
 ========

regex_match $A' \to A$
nn candidate a small extension
nn a small extension
new_annotation [(226564, 226574, 'VAR'), (226578, 226595, 'TYPE')]
===== sent This follows from our assumption on
infinitesimal automorphisms and Lemma \ref{lemma-lifted-automorphisms-torsor}.
\end{proof}

\begin{lemma}
\label{lemma-easy-check-smooth}
Let $f : \mathcal{F} \to \mathcal{G}$ be a map of deformation
categories. ========

regex_match $f : \mathcal{F} \to \mathcal{G}$
nn candidate \mathcal{F
nn candidate a map
nn a map
new_annotation [(228430, 228463, 'VAR'), (228467, 228472, 'TYPE')]
===== sent Let $x_0 \in \Ob(\mathcal{F}(k))$ with image
$y_0 \in \Ob(\mathcal{G}(k))$. If
\begin{enumerate}
\item the map $T\mathcal{F} \to T\mathcal{G}$ is surjective, and
\item for every small extension $A' \to A$ in $\mathcal{C}_\Lambda$
and $x \in \mathcal{F}(A)$ with image $y \in ========

new_annotation []
===== sent Let $A' \to A$ be a small extension. ========

regex_match $A' \to A$
nn candidate a small extension
nn a small extension
new_annotation [(228973, 228983, 'VAR'), (228987, 229004, 'TYPE')]
===== sent Let $x \in \mathcal{F}(A)$.
Let $y' \to f(x)$ be a morphism in $\mathcal{G}$ over $A' \to A$.
 ========

new_annotation []
===== sent Let $x \in \mathcal{F}(A)$.
Let $y' \to f(x)$ be a morphism in $\mathcal{G}$ over $A' \to A$.
 ========

regex_match $y' \to f(x)$
nn candidate $y
nn candidate a morphism
nn a morphism
new_annotation [(229038, 229051, 'VAR'), (229055, 229065, 'TYPE')]
===== sent \end{proof}

\begin{lemma}
\label{lemma-map-between-smooth}
Let $\mathcal{F} \to \mathcal{G} \to \mathcal{H}$ be maps of categories
cofibred in groupoids over $\mathcal{C}_\Lambda$. ========

new_annotation []
===== sent Let $A' \to A$ be a small extension in $\mathcal{C}_\Lambda$
and let $x \in \mathcal{F}(A)$ with image $y \in ========

regex_match $A' \to A$
nn candidate a small extension
nn a small extension
new_annotation [(230023, 230033, 'VAR'), (230037, 230054, 'TYPE')]
===== sent Let $A' \to A$ be a small extension in $\mathcal{C}_\Lambda$
and let $x \in \mathcal{F}(A)$ with image $y \in ========

new_annotation []
===== sent \begin{definition}
\label{definition-groupoid-in-functors}
Let $\mathcal{C}$ be a category. ========

regex_match $\mathcal{C}$
nn candidate a category
nn a category
new_annotation [(231048, 231061, 'VAR'), (231065, 231075, 'TYPE')]
===== sent \end{remark}

\begin{remark}
\label{remark-identity-inverse}
Let $(U, R, s, t, c)$ be a groupoid in functors on a category $\mathcal{C}$.
There are unique morphisms $e : U \to R$ and $i : R \to R$ such that
for every object $T$ of $\mathcal{C}$, $e: U(T) \to R(T)$ sends
$x \in U(T)$ to the identity morphism on $x$ and $i: R(T) \to R(T)$ sends
$a \in U(T)$ to the inverse of $a$ in the groupoid category
$(U(T), R(T), s, t, c)$. We will sometimes refer to $s$, $t$, $c$, $e$,
and $i$ as ``source'', ``target'', ``composition'', ``identity'', and
``inverse''.
 ========

regex_match $(U, R, s, t, c)$
nn candidate U
nn candidate R
nn candidate a groupoid
nn a groupoid
new_annotation [(232812, 232829, 'VAR'), (232833, 232843, 'TYPE')]
===== sent \end{remark}

\begin{definition}
\label{definition-representable}
Let $\mathcal{C}$ be a category. ========

regex_match $\mathcal{C}$
nn candidate a category
nn a category
new_annotation [(233377, 233390, 'VAR'), (233394, 233404, 'TYPE')]
===== sent \end{remark}

\begin{remark}
\label{remark-simplify-terminology}
We will say ``{\it let $(\underline{U}, \underline{R}, s, t, c)$ be a
groupoid in functors on $\mathcal{C}$}'' to mean that we have
a representable groupoid in functors. ========

new_annotation []
===== sent This will be
relevant below in situations where we restrict from $\widehat{\mathcal
C}_\Lambda$ to $\mathcal{C}_\Lambda$.

\begin{definition}
\label{definition-restricting-groupoids-in-functors}
Let $(U, R, s, t, c)$ be a groupoid in functors on a category $\mathcal{C}$.
Let $\mathcal{C}'$ be a subcategory of $\mathcal{C}$. The
{\it restriction $(U, R, s, t, c)|_{\mathcal{C}'}$ of $(U, R, s, t, c)$
to $\mathcal{C}'$} is the groupoid
in functors on $\mathcal{C}'$ given by $(U|_{\mathcal{C}'}, R|_{\mathcal
C'}, s|_{\mathcal{C}'}, t|_{\mathcal{C}'}, c|_{\mathcal{C}'})$.
\end{definition}

\begin{remark}
\label{remark-notation-restriction}
 ========

regex_match $(U, R, s, t, c)$
nn candidate U
nn candidate R
nn candidate a groupoid
nn a groupoid
new_annotation [(235253, 235270, 'VAR'), (235274, 235284, 'TYPE')]
===== sent This will be
relevant below in situations where we restrict from $\widehat{\mathcal
C}_\Lambda$ to $\mathcal{C}_\Lambda$.

\begin{definition}
\label{definition-restricting-groupoids-in-functors}
Let $(U, R, s, t, c)$ be a groupoid in functors on a category $\mathcal{C}$.
Let $\mathcal{C}'$ be a subcategory of $\mathcal{C}$. The
{\it restriction $(U, R, s, t, c)|_{\mathcal{C}'}$ of $(U, R, s, t, c)$
to $\mathcal{C}'$} is the groupoid
in functors on $\mathcal{C}'$ given by $(U|_{\mathcal{C}'}, R|_{\mathcal
C'}, s|_{\mathcal{C}'}, t|_{\mathcal{C}'}, c|_{\mathcal{C}'})$.
\end{definition}

\begin{remark}
\label{remark-notation-restriction}
 ========

regex_match $\mathcal{C}'$
nn candidate a subcategory
nn a subcategory
new_annotation [(235330, 235344, 'VAR'), (235348, 235361, 'TYPE')]
===== sent In the situation of Definition
\ref{definition-restricting-groupoids-in-functors}, we often denote
$s|_{\mathcal{C}'}, t|_{\mathcal{C}'}, c|_{\mathcal{C}'}$ simply by $s, t, c$.
\end{remark}

\begin{definition}
\label{definition-quotient}
Let $(U, R, s, t, c)$ be a groupoid in functors on a category $\mathcal{C}$.
\begin{enumerate}
\item The assignment $T \mapsto  (U(T), R(T), s, t, c)$ determines a functor
$\mathcal{C} \to \textit{Groupoids}$. The {\it quotient category
cofibered in groupoids $[U/R] \to \mathcal{C}$} is the category
cofibered in groupoids over $\mathcal{C}$ associated to this functor (as in
Remarks \ref{remarks-cofibered-groupoids}
(\ref{item-construction-associated-cofibered-groupoid})).
 ========

regex_match $(U, R, s, t, c)$
nn candidate U
nn candidate R
nn candidate a groupoid
nn a groupoid
new_annotation [(235940, 235957, 'VAR'), (235961, 235971, 'TYPE')]
===== sent In this section we discuss groupoids in functors on $\mathcal{C}_\Lambda$.
Our eventual goal is to show that prorepresentable groupoids
in functors on $\mathcal{C}_\Lambda$ serve as ``presentations''
for well-behaved deformation categories in the same way that
smooth groupoids in algebraic spaces serve as presentations for
algebraic stacks,
cf.\ Algebraic Stacks, Section \ref{algebraic-section-stack-to-presentation}.

\begin{definition}
\label{definition-prorepresentable-groupoid-in-functors}
A groupoid in functors on $\mathcal{C}_\Lambda$ is {\it prorepresentable}
if it is isomorphic to
$(\underline{R_0}, \underline{R_1}, s, t, c)|_{\mathcal{C}_\Lambda}$
for some representable groupoid in functors
$(\underline{R_0}, \underline{R_1}, s, t, c)$ on the category
$\widehat{\mathcal{C}}_\Lambda$.
\end{definition}

\noindent
Let $(U, R, s, t, c)$ be a groupoid in functors on $\mathcal{C}_\Lambda$.
Taking completions, we get a quintuple
$(\widehat{U}, \widehat{R}, \widehat{s}, \widehat{t}, \widehat{c})$. By
Remark \ref{remark-completion-restriction-cofset-adjoint}
completion as a functor on $\text{CofSet}(\mathcal{C}_\Lambda)$ is a right
adjoint, so it commutes with limits. ========

regex_match $(U, R, s, t, c)$
nn candidate U
nn candidate R
nn candidate a groupoid
nn a groupoid
new_annotation [(237793, 237810, 'VAR'), (237814, 237824, 'TYPE')]
===== sent In particular, there is a canonical
isomorphism
$$
\widehat{R \times_{s, U, t} R}
\longrightarrow
\widehat{R} \times_{\widehat{s}, \widehat{U}, \widehat{t}} \widehat{R},
$$
so $\widehat{c}$ can be regarded as a functor
$\widehat{R} \times_{\widehat{s}, \widehat{U}, \widehat{t}} \widehat{R} \to
\widehat{R}$. Then
$(\widehat{U}, \widehat{R}, \widehat{s}, \widehat{t}, \widehat{c})$
is a groupoid in functors on $\widehat{\mathcal{C}}_\Lambda$, with
identity and inverse morphisms being the completions of those of
$(U, R, s, t, c)$.

\begin{definition}
\label{definition-completion-groupoid-in-functors}
Let $(U, R, s, t, c)$ be a groupoid in functors on $\mathcal{C}_\Lambda$.
The {\it completion $( ========

regex_match $(U, R, s, t, c)$
nn candidate U
nn candidate R
nn candidate a groupoid
nn a groupoid
new_annotation [(238752, 238769, 'VAR'), (238773, 238783, 'TYPE')]
===== sent \end{definition}

\begin{remark}
\label{remark-groupoid-in-functors-complete-restrict}
Let $(U, R, s, t, c)$ be a groupoid in functors on $\mathcal{C}_\Lambda$.
Then there is a canonical isomorphism
$(U, R, s, t, c)^{\wedge}|_{\mathcal{C}_\Lambda} \cong (U, R, s, t, c)$, see
Remark \ref{remark-restrict-completion}.
On the other hand, let $(U, R, s, t, c)$ be a groupoid in functors on
$\widehat{\mathcal{C}}_\Lambda$ such that
$U, R : \widehat{\mathcal{C}}_\Lambda \to \textit{Sets}$
both commute with limits, e.g.\ if $U, R$ are representable.
 ========

regex_match $(U, R, s, t, c)$
nn candidate U
nn candidate R
nn candidate a groupoid
nn a groupoid
new_annotation [(239130, 239147, 'VAR'), (239151, 239161, 'TYPE')]
===== sent \end{definition}

\begin{remark}
\label{remark-groupoid-in-functors-complete-restrict}
Let $(U, R, s, t, c)$ be a groupoid in functors on $\mathcal{C}_\Lambda$.
Then there is a canonical isomorphism
$(U, R, s, t, c)^{\wedge}|_{\mathcal{C}_\Lambda} \cong (U, R, s, t, c)$, see
Remark \ref{remark-restrict-completion}.
On the other hand, let $(U, R, s, t, c)$ be a groupoid in functors on
$\widehat{\mathcal{C}}_\Lambda$ such that
$U, R : \widehat{\mathcal{C}}_\Lambda \to \textit{Sets}$
both commute with limits, e.g.\ if $U, R$ are representable.
 ========

regex_match $(U, R, s, t, c)$
nn candidate U
nn candidate R
nn candidate a groupoid
nn a groupoid
new_annotation [(239379, 239396, 'VAR'), (239400, 239410, 'TYPE')]
===== sent \label{lemma-groupoid-in-functors-prorep-equivalences}
Let $(U, R, s, t, c)$ be a groupoid in functors on $\mathcal{C}_\Lambda$.
\begin{enumerate}
\item $(U, R, s, t, c)$ is prorepresentable if and only if its completion is
representable as a groupoid in functors on $\widehat{\mathcal{C}}_\Lambda$.
\item $( ========

regex_match $(U, R, s, t, c)$
nn candidate U
nn candidate R
nn candidate a groupoid
nn a groupoid
new_annotation [(239862, 239879, 'VAR'), (239883, 239893, 'TYPE')]
===== sent \begin{definition}
\label{definition-smooth-groupoid-in-functors}
Let $(U, R, s, t, c)$ be a groupoid in functors on $\mathcal{C}_\Lambda$. We
say $(U, R, s, t, c)$ is {\it smooth} if $s, t: R \to U$ are smooth.
 ========

regex_match $(U, R, s, t, c)$
nn candidate U
nn candidate R
nn candidate a groupoid
nn a groupoid
new_annotation [(241895, 241912, 'VAR'), (241916, 241926, 'TYPE')]
===== sent \end{definition}

\begin{remark}
\label{remark-smooth-groupoid-in-functors-warning}
We note that this terminology is potentially confusing:
if $(U, R, s, t, c)$ is a smooth groupoid in functors, then the quotient
$[U/R]$ need not be a smooth category cofibred in groupoids as defined in
Definition \ref{definition-cofibered-groupoid-projection-smooth}.
However smoothness of $(U, R, s, t, c)$ does imply (in fact is equivalent to)
smoothness of the quotient morphism $U \to [U/R]$ as we shall
see in
Lemma \ref{lemma-smooth-quotient-morphism}.
\end{remark}

\begin{remark}
\label{remark-smooth-power-series-prorepresentable-smooth-groupoid-in-functors}
Let $(\underline{R_0}, \underline{R_1}, s, t, c)|_{\mathcal{C}_\Lambda}$
be a prorepresentable groupoid in functors on $\mathcal{C}_\Lambda$.
Then $(\underline{R_0}, \underline{R_1}, s, t, c)|_{\mathcal{C}_\Lambda}$
is smooth if and only if $R_1$ is a power series over $R_0$ via both $s$
and $t$. ========

new_annotation []
===== sent This follows from
Lemma \ref{lemma-smooth-morphism-power-series}.
\end{remark}

\begin{lemma}
\label{lemma-smooth-quotient-morphism}
Let $(U, R, s, t, c)$ be a groupoid in functors on $\mathcal{C}_\Lambda$.
The following are equivalent:
\begin{enumerate}
\item ========

regex_match $(U, R, s, t, c)$
nn candidate U
nn candidate R
nn candidate a groupoid
nn a groupoid
new_annotation [(243125, 243142, 'VAR'), (243146, 243156, 'TYPE')]
===== sent Smoothness of $s: R \to U$ amounts to the following
condition: If $f: B \to A$ is a surjective ring map in
$\mathcal{C}_\Lambda$, $a \in R(A)$, and $y \in U(B)$ such that
$s(a) = U(f)(y)$, then there exists $a' \in R(B)$ such that
$R(f)(a') = a$ and $s(a') = y$.
\item[(4)] Smoothness of $U \to [U/R]$ amounts to the following
condition: If $f: B \to A$ is a surjective ring map in
$\mathcal{C}_\Lambda$ and $(f, a) : (B, y) \to (A, x)$
is a morphism of $[U/R]$, then there exists $x' \in U(B)$ and
$b \in R(B)$ with $s(b) = x'$, $t(b) = y$
such that $c(a, R(f)(b)) = e(x)$.  Here $e : U \to R$ denotes the
identity and the notation $(f, a)$ is as in Remarks
\ref{remarks-cofibered-groupoids}
(\ref{item-construction-associated-cofibered-groupoid});
in particular $a \in R(A)$ with $s(a) = U(f)(y)$ and $t(a) = x$.
\end{enumerate}
If (4) holds and $f, a, y$ as in (2) are given, let $x = t(a)$ so that
we have a morphism $(f, a): (B, y) \to (A, x)$.
Then (4) produces $x'$ and $b$, and $a' = i(b)$ satisfies the requirements
of (2). ========

new_annotation []
===== sent Conversely, assume (2) holds and let $(f, a): (B, y) \to (A, x)$
as in (4) be given. ========

new_annotation []
===== sent \begin{lemma}
\label{lemma-smooth-RS-groupoid-in-functors-quotient}
Let $(U, R, s, t, c)$ be a smooth groupoid in functors on $\mathcal{C}_\Lambda$.
Assume $U$ and $R$ satisfy (RS). ========

regex_match $(U, R, s, t, c)$
nn candidate U
nn candidate R
nn candidate a smooth groupoid
nn a smooth groupoid
new_annotation [(245505, 245522, 'VAR'), (245526, 245543, 'TYPE')]
===== sent \medskip\noindent
Let $a = c(i(a_1), a_2)$, where $i: R \to R$ is the inverse morphism.
 ========

new_annotation []
===== sent By the assumption that $U$ satisfies (RS), we have an identification
$U(A_1) \times_{U(A)} U(A_2) = U(A_1 \times_A A_2)$. Let us denote
$x_1 \times t(\widetilde{a}) \in U(A_1 \times_A A_2)$ the element
corresponding to $(x_1, t(\widetilde{a})) \in U(A_1) \times_{U(A)} U(A_2)$.
Let $p_1, p_2$ be the projections of $A_1 \times_A A_2$.  We claim
$$
\xymatrix{
(A_1 \times_A A_2, x_1 \times t(\widetilde{a})) \ar[d]_{(p_1, e(x_1))}
\ar[rr]_-{(p_2, i(\widetilde{a}))} & & (A_2, x_2) \ar[d]^{(f_2, a_2)} \\
(A_1, x_1) \ar[rr]^{(f_1, a_1)} & & (A, x)
}
$$
is a fiber square in $[U/R]$. (Note $e: U \to R$ denotes the identity.)

\medskip\noindent
The diagram is commutative because
$c(a_2, R(f_2)(i(\widetilde{a}))) = c(a_2, i(a)) = a_1$.
To check it is a fiber square, let
$$
\xymatrix{
(B, z) \ar[d]_{(g_1, b_1)} \ar[rr]_{(g_2, b_2)} & & (A_2, x_2)
\ar[d]^{(f_2, a_2)} \\
(A_1, x_1) \ar[rr]^{(f_1, a_1)} & & (A, x)
}
$$
be a commutative diagram in $[U/R]$. We will show there is a unique morphism
$(g, b) : (B, z) \to (A_1 \times_A A_2, x_1 \times t(\widetilde{a}))$
compatible with the morphisms to $(A_1, x_1)$ and $(A_2, x_2)$.
We must take $g = (g_1, g_2) : B \to A_1 \times_A A_2$.
 ========

regex_match $p_1, p_2$
nn candidate the projections
nn the projections
new_annotation [(247145, 247155, 'VAR'), (247159, 247174, 'TYPE')]
===== sent \end{proof}

\begin{lemma}
\label{lemma-deformation-groupoid-quotient}
Let $(U, R, s, t, c)$ be a smooth groupoid in functors on $\mathcal{C}_\Lambda$.
Assume $U$ and $R$ are deformation functors. ========

regex_match $(U, R, s, t, c)$
nn candidate U
nn candidate R
nn candidate a smooth groupoid
nn a smooth groupoid
new_annotation [(248571, 248588, 'VAR'), (248592, 248609, 'TYPE')]
===== sent \begin{definition}
\label{definition-presentation}
Let $\mathcal{F}$ be a category cofibered in groupoids over a category
$\mathcal{C}$. Let $(U, R, s, t, c)$ be a groupoid in functors on
$\mathcal{C}$. ========

regex_match $\mathcal{F}$
nn candidate a category
nn a category
new_annotation [(249603, 249616, 'VAR'), (249620, 249630, 'TYPE')]
===== sent \begin{definition}
\label{definition-presentation}
Let $\mathcal{F}$ be a category cofibered in groupoids over a category
$\mathcal{C}$. Let $(U, R, s, t, c)$ be a groupoid in functors on
$\mathcal{C}$. ========

regex_match $(U, R, s, t, c)$
nn candidate U
nn candidate R
nn candidate a groupoid
nn a groupoid
new_annotation [(249689, 249706, 'VAR'), (249710, 249720, 'TYPE')]
===== sent \begin{lemma}
\label{lemma-presentation-construction}
Let $\mathcal{F}$ be category cofibered in groupoids over a category
$\mathcal{C}$. Let $U : \mathcal{C} \to \textit{Sets}$ be a functor.
 ========

new_annotation []
===== sent \begin{lemma}
\label{lemma-presentation-construction}
Let $\mathcal{F}$ be category cofibered in groupoids over a category
$\mathcal{C}$. Let $U : \mathcal{C} \to \textit{Sets}$ be a functor.
 ========

regex_match $U : \mathcal{C} \to \textit{Sets}$
nn candidate $U
nn candidate a functor
nn a functor
new_annotation [(250158, 250193, 'VAR'), (250197, 250206, 'TYPE')]
===== sent Let $f : U \to \mathcal{F}$ be a morphism of categories cofibered in groupoids
over $\mathcal{C}$. Define $R, s, t, c$ as follows:
\begin{enumerate}
\item $R : \mathcal{C} \to \textit{Sets}$ is the functor
$U \times_{f, \mathcal{F}, f} U$.
\item $t, s : R \to U$ are the first and second projections,
respectively.
 ========

regex_match $f : U \to \mathcal{F}$
nn candidate U \to
nn candidate a morphism
nn a morphism
new_annotation [(250212, 250235, 'VAR'), (250239, 250249, 'TYPE')]
===== sent \end{proof}

\begin{lemma}
\label{lemma-presentation-morphism}
Let $\mathcal{F}$ be category cofibered in groupoids over a category
$\mathcal{C}$. Let $U : \mathcal{C} \to \textit{Sets}$ be a functor.
 ========

new_annotation []
===== sent \end{proof}

\begin{lemma}
\label{lemma-presentation-morphism}
Let $\mathcal{F}$ be category cofibered in groupoids over a category
$\mathcal{C}$. Let $U : \mathcal{C} \to \textit{Sets}$ be a functor.
 ========

regex_match $U : \mathcal{C} \to \textit{Sets}$
nn candidate $U
nn candidate a functor
nn a functor
new_annotation [(251085, 251120, 'VAR'), (251124, 251133, 'TYPE')]
===== sent Let $f : U \to \mathcal{F}$ be a morphism of categories cofibered in groupoids
over $\mathcal{C}$. ========

regex_match $f : U \to \mathcal{F}$
nn candidate U \to
nn candidate a morphism
nn a morphism
new_annotation [(251139, 251162, 'VAR'), (251166, 251176, 'TYPE')]
===== sent Let $(U, R, s, t, c)$ be the groupoid in functors on
$\mathcal{C}$ constructed from $f : U \to \mathcal{F}$ in
Lemma \ref{lemma-presentation-construction}.
Then there is a natural morphism $[f] : [U/R] \to \mathcal{F}$ such that:
\begin{enumerate}
\item $[f]: [U/R] \to \mathcal{F}$ is fully faithful ========

regex_match $(U, R, s, t, c)$
nn candidate U
nn candidate R
nn candidate the groupoid
nn the groupoid
new_annotation [(251238, 251255, 'VAR'), (251259, 251271, 'TYPE')]
===== sent \begin{lemma}
\label{lemma-smooth-groupoid-in-functors-construction}
Let $\mathcal{F}$ be a category cofibered in groupoids over
$\mathcal{C}_\Lambda$. Let $U : \mathcal{C}_\Lambda \to \textit{Sets}$
be a functor. ========

regex_match $\mathcal{F}$
nn candidate a category
nn a category
new_annotation [(252113, 252126, 'VAR'), (252130, 252140, 'TYPE')]
===== sent \begin{lemma}
\label{lemma-smooth-groupoid-in-functors-construction}
Let $\mathcal{F}$ be a category cofibered in groupoids over
$\mathcal{C}_\Lambda$. Let $U : \mathcal{C}_\Lambda \to \textit{Sets}$
be a functor. ========

new_annotation []
===== sent Let $f : U \to \mathcal{F}$ be a smooth morphism of
categories cofibered in groupoids. ========

regex_match $f : U \to \mathcal{F}$
nn candidate $
nn candidate U \to
nn candidate a smooth morphism
nn a smooth morphism
new_annotation [(252258, 252281, 'VAR'), (252285, 252302, 'TYPE')]
===== sent \end{proof}

\begin{lemma}
\label{lemma-deformation-functor-diagonal}
Let $\mathcal{F}$ be a deformation category.
 ========

regex_match $\mathcal{F}$
nn candidate a deformation category
nn a deformation category
new_annotation [(253496, 253509, 'VAR'), (253513, 253535, 'TYPE')]
===== sent Let $U : \mathcal{C}_\Lambda \to \textit{Sets}$ be a deformation functor.
 ========

regex_match $U : \mathcal{C}_\Lambda \to \textit{Sets}$
nn candidate \mathcal{C}_\Lambda \to
nn candidate a deformation functor
nn a deformation functor
new_annotation [(253541, 253584, 'VAR'), (253588, 253609, 'TYPE')]
===== sent Let $f: U \to \mathcal{F}$ be a morphism of categories cofibered in groupoids.
 ========

regex_match $f: U \to \mathcal{F}$
nn candidate U \to
nn candidate a morphism
nn a morphism
new_annotation [(253615, 253637, 'VAR'), (253641, 253651, 'TYPE')]
===== sent \begin{lemma}
\label{lemma-prorepresentable-groupoid-in-functors-construction}
Let $\mathcal{F}$ be a deformation category.
 ========

regex_match $\mathcal{F}$
nn candidate a deformation category
nn a deformation category
new_annotation [(254157, 254170, 'VAR'), (254174, 254196, 'TYPE')]
===== sent Let $U : \mathcal{C}_\Lambda \to \textit{Sets}$ be a prorepresentable functor.
 ========

regex_match $U : \mathcal{C}_\Lambda \to \textit{Sets}$
nn candidate \mathcal{C}_\Lambda \to
nn candidate a prorepresentable functor
nn a prorepresentable functor
new_annotation [(254202, 254245, 'VAR'), (254249, 254275, 'TYPE')]
===== sent Let $f : U \to \mathcal{F}$ be a morphism of categories cofibered in groupoids.
 ========

regex_match $f : U \to \mathcal{F}$
nn candidate U \to
nn candidate a morphism
nn a morphism
new_annotation [(254281, 254304, 'VAR'), (254308, 254318, 'TYPE')]
===== sent Let $(U, R, s, t, c)$ be the groupoid in functors on $\mathcal{C}_\Lambda$
constructed from $f : U \to \mathcal{F}$ in
Lemma \ref{lemma-presentation-construction}. If
$\dim_k \text{Inf}(\mathcal{F}) < \infty$, then
$(U, R, s, t, c)$ is prorepresentable.
 ========

regex_match $(U, R, s, t, c)$
nn candidate U
nn candidate R
nn candidate the groupoid
nn the groupoid
new_annotation [(254361, 254378, 'VAR'), (254382, 254394, 'TYPE')]
===== sent \end{proof}

\begin{theorem}
\label{theorem-presentation-deformation-groupoid}
Let $\mathcal{F}$ be a category cofibered in groupoids over
$\mathcal{C}_\Lambda$. Then $\mathcal{F}$ admits a presentation by a
smooth prorepresentable groupoid in functors on $\mathcal{C}_\Lambda$
if and only if the following conditions hold:
\begin{enumerate}
\item $\mathcal{F}$ is a deformation category.
 ========

regex_match $\mathcal{F}$
nn candidate a category
nn a category
new_annotation [(255728, 255741, 'VAR'), (255745, 255755, 'TYPE')]
===== sent Let $(U, R, s, t, c)$ be the groupoid in functors constructed in
Lemma \ref{lemma-presentation-construction}
from the map $\underline{\xi}$. By
Lemma \ref{lemma-smooth-groupoid-in-functors-construction}
we see that $(U, R, s, t, c)$ is a smooth groupoid in functors and that
$[U/R] \to \mathcal{F}$ is an equivalence. ========

regex_match $(U, R, s, t, c)$
nn candidate U
nn candidate R
nn candidate the groupoid
nn the groupoid
new_annotation [(256928, 256945, 'VAR'), (256949, 256961, 'TYPE')]
===== sent \begin{definition}
\label{definition-minimal-groupoid-in-functors}
Let $(U, R, s, t, c)$ be a smooth prorepresentable groupoid in functors
on $\mathcal{C}_\Lambda$.
\begin{enumerate}
\item We say $(U, R, s, t, c)$ is {\it normalized} if the groupoid
$(U(k[\epsilon]), R(k[\epsilon]), s, t, c)$ is totally disconnected,
i.e., there are no morphisms between distinct objects.
 ========

regex_match $(U, R, s, t, c)$
nn candidate U
nn candidate R
nn candidate a smooth prorepresentable groupoid
nn a smooth prorepresentable groupoid
new_annotation [(258102, 258119, 'VAR'), (258123, 258157, 'TYPE')]
===== sent \begin{lemma}
\label{lemma-characterize-minimal-groupoid-in-functors}
Let $(U, R, s, t, c)$ be a smooth prorepresentable groupoid in
functors on $\mathcal{C}_\Lambda$.
\begin{enumerate}
\item $(U, R, s, t, c)$ is normalized if and only if the morphism
$U \to [U/R]$ induces an isomorphism on tangent spaces, and
\item $(U, R, s, t, c)$ is minimal if and only if the kernel of
$TU \to T[U/R]$ is contained in the image of
$\text{Der}_\Lambda(k, k) \to TU$.
\end{enumerate}
\end{lemma}

\begin{proof}
Part (1) follows immediately from the definitions.
 ========

regex_match $(U, R, s, t, c)$
nn candidate U
nn candidate R
nn candidate a smooth prorepresentable groupoid
nn a smooth prorepresentable groupoid
new_annotation [(258982, 258999, 'VAR'), (259003, 259037, 'TYPE')]
===== sent \begin{lemma}
\label{lemma-surjective-morphism-prorepresentable-functor}
Let $U: \mathcal{C}_\Lambda \to \textit{Sets}$ be a
prorepresentable functor. ========

new_annotation []
===== sent Let $\varphi : U \to U$ be a morphism such
that $d\varphi : TU \to TU$ is an isomorphism.   ========

regex_match $\varphi : U \to U$
nn candidate $\varphi
nn candidate a morphism
nn a morphism
new_annotation [(260581, 260600, 'VAR'), (260604, 260614, 'TYPE')]
===== sent \end{proof}

\begin{lemma}
\label{lemma-minimal-prorepresentable-groupoid-autoequivalence}
Let $(U, R, s, t, c)$ be a minimal smooth prorepresentable groupoid in
functors on $\mathcal{C}_\Lambda$. If $\varphi : [U/R] \to [U/R]$ is an
equivalence of categories cofibered in groupoids, then $\varphi$ is an
isomorphism.
 ========

regex_match $(U, R, s, t, c)$
nn candidate U
nn candidate R
nn candidate a minimal smooth prorepresentable groupoid
nn a minimal smooth prorepresentable groupoid
new_annotation [(261576, 261593, 'VAR'), (261597, 261639, 'TYPE')]
===== sent \end{proof}

\begin{lemma}
\label{lemma-minimal-prorepresentable-groupoid-equivalence}
Let $(U, R, s, t, c)$ and $(U', R', s', t', c')$ be minimal smooth
prorepresentable groupoids in functors on $\mathcal{C}_\Lambda$. If
$\varphi : [U/R] ========

new_annotation []
===== sent Let $\psi : [U'/R'] \to [U/R]$ be a quasi-inverse to $\varphi$.
Then $\psi \circ \varphi$ and $\varphi \circ \psi$ are isomorphisms by
Lemma \ref{lemma-minimal-prorepresentable-groupoid-autoequivalence},
hence $\varphi$ and $\psi$ are isomorphisms.
 ========

regex_match $\psi : [U'/R'] \to [U/R]$
nn candidate $\psi
nn candidate a quasi-inverse
nn a quasi-inverse
new_annotation [(263362, 263388, 'VAR'), (263392, 263407, 'TYPE')]
===== sent \begin{lemma}
\label{lemma-minimal-groupoid-in-functors-construction}
Let $\mathcal{F}$ be a deformation category such that
$\dim_k T\mathcal{F} <\infty$ and
$\dim_k \text{Inf}(\mathcal{F}) < \infty$.
Then there exists a minimal versal formal object $\xi$ of $\mathcal{F}$.
Say $\xi$ lies over $R ========

regex_match $\mathcal{F}$
nn candidate a deformation category
nn a deformation category
new_annotation [(263793, 263806, 'VAR'), (263810, 263832, 'TYPE')]
===== sent \in \Ob(\widehat{\mathcal{C}}_\Lambda)$.
Let $U = \underline{R}|_{\mathcal{C}_\Lambda}$.
Let $f = \underline{\xi} : U \to \mathcal{F}$ be the associated
morphism. ========

new_annotation []
===== sent \in \Ob(\widehat{\mathcal{C}}_\Lambda)$.
Let $U = \underline{R}|_{\mathcal{C}_\Lambda}$.
Let $f = \underline{\xi} : U \to \mathcal{F}$ be the associated
morphism. ========

regex_match $f = \underline{\xi} : U \to \mathcal{F}$
nn candidate the associated
morphism
nn the associated
morphism
new_annotation [(264109, 264150, 'VAR'), (264154, 264177, 'TYPE')]
===== sent Let $(U, R, s, t, c)$ be the groupoid in functors on
$\mathcal{C}_\Lambda$ constructed from $f : U \to \mathcal{F}$ in
Lemma \ref{lemma-presentation-construction}.
 ========

regex_match $(U, R, s, t, c)$
nn candidate U
nn candidate R
nn candidate the groupoid
nn the groupoid
new_annotation [(264183, 264200, 'VAR'), (264204, 264216, 'TYPE')]
===== sent Let $(U, R, s, t, c)$ be the groupoid in functors constructed in
Lemma \ref{lemma-presentation-construction}
from the map $\underline{\xi}$. By
Lemma \ref{lemma-smooth-groupoid-in-functors-construction}
we see that $(U, R, s, t, c)$ is a smooth groupoid in functors and that
$[U/R] \to \mathcal{F}$ is an equivalence. ========

regex_match $(U, R, s, t, c)$
nn candidate U
nn candidate R
nn candidate the groupoid
nn the groupoid
new_annotation [(265018, 265035, 'VAR'), (265039, 265051, 'TYPE')]
===== sent \begin{lemma}
\label{lemma-minimal-presentations-equivalent}
Let $\mathcal{F}$ be category cofibered in groupoids over
$\mathcal{C}_\Lambda$. Assume there exist presentations of
$\mathcal{F}$ by minimal smooth prorepresentable groupoids
in functors $(U, R, s, t, c)$ and $(U', R', s', t', c')$.
Then $(U, R, s, t, c)$ and $(U', R', s', t', c')$ are isomorphic.
 ========

new_annotation []
===== sent \begin{theorem}
\label{theorem-minimal-smooth-prorepresentable-presentations}
Let $\mathcal{F}$ be a category cofibered in groupoids over
$\mathcal{C}_\Lambda$. Consider the following conditions
\begin{enumerate}
\item $\mathcal{F}$ admits a presentation by a normalized
smooth prorepresentable groupoid in functors on $\mathcal{C}_\Lambda$,
\item $\mathcal{F}$ admits a presentation by a
smooth prorepresentable groupoid in functors on $\mathcal{C}_\Lambda$,
\item $\mathcal{F}$ admits a presentation by a minimal
smooth prorepresentable groupoid in functors on $\mathcal{C}_\Lambda$, and
\item $\mathcal{F}$ satisfies the following conditions
\begin{enumerate}
\item $\mathcal{F}$ is a deformation category.
 ========

regex_match $\mathcal{F}$
nn candidate a category
nn a category
new_annotation [(266531, 266544, 'VAR'), (266548, 266558, 'TYPE')]
===== sent \end{lemma}

\begin{proof}
Namely, if $(r, h, k)$ is a formal homotopy between $f_1$ and $f_2$, then
$(r, h, g \circ k)$ is a formal homotopy between $g \circ f_1$ and
$g \circ f_2$.
\end{proof}

\begin{lemma}
\label{lemma-versal-unique-up-to-homotopy}
Let $\mathcal{F}$ be a deformation category over $\mathcal{C}_\Lambda$
with $\dim_k T\mathcal{F} < \infty$ and
$\dim_k \text{Inf}(\mathcal{F}) < \infty$. Let $\xi$ be a versal formal
object lying over $R$. Let $\eta$ be a formal object lying over $S$.
Then any two maps
$$
f, g : R \to S
$$
such that $f_*\xi \cong \eta \cong g_*\xi$ are formally homotopic.
\end{lemma}

\begin{proof}
 ========

regex_match $\mathcal{F}$
nn candidate a deformation category
nn a deformation category
new_annotation [(271578, 271591, 'VAR'), (271595, 271617, 'TYPE')]
===== sent \end{lemma}

\begin{proof}
Namely, if $(r, h, k)$ is a formal homotopy between $f_1$ and $f_2$, then
$(r, h, g \circ k)$ is a formal homotopy between $g \circ f_1$ and
$g \circ f_2$.
\end{proof}

\begin{lemma}
\label{lemma-versal-unique-up-to-homotopy}
Let $\mathcal{F}$ be a deformation category over $\mathcal{C}_\Lambda$
with $\dim_k T\mathcal{F} < \infty$ and
$\dim_k \text{Inf}(\mathcal{F}) < \infty$. Let $\xi$ be a versal formal
object lying over $R$. Let $\eta$ be a formal object lying over $S$.
Then any two maps
$$
f, g : R \to S
$$
such that $f_*\xi \cong \eta \cong g_*\xi$ are formally homotopic.
\end{lemma}

\begin{proof}
 ========

regex_match $\xi$
nn candidate a versal formal
object
nn a versal formal
object
new_annotation [(271732, 271737, 'VAR'), (271741, 271763, 'TYPE')]
===== sent \end{lemma}

\begin{proof}
Namely, if $(r, h, k)$ is a formal homotopy between $f_1$ and $f_2$, then
$(r, h, g \circ k)$ is a formal homotopy between $g \circ f_1$ and
$g \circ f_2$.
\end{proof}

\begin{lemma}
\label{lemma-versal-unique-up-to-homotopy}
Let $\mathcal{F}$ be a deformation category over $\mathcal{C}_\Lambda$
with $\dim_k T\mathcal{F} < \infty$ and
$\dim_k \text{Inf}(\mathcal{F}) < \infty$. Let $\xi$ be a versal formal
object lying over $R$. Let $\eta$ be a formal object lying over $S$.
Then any two maps
$$
f, g : R \to S
$$
such that $f_*\xi \cong \eta \cong g_*\xi$ are formally homotopic.
\end{lemma}

\begin{proof}
 ========

regex_match $\eta$
nn candidate a formal object
nn a formal object
new_annotation [(271784, 271790, 'VAR'), (271794, 271809, 'TYPE')]
===== sent Let $\Lambda$ be a Noetherian ring and
let $\Lambda \to k$ be a finite ring map where $k$ is a field. ========

regex_match $\Lambda$
nn candidate a Noetherian ring
nn a Noetherian ring
new_annotation [(274161, 274170, 'VAR'), (274174, 274191, 'TYPE')]
===== sent Let $\Lambda$ be a Noetherian ring and
let $\Lambda \to k$ be a finite ring map where $k$ is a field. ========

regex_match $\Lambda \to k$
nn candidate a finite ring map
nn a finite ring map
new_annotation [(274200, 274215, 'VAR'), (274219, 274236, 'TYPE')]
===== sent However, since in this section we will discuss what happen when we change
$k$ we will instead use the notation $\mathcal{C}_{\Lambda, k}$ to
indicate the dependence on $k$.

\begin{situation}
\label{situation-change-of-fields}
Let $\Lambda$ be a Noetherian ring and let $\Lambda \to k \to l$ be a finite
ring maps where $k$ and $l$ are fields. ========

regex_match $\Lambda$
nn candidate a Noetherian ring
nn a Noetherian ring
new_annotation [(274700, 274709, 'VAR'), (274713, 274730, 'TYPE')]
===== sent However, since in this section we will discuss what happen when we change
$k$ we will instead use the notation $\mathcal{C}_{\Lambda, k}$ to
indicate the dependence on $k$.

\begin{situation}
\label{situation-change-of-fields}
Let $\Lambda$ be a Noetherian ring and let $\Lambda \to k \to l$ be a finite
ring maps where $k$ and $l$ are fields. ========

regex_match $\Lambda \to k \to l$
nn candidate a finite
ring maps
nn a finite
ring maps
new_annotation [(274739, 274760, 'VAR'), (274764, 274782, 'TYPE')]
===== sent \end{lemma}

\begin{proof}
Let $x_0 \in \Ob(\mathcal{F}(k))$ and denote $x_{l, 0}$ the corresponding
object of $\mathcal{F}_{l/k}$ over $l$. Recall that
$\text{Inf}(\mathcal{F}) = \text{Inf}_{x_0}(\mathcal{F})$ and
$\text{Inf}(\mathcal{F}_{l/k}) = \text{Inf}_{x_{l, 0}}(\mathcal{F}_{l/k})$,
see Remark \ref{remark-trivial-aut-point}.
Recall that the vector space structure on
$\text{Inf}_{x_0}(\mathcal{F})$ comes from identifying it with the tangent
space of the functor $\mathit{Aut}(x_0)$ which is defined on
the category $\mathcal{C}_{k, k}$ of Artinian local $k$-algebras with
residue field $k$. Similarly, $\text{Inf}_{x_{l, 0}}(\mathcal{F}_{l/k})$
is the tangent space of $\mathit{Aut}(x_{l, 0})$ which is defined on the
category $\mathcal{C}_{l, l}$ of Artinian local $l$-algebras with residue
field $l$. Unwinding the definitions we see that
$\mathit{Aut}(x_{l, 0})$ is the restriction of $\mathit{Aut}(x_0)_{l/k}$
(which lives on $\mathcal{C}_{k, l}$) to $\mathcal{C}_{l, l}$. Since
there is no difference between the tangent space of
$\mathit{Aut}(x_0)_{l/k}$ seen as a functor on $\mathcal{C}_{k, l}$ or
$\mathcal{C}_{l, l}$, the lemma follows from 
Lemma \ref{lemma-tangent-space-change-of-field}
and the fact that $\mathit{Aut}(x_0)$ satisfies (RS) by
Lemma \ref{lemma-Aut-functor-RS} (whence we have (S2) by
Lemma \ref{lemma-RS-implies-S1-S2}).
 ========

new_annotation []
===== sent With notation and assumptions as in Situation \ref{situation-change-of-fields}.
Let $\xi$ be a versal formal object for $\mathcal{F}$ lying over
$R \in \Ob(\widehat{\mathcal{C}}_{\Lambda, k})$. Then there exist
\begin{enumerate}
\item an $S \in \Ob(\widehat{\mathcal{C}}_{\Lambda, l})$
and a local $\Lambda$-algebra homomorphism $R \to S$ which is
formally smooth in the $\mathfrak m_S$-adic topology and induces
the given field extension $l/k$ on residue fieds, and
\item a versal formal object of $\mathcal{F}_{l/k}$ lying over $S$.
\end{enumerate}
\end{lemma}

\begin{proof}
Construction of $S$. Choose a surjection $R[x_1, \ldots, x_n] \to l$
of $R$-algebras. ========

regex_match $\xi$
nn candidate a versal formal object
nn a versal formal object
new_annotation [(282524, 282529, 'VAR'), (282533, 282555, 'TYPE')]
file:  spaces-more-cohomology.tex
===== sent \medskip\noindent
Let $S$ be a scheme and let $X$ be an algebraic space over $S$.
In this chapter and the following we will write $X \times_S X$
for the product of $X$ with itself (in the category of algebraic
spaces over $S$), instead of $X \times X$.







\section{Transporting results from schemes}
\label{section-api}

\noindent
 ========

regex_match $S$
nn candidate a scheme
nn a scheme
new_annotation [(1051, 1054, 'VAR'), (1058, 1066, 'TYPE')]
===== sent \medskip\noindent
Let $S$ be a scheme and let $X$ be an algebraic space over $S$.
In this chapter and the following we will write $X \times_S X$
for the product of $X$ with itself (in the category of algebraic
spaces over $S$), instead of $X \times X$.







\section{Transporting results from schemes}
\label{section-api}

\noindent
 ========

regex_match $X$
nn candidate $X$
nn candidate an algebraic space
nn an algebraic space
new_annotation [(1075, 1078, 'VAR'), (1082, 1100, 'TYPE')]
===== sent For quasi-coherent modules more is true
(because \'etale cohomology of a quasi-coherent module
over a scheme agrees with Zariski cohomology) and this
has already been discussed in Cohomology of Spaces, Section
\ref{spaces-cohomology-section-higher-direct-image}.

\medskip\noindent
Let $S$ be a scheme. ========

regex_match $S$
nn candidate a scheme
nn a scheme
new_annotation [(1811, 1814, 'VAR'), (1818, 1826, 'TYPE')]
===== sent Let $X$ be an algebraic space over $S$.
Now suppose that $X$ is representable by the scheme $X_0$
(awkward but temporary notation; we usually just say ``$X$
is a scheme''). ========

regex_match $X$
nn candidate an algebraic space
nn an algebraic space
new_annotation [(1832, 1835, 'VAR'), (1839, 1857, 'TYPE')]
===== sent \begin{lemma}
\label{lemma-compare-cohomology-other-topologies}
Let $S$ be a scheme. ========

regex_match $S$
nn candidate a scheme
nn a scheme
new_annotation [(3600, 3603, 'VAR'), (3607, 3615, 'TYPE')]
===== sent Let $\tau \in \{\etale, fppf, ph\}$ (add more here).
 ========

new_annotation []
===== sent \begin{lemma}
\label{lemma-surjective-proper}
Let $S$ be a scheme. ========

regex_match $S$
nn candidate a scheme
nn a scheme
new_annotation [(4395, 4398, 'VAR'), (4402, 4410, 'TYPE')]
===== sent Let $f : Y \to X$ be a surjective proper morphism
of algebraic spaces over $S$. Let $\mathcal{F}$ be a sheaf on $X_\etale$.
 ========

regex_match $f : Y \to X$
nn candidate $
nn candidate : Y \to X$
nn candidate a surjective proper morphism
nn a surjective proper morphism
new_annotation [(4416, 4429, 'VAR'), (4433, 4461, 'TYPE')]
===== sent Let $f : Y \to X$ be a surjective proper morphism
of algebraic spaces over $S$. Let $\mathcal{F}$ be a sheaf on $X_\etale$.
 ========

regex_match $\mathcal{F}$
nn candidate a sheaf
nn a sheaf
new_annotation [(4496, 4509, 'VAR'), (4513, 4520, 'TYPE')]
===== sent \end{proof}

\begin{lemma}
\label{lemma-h0-proper-over-henselian-pair}
Let $(A, I)$ be a henselian pair. ========

regex_match $(A, I)$
nn candidate $(A
nn candidate a henselian pair
nn a henselian pair
new_annotation [(6047, 6055, 'VAR'), (6059, 6075, 'TYPE')]
===== sent Let $X$ be an algebraic space over $A$
such that the structure morphism $f : X \to \Spec(A)$ is proper.
 ========

regex_match $X$
nn candidate an algebraic space
nn an algebraic space
new_annotation [(6081, 6084, 'VAR'), (6088, 6106, 'TYPE')]
===== sent Let $i : X_0 \to X$ be the inclusion of $X \times_{\Spec(A)} \Spec(A/I)$.
 ========

regex_match $i : X_0 \to X$
nn candidate X_0 \to
nn candidate the inclusion
nn the inclusion
new_annotation [(6185, 6200, 'VAR'), (6204, 6217, 'TYPE')]
===== sent \end{proof}

\begin{lemma}
\label{lemma-h0-proper-over-henselian-local}
Let $A$ be a henselian local ring. ========

regex_match $A$
nn candidate a henselian local ring
nn a henselian local ring
new_annotation [(7333, 7336, 'VAR'), (7340, 7362, 'TYPE')]
===== sent Let $X$ be an algebraic space
over $A$ such that $f : X \to \Spec(A)$
be a proper morphism. ========

regex_match $X$
nn candidate an algebraic space
nn an algebraic space
new_annotation [(7368, 7371, 'VAR'), (7375, 7393, 'TYPE')]
===== sent Let $X_0 \subset X$ be the fibre of
$f$ over the closed point. ========

regex_match $X_0 \subset X$
nn candidate $
nn candidate the fibre
nn the fibre
new_annotation [(7460, 7475, 'VAR'), (7479, 7488, 'TYPE')]
===== sent For any sheaf $\mathcal{F}$ on $X_\etale$ we
have $\Gamma(X, \mathcal{F}) = \Gamma(X_0, \mathcal{F}|_{X_0})$.
\end{lemma}

\begin{proof}
This is a special case of Lemma \ref{lemma-h0-proper-over-henselian-pair}.
\end{proof}

\begin{lemma}
\label{lemma-proper-base-change-f-star}
Let $S$ be a scheme. ========

regex_match $S$
nn candidate a scheme
nn a scheme
new_annotation [(7802, 7805, 'VAR'), (7809, 7817, 'TYPE')]
===== sent Let $f : X \to Y$ and $g : Y' \to Y$
be a morphisms of algebraic spaces over $S$. Assume $f$ is proper.
Set $X' = Y' \times_Y X$ with projections $f' : X' \to Y'$ and $g' : X' \to X$.
Let $\mathcal{F}$ be any sheaf on $X_\etale$. Then
$g^{-1}f_*\mathcal{F} = f'_*(g')^{-1}\mathcal{F}$.
\end{lemma}

\begin{proof}
The question is \'etale local on $Y'$. Choose a scheme $V$ and a surjective
\'etale morphism ========

new_annotation []
===== sent Let $f : X \to Y$ and $g : Y' \to Y$
be a morphisms of algebraic spaces over $S$. Assume $f$ is proper.
Set $X' = Y' \times_Y X$ with projections $f' : X' \to Y'$ and $g' : X' \to X$.
Let $\mathcal{F}$ be any sheaf on $X_\etale$. Then
$g^{-1}f_*\mathcal{F} = f'_*(g')^{-1}\mathcal{F}$.
\end{lemma}

\begin{proof}
The question is \'etale local on $Y'$. Choose a scheme $V$ and a surjective
\'etale morphism ========

new_annotation []
===== sent Let $S$ be a scheme.
 ========

regex_match $S$
nn candidate a scheme
nn a scheme
new_annotation [(9541, 9544, 'VAR'), (9548, 9556, 'TYPE')]
===== sent Let $f : Y \to X$ be a morphism of algebraic spaces over $S$. Let
$\overline{x} : \Spec(k) \to ========

regex_match $f : Y \to X$
nn candidate : Y \to X$
nn candidate a morphism
nn a morphism
new_annotation [(9562, 9575, 'VAR'), (9579, 9589, 'TYPE')]
===== sent In the following we will consider the set
$\Gamma(Y_{\overline{x}}, \mathcal{F}_{\overline{x}})$.

\begin{lemma}
\label{lemma-proper-pushforward-stalk}
Let $S$ be a scheme.
 ========

regex_match $S$
nn candidate a scheme
nn a scheme
new_annotation [(10185, 10188, 'VAR'), (10192, 10200, 'TYPE')]
===== sent Let $f : Y \to X$ be a proper morphism of algebraic spaces over $S$. Let
$\overline{x} \to X$ be a geometric point.
 ========

regex_match $f : Y \to X$
nn candidate Y \to X$
nn candidate a proper morphism
nn a proper morphism
new_annotation [(10206, 10219, 'VAR'), (10223, 10240, 'TYPE')]
===== sent This is a special case of Lemma \ref{lemma-proper-base-change-f-star}.
\end{proof}

\begin{theorem}
\label{theorem-proper-base-change}
Let $S$ be a scheme. ========

regex_match $S$
nn candidate a scheme
nn a scheme
new_annotation [(10665, 10668, 'VAR'), (10672, 10680, 'TYPE')]
===== sent Let $\mathcal{F}$ be an abelian torsion sheaf on $X_\etale$.
 ========

regex_match $\mathcal{F}$
nn candidate an abelian torsion sheaf
nn an abelian torsion sheaf
new_annotation [(10842, 10855, 'VAR'), (10859, 10883, 'TYPE')]
===== sent \medskip\noindent
For every $n \geq 1$ let $\mathcal{F}[n]$ be the subsheaf of sections
of $\mathcal{F}$ annihilated by $n$. ========

regex_match $\mathcal{F}[n]$
nn candidate the subsheaf
nn the subsheaf
new_annotation [(11525, 11541, 'VAR'), (11545, 11557, 'TYPE')]
===== sent Let $\mathcal{F} \to \mathcal{I}^\bullet$ be a resolution by
injective sheaves of $\mathbf{Z}/n\mathbf{Z}$-modules.
 ========

regex_match $\mathcal{F} \to \mathcal{I}^\bullet$
nn candidate $\mathcal{F
nn candidate a resolution
nn a resolution
new_annotation [(11912, 11949, 'VAR'), (11953, 11965, 'TYPE')]
===== sent \end{proof}

\begin{lemma}
\label{lemma-proper-base-change}
Let $S$ be a scheme. ========

regex_match $S$
nn candidate a scheme
nn a scheme
new_annotation [(13738, 13741, 'VAR'), (13745, 13753, 'TYPE')]
===== sent Let $E \in D^+(X_\etale)$ have torsion cohomology sheaves.
 ========

new_annotation []
===== sent \end{proof}

\begin{lemma}
\label{lemma-proper-base-change-stalk}
Let $S$ be a scheme.
 ========

regex_match $S$
nn candidate a scheme
nn a scheme
new_annotation [(14555, 14558, 'VAR'), (14562, 14570, 'TYPE')]
===== sent Let $f : X \to Y$ be a proper morphism of algebraic spaces.
 ========

regex_match $f : X \to Y$
nn candidate X \to
nn candidate a proper morphism
nn a proper morphism
new_annotation [(14576, 14589, 'VAR'), (14593, 14610, 'TYPE')]
===== sent Let $\overline{y} \to Y$ be a geometric point.
 ========

regex_match $\overline{y} \to Y$
nn candidate $\overline{y
nn candidate a geometric point
nn a geometric point
new_annotation [(14636, 14656, 'VAR'), (14660, 14677, 'TYPE')]
===== sent Since pulling back by $\overline{y} \to Y$ produces the
stalk of $\mathcal{F}$, the first statement of the lemma
is a special case of Theorem \ref{theorem-proper-base-change}.
The second one is a special case of Lemma \ref{lemma-proper-base-change}.
\end{proof}

\begin{lemma}
\label{lemma-base-change-separably-closed}
Let $k \subset k'$ be an extension of separably closed fields.
 ========

regex_match $k \subset k'$
nn candidate an extension
nn an extension
new_annotation [(15514, 15528, 'VAR'), (15532, 15544, 'TYPE')]
===== sent Let $X$ be a proper algebraic space over $k$.
Let $\mathcal{F}$ be a torsion abelian sheaf on $X$.
 ========

regex_match $X$
nn candidate a proper algebraic space
nn a proper algebraic space
new_annotation [(15577, 15580, 'VAR'), (15584, 15608, 'TYPE')]
===== sent Let $X$ be a proper algebraic space over $k$.
Let $\mathcal{F}$ be a torsion abelian sheaf on $X$.
 ========

regex_match $\mathcal{F}$
nn candidate a torsion abelian sheaf
nn a torsion abelian sheaf
new_annotation [(15623, 15636, 'VAR'), (15640, 15663, 'TYPE')]
===== sent Then the map $H^q_\etale(X, \mathcal{F}) \to
H^q_\etale(X_{k'}, \mathcal{F}|_{X_{k'}})$ is an isomorphism
for $q \geq 0$.
\end{lemma}

\begin{proof}
This is a special case of Theorem \ref{theorem-proper-base-change}.
\end{proof}










\section{Comparing big and small topoi}
\label{section-compare}

\noindent
Let $S$ be a scheme and let $X$ be an algebraic space over $S$.
In Topologies on Spaces, Lemma
\ref{spaces-topologies-lemma-at-the-bottom-etale}
we have introduced comparison morphisms
$\pi_X : (\textit{Spaces}/X)_\etale \to X_{spaces, \etale}$ and
$i_X : \Sh(X_\etale) \to \Sh((\textit{Spaces}/X)_\etale)$
with $\pi_X \circ i_X = \text{id}$ as morphisms of topoi and
$\pi_{X, *} = i_X^{-1}$.
More generally, if $f : Y \to X$ is an object of $(\textit{Spaces}/X)_\etale$,
then there is a morphism
$i_f : \Sh(Y_\etale) \to \Sh((\textit{Spaces}/X)_\etale)$
such that $f_{small} = \pi_X \circ i_f$, see
Topologies on Spaces, Lemmas \ref{spaces-topologies-lemma-put-in-T-etale} and
\ref{spaces-topologies-lemma-morphism-big-small-etale}. In
Topologies on Spaces, ========

regex_match $S$
nn candidate a scheme
nn a scheme
new_annotation [(15990, 15993, 'VAR'), (15997, 16005, 'TYPE')]
===== sent Then the map $H^q_\etale(X, \mathcal{F}) \to
H^q_\etale(X_{k'}, \mathcal{F}|_{X_{k'}})$ is an isomorphism
for $q \geq 0$.
\end{lemma}

\begin{proof}
This is a special case of Theorem \ref{theorem-proper-base-change}.
\end{proof}










\section{Comparing big and small topoi}
\label{section-compare}

\noindent
Let $S$ be a scheme and let $X$ be an algebraic space over $S$.
In Topologies on Spaces, Lemma
\ref{spaces-topologies-lemma-at-the-bottom-etale}
we have introduced comparison morphisms
$\pi_X : (\textit{Spaces}/X)_\etale \to X_{spaces, \etale}$ and
$i_X : \Sh(X_\etale) \to \Sh((\textit{Spaces}/X)_\etale)$
with $\pi_X \circ i_X = \text{id}$ as morphisms of topoi and
$\pi_{X, *} = i_X^{-1}$.
More generally, if $f : Y \to X$ is an object of $(\textit{Spaces}/X)_\etale$,
then there is a morphism
$i_f : \Sh(Y_\etale) \to \Sh((\textit{Spaces}/X)_\etale)$
such that $f_{small} = \pi_X \circ i_f$, see
Topologies on Spaces, Lemmas \ref{spaces-topologies-lemma-put-in-T-etale} and
\ref{spaces-topologies-lemma-morphism-big-small-etale}. In
Topologies on Spaces, ========

regex_match $X$
nn candidate $X$
nn candidate an algebraic space
nn an algebraic space
new_annotation [(16014, 16017, 'VAR'), (16021, 16039, 'TYPE')]
===== sent \begin{lemma}
\label{lemma-describe-pullback}
Let $S$ be a scheme. ========

regex_match $S$
nn candidate a scheme
nn a scheme
new_annotation [(18015, 18018, 'VAR'), (18022, 18030, 'TYPE')]
===== sent Let $X$ be an algebraic space over $S$.
Let $\mathcal{F}$ be a sheaf on $X_\etale$. Then
$\pi_X^{-1}\mathcal{F}$ is given by the rule
$$
(\pi_X^{-1}\mathcal{F})(Y) = \Gamma(Y_\etale, f_{small}^{-1}\mathcal{F})
$$
for $f : Y \to X$ in $(\textit{Spaces}/X)_\etale$.
 ========

regex_match $X$
nn candidate an algebraic space
nn an algebraic space
new_annotation [(18036, 18039, 'VAR'), (18043, 18061, 'TYPE')]
===== sent Let $X$ be an algebraic space over $S$.
Let $\mathcal{F}$ be a sheaf on $X_\etale$. Then
$\pi_X^{-1}\mathcal{F}$ is given by the rule
$$
(\pi_X^{-1}\mathcal{F})(Y) = \Gamma(Y_\etale, f_{small}^{-1}\mathcal{F})
$$
for $f : Y \to X$ in $(\textit{Spaces}/X)_\etale$.
 ========

regex_match $\mathcal{F}$
nn candidate a sheaf
nn a sheaf
new_annotation [(18076, 18089, 'VAR'), (18093, 18100, 'TYPE')]
===== sent This we have seen in Lemma \ref{lemma-surjective-proper}.

\medskip\noindent
The case of smooth, syntomic, fppf coverings follows from the case
of ph coverings by Topologies on Spaces, Lemma
\ref{spaces-topologies-lemma-zariski-etale-smooth-syntomic-fppf-ph}.

\medskip\noindent
Let $\mathcal{U} = \{U_i \to U\}_{i \in I}$ be an fpqc covering of algebraic
spaces over $X$. Let $s_i \in (\pi_X^{-1}\mathcal{F})(U_i)$ be sections
which agree over $U_i \times_U U_j$. We have to prove there exists a unique
$s \in (\pi_X^{-1}\mathcal{F})(U)$ restricting to $s_i$ over $U_i$.
Case I: $U$ and $U_i$ are schemes. ========

regex_match $\mathcal{U} = \{U_i \to U\}_{i \in I}$
nn candidate $\mathcal{U
nn candidate an fpqc
nn an fpqc
new_annotation [(19359, 19398, 'VAR'), (19402, 19409, 'TYPE')]
===== sent This we have seen in Lemma \ref{lemma-surjective-proper}.

\medskip\noindent
The case of smooth, syntomic, fppf coverings follows from the case
of ph coverings by Topologies on Spaces, Lemma
\ref{spaces-topologies-lemma-zariski-etale-smooth-syntomic-fppf-ph}.

\medskip\noindent
Let $\mathcal{U} = \{U_i \to U\}_{i \in I}$ be an fpqc covering of algebraic
spaces over $X$. Let $s_i \in (\pi_X^{-1}\mathcal{F})(U_i)$ be sections
which agree over $U_i \times_U U_j$. We have to prove there exists a unique
$s \in (\pi_X^{-1}\mathcal{F})(U)$ restricting to $s_i$ over $U_i$.
Case I: $U$ and $U_i$ are schemes. ========

new_annotation []
===== sent Let $W \to U$ be a surjective \'etale
morphism, where $W$ is a scheme. ========

regex_match $W \to U$
nn candidate $
nn candidate a surjective \'etale
morphism
nn a surjective \'etale
morphism
new_annotation [(20106, 20115, 'VAR'), (20119, 20148, 'TYPE')]
===== sent Then $\mathcal{W} = \{U_i \times_U W \to W\}$
is an fpqc covering (by algebraic spaces) of the scheme $W$.
By case II the result hold for $\mathcal{W}$.
We omit the verification that this implies the result for $\mathcal{U}$.
\end{proof}

\begin{lemma}
\label{lemma-compare-injectives}
Let $S$ be a scheme.
 ========

regex_match $S$
nn candidate a scheme
nn a scheme
new_annotation [(20463, 20466, 'VAR'), (20470, 20478, 'TYPE')]
===== sent Let $Y \to X$ be a morphism of $(\textit{Spaces}/S)_\etale$.
\begin{enumerate}
\item If $\mathcal{I}$ is injective in
$\textit{Ab}((\textit{Spaces}/X)_\etale)$, then
\begin{enumerate}
\item $i_f^{-1}\mathcal{I}$ is injective in $\textit{Ab}(Y_\etale)$,
\item $\mathcal{I}|_{X_\etale}$ is injective in $\textit{Ab}(X_\etale)$,
\end{enumerate}
\item If $\mathcal{I}^\bullet$ is a K-injective complex
in $\textit{Ab}((\textit{Spaces}/X)_\etale)$, then
\begin{enumerate}
\item $i_f^{-1}\mathcal{I}^\bullet$ is a K-injective complex in
$\textit{Ab}(Y_\etale)$,
\item $\mathcal{I}^\bullet|_{X_\etale}$ is a K-injective complex in
$\textit{Ab}(X_\etale)$,
\end{enumerate}
\end{enumerate}
The corresponding statements for modules do not hold.
 ========

regex_match $Y \to X$
nn candidate $Y
nn candidate a morphism
nn a morphism
new_annotation [(20484, 20493, 'VAR'), (20497, 20507, 'TYPE')]
===== sent \medskip\noindent
To see a counter example for the case of modules we refer to
\'Etale Cohomology, Lemma \ref{etale-cohomology-lemma-compare-injectives}.
\end{proof}

\noindent
Let $S$ be a scheme.
 ========

regex_match $S$
nn candidate a scheme
nn a scheme
new_annotation [(22681, 22684, 'VAR'), (22688, 22696, 'TYPE')]
===== sent Let $f : Y \to X$ be a morphism of algebraic spaces over $S$.
The commutative diagram of
Topologies on Spaces, Lemma
\ref{spaces-topologies-lemma-morphism-big-small-etale} (3)
leads to a commutative diagram of ringed sites
$$
\xymatrix{
(Y_{spaces, \etale}, \mathcal{O}_Y) \ar[d]_{f_{spaces, \etale}} &
((\textit{Spaces}/Y)_\etale, \mathcal{O}) \ar[d]^{f_{big}} \ar[l]^{\pi_Y} \\
(X_{spaces, \etale}, \mathcal{O}_X) &
((\textit{Spaces}/X)_\etale, \mathcal{O}) \ar[l]_{\pi_X}
}
$$
as one easily sees by writing out the definitions of
$f_{small}^\sharp$, $f_{big}^\sharp$, $\pi_X^\sharp$, and ========

regex_match $f : Y \to X$
nn candidate : Y \to X$
nn candidate a morphism
nn a morphism
new_annotation [(22702, 22715, 'VAR'), (22719, 22729, 'TYPE')]
===== sent $\pi_Y^\sharp$.
In particular this means that
\begin{equation}
\label{equation-compare-big-small}
(f_{big, *}\mathcal{F})|_{X_\etale} =
f_{small, *}(\mathcal{F}|_{Y_\etale})
\end{equation}
for any sheaf $\mathcal{F}$ on $(\textit{Spaces}/Y)_\etale$ and if
$\mathcal{F}$ is a sheaf of $\mathcal{O}$-modules, then
(\ref{equation-compare-big-small})
is an isomorphism of $\mathcal{O}_X$-modules on $X_\etale$.

\begin{lemma}
\label{lemma-compare-higher-direct-image}
Let $S$ be a scheme.
 ========

regex_match $S$
nn candidate a scheme
nn a scheme
new_annotation [(23757, 23760, 'VAR'), (23764, 23772, 'TYPE')]
===== sent Let $f : Y \to X$ be a morphism of algebraic spaces over $S$.
\begin{enumerate}
\item For $K$ in $D((\textit{Spaces}/Y)_\etale)$ we have
$
(Rf_{big, *}K)|_{X_\etale} = Rf_{small, *}(K|_{Y_\etale})
$
in $D(X_\etale)$.
\item For $K$ in $D((\textit{Spaces}/Y)_\etale, \mathcal{O})$ we have
$
(Rf_{big, *}K)|_{X_\etale} = Rf_{small, *}(K|_{Y_\etale})
$
in $D(\textit{Mod}(X_\etale, \mathcal{O}_X))$.
\end{enumerate}
More generally, let $g : X' \to X$ be an object of
$(\textit{Spaces}/X)_\etale$. Consider the fibre product
$$
\xymatrix{
Y' \ar[r]_{g'} \ar[d]_{f'} & Y \ar[d]^f \\
X' \ar[r]^g & X
}
$$
Then
\begin{enumerate}
\item[(3)] For $K$ in $D((\textit{Spaces}/Y)_\etale)$ we have
$i_g^{-1}(Rf_{big, *}K) = Rf'_{small, *}(i_{g'}^{-1}K)$
in $D(X'_\etale)$.
\item[(4)] For $K$ in $D((\textit{Spaces}/Y)_\etale, \mathcal{O})$ we have
$i_g^*(Rf_{big, *}K) = Rf'_{small, *}(i_{g'}^*K)$
in $D(\textit{Mod}(X'_\etale, \mathcal{O}_{X'}))$.
\item[(5)] For $K$ in $D((\textit{Spaces}/Y)_\etale)$ we have
$g_{big}^{-1}(Rf_{big, *}K) = Rf'_{small, *}((g'_{big})^{-1}K)$
in $D((\textit{Spaces}/X')_\etale)$.
\item[(6)] For $K$ in $D((\textit{Spaces}/Y)_\etale, \mathcal{O})$ we have
$g_{big}^*(Rf_{big, *}K) = Rf'_{small, *}((g'_{big})^*K)$
in $D(\textit{Mod}(X'_\etale, \mathcal{O}_{X'}))$.
\end{enumerate}
\end{lemma}

\begin{proof}
Part (1) follows from
Lemma \ref{lemma-compare-injectives}
and (\ref{equation-compare-big-small})
on choosing a K-injective complex of abelian sheaves representing $K$.

\medskip\noindent
Part (3) follows from Lemma \ref{lemma-compare-injectives}
and Topologies, Lemma
\ref{topologies-lemma-morphism-big-small-cartesian-diagram-etale}
on choosing a K-injective complex of abelian sheaves representing $K$.

\medskip\noindent
Part (5) follows from
Cohomology on Sites, Lemmas \ref{sites-cohomology-lemma-cohomology-of-open} and
\ref{sites-cohomology-lemma-restrict-K-injective-to-open}
and Topologies, Lemma
\ref{topologies-lemma-morphism-big-small-cartesian-diagram-etale}
on choosing a K-injective complex of abelian sheaves representing $K$.

\medskip\noindent
Part (6): Observe that $g_{big}$ and $g'_{big}$ are localizations
and hence $g_{big}^{-1} = g_{big}^*$ and $(g'_{big})^{-1} = (g'_{big})^*$
are the restriction functors. ========

regex_match $f : Y \to X$
nn candidate : Y \to X$
nn candidate a morphism
nn a morphism
new_annotation [(23778, 23791, 'VAR'), (23795, 23805, 'TYPE')]
===== sent Let $f : Y \to X$ be a morphism of algebraic spaces over $S$.
\begin{enumerate}
\item For $K$ in $D((\textit{Spaces}/Y)_\etale)$ we have
$
(Rf_{big, *}K)|_{X_\etale} = Rf_{small, *}(K|_{Y_\etale})
$
in $D(X_\etale)$.
\item For $K$ in $D((\textit{Spaces}/Y)_\etale, \mathcal{O})$ we have
$
(Rf_{big, *}K)|_{X_\etale} = Rf_{small, *}(K|_{Y_\etale})
$
in $D(\textit{Mod}(X_\etale, \mathcal{O}_X))$.
\end{enumerate}
More generally, let $g : X' \to X$ be an object of
$(\textit{Spaces}/X)_\etale$. Consider the fibre product
$$
\xymatrix{
Y' \ar[r]_{g'} \ar[d]_{f'} & Y \ar[d]^f \\
X' \ar[r]^g & X
}
$$
Then
\begin{enumerate}
\item[(3)] For $K$ in $D((\textit{Spaces}/Y)_\etale)$ we have
$i_g^{-1}(Rf_{big, *}K) = Rf'_{small, *}(i_{g'}^{-1}K)$
in $D(X'_\etale)$.
\item[(4)] For $K$ in $D((\textit{Spaces}/Y)_\etale, \mathcal{O})$ we have
$i_g^*(Rf_{big, *}K) = Rf'_{small, *}(i_{g'}^*K)$
in $D(\textit{Mod}(X'_\etale, \mathcal{O}_{X'}))$.
\item[(5)] For $K$ in $D((\textit{Spaces}/Y)_\etale)$ we have
$g_{big}^{-1}(Rf_{big, *}K) = Rf'_{small, *}((g'_{big})^{-1}K)$
in $D((\textit{Spaces}/X')_\etale)$.
\item[(6)] For $K$ in $D((\textit{Spaces}/Y)_\etale, \mathcal{O})$ we have
$g_{big}^*(Rf_{big, *}K) = Rf'_{small, *}((g'_{big})^*K)$
in $D(\textit{Mod}(X'_\etale, \mathcal{O}_{X'}))$.
\end{enumerate}
\end{lemma}

\begin{proof}
Part (1) follows from
Lemma \ref{lemma-compare-injectives}
and (\ref{equation-compare-big-small})
on choosing a K-injective complex of abelian sheaves representing $K$.

\medskip\noindent
Part (3) follows from Lemma \ref{lemma-compare-injectives}
and Topologies, Lemma
\ref{topologies-lemma-morphism-big-small-cartesian-diagram-etale}
on choosing a K-injective complex of abelian sheaves representing $K$.

\medskip\noindent
Part (5) follows from
Cohomology on Sites, Lemmas \ref{sites-cohomology-lemma-cohomology-of-open} and
\ref{sites-cohomology-lemma-restrict-K-injective-to-open}
and Topologies, Lemma
\ref{topologies-lemma-morphism-big-small-cartesian-diagram-etale}
on choosing a K-injective complex of abelian sheaves representing $K$.

\medskip\noindent
Part (6): Observe that $g_{big}$ and $g'_{big}$ are localizations
and hence $g_{big}^{-1} = g_{big}^*$ and $(g'_{big})^{-1} = (g'_{big})^*$
are the restriction functors. ========

regex_match $g : X' \to X$
nn candidate let $g
nn candidate an object
nn an object
new_annotation [(24206, 24220, 'VAR'), (24224, 24233, 'TYPE')]
===== sent \medskip\noindent
Part (4) follows from part (6) and part (2) applied to $f' : Y' \to X'$.
\end{proof}

\noindent
Let $S$ be a scheme. ========

regex_match $S$
nn candidate a scheme
nn a scheme
new_annotation [(26875, 26878, 'VAR'), (26882, 26890, 'TYPE')]
===== sent Let $X$ be an algebraic space over $S$.
Let $\mathcal{H}$ be an abelian sheaf on
$(\textit{Spaces}/X)_\etale$. Recall that $H^n_\etale(U, \mathcal{H})$
denotes the cohomology of $\mathcal{H}$ over an object
$U$ of $(\textit{Spaces}/X)_\etale$.

\begin{lemma}
\label{lemma-compare-cohommology}
Let $S$ be a scheme.
 ========

regex_match $X$
nn candidate an algebraic space
nn an algebraic space
new_annotation [(26896, 26899, 'VAR'), (26903, 26921, 'TYPE')]
===== sent Let $X$ be an algebraic space over $S$.
Let $\mathcal{H}$ be an abelian sheaf on
$(\textit{Spaces}/X)_\etale$. Recall that $H^n_\etale(U, \mathcal{H})$
denotes the cohomology of $\mathcal{H}$ over an object
$U$ of $(\textit{Spaces}/X)_\etale$.

\begin{lemma}
\label{lemma-compare-cohommology}
Let $S$ be a scheme.
 ========

regex_match $\mathcal{H}$
nn candidate an abelian sheaf
nn an abelian sheaf
new_annotation [(26936, 26949, 'VAR'), (26953, 26969, 'TYPE')]
===== sent Let $X$ be an algebraic space over $S$.
Let $\mathcal{H}$ be an abelian sheaf on
$(\textit{Spaces}/X)_\etale$. Recall that $H^n_\etale(U, \mathcal{H})$
denotes the cohomology of $\mathcal{H}$ over an object
$U$ of $(\textit{Spaces}/X)_\etale$.

\begin{lemma}
\label{lemma-compare-cohommology}
Let $S$ be a scheme.
 ========

regex_match $S$
nn candidate a scheme
nn a scheme
new_annotation [(27189, 27192, 'VAR'), (27196, 27204, 'TYPE')]
===== sent Let $f : Y \to X$ be a morphism of algebraic spaces over $S$. ========

regex_match $f : Y \to X$
nn candidate : Y \to X$
nn candidate a morphism
nn a morphism
new_annotation [(27210, 27223, 'VAR'), (27227, 27237, 'TYPE')]
===== sent \end{proof}

\begin{lemma}
\label{lemma-cohomological-descent-etale}
Let $S$ be a scheme. ========

regex_match $S$
nn candidate a scheme
nn a scheme
new_annotation [(28525, 28528, 'VAR'), (28532, 28540, 'TYPE')]
===== sent Let $X$ be an algebraic space over $S$.
For $K \in D(X_\etale)$ the map
$$
K \longrightarrow R\pi_{X, *}\pi_X^{-1}K
$$
is an isomorphism where
$\pi_X : \Sh((\textit{Spaces}/X)_\etale) \to \Sh(X_\etale)$ is as above.
 ========

regex_match $X$
nn candidate an algebraic space
nn an algebraic space
new_annotation [(28546, 28549, 'VAR'), (28553, 28571, 'TYPE')]
===== sent \end{proof}

\begin{lemma}
\label{lemma-compare-higher-direct-image-proper}
Let $S$ be a scheme.
 ========

regex_match $S$
nn candidate a scheme
nn a scheme
new_annotation [(29026, 29029, 'VAR'), (29033, 29041, 'TYPE')]
===== sent Let $f : Y \to X$ be a proper morphism of algebraic spaces over $S$.
Then we have
\begin{enumerate}
\item $\pi_X^{-1} \circ f_{small, *} = f_{big, *} \circ \pi_Y^{-1}$
as functors $\Sh(Y_\etale) \to \Sh((\textit{Spaces}/X)_\etale)$,
\item $\pi_X^{-1}Rf_{small, *}K = Rf_{big, *}\pi_Y^{-1}K$
for $K$ in $D^+(Y_\etale)$ whose cohomology sheaves are torsion, and
\item $\pi_X^{-1}Rf_{small, *}K = Rf_{big, *}\pi_Y^{-1}K$
for all $K$ in $D(Y_\etale)$ if $f$ is finite.
 ========

regex_match $f : Y \to X$
nn candidate : Y \to X$
nn candidate a proper morphism
nn a proper morphism
new_annotation [(29047, 29060, 'VAR'), (29064, 29081, 'TYPE')]
===== sent Let $\mathcal{F}$ be a sheaf on $Y_\etale$.
Let $g : X' \to X$ be an object of $(\textit{Spaces}/X)_\etale$.
Consider the fibre product
$$
\xymatrix{
Y' \ar[r]_{f'} \ar[d]_{g'} & X' \ar[d]^g \\
Y \ar[r]^f & X
}
$$
Then we have
$$
(f_{big, *}\pi_Y^{-1}\mathcal{F})(X') =
(\pi_Y^{-1}\mathcal{F})(Y') =
((g'_{small})^{-1}\mathcal{F})(Y')  =
(f'_{small, *}(g'_{small})^{-1}\mathcal{F})(X')
$$
the second equality by Lemma \ref{lemma-describe-pullback}.
On the other hand
$$
(\pi_X^{-1}f_{small, *}\mathcal{F})(X') =
(g_{small}^{-1}f_{small, *}\mathcal{F})(X')
$$
again by Lemma \ref{lemma-describe-pullback}.
 ========

regex_match $\mathcal{F}$
nn candidate a sheaf
nn a sheaf
new_annotation [(29569, 29582, 'VAR'), (29586, 29593, 'TYPE')]
===== sent Let $\mathcal{F}$ be a sheaf on $Y_\etale$.
Let $g : X' \to X$ be an object of $(\textit{Spaces}/X)_\etale$.
Consider the fibre product
$$
\xymatrix{
Y' \ar[r]_{f'} \ar[d]_{g'} & X' \ar[d]^g \\
Y \ar[r]^f & X
}
$$
Then we have
$$
(f_{big, *}\pi_Y^{-1}\mathcal{F})(X') =
(\pi_Y^{-1}\mathcal{F})(Y') =
((g'_{small})^{-1}\mathcal{F})(Y')  =
(f'_{small, *}(g'_{small})^{-1}\mathcal{F})(X')
$$
the second equality by Lemma \ref{lemma-describe-pullback}.
On the other hand
$$
(\pi_X^{-1}f_{small, *}\mathcal{F})(X') =
(g_{small}^{-1}f_{small, *}\mathcal{F})(X')
$$
again by Lemma \ref{lemma-describe-pullback}.
 ========

regex_match $g : X' \to X$
nn candidate $g
nn candidate an object
nn an object
new_annotation [(29613, 29627, 'VAR'), (29631, 29640, 'TYPE')]
===== sent There is a canonical base change map
$\pi_X^{-1}Rf_{small, *}K \to Rf_{big, *}\pi_Y^{-1}K$
for any $K$ in $D(Y_\etale)$, see
Cohomology on Sites, Remark \ref{sites-cohomology-remark-base-change}.
To prove it is an isomorphism, it suffices to prove the pull back of
the base change map by $i_g : \Sh(X'_\etale) \to \Sh((\Sch/X)_\etale)$
is an isomorphism for any object $g : X' \to X$ of $(\Sch/X)_\etale$.
Let $T', g', f'$ be as in the previous paragraph.
 ========

new_annotation []
===== sent \end{proof}







\section{Comparing fppf and \'etale topologies}
\label{section-fppf-etale}

\noindent
This section is the analogue of
\'Etale Cohomology, Section \ref{etale-cohomology-section-fppf-etale}.

\medskip\noindent
Let $S$ be a scheme. ========

regex_match $S$
nn candidate a scheme
nn a scheme
new_annotation [(32330, 32333, 'VAR'), (32337, 32345, 'TYPE')]
===== sent Let $X$ be an algebraic space over $S$.
On the category $\textit{Spaces}/X$ ========

regex_match $X$
nn candidate an algebraic space
nn an algebraic space
new_annotation [(32351, 32354, 'VAR'), (32358, 32376, 'TYPE')]
===== sent Let $S$ be a scheme. ========

regex_match $S$
nn candidate a scheme
nn a scheme
new_annotation [(33517, 33520, 'VAR'), (33524, 33532, 'TYPE')]
===== sent Let $X$ be an algebraic space over $S$.
\begin{enumerate}
\item ========

regex_match $X$
nn candidate an algebraic space
nn an algebraic space
new_annotation [(33538, 33541, 'VAR'), (33545, 33563, 'TYPE')]
===== sent \begin{lemma}
\label{lemma-cohomological-descent-etale-fppf}
Let $S$ be a scheme. ========

regex_match $S$
nn candidate a scheme
nn a scheme
new_annotation [(36785, 36788, 'VAR'), (36792, 36800, 'TYPE')]
===== sent Let $X$ be an algebraic space over $S$.
For $K \in D^+(X_\etale)$ the maps
$$
\pi_X^{-1}K \longrightarrow R\epsilon_{X, *}a_X^{-1}K
\quad\text{and}\quad
K \longrightarrow Ra_{X, *}a_X^{-1}K
$$
are isomorphisms with
$a_X : \Sh((\textit{Spaces}/X)_{fppf}) \to \Sh(X_\etale)$ as above.
 ========

regex_match $X$
nn candidate an algebraic space
nn an algebraic space
new_annotation [(36806, 36809, 'VAR'), (36813, 36831, 'TYPE')]
===== sent \end{proof}

\begin{lemma}
\label{lemma-compare-cohomology-etale-fppf}
Let $S$ be a scheme and let $X$ be an algebraic space over $S$.
With $a_X : \Sh((\textit{Spaces}/X)_{fppf}) \to \Sh(X_\etale)$
as above:
\begin{enumerate}
\item $H^q(X_\etale, \mathcal{F}) = H^q_{fppf}(X, a_X^{-1}\mathcal{F})$
for an abelian sheaf $\mathcal{F}$ on $X_\etale$,
\item $H^q(X_\etale, K) = H^q_{fppf}(X, a_X^{-1}K)$ for $K \in D^+(X_\etale)$.
\end{enumerate}
Example: if $A$ is an abelian group, then
$H^q_\etale(X, \underline{A}) = H^q_{fppf}(X, \underline{A})$.
\end{lemma}

\begin{proof}
This follows from Lemma \ref{lemma-cohomological-descent-etale-fppf}
by Cohomology on Sites, Remark \ref{sites-cohomology-remark-before-Leray}.
\end{proof}

\begin{lemma}
\label{lemma-push-pull-fppf-etale}
Let $S$ be a scheme.
 ========

regex_match $S$
nn candidate a scheme
nn a scheme
new_annotation [(38493, 38496, 'VAR'), (38500, 38508, 'TYPE')]
===== sent \end{proof}

\begin{lemma}
\label{lemma-compare-cohomology-etale-fppf}
Let $S$ be a scheme and let $X$ be an algebraic space over $S$.
With $a_X : \Sh((\textit{Spaces}/X)_{fppf}) \to \Sh(X_\etale)$
as above:
\begin{enumerate}
\item $H^q(X_\etale, \mathcal{F}) = H^q_{fppf}(X, a_X^{-1}\mathcal{F})$
for an abelian sheaf $\mathcal{F}$ on $X_\etale$,
\item $H^q(X_\etale, K) = H^q_{fppf}(X, a_X^{-1}K)$ for $K \in D^+(X_\etale)$.
\end{enumerate}
Example: if $A$ is an abelian group, then
$H^q_\etale(X, \underline{A}) = H^q_{fppf}(X, \underline{A})$.
\end{lemma}

\begin{proof}
This follows from Lemma \ref{lemma-cohomological-descent-etale-fppf}
by Cohomology on Sites, Remark \ref{sites-cohomology-remark-before-Leray}.
\end{proof}

\begin{lemma}
\label{lemma-push-pull-fppf-etale}
Let $S$ be a scheme.
 ========

regex_match $X$
nn candidate $X$
nn candidate an algebraic space
nn an algebraic space
new_annotation [(38517, 38520, 'VAR'), (38524, 38542, 'TYPE')]
===== sent \end{proof}

\begin{lemma}
\label{lemma-compare-cohomology-etale-fppf}
Let $S$ be a scheme and let $X$ be an algebraic space over $S$.
With $a_X : \Sh((\textit{Spaces}/X)_{fppf}) \to \Sh(X_\etale)$
as above:
\begin{enumerate}
\item $H^q(X_\etale, \mathcal{F}) = H^q_{fppf}(X, a_X^{-1}\mathcal{F})$
for an abelian sheaf $\mathcal{F}$ on $X_\etale$,
\item $H^q(X_\etale, K) = H^q_{fppf}(X, a_X^{-1}K)$ for $K \in D^+(X_\etale)$.
\end{enumerate}
Example: if $A$ is an abelian group, then
$H^q_\etale(X, \underline{A}) = H^q_{fppf}(X, \underline{A})$.
\end{lemma}

\begin{proof}
This follows from Lemma \ref{lemma-cohomological-descent-etale-fppf}
by Cohomology on Sites, Remark \ref{sites-cohomology-remark-before-Leray}.
\end{proof}

\begin{lemma}
\label{lemma-push-pull-fppf-etale}
Let $S$ be a scheme.
 ========

regex_match $S$
nn candidate a scheme
nn a scheme
new_annotation [(39203, 39206, 'VAR'), (39210, 39218, 'TYPE')]
===== sent Let $f : X \to Y$ be a morphism of algebraic spaces over $S$.
Then there are commutative diagrams of topoi
$$
\xymatrix{
\Sh((\textit{Spaces}/X)_{fppf}) \ar[rr]_{f_{big, fppf}} \ar[d]_{\epsilon_X} & &
\Sh((\textit{Spaces}/Y)_{fppf}) \ar[d]^{\epsilon_Y} \\
 ========

regex_match $f : X \to Y$
nn candidate X \to
nn candidate a morphism
nn a morphism
new_annotation [(39224, 39237, 'VAR'), (39241, 39251, 'TYPE')]
===== sent \end{proof}

\begin{lemma}
\label{lemma-finite-push-pull-fppf-etale}
In Lemma \ref{lemma-push-pull-fppf-etale} if $f$ is finite, then
$a_Y^{-1}(Rf_{small, *}K) = Rf_{big, fppf, *}(a_X^{-1}K)$
for $K$ in $D^+(X_\etale)$.
\end{lemma}

\begin{proof}
Let $V \to Y$ be a surjective \'etale morphism where $V$ is a scheme.
 ========

regex_match $V \to Y$
nn candidate $V
nn candidate a surjective \'etale morphism
nn a surjective \'etale morphism
new_annotation [(43422, 43431, 'VAR'), (43435, 43464, 'TYPE')]
===== sent Let $f_i : X_i \to Y_i$ be the base change of $f$.
Denote $f_{ij} = f_i \times f_j : X_i \times_X X_j  \to Y_i \times_Y Y_j$.
Claim: if the lemma is true for $f_i$ and $f_{ij}$ for all $i, j$, then
the lemma is true for $f$. To see this, note that the given \'etale covering
determines an \'etale covering of the final object in each of
the four sites $Y_\etale, X_\etale, (\Sch/Y)_{fppf}, (\Sch/X)_{fppf}$.
 ========

regex_match $f_i : X_i \to Y_i$
nn candidate $f_i
nn candidate X_i \to
nn candidate the base change
nn the base change
new_annotation [(45128, 45147, 'VAR'), (45151, 45166, 'TYPE')]
===== sent \medskip\noindent
Let $S$ be a scheme. ========

regex_match $S$
nn candidate a scheme
nn a scheme
new_annotation [(47149, 47152, 'VAR'), (47156, 47164, 'TYPE')]
===== sent Let $X$ be an algebraic space over $S$.
The morphisms of sites $\epsilon_X$, $\pi_X$, and
their composition $a_X$ introduced in Section \ref{section-fppf-etale}
have natural enhancements to morphisms of ringed sites. ========

regex_match $X$
nn candidate an algebraic space
nn an algebraic space
new_annotation [(47170, 47173, 'VAR'), (47177, 47195, 'TYPE')]
===== sent \begin{lemma}
\label{lemma-review-quasi-coherent}
Let $S$ be a scheme. ========

regex_match $S$
nn candidate a scheme
nn a scheme
new_annotation [(48013, 48016, 'VAR'), (48020, 48028, 'TYPE')]
===== sent Let $X$ be an algebraic space over $S$.
Let $\mathcal{F}$ be a quasi-coherent $\mathcal{O}_X$-module.
 ========

regex_match $X$
nn candidate an algebraic space
nn an algebraic space
new_annotation [(48034, 48037, 'VAR'), (48041, 48059, 'TYPE')]
===== sent Let $X$ be an algebraic space over $S$.
Let $\mathcal{F}$ be a quasi-coherent $\mathcal{O}_X$-module.
 ========

regex_match $\mathcal{F}$
nn candidate a quasi-coherent $\mathcal{O}_X$-module
nn a quasi-coherent $\mathcal{O}_X$-module
new_annotation [(48074, 48087, 'VAR'), (48091, 48130, 'TYPE')]
===== sent \end{proof}

\begin{lemma}
\label{lemma-cohomological-descent-etale-fppf-modules}
Let $S$ be a scheme. ========

regex_match $S$
nn candidate a scheme
nn a scheme
new_annotation [(51640, 51643, 'VAR'), (51647, 51655, 'TYPE')]
===== sent Let $X$ be an algebraic space over $S$.
For $\mathcal{F}$ a quasi-coherent $\mathcal{O}_X$-module the maps
$$
\pi_X^*\mathcal{F} \longrightarrow R\epsilon_{X, *}(a_X^*\mathcal{F})
\quad\text{and}\quad
\mathcal{F} \longrightarrow Ra_{X, *}(a_X^*\mathcal{F})
$$
are isomorphisms.
 ========

regex_match $X$
nn candidate an algebraic space
nn an algebraic space
new_annotation [(51661, 51664, 'VAR'), (51668, 51686, 'TYPE')]
===== sent \end{lemma}

\begin{proof}
This is an immediate consequence of parts (6) and (7) of
Lemma \ref{lemma-review-quasi-coherent}.
\end{proof}

\begin{lemma}
\label{lemma-vanishing-adequate}
Let $S$ be a scheme. ========

regex_match $S$
nn candidate a scheme
nn a scheme
new_annotation [(52124, 52127, 'VAR'), (52131, 52139, 'TYPE')]
===== sent Let $X$ be an algebraic space over $S$.
Let $\mathcal{F}_1 \to \mathcal{F}_2 \to \mathcal{F}_3$
be a complex of quasi-coherent $\mathcal{O}_X$-modules.
 ========

regex_match $X$
nn candidate an algebraic space
nn an algebraic space
new_annotation [(52145, 52148, 'VAR'), (52152, 52170, 'TYPE')]
===== sent Let $X$ be an algebraic space over $S$.
Let $\mathcal{F}_1 \to \mathcal{F}_2 \to \mathcal{F}_3$
be a complex of quasi-coherent $\mathcal{O}_X$-modules.
 ========

new_annotation []
===== sent Next, let $g : V \to U$ be a flat
morphism in $(\textit{Spaces}/X)_\etale$. Since
$$
i_{f \circ g}^* \circ \pi_X^* = (f \circ g)^* = g^* \circ f^*
$$
as morphisms of sites $V_\etale \to X_\etale$ and since $g$ is flat
hence $g^*$ is exact, we obtain
$$
\mathcal{H}_\etale|_{V_\etale} =
g^*\left(\mathcal{H}_\etale|_{U_\etale}\right)
$$
With these preparations we are ready to prove the lemma.

\medskip\noindent
Let $\mathcal{U} = \{g_i : U_i ========

regex_match $g : V \to U$
nn candidate V \to
nn candidate a flat
morphism
nn a flat
morphism
new_annotation [(53773, 53786, 'VAR'), (53790, 53805, 'TYPE')]
===== sent Next, let $g : V \to U$ be a flat
morphism in $(\textit{Spaces}/X)_\etale$. Since
$$
i_{f \circ g}^* \circ \pi_X^* = (f \circ g)^* = g^* \circ f^*
$$
as morphisms of sites $V_\etale \to X_\etale$ and since $g$ is flat
hence $g^*$ is exact, we obtain
$$
\mathcal{H}_\etale|_{V_\etale} =
g^*\left(\mathcal{H}_\etale|_{U_\etale}\right)
$$
With these preparations we are ready to prove the lemma.

\medskip\noindent
Let $\mathcal{U} = \{g_i : U_i ========

new_annotation []
===== sent In particular
$\mathcal{H}_\etale = \epsilon_{X, *}\mathcal{H}_{fppf}$.

\medskip\noindent
Finally, to prove the vanishing, we use
Cohomology on Sites, Lemma \ref{sites-cohomology-lemma-cech-vanish-collection}.
We let $\mathcal{B}$ be the affine objects of
$(\textit{Spaces}/X)_{fppf}$ and we let
$\text{Cov}$ be the set of finite fppf coverings
$\mathcal{U} = \{U_i \to U\}_{i = 1, \ldots, n}$ with $U$, $U_i$ affine.
We have
$$
{\check H}^p(\mathcal{U}, \mathcal{H}_\etale) =
{\check H}^p(\mathcal{U}, \left(\mathcal{H}_\etale|_{U_\etale}\right)^a)
$$
because the values of $\mathcal{H}_\etale$ on the
affine schemes $U_{i_0} \times_U \ldots \times_U U_{i_p}$
flat over $U$ agree with the values of the pullback
of the quasi-coherent module $\mathcal{H}_\etale|_{U_\etale}$ by
the first paragraph. ========

regex_match $\mathcal{B}$
nn candidate the affine objects
nn the affine objects
new_annotation [(54838, 54851, 'VAR'), (54855, 54873, 'TYPE')]
===== sent \end{proof}

\begin{lemma}
\label{lemma-cohomological-descent-etale-fppf-modules-unbounded}
Let $S$ be a scheme. ========

regex_match $S$
nn candidate a scheme
nn a scheme
new_annotation [(55644, 55647, 'VAR'), (55651, 55659, 'TYPE')]
===== sent Let $X$ be an algebraic space over $S$.
For $K \in D_\QCoh(\mathcal{O}_X)$ the maps
$$
L\pi_X^*K \longrightarrow R\epsilon_{X, *}(La_X^*\mathcal{F})
\quad\text{and}\quad
K \longrightarrow Ra_{X, *}(La_X^*K)
$$
are isomorphisms. ========

regex_match $X$
nn candidate an algebraic space
nn an algebraic space
new_annotation [(55665, 55668, 'VAR'), (55672, 55690, 'TYPE')]
===== sent You can avoid this step by using more about cohomology of the terms
of the complex $\tau_{\geq -n}a_X^*\mathcal{K}^\bullet$ proved
in Lemma \ref{lemma-vanishing-adequate} as this will prove
directly that $R\epsilon_{X, *}(\tau_{\geq -n}(a_X^*\mathcal{K}^\bullet)) =
\tau_{\geq -n}(\pi_X^*\mathcal{K}^\bullet)$.

\medskip\noindent
The equality $K = Ra_{X, *}(La_X^*\mathcal{F})$ is
proved in exactly the same way using in the final step that
$K = R\lim \tau_{\geq -n}K$ by
Derived Categories of Spaces, Lemma \ref{spaces-perfect-lemma-nice-K-injective}.
\end{proof}








\section{Comparing ph and \'etale topologies}
\label{section-ph-etale}

\noindent
This section is the analogue of
\'Etale Cohomology, Section \ref{etale-cohomology-section-ph-etale}.

\medskip\noindent
Let $S$ be a scheme. ========

regex_match $S$
nn candidate a scheme
nn a scheme
new_annotation [(60816, 60819, 'VAR'), (60823, 60831, 'TYPE')]
===== sent Let $X$ be an algebraic space over $S$.
On the category $\textit{Spaces}/X$ ========

regex_match $X$
nn candidate an algebraic space
nn an algebraic space
new_annotation [(60837, 60840, 'VAR'), (60844, 60862, 'TYPE')]
===== sent Let $S$ be a scheme. ========

regex_match $S$
nn candidate a scheme
nn a scheme
new_annotation [(62133, 62136, 'VAR'), (62140, 62148, 'TYPE')]
===== sent Let $X$ be an algebraic space over $S$.
\begin{enumerate}
\item ========

regex_match $X$
nn candidate an algebraic space
nn an algebraic space
new_annotation [(62154, 62157, 'VAR'), (62161, 62179, 'TYPE')]
===== sent \begin{lemma}
\label{lemma-cohomological-descent-etale-ph}
Let $S$ be a scheme. ========

regex_match $S$
nn candidate a scheme
nn a scheme
new_annotation [(65397, 65400, 'VAR'), (65404, 65412, 'TYPE')]
===== sent Let $X$ be an algebraic space over $S$.
For $K \in D^+(X_\etale)$ with torsion cohomology sheaves the maps
$$
\pi_X^{-1}K \longrightarrow R\epsilon_{X, *}a_X^{-1}K
\quad\text{and}\quad
K \longrightarrow Ra_{X, *}a_X^{-1}K
$$
are isomorphisms with
$a_X : \Sh((\textit{Spaces}/X)_{ph}) \to \Sh(X_\etale)$ as above.
 ========

regex_match $X$
nn candidate an algebraic space
nn an algebraic space
new_annotation [(65418, 65421, 'VAR'), (65425, 65443, 'TYPE')]
===== sent \end{proof}

\begin{lemma}
\label{lemma-compare-cohomology-etale-ph}
Let $S$ be a scheme and let $X$ be an algebraic space over $S$.
With $a_X : \Sh((\textit{Spaces}/X)_{ph}) \to \Sh(X_\etale)$
as above:
\begin{enumerate}
\item $H^q(X_\etale, \mathcal{F}) = H^q_{ph}(X, a_X^{-1}\mathcal{F})$
for a torsion abelian sheaf $\mathcal{F}$ on $X_\etale$,
\item $H^q(X_\etale, K) = H^q_{ph}(X, a_X^{-1}K)$ for $K \in D^+(X_\etale)$
with torsion cohomology sheaves
\end{enumerate}
Example: if $A$ is a torsion abelian group, then
$H^q_\etale(X, \underline{A}) = H^q_{ph}(X, \underline{A})$.
\end{lemma}

\begin{proof}
This follows from Lemma \ref{lemma-cohomological-descent-etale-ph}
by Cohomology on Sites, Remark \ref{sites-cohomology-remark-before-Leray}.
\end{proof}

\begin{lemma}
\label{lemma-push-pull-ph-etale}
Let $S$ be a scheme.
 ========

regex_match $S$
nn candidate a scheme
nn a scheme
new_annotation [(67285, 67288, 'VAR'), (67292, 67300, 'TYPE')]
===== sent \end{proof}

\begin{lemma}
\label{lemma-compare-cohomology-etale-ph}
Let $S$ be a scheme and let $X$ be an algebraic space over $S$.
With $a_X : \Sh((\textit{Spaces}/X)_{ph}) \to \Sh(X_\etale)$
as above:
\begin{enumerate}
\item $H^q(X_\etale, \mathcal{F}) = H^q_{ph}(X, a_X^{-1}\mathcal{F})$
for a torsion abelian sheaf $\mathcal{F}$ on $X_\etale$,
\item $H^q(X_\etale, K) = H^q_{ph}(X, a_X^{-1}K)$ for $K \in D^+(X_\etale)$
with torsion cohomology sheaves
\end{enumerate}
Example: if $A$ is a torsion abelian group, then
$H^q_\etale(X, \underline{A}) = H^q_{ph}(X, \underline{A})$.
\end{lemma}

\begin{proof}
This follows from Lemma \ref{lemma-cohomological-descent-etale-ph}
by Cohomology on Sites, Remark \ref{sites-cohomology-remark-before-Leray}.
\end{proof}

\begin{lemma}
\label{lemma-push-pull-ph-etale}
Let $S$ be a scheme.
 ========

regex_match $X$
nn candidate $X$
nn candidate an algebraic space
nn an algebraic space
new_annotation [(67309, 67312, 'VAR'), (67316, 67334, 'TYPE')]
===== sent \end{proof}

\begin{lemma}
\label{lemma-compare-cohomology-etale-ph}
Let $S$ be a scheme and let $X$ be an algebraic space over $S$.
With $a_X : \Sh((\textit{Spaces}/X)_{ph}) \to \Sh(X_\etale)$
as above:
\begin{enumerate}
\item $H^q(X_\etale, \mathcal{F}) = H^q_{ph}(X, a_X^{-1}\mathcal{F})$
for a torsion abelian sheaf $\mathcal{F}$ on $X_\etale$,
\item $H^q(X_\etale, K) = H^q_{ph}(X, a_X^{-1}K)$ for $K \in D^+(X_\etale)$
with torsion cohomology sheaves
\end{enumerate}
Example: if $A$ is a torsion abelian group, then
$H^q_\etale(X, \underline{A}) = H^q_{ph}(X, \underline{A})$.
\end{lemma}

\begin{proof}
This follows from Lemma \ref{lemma-cohomological-descent-etale-ph}
by Cohomology on Sites, Remark \ref{sites-cohomology-remark-before-Leray}.
\end{proof}

\begin{lemma}
\label{lemma-push-pull-ph-etale}
Let $S$ be a scheme.
 ========

regex_match $S$
nn candidate a scheme
nn a scheme
new_annotation [(68028, 68031, 'VAR'), (68035, 68043, 'TYPE')]
===== sent Let $f : X \to Y$ be a morphism of algebraic spaces over $S$.
Then there are commutative diagrams of topoi
$$
\xymatrix{
\Sh((\textit{Spaces}/X)_{ph}) \ar[rr]_{f_{big, ph}} \ar[d]_{\epsilon_X} & &
\Sh((\textit{Spaces}/Y)_{ph}) \ar[d]^{\epsilon_Y} \\
 ========

regex_match $f : X \to Y$
nn candidate X \to
nn candidate a morphism
nn a morphism
new_annotation [(68049, 68062, 'VAR'), (68066, 68076, 'TYPE')]
file:  divisors.tex
===== sent A basic reference is \cite{EGA}.



\section{Associated points}
\label{section-associated}

\noindent
Let $R$ be a ring and let $M$ be an $R$-module.
 ========

regex_match $R$
nn candidate a ring
nn a ring
new_annotation [(401, 404, 'VAR'), (408, 414, 'TYPE')]
===== sent A basic reference is \cite{EGA}.



\section{Associated points}
\label{section-associated}

\noindent
Let $R$ be a ring and let $M$ be an $R$-module.
 ========

regex_match $M$
nn candidate $M$
nn candidate an $R$-module
nn an $R$-module
new_annotation [(423, 426, 'VAR'), (430, 443, 'TYPE')]
===== sent Here is the definition of associated points
for quasi-coherent sheaves on schemes
as given in \cite[IV Definition 3.1.1]{EGA}.

\begin{definition}
\label{definition-associated}
Let $X$ be a scheme.
 ========

regex_match $X$
nn candidate a scheme
nn a scheme
new_annotation [(828, 831, 'VAR'), (835, 843, 'TYPE')]
===== sent Let $\mathcal{F}$ be a quasi-coherent sheaf on $X$.
\begin{enumerate}
\item We say $x \in X$ is {\it associated} to $\mathcal{F}$
if the maximal ideal
$\mathfrak m_x$ is associated to the $\mathcal{O}_{X, x}$-module
$\mathcal{F}_x$.
\item We denote $\text{Ass}(\mathcal{F})$ or $\text{Ass}_X(\mathcal{F})$
the set of associated points of $\mathcal{F}$.
\item ========

regex_match $\mathcal{F}$
nn candidate a quasi-coherent sheaf
nn a quasi-coherent sheaf
new_annotation [(849, 862, 'VAR'), (866, 888, 'TYPE')]
===== sent \begin{lemma}
\label{lemma-associated-affine-open}
Let $X$ be a scheme. ========

regex_match $X$
nn candidate a scheme
nn a scheme
new_annotation [(1936, 1939, 'VAR'), (1943, 1951, 'TYPE')]
===== sent Let $\mathcal{F}$ be a quasi-coherent sheaf on $X$.
Let $\Spec(A) = U \subset X$ be an affine open, and set
$M = \Gamma(U, \mathcal{F})$.
Let $x \in U$, and let $\mathfrak p \subset A$ be the corresponding prime.
 ========

regex_match $\mathcal{F}$
nn candidate a quasi-coherent sheaf
nn a quasi-coherent sheaf
new_annotation [(1957, 1970, 'VAR'), (1974, 1996, 'TYPE')]
===== sent Let $\mathcal{F}$ be a quasi-coherent sheaf on $X$.
Let $\Spec(A) = U \subset X$ be an affine open, and set
$M = \Gamma(U, \mathcal{F})$.
Let $x \in U$, and let $\mathfrak p \subset A$ be the corresponding prime.
 ========

regex_match $\Spec(A) = U \subset X$
nn candidate = U
nn candidate an affine
nn an affine
new_annotation [(2009, 2033, 'VAR'), (2037, 2046, 'TYPE')]
===== sent Let $\mathcal{F}$ be a quasi-coherent sheaf on $X$.
Let $\Spec(A) = U \subset X$ be an affine open, and set
$M = \Gamma(U, \mathcal{F})$.
Let $x \in U$, and let $\mathfrak p \subset A$ be the corresponding prime.
 ========

new_annotation []
===== sent Let $\mathcal{F}$ be a quasi-coherent sheaf on $X$.
Let $\Spec(A) = U \subset X$ be an affine open, and set
$M = \Gamma(U, \mathcal{F})$.
Let $x \in U$, and let $\mathfrak p \subset A$ be the corresponding prime.
 ========

regex_match $\mathfrak p \subset A$
nn candidate p \subset
nn candidate the corresponding prime
nn the corresponding prime
new_annotation [(2114, 2137, 'VAR'), (2141, 2164, 'TYPE')]
===== sent \end{proof}

\begin{lemma}
\label{lemma-ass-support}
Let $X$ be a scheme.
 ========

regex_match $X$
nn candidate a scheme
nn a scheme
new_annotation [(4009, 4012, 'VAR'), (4016, 4024, 'TYPE')]
===== sent Let $\mathcal{F}$ be a quasi-coherent $\mathcal{O}_X$-module.
 ========

regex_match $\mathcal{F}$
nn candidate a quasi-coherent $\mathcal{O}_X$-module
nn a quasi-coherent $\mathcal{O}_X$-module
new_annotation [(4030, 4043, 'VAR'), (4047, 4086, 'TYPE')]
===== sent \end{proof}

\begin{lemma}
\label{lemma-ses-ass}
Let $X$ be a scheme.
 ========

regex_match $X$
nn candidate a scheme
nn a scheme
new_annotation [(4273, 4276, 'VAR'), (4280, 4288, 'TYPE')]
===== sent Let $0 \to \mathcal{F}_1 \to \mathcal{F}_2 \to \mathcal{F}_3 \to 0$
be a short exact sequence of quasi-coherent sheaves on $X$.
Then
$\text{Ass}(\mathcal{F}_2) \subset
\text{Ass}(\mathcal{F}_1) \cup \text{Ass}(\mathcal{F}_3)$
and
$\text{Ass}(\mathcal{F}_1) \subset \text{Ass}(\mathcal{F}_2)$.
\end{lemma}

\begin{proof}
For every point $x \in X$ the sequence of stalks
$0 \to \mathcal{F}_{1, x} \to \mathcal{F}_{2, x} \to \mathcal{F}_{3, x} \to 0$
is a short exact sequence of $\mathcal{O}_{X, x}$-modules.
 ========

new_annotation []
===== sent Hence the lemma follows from
Algebra, Lemma \ref{algebra-lemma-ass}.
\end{proof}

\begin{lemma}
\label{lemma-finite-ass}
Let $X$ be a locally Noetherian scheme.
 ========

regex_match $X$
nn candidate a locally Noetherian scheme
nn a locally Noetherian scheme
new_annotation [(4922, 4925, 'VAR'), (4929, 4956, 'TYPE')]
===== sent Let $\mathcal{F}$ be a coherent $\mathcal{O}_X$-module.
 ========

regex_match $\mathcal{F}$
nn candidate a coherent $\mathcal{O}_X$-module
nn a coherent $\mathcal{O}_X$-module
new_annotation [(4962, 4975, 'VAR'), (4979, 5012, 'TYPE')]
===== sent X$.
\end{lemma}

\begin{proof}
This is true because the set of associated primes of a finite module over
a Noetherian ring is finite, see
Algebra, Lemma \ref{algebra-lemma-finite-ass}.
To translate from schemes to algebra use that $U$ is a finite union of
affine opens, each of these opens is the spectrum of a Noetherian ring
(Properties, Lemma \ref{properties-lemma-locally-Noetherian}),
$\mathcal{F}$ corresponds to a finite module over this ring
(Cohomology of Schemes, Lemma \ref{coherent-lemma-coherent-Noetherian}),
and finally use
Lemma \ref{lemma-associated-affine-open}.
\end{proof}

\begin{lemma}
\label{lemma-ass-zero}
Let $X$ be a locally Noetherian scheme. ========

regex_match $X$
nn candidate a locally Noetherian scheme
nn a locally Noetherian scheme
new_annotation [(5737, 5740, 'VAR'), (5744, 5771, 'TYPE')]
===== sent Let $\mathcal{F}$ be a
quasi-coherent $\mathcal{O}_X$-module. ========

new_annotation []
===== sent Conversely, if $\text{Ass}(\mathcal{F}) = \emptyset$,
then $\mathcal{F} = 0$ by
Algebra, Lemma \ref{algebra-lemma-ass-zero}.
To translate from schemes to algebra, restrict to any affine and use
Lemma \ref{lemma-associated-affine-open}.
\end{proof}

\begin{example}
\label{example-no-associated-prime}
Let $k$ be a field. The ring $R = k[x_1, x_2, x_3, \ldots]/(x_i^2)$
is local with locally nilpotent maximal ideal $\mathfrak m$.
There exists no element of $R$ which has annihilator $\mathfrak ========

regex_match $k$
nn candidate a field
nn a field
new_annotation [(6327, 6330, 'VAR'), (6334, 6341, 'TYPE')]
===== sent \end{example}

\begin{lemma}
\label{lemma-restriction-injective-open-contains-ass}
Let $X$ be a locally Noetherian scheme. ========

regex_match $X$
nn candidate a locally Noetherian scheme
nn a locally Noetherian scheme
new_annotation [(6719, 6722, 'VAR'), (6726, 6753, 'TYPE')]
===== sent Let $\mathcal{F}$ be a quasi-coherent
$\mathcal{O}_X$-module. ========

regex_match $\mathcal{F}$
new_annotation []
===== sent \end{lemma}

\begin{proof}
Let $s \in \Gamma(X, \mathcal{F})$ be a section which restricts to zero on $U$.
Let $\mathcal{F}' \subset \mathcal{F}$ be the image of the map
$\mathcal{O}_X \to \mathcal{F}$ defined by $s$. Then
$\text{Supp}(\mathcal{F}') \cap U ========

regex_match $s \in \Gamma(X, \mathcal{F})$
nn candidate a section
nn a section
new_annotation [(6977, 7007, 'VAR'), (7011, 7020, 'TYPE')]
===== sent \end{lemma}

\begin{proof}
Let $s \in \Gamma(X, \mathcal{F})$ be a section which restricts to zero on $U$.
Let $\mathcal{F}' \subset \mathcal{F}$ be the image of the map
$\mathcal{O}_X \to \mathcal{F}$ defined by $s$. Then
$\text{Supp}(\mathcal{F}') \cap U ========

regex_match $\mathcal{F}' \subset \mathcal{F}$
nn candidate the image
nn the image
new_annotation [(7057, 7091, 'VAR'), (7095, 7104, 'TYPE')]
===== sent Hence $\mathcal{F}' = 0$ by Lemma \ref{lemma-ass-zero}.
\end{proof}

\begin{lemma}
\label{lemma-minimal-support-in-ass}
Let $X$ be a locally Noetherian scheme.
 ========

regex_match $X$
nn candidate a locally Noetherian scheme
nn a locally Noetherian scheme
new_annotation [(7605, 7608, 'VAR'), (7612, 7639, 'TYPE')]
===== sent Let $\mathcal{F}$ be a quasi-coherent $\mathcal{O}_X$-module.
 ========

regex_match $\mathcal{F}$
nn candidate a quasi-coherent $\mathcal{O}_X$-module
nn a quasi-coherent $\mathcal{O}_X$-module
new_annotation [(7645, 7658, 'VAR'), (7662, 7701, 'TYPE')]
===== sent Let $x \in \text{Supp}(\mathcal{F})$ be a point in the support
of $\mathcal{F}$ which is not a specialization of another point of
$\text{Supp}(\mathcal{F})$. Then $x \in ========

regex_match $x \in \text{Supp}(\mathcal{F})$
nn candidate a point
nn a point
new_annotation [(7707, 7739, 'VAR'), (7743, 7750, 'TYPE')]
===== sent \end{proof}

\noindent
The following lemma is the analogue of
More on Algebra, Lemma \ref{more-algebra-lemma-check-injective-on-ass}.

\begin{lemma}
\label{lemma-check-injective-on-ass}
Let $X$ be a locally Noetherian scheme. ========

regex_match $X$
nn candidate a locally Noetherian scheme
nn a locally Noetherian scheme
new_annotation [(8696, 8699, 'VAR'), (8703, 8730, 'TYPE')]
===== sent The assumptions imply that $\text{Ass}(\Ker(\varphi)) = \emptyset$
and hence $\Ker(\varphi) = 0$ by Lemma \ref{lemma-ass-zero}.
\end{proof}

\begin{lemma}
\label{lemma-check-isomorphism-via-depth-and-ass}
Let $X$ be a locally Noetherian scheme. ========

regex_match $X$
nn candidate a locally Noetherian scheme
nn a locally Noetherian scheme
new_annotation [(9298, 9301, 'VAR'), (9305, 9332, 'TYPE')]
===== sent \end{proof}





\section{Morphisms and associated points}
\label{section-morphisms-associated}

\begin{lemma}
\label{lemma-bourbaki}
Let $f : X \to S$ be a morphism of schemes.
 ========

regex_match $f : X \to S$
nn candidate X \to
nn candidate a morphism
nn a morphism
new_annotation [(10040, 10053, 'VAR'), (10057, 10067, 'TYPE')]
===== sent Let $\mathcal{F}$ be a quasi-coherent sheaf on $X$ which is flat over $S$.
Let $\mathcal{G}$ be a quasi-coherent sheaf on $S$.
 ========

regex_match $\mathcal{F}$
nn candidate a quasi-coherent sheaf
nn a quasi-coherent sheaf
new_annotation [(10084, 10097, 'VAR'), (10101, 10123, 'TYPE')]
===== sent Let $\mathcal{F}$ be a quasi-coherent sheaf on $X$ which is flat over $S$.
Let $\mathcal{G}$ be a quasi-coherent sheaf on $S$.
 ========

regex_match $\mathcal{G}$
nn candidate a quasi-coherent sheaf
nn a quasi-coherent sheaf
new_annotation [(10159, 10172, 'VAR'), (10176, 10198, 'TYPE')]
===== sent \end{lemma}

\begin{proof}
Let $x \in X$ and let $s = f(x) \in S$.
Set $B = \mathcal{O}_{X, x}$, $A = \mathcal{O}_{S, s}$,
$N = \mathcal{F}_x$, and $M = \mathcal{G}_s$.
Note that the stalk of $\mathcal{F} \otimes_{\mathcal{O}_X} f^*\mathcal{G}$
at $x$ is equal to the $B$-module $M \otimes_A N$. Hence
$x \in \text{Ass}_X(\mathcal{F} \otimes_{\mathcal{O}_X} f^*\mathcal{G})$
if and only if $\mathfrak m_B$ is in $\text{Ass}_B(M \otimes_A N)$.
Similarly $s \in \text{Ass}_S(\mathcal{G})$ and
$x \in \text{Ass}_{X_s}(\mathcal{F}_s)$ if and only if
$\mathfrak m_A \in \text{Ass}_A(M)$ and
$\mathfrak m_B/\mathfrak m_A B \in
\text{Ass}_{B \otimes \kappa(\mathfrak m_A)}(N \otimes \kappa(\mathfrak m_A))$.
 ========

new_annotation []
===== sent \end{lemma}

\begin{proof}
Let $x \in X$ and let $s = f(x) \in S$.
Set $B = \mathcal{O}_{X, x}$, $A = \mathcal{O}_{S, s}$,
$N = \mathcal{F}_x$, and $M = \mathcal{G}_s$.
Note that the stalk of $\mathcal{F} \otimes_{\mathcal{O}_X} f^*\mathcal{G}$
at $x$ is equal to the $B$-module $M \otimes_A N$. Hence
$x \in \text{Ass}_X(\mathcal{F} \otimes_{\mathcal{O}_X} f^*\mathcal{G})$
if and only if $\mathfrak m_B$ is in $\text{Ass}_B(M \otimes_A N)$.
Similarly $s \in \text{Ass}_S(\mathcal{G})$ and
$x \in \text{Ass}_{X_s}(\mathcal{F}_s)$ if and only if
$\mathfrak m_A \in \text{Ass}_A(M)$ and
$\mathfrak m_B/\mathfrak m_A B \in
\text{Ass}_{B \otimes \kappa(\mathfrak m_A)}(N \otimes \kappa(\mathfrak m_A))$.
 ========

new_annotation []
===== sent \noindent
Let $R$ be a ring and let $M$ be an $R$-module.
 ========

regex_match $R$
nn candidate a ring
nn a ring
new_annotation [(11295, 11298, 'VAR'), (11302, 11308, 'TYPE')]
===== sent \noindent
Let $R$ be a ring and let $M$ be an $R$-module.
 ========

regex_match $M$
nn candidate $M$
nn candidate an $R$-module
nn an $R$-module
new_annotation [(11317, 11320, 'VAR'), (11324, 11337, 'TYPE')]
===== sent Here is the definition of embedded associated points
for quasi-coherent sheaves on schemes
as given in \cite[IV Definition 3.1.1]{EGA}.

\begin{definition}
\label{definition-embedded}
Let $X$ be a scheme.
 ========

regex_match $X$
nn candidate a scheme
nn a scheme
new_annotation [(11775, 11778, 'VAR'), (11782, 11790, 'TYPE')]
===== sent Let $\mathcal{F}$ be a quasi-coherent sheaf on $X$.
\begin{enumerate}
\item An {\it embedded associated point} of $\mathcal{F}$
is an associated point which is not maximal among the
associated points of $\mathcal{F}$, i.e., it is the specialization
of another associated point of $\mathcal{F}$.
\item A point $x$ of $X$ is called an {\it embedded point}
if $x$ is an embedded associated point of $\mathcal{O}_X$.
\item ========

regex_match $\mathcal{F}$
nn candidate a quasi-coherent sheaf
nn a quasi-coherent sheaf
new_annotation [(11796, 11809, 'VAR'), (11813, 11835, 'TYPE')]
===== sent \begin{lemma}
\label{lemma-embedded}
Let $X$ be a locally Noetherian scheme.
 ========

regex_match $X$
nn candidate a locally Noetherian scheme
nn a locally Noetherian scheme
new_annotation [(12503, 12506, 'VAR'), (12510, 12537, 'TYPE')]
===== sent Let $\mathcal{F}$ be a coherent $\mathcal{O}_X$-module.
 ========

regex_match $\mathcal{F}$
nn candidate a coherent $\mathcal{O}_X$-module
nn a coherent $\mathcal{O}_X$-module
new_annotation [(12543, 12556, 'VAR'), (12560, 12593, 'TYPE')]
===== sent \end{proof}

\begin{lemma}
\label{lemma-S1-no-embedded}
Let $X$ be a locally Noetherian scheme.
 ========

regex_match $X$
nn candidate a locally Noetherian scheme
nn a locally Noetherian scheme
new_annotation [(13671, 13674, 'VAR'), (13678, 13705, 'TYPE')]
===== sent Let $\mathcal{F}$ be a coherent sheaf on $X$.
 ========

regex_match $\mathcal{F}$
nn candidate a coherent sheaf
nn a coherent sheaf
new_annotation [(13711, 13724, 'VAR'), (13728, 13744, 'TYPE')]
===== sent \end{proof}

\begin{lemma}
\label{lemma-noetherian-dim-1-CM-no-embedded-points}
Let $X$ be a locally Noetherian scheme of dimension $\leq 1$.
The following are equivalent
\begin{enumerate}
\item $X$ is Cohen-Macaulay, and
\item $X$ has no embedded points.
 ========

regex_match $X$
nn candidate a locally Noetherian scheme
nn a locally Noetherian scheme
new_annotation [(14169, 14172, 'VAR'), (14176, 14203, 'TYPE')]
===== sent \end{proof}

\begin{lemma}
\label{lemma-scheme-theoretically-dense-contain-embedded-points}
Let $X$ be a locally Noetherian scheme. ========

regex_match $X$
nn candidate a locally Noetherian scheme
nn a locally Noetherian scheme
new_annotation [(14547, 14550, 'VAR'), (14554, 14581, 'TYPE')]
===== sent Let $U \subset X$ be an
open subscheme. ========

new_annotation []
===== sent Assume (2), i.e., every associated prime $\mathfrak p$ of $A$
corresponds to a prime of $A_{f_i}$ for some $i$. Then
$A \to A_{f_1} \times \ldots \times A_{f_n}$ is injective because
$A \to \prod_{\mathfrak p \in \text{Ass}(A)} A_\mathfrak p$ is injective
by Algebra, Lemma \ref{algebra-lemma-zero-at-ass-zero}.
\end{proof}

\begin{lemma}
\label{lemma-remove-embedded-points}
Let $X$ be a locally Noetherian scheme.
Let $\mathcal{F}$ be a coherent sheaf on $X$.
The set of coherent subsheaves
$$
\{
\mathcal{K} \subset \mathcal{F}
\mid
 ========

regex_match $X$
nn candidate a locally Noetherian scheme
nn a locally Noetherian scheme
new_annotation [(16195, 16198, 'VAR'), (16202, 16229, 'TYPE')]
===== sent Assume (2), i.e., every associated prime $\mathfrak p$ of $A$
corresponds to a prime of $A_{f_i}$ for some $i$. Then
$A \to A_{f_1} \times \ldots \times A_{f_n}$ is injective because
$A \to \prod_{\mathfrak p \in \text{Ass}(A)} A_\mathfrak p$ is injective
by Algebra, Lemma \ref{algebra-lemma-zero-at-ass-zero}.
\end{proof}

\begin{lemma}
\label{lemma-remove-embedded-points}
Let $X$ be a locally Noetherian scheme.
Let $\mathcal{F}$ be a coherent sheaf on $X$.
The set of coherent subsheaves
$$
\{
\mathcal{K} \subset \mathcal{F}
\mid
 ========

regex_match $\mathcal{F}$
nn candidate a coherent sheaf
nn a coherent sheaf
new_annotation [(16235, 16248, 'VAR'), (16252, 16268, 'TYPE')]
===== sent This follows from
Algebra, Lemmas \ref{algebra-lemma-remove-embedded-primes} and
\ref{algebra-lemma-remove-embedded-primes-localize}.
Note that $U$ can be taken as the complement of the closure
of the set of embedded associated points of $\mathcal{F}$.
\end{proof}

\begin{lemma}
\label{lemma-no-embedded-points-endos}
Let $X$ be a locally Noetherian scheme.
 ========

regex_match $X$
nn candidate a locally Noetherian scheme
nn a locally Noetherian scheme
new_annotation [(17221, 17224, 'VAR'), (17228, 17255, 'TYPE')]
===== sent Let $\mathcal{F}$ be a coherent $\mathcal{O}_X$-module
without embedded associated points. ========

regex_match $\mathcal{F}$
nn candidate embedded associated points
nn embedded associated points
new_annotation [(17261, 17274, 'VAR'), (17320, 17346, 'TYPE')]
===== sent \begin{proof}
Some of the statements we have seen in the proof of
Cohomology of Schemes, Lemma \ref{coherent-lemma-coherent-support-closed}.
The others follow from
Algebra, Lemma \ref{algebra-lemma-no-embedded-primes-endos}.
\end{proof}



\section{Weakly associated points}
\label{section-weakly-associated}

\noindent
Let $R$ be a ring and let $M$ be an $R$-module.
 ========

regex_match $R$
nn candidate a ring
nn a ring
new_annotation [(18122, 18125, 'VAR'), (18129, 18135, 'TYPE')]
===== sent \begin{proof}
Some of the statements we have seen in the proof of
Cohomology of Schemes, Lemma \ref{coherent-lemma-coherent-support-closed}.
The others follow from
Algebra, Lemma \ref{algebra-lemma-no-embedded-primes-endos}.
\end{proof}



\section{Weakly associated points}
\label{section-weakly-associated}

\noindent
Let $R$ be a ring and let $M$ be an $R$-module.
 ========

regex_match $M$
nn candidate $M$
nn candidate an $R$-module
nn an $R$-module
new_annotation [(18144, 18147, 'VAR'), (18151, 18164, 'TYPE')]
===== sent Algebra, Lemma \ref{algebra-lemma-weakly-ass-local}.

\begin{definition}
\label{definition-weakly-associated}
Let $X$ be a scheme.
 ========

regex_match $X$
nn candidate a scheme
nn a scheme
new_annotation [(18746, 18749, 'VAR'), (18753, 18761, 'TYPE')]
===== sent Let $\mathcal{F}$ be a quasi-coherent sheaf on $X$.
\begin{enumerate}
\item We say $x \in X$ is {\it weakly associated} to $\mathcal{F}$
if the maximal ideal $\mathfrak m_x$ is weakly associated to the
$\mathcal{O}_{X, x}$-module $\mathcal{F}_x$.
\item We denote $\text{WeakAss}(\mathcal{F})$ the set of weakly associated
points of $\mathcal{F}$.
\item The {\it weakly associated points of $X$} are the weakly associated
points of $\mathcal{O}_X$.
\end{enumerate}
\end{definition}

\noindent
 ========

regex_match $\mathcal{F}$
nn candidate a quasi-coherent sheaf
nn a quasi-coherent sheaf
new_annotation [(18767, 18780, 'VAR'), (18784, 18806, 'TYPE')]
===== sent \begin{lemma}
\label{lemma-weakly-associated-affine-open}
Let $X$ be a scheme. ========

regex_match $X$
nn candidate a scheme
nn a scheme
new_annotation [(19458, 19461, 'VAR'), (19465, 19473, 'TYPE')]
===== sent Let $\mathcal{F}$ be a quasi-coherent sheaf on $X$.
Let $\Spec(A) = U \subset X$ be an affine open, and set
$M = \Gamma(U, \mathcal{F})$.
Let $x \in U$, and let $\mathfrak p \subset A$ be the corresponding prime.
 ========

regex_match $\mathcal{F}$
nn candidate a quasi-coherent sheaf
nn a quasi-coherent sheaf
new_annotation [(19479, 19492, 'VAR'), (19496, 19518, 'TYPE')]
===== sent Let $\mathcal{F}$ be a quasi-coherent sheaf on $X$.
Let $\Spec(A) = U \subset X$ be an affine open, and set
$M = \Gamma(U, \mathcal{F})$.
Let $x \in U$, and let $\mathfrak p \subset A$ be the corresponding prime.
 ========

regex_match $\Spec(A) = U \subset X$
nn candidate U
nn candidate an affine
nn an affine
new_annotation [(19531, 19555, 'VAR'), (19559, 19568, 'TYPE')]
===== sent Let $\mathcal{F}$ be a quasi-coherent sheaf on $X$.
Let $\Spec(A) = U \subset X$ be an affine open, and set
$M = \Gamma(U, \mathcal{F})$.
Let $x \in U$, and let $\mathfrak p \subset A$ be the corresponding prime.
 ========

new_annotation []
===== sent Let $\mathcal{F}$ be a quasi-coherent sheaf on $X$.
Let $\Spec(A) = U \subset X$ be an affine open, and set
$M = \Gamma(U, \mathcal{F})$.
Let $x \in U$, and let $\mathfrak p \subset A$ be the corresponding prime.
 ========

regex_match $\mathfrak p \subset A$
nn candidate p \subset
nn candidate the corresponding prime
nn the corresponding prime
new_annotation [(19636, 19659, 'VAR'), (19663, 19686, 'TYPE')]
===== sent is weakly associated to $M$, and
\item $x$ is weakly associated to $\mathcal{F}$.
\end{enumerate}
\end{lemma}

\begin{proof}
This follows from
Algebra, Lemma \ref{algebra-lemma-weakly-ass-local}.
\end{proof}

\begin{lemma}
\label{lemma-weakly-ass-support}
Let $X$ be a scheme.
 ========

regex_match $X$
nn candidate a scheme
nn a scheme
new_annotation [(20015, 20018, 'VAR'), (20022, 20030, 'TYPE')]
===== sent Let $\mathcal{F}$ be a quasi-coherent $\mathcal{O}_X$-module.
 ========

regex_match $\mathcal{F}$
nn candidate a quasi-coherent $\mathcal{O}_X$-module
nn a quasi-coherent $\mathcal{O}_X$-module
new_annotation [(20036, 20049, 'VAR'), (20053, 20092, 'TYPE')]
===== sent \end{proof}

\begin{lemma}
\label{lemma-ses-weakly-ass}
Let $X$ be a scheme.
 ========

regex_match $X$
nn candidate a scheme
nn a scheme
new_annotation [(20326, 20329, 'VAR'), (20333, 20341, 'TYPE')]
===== sent Let $0 \to \mathcal{F}_1 \to \mathcal{F}_2 \to \mathcal{F}_3 \to 0$
be a short exact sequence of quasi-coherent sheaves on $X$.
Then
$\text{WeakAss}(\mathcal{F}_2) \subset
\text{WeakAss}(\mathcal{F}_1) \cup \text{WeakAss}(\mathcal{F}_3)$
and
$\text{WeakAss}(\mathcal{F}_1) \subset \text{WeakAss}(\mathcal{F}_2)$.
\end{lemma}

\begin{proof}
For every point $x \in X$ the sequence of stalks
$0 \to \mathcal{F}_{1, x} \to \mathcal{F}_{2, x} \to \mathcal{F}_{3, x} \to 0 ========

new_annotation []
===== sent Hence the lemma follows from
Algebra, Lemma \ref{algebra-lemma-weakly-ass}.
\end{proof}

\begin{lemma}
\label{lemma-weakly-ass-zero}
Let $X$ be a scheme.
 ========

regex_match $X$
nn candidate a scheme
nn a scheme
new_annotation [(21007, 21010, 'VAR'), (21014, 21022, 'TYPE')]
===== sent Let $\mathcal{F}$ be a quasi-coherent $\mathcal{O}_X$-module.
 ========

regex_match $\mathcal{F}$
nn candidate a quasi-coherent $\mathcal{O}_X$-module
nn a quasi-coherent $\mathcal{O}_X$-module
new_annotation [(21028, 21041, 'VAR'), (21045, 21084, 'TYPE')]
===== sent Then
$$
\mathcal{F} = (0) \Leftrightarrow \text{WeakAss}(\mathcal{F}) = \emptyset
$$
\end{lemma}

\begin{proof}
Follows from
Lemma \ref{lemma-weakly-associated-affine-open}
and
Algebra, Lemma \ref{algebra-lemma-weakly-ass-zero}
\end{proof}

\begin{lemma}
\label{lemma-restriction-injective-open-contains-weakly-ass}
Let $X$ be a scheme. ========

regex_match $X$
nn candidate a scheme
nn a scheme
new_annotation [(21406, 21409, 'VAR'), (21413, 21421, 'TYPE')]
===== sent Let $\mathcal{F}$ be a quasi-coherent
$\mathcal{O}_X$-module. ========

regex_match $\mathcal{F}$
new_annotation []
===== sent \end{lemma}

\begin{proof}
Let $s \in \Gamma(X, \mathcal{F})$ be a section which restricts to zero on $U$.
Let $\mathcal{F}' \subset \mathcal{F}$ be the image of the map
$\mathcal{O}_X \to \mathcal{F}$ defined by $s$. Then
$\text{Supp}(\mathcal{F}') \cap U ========

regex_match $s \in \Gamma(X, \mathcal{F})$
nn candidate a section
nn a section
new_annotation [(21649, 21679, 'VAR'), (21683, 21692, 'TYPE')]
===== sent \end{lemma}

\begin{proof}
Let $s \in \Gamma(X, \mathcal{F})$ be a section which restricts to zero on $U$.
Let $\mathcal{F}' \subset \mathcal{F}$ be the image of the map
$\mathcal{O}_X \to \mathcal{F}$ defined by $s$. Then
$\text{Supp}(\mathcal{F}') \cap U ========

regex_match $\mathcal{F}' \subset \mathcal{F}$
nn candidate the image
nn the image
new_annotation [(21729, 21763, 'VAR'), (21767, 21776, 'TYPE')]
===== sent Hence $\mathcal{F}' = 0$ by Lemma \ref{lemma-weakly-ass-zero}.
\end{proof}

\begin{lemma}
\label{lemma-minimal-support-in-weakly-ass}
Let $X$ be a scheme.
 ========

regex_match $X$
nn candidate a scheme
nn a scheme
new_annotation [(22317, 22320, 'VAR'), (22324, 22332, 'TYPE')]
===== sent Let $\mathcal{F}$ be a quasi-coherent $\mathcal{O}_X$-module.
 ========

regex_match $\mathcal{F}$
nn candidate a quasi-coherent $\mathcal{O}_X$-module
nn a quasi-coherent $\mathcal{O}_X$-module
new_annotation [(22338, 22351, 'VAR'), (22355, 22394, 'TYPE')]
===== sent Let $x \in \text{Supp}(\mathcal{F})$ be a point in the support
of $\mathcal{F}$ which is not a specialization of another point of
$\text{Supp}(\mathcal{F})$. ========

regex_match $x \in \text{Supp}(\mathcal{F})$
nn candidate a point
nn a point
new_annotation [(22400, 22432, 'VAR'), (22436, 22443, 'TYPE')]
===== sent \end{proof}

\begin{lemma}
\label{lemma-ass-weakly-ass}
Let $X$ be a scheme.
 ========

regex_match $X$
nn candidate a scheme
nn a scheme
new_annotation [(23302, 23305, 'VAR'), (23309, 23317, 'TYPE')]
===== sent Let $\mathcal{F}$ be a quasi-coherent $\mathcal{O}_X$-module.
 ========

regex_match $\mathcal{F}$
nn candidate a quasi-coherent $\mathcal{O}_X$-module
nn a quasi-coherent $\mathcal{O}_X$-module
new_annotation [(23323, 23336, 'VAR'), (23340, 23379, 'TYPE')]
===== sent $$
In particular, if $X$ is locally Noetherian, then
$\text{Ass}(\mathcal{F}) = \text{WeakAss}(\mathcal{F})$.
\end{lemma}

\begin{proof}
See
Algebra, Lemma \ref{algebra-lemma-ass-weakly-ass}.
\end{proof}

\begin{lemma}
\label{lemma-weakass-pushforward}
Let $f : X \to S$ be a quasi-compact and quasi-separated morphism of schemes.
 ========

regex_match $f : X \to S$
nn candidate X \to
nn candidate a quasi-compact and quasi-separated morphism
nn a quasi-compact and quasi-separated morphism
new_annotation [(23801, 23814, 'VAR'), (23818, 23862, 'TYPE')]
===== sent Let $\mathcal{F}$ be a quasi-coherent $\mathcal{O}_X$-module.
 ========

regex_match $\mathcal{F}$
nn candidate a quasi-coherent $\mathcal{O}_X$-module
nn a quasi-coherent $\mathcal{O}_X$-module
new_annotation [(23879, 23892, 'VAR'), (23896, 23935, 'TYPE')]
===== sent Let $s \in S$ be a point which is not in the image of $f$. Then
$s$ is not weakly associated to $f_*\mathcal{F}$.
\end{lemma}

\begin{proof}
The question is local so we may assume $X = \Spec(A)$.
By Schemes, Lemma \ref{schemes-lemma-push-forward-quasi-coherent}
the sheaf $f_*\mathcal{F}$ is quasi-coherent, say corresponding
to the $A$-module $M$.
Say $s$ corresponds to $\mathfrak p \subset A$.
 ========

regex_match $s \in S$
nn candidate $s
nn candidate a point
nn a point
new_annotation [(23941, 23950, 'VAR'), (23954, 23961, 'TYPE')]
===== sent Hence for any nonzero element $m$ of the stalk $M_\mathfrak p$
there exists an $i$ such that $a_i^n m$ is nonzero for all $n \geq 0$.
Thus $\mathfrak pA_\mathfrak p$ is not weakly associated to $M_\mathfrak p$.
\end{proof}

\begin{lemma}
\label{lemma-check-injective-on-weakass}
Let $X$ be a scheme. ========

regex_match $X$
nn candidate a scheme
nn a scheme
new_annotation [(24942, 24945, 'VAR'), (24949, 24957, 'TYPE')]
===== sent Let $\varphi ========

new_annotation []
===== sent The assumptions imply that $\text{WeakAss}(\Ker(\varphi)) = \emptyset$
and hence $\Ker(\varphi) = 0$ by Lemma \ref{lemma-weakly-ass-zero}.
\end{proof}



\section{Morphisms and weakly associated points}
\label{section-morphisms-weakly-associated}

\begin{lemma}
\label{lemma-weakly-ass-reverse-functorial}
Let $f : X \to S$ be an affine morphism of schemes.
 ========

regex_match $f : X \to S$
nn candidate X \to
nn candidate an affine morphism
nn an affine morphism
new_annotation [(25630, 25643, 'VAR'), (25647, 25665, 'TYPE')]
===== sent Let $\mathcal{F}$ be a quasi-coherent $\mathcal{O}_X$-module.
 ========

regex_match $\mathcal{F}$
nn candidate a quasi-coherent $\mathcal{O}_X$-module
nn a quasi-coherent $\mathcal{O}_X$-module
new_annotation [(25682, 25695, 'VAR'), (25699, 25738, 'TYPE')]
===== sent Hence the lemma follows from
Algebra, Lemma \ref{algebra-lemma-weakly-ass-reverse-functorial}.
\end{proof}

\begin{lemma}
\label{lemma-ass-functorial-equal}
Let $f : X \to S$ be an affine morphism of schemes.
 ========

regex_match $f : X \to S$
nn candidate X \to
nn candidate an affine morphism
nn an affine morphism
new_annotation [(26452, 26465, 'VAR'), (26469, 26487, 'TYPE')]
===== sent Let $\mathcal{F}$ be a quasi-coherent $\mathcal{O}_X$-module.
 ========

regex_match $\mathcal{F}$
nn candidate a quasi-coherent $\mathcal{O}_X$-module
nn a quasi-coherent $\mathcal{O}_X$-module
new_annotation [(26504, 26517, 'VAR'), (26521, 26560, 'TYPE')]
===== sent \end{proof}

\begin{lemma}
\label{lemma-weakly-associated-finite}
Let $f : X \to S$ be a finite morphism of schemes.
 ========

regex_match $f : X \to S$
nn candidate X \to
nn candidate a finite morphism
nn a finite morphism
new_annotation [(27885, 27898, 'VAR'), (27902, 27919, 'TYPE')]
===== sent Let $\mathcal{F}$ be a quasi-coherent $\mathcal{O}_X$-module.
 ========

regex_match $\mathcal{F}$
nn candidate a quasi-coherent $\mathcal{O}_X$-module
nn a quasi-coherent $\mathcal{O}_X$-module
new_annotation [(27936, 27949, 'VAR'), (27953, 27992, 'TYPE')]
===== sent Hence the lemma follows from
Algebra, Lemma \ref{algebra-lemma-weakly-ass-finite-ring-map}.
\end{proof}

\begin{lemma}
\label{lemma-weakly-ass-pullback}
Let $f : X \to S$ be a morphism of schemes. ========

regex_match $f : X \to S$
nn candidate X \to
nn candidate a morphism
nn a morphism
new_annotation [(28689, 28702, 'VAR'), (28706, 28716, 'TYPE')]
===== sent Let $\mathcal{G}$ be a
quasi-coherent $\mathcal{O}_S$-module. ========

new_annotation []
===== sent Let $x \in X$ with $s = f(x)$.
If $f$ is flat at $x$, the point $x$ is a generic point of the fibre $X_s$, and
$s \in \text{WeakAss}_S(\mathcal{G})$, then
$x \in \text{WeakAss}(f^*\mathcal{G})$.
\end{lemma}

 ========

new_annotation []
===== sent \begin{proof}
Let $A = \mathcal{O}_{S, s}$, $B = \mathcal{O}_{X, x}$, and
$M = \mathcal{G}_s$. Let $m \in M$ be an element whose annihilator
$I = \{a \in A \mid am = 0\}$ has radical $\mathfrak m_A$. Then
$m \otimes 1$ has annihilator $I B$ as $A \to B$ is
faithfully flat. ========

new_annotation []
===== sent \begin{proof}
Let $A = \mathcal{O}_{S, s}$, $B = \mathcal{O}_{X, x}$, and
$M = \mathcal{G}_s$. Let $m \in M$ be an element whose annihilator
$I = \{a \in A \mid am = 0\}$ has radical $\mathfrak m_A$. Then
$m \otimes 1$ has annihilator $I B$ as $A \to B$ is
faithfully flat. ========

regex_match $m \in M$
nn candidate $m
nn candidate an element
nn an element
new_annotation [(29098, 29107, 'VAR'), (29111, 29121, 'TYPE')]
===== sent Let $K/k$ be a field extension. ========

regex_match $K/k$
nn candidate a field extension
nn a field extension
new_annotation [(29637, 29642, 'VAR'), (29646, 29663, 'TYPE')]
===== sent Let $X$ be a scheme over $k$.
Let $\mathcal{F}$ be a quasi-coherent $\mathcal{O}_X$-module.
 ========

regex_match $X$
nn candidate a scheme
nn a scheme
new_annotation [(29669, 29672, 'VAR'), (29676, 29684, 'TYPE')]
===== sent Let $X$ be a scheme over $k$.
Let $\mathcal{F}$ be a quasi-coherent $\mathcal{O}_X$-module.
 ========

regex_match $\mathcal{F}$
nn candidate a quasi-coherent $\mathcal{O}_X$-module
nn a quasi-coherent $\mathcal{O}_X$-module
new_annotation [(29699, 29712, 'VAR'), (29716, 29755, 'TYPE')]
===== sent Let $y \in X_K$ with image $x \in X$. If $y$ is a weakly
associated point of the pullback $\mathcal{F}_K$, then $x$
is a weakly associated point of $\mathcal{F}$.
\end{lemma}

\begin{proof}
This is the translation of
Algebra, Lemma \ref{algebra-lemma-weakly-ass-change-fields}
into the language of schemes.
 ========

new_annotation []
===== sent Let $A \to B$ be a ring map. ========

regex_match $A \to B$
nn candidate a ring map
nn a ring map
new_annotation [(30161, 30170, 'VAR'), (30174, 30184, 'TYPE')]
===== sent Let $N$ be a $B$-module. ========

regex_match $N$
nn candidate a $B$-module
nn a $B$-module
new_annotation [(30190, 30193, 'VAR'), (30197, 30209, 'TYPE')]
===== sent \begin{definition}
\label{definition-relative-assassin}
Let $f : X \to S$ be a morphism of schemes.
 ========

regex_match $f : X \to S$
nn candidate X \to
nn candidate a morphism
nn a morphism
new_annotation [(30595, 30608, 'VAR'), (30612, 30622, 'TYPE')]
===== sent Let $\mathcal{F}$ be a quasi-coherent $\mathcal{O}_X$-module.
 ========

regex_match $\mathcal{F}$
nn candidate a quasi-coherent $\mathcal{O}_X$-module
nn a quasi-coherent $\mathcal{O}_X$-module
new_annotation [(30639, 30652, 'VAR'), (30656, 30695, 'TYPE')]
===== sent \begin{lemma}
\label{lemma-relative-assassin-affine-open}
Let $f : X \to S$ be a morphism of schemes.
 ========

regex_match $f : X \to S$
nn candidate X \to
nn candidate a morphism
nn a morphism
new_annotation [(31497, 31510, 'VAR'), (31514, 31524, 'TYPE')]
===== sent Let $\mathcal{F}$ be a quasi-coherent sheaf on $X$.
Let $U \subset X$ and $V \subset S$ be affine opens
with $f(U) \subset V$. Write $U = \Spec(A)$, $V = \Spec(R)$, and set
$M = \Gamma(U, \mathcal{F})$.
Let $x \in U$, and let $\mathfrak p \subset A$ be the corresponding prime.
 ========

regex_match $\mathcal{F}$
nn candidate a quasi-coherent sheaf
nn a quasi-coherent sheaf
new_annotation [(31541, 31554, 'VAR'), (31558, 31580, 'TYPE')]
===== sent Let $\mathcal{F}$ be a quasi-coherent sheaf on $X$.
Let $U \subset X$ and $V \subset S$ be affine opens
with $f(U) \subset V$. Write $U = \Spec(A)$, $V = \Spec(R)$, and set
$M = \Gamma(U, \mathcal{F})$.
Let $x \in U$, and let $\mathfrak p \subset A$ be the corresponding prime.
 ========

new_annotation []
===== sent Let $\mathcal{F}$ be a quasi-coherent sheaf on $X$.
Let $U \subset X$ and $V \subset S$ be affine opens
with $f(U) \subset V$. Write $U = \Spec(A)$, $V = \Spec(R)$, and set
$M = \Gamma(U, \mathcal{F})$.
Let $x \in U$, and let $\mathfrak p \subset A$ be the corresponding prime.
 ========

new_annotation []
===== sent Let $\mathcal{F}$ be a quasi-coherent sheaf on $X$.
Let $U \subset X$ and $V \subset S$ be affine opens
with $f(U) \subset V$. Write $U = \Spec(A)$, $V = \Spec(R)$, and set
$M = \Gamma(U, \mathcal{F})$.
Let $x \in U$, and let $\mathfrak p \subset A$ be the corresponding prime.
 ========

regex_match $\mathfrak p \subset A$
nn candidate p \subset
nn candidate the corresponding prime
nn the corresponding prime
new_annotation [(31763, 31786, 'VAR'), (31790, 31813, 'TYPE')]
===== sent \end{lemma}

\begin{proof}
The set $\text{Ass}_{A/R}(M)$ is defined in
Algebra, Definition \ref{algebra-definition-relative-assassin}.
Choose a pair $(\mathfrak p, x)$. Let $s = f(x)$.
Let $\mathfrak r \subset R$ be the prime lying under $\mathfrak p$,
i.e., the prime corresponding to $s$.
Let $\mathfrak p' \subset A \otimes_R \kappa(\mathfrak r)$
be the prime whose inverse image is $\mathfrak p$, i.e.,
the prime corresponding to $x$ viewed as a point of its fibre $X_s$.
Then $\mathfrak p \in \text{Ass}_{A/R}(M)$ if and only if
$\mathfrak p'$ is an associated prime of
$M \otimes_R \kappa(\mathfrak r)$, see
Algebra, Lemma \ref{algebra-lemma-compare-relative-assassins}.
Note that the ring $A \otimes_R \kappa(\mathfrak r)$ corresponds to $U_s$
and the module $M \otimes_R \kappa(\mathfrak r)$ corresponds to the
quasi-coherent sheaf $\mathcal{F}_s|_{U_s}$.
Hence $x$ is an associated point of $\mathcal{F}_s$
by Lemma \ref{lemma-associated-affine-open}.
The reverse implication holds if $\mathfrak p'$ is finitely generated
which is how the last sentence is seen to be true.
 ========

new_annotation []
===== sent \end{lemma}

\begin{proof}
The set $\text{Ass}_{A/R}(M)$ is defined in
Algebra, Definition \ref{algebra-definition-relative-assassin}.
Choose a pair $(\mathfrak p, x)$. Let $s = f(x)$.
Let $\mathfrak r \subset R$ be the prime lying under $\mathfrak p$,
i.e., the prime corresponding to $s$.
Let $\mathfrak p' \subset A \otimes_R \kappa(\mathfrak r)$
be the prime whose inverse image is $\mathfrak p$, i.e.,
the prime corresponding to $x$ viewed as a point of its fibre $X_s$.
Then $\mathfrak p \in \text{Ass}_{A/R}(M)$ if and only if
$\mathfrak p'$ is an associated prime of
$M \otimes_R \kappa(\mathfrak r)$, see
Algebra, Lemma \ref{algebra-lemma-compare-relative-assassins}.
Note that the ring $A \otimes_R \kappa(\mathfrak r)$ corresponds to $U_s$
and the module $M \otimes_R \kappa(\mathfrak r)$ corresponds to the
quasi-coherent sheaf $\mathcal{F}_s|_{U_s}$.
Hence $x$ is an associated point of $\mathcal{F}_s$
by Lemma \ref{lemma-associated-affine-open}.
The reverse implication holds if $\mathfrak p'$ is finitely generated
which is how the last sentence is seen to be true.
 ========

regex_match $\mathfrak r \subset R$
nn candidate r \subset
nn candidate the prime lying
nn the prime lying
new_annotation [(32288, 32311, 'VAR'), (32315, 32330, 'TYPE')]
===== sent \end{lemma}

\begin{proof}
The set $\text{Ass}_{A/R}(M)$ is defined in
Algebra, Definition \ref{algebra-definition-relative-assassin}.
Choose a pair $(\mathfrak p, x)$. Let $s = f(x)$.
Let $\mathfrak r \subset R$ be the prime lying under $\mathfrak p$,
i.e., the prime corresponding to $s$.
Let $\mathfrak p' \subset A \otimes_R \kappa(\mathfrak r)$
be the prime whose inverse image is $\mathfrak p$, i.e.,
the prime corresponding to $x$ viewed as a point of its fibre $X_s$.
Then $\mathfrak p \in \text{Ass}_{A/R}(M)$ if and only if
$\mathfrak p'$ is an associated prime of
$M \otimes_R \kappa(\mathfrak r)$, see
Algebra, Lemma \ref{algebra-lemma-compare-relative-assassins}.
Note that the ring $A \otimes_R \kappa(\mathfrak r)$ corresponds to $U_s$
and the module $M \otimes_R \kappa(\mathfrak r)$ corresponds to the
quasi-coherent sheaf $\mathcal{F}_s|_{U_s}$.
Hence $x$ is an associated point of $\mathcal{F}_s$
by Lemma \ref{lemma-associated-affine-open}.
The reverse implication holds if $\mathfrak p'$ is finitely generated
which is how the last sentence is seen to be true.
 ========

new_annotation []
===== sent \end{proof}

\begin{lemma}
\label{lemma-base-change-relative-assassin}
Let $f : X \to S$ be a morphism of schemes.
 ========

regex_match $f : X \to S$
nn candidate X \to
nn candidate a morphism
nn a morphism
new_annotation [(33256, 33269, 'VAR'), (33273, 33283, 'TYPE')]
===== sent Let $\mathcal{F}$ be a quasi-coherent $\mathcal{O}_X$-module.
 ========

regex_match $\mathcal{F}$
nn candidate a quasi-coherent $\mathcal{O}_X$-module
nn a quasi-coherent $\mathcal{O}_X$-module
new_annotation [(33300, 33313, 'VAR'), (33317, 33356, 'TYPE')]
===== sent Let $g : S' \to S$ be a morphism of schemes.
 ========

regex_match $g : S' \to S$
nn candidate a morphism
nn a morphism
new_annotation [(33362, 33376, 'VAR'), (33380, 33390, 'TYPE')]
===== sent S' \ar[r]^g & S
}
$$
and set $\mathcal{F}' = (g')^*\mathcal{F}$. Let $x' \in X'$ be a point
with images $x \in X$, $s' \in S'$ and $s \in S$.
Assume $f$ locally of finite type.
 ========

regex_match $x' \in X'$
nn candidate x
nn candidate a point
nn a point
new_annotation [(33555, 33566, 'VAR'), (33570, 33577, 'TYPE')]
===== sent In fact, this holds more generally for any morphism
$g : S' \to S$ such that all the field extensions
$\kappa(s) \subset \kappa(s')$ are algebraic, because in this case all
prime ideals of $\kappa(s') \otimes_{\kappa(s)} \kappa(x)$ are
maximal (and minimal) primes, see
Algebra, Lemma \ref{algebra-lemma-integral-over-field}.
\end{remark}




\section{Relative weak assassin}
\label{section-relative-weak-assassin}

\begin{definition}
\label{definition-relative-weak-assassin}
Let $f : X \to S$ be a morphism of schemes.
 ========

regex_match $f : X \to S$
nn candidate X \to
nn candidate a morphism
nn a morphism
new_annotation [(36171, 36184, 'VAR'), (36188, 36198, 'TYPE')]
===== sent Let $\mathcal{F}$ be a quasi-coherent $\mathcal{O}_X$-module.
 ========

regex_match $\mathcal{F}$
nn candidate a quasi-coherent $\mathcal{O}_X$-module
nn a quasi-coherent $\mathcal{O}_X$-module
new_annotation [(36215, 36228, 'VAR'), (36232, 36271, 'TYPE')]
===== sent The {\it relative weak assassin of $\mathcal{F}$ in $X$ over $S$}
is the set
$$
\text{WeakAss}_{X/S}(\mathcal{F}) =
\bigcup\nolimits_{s \in S} \text{WeakAss}(\mathcal{F}_s)
$$
where $\mathcal{F}_s = (X_s \to X)^*\mathcal{F}$ is the restriction
of $\mathcal{F}$ to the fibre of $f$ at $s$.
\end{definition}

\begin{lemma}
\label{lemma-relative-weak-assassin-assassin-finite-type}
Let $f : X \to S$ be a morphism of schemes which is locally of finite type.
 ========

regex_match $f : X \to S$
nn candidate X \to
nn candidate a morphism
nn a morphism
new_annotation [(36656, 36669, 'VAR'), (36673, 36683, 'TYPE')]
===== sent Let $\mathcal{F}$ be a quasi-coherent $\mathcal{O}_X$-module.
 ========

regex_match $\mathcal{F}$
nn candidate a quasi-coherent $\mathcal{O}_X$-module
nn a quasi-coherent $\mathcal{O}_X$-module
new_annotation [(36732, 36745, 'VAR'), (36749, 36788, 'TYPE')]
===== sent Then $\text{WeakAss}_{X/S}(\mathcal{F}) = \text{Ass}_{X/S}(\mathcal{F})$.
\end{lemma}

\begin{proof}
This is true because the fibres of $f$ are locally Noetherian schemes,
and associated and weakly associated points agree on locally Noetherian
schemes, see
Lemma \ref{lemma-ass-weakly-ass}.
\end{proof}

\begin{lemma}
\label{lemma-relative-weak-assassin-finite}
Let $f : X \to S$ be a morphism of schemes.
 ========

regex_match $f : X \to S$
nn candidate X \to
nn candidate a morphism
nn a morphism
new_annotation [(37156, 37169, 'VAR'), (37173, 37183, 'TYPE')]
===== sent Let $i : Z \to X$ be a finite morphism.
 ========

regex_match $i : Z \to X$
nn candidate Z \to
nn candidate a finite morphism
nn a finite morphism
new_annotation [(37200, 37213, 'VAR'), (37217, 37234, 'TYPE')]
===== sent Let $\mathcal{F}$ be a quasi-coherent $\mathcal{O}_Z$-module.
 ========

regex_match $\mathcal{F}$
nn candidate a quasi-coherent $\mathcal{O}_Z$-module
nn a quasi-coherent $\mathcal{O}_Z$-module
new_annotation [(37240, 37253, 'VAR'), (37257, 37296, 'TYPE')]
===== sent Let $i_s : Z_s \to X_s$ be the induced morphism between fibres.
 ========

regex_match $i_s : Z_s \to X_s$
nn candidate the induced morphism
nn the induced morphism
new_annotation [(37413, 37432, 'VAR'), (37436, 37456, 'TYPE')]
===== sent \end{proof}





\section{Fitting ideals}
\label{section-fitting-ideals}

\noindent
This section is the continuation of the discussion in
More on Algebra, Section \ref{more-algebra-section-fitting-ideals}.
Let $S$ be a scheme. ========

regex_match $S$
nn candidate a scheme
nn a scheme
new_annotation [(37915, 37918, 'VAR'), (37922, 37930, 'TYPE')]
===== sent Let $\mathcal{F}$ be a
finite type quasi-coherent $\mathcal{O}_S$-module.
 ========

new_annotation []
===== sent \begin{lemma}
\label{lemma-base-change-fitting-ideal}
Let $f : T \to S$ be a morphism of schemes.
 ========

regex_match $f : T \to S$
nn candidate T \to
nn candidate a morphism
nn a morphism
new_annotation [(39376, 39389, 'VAR'), (39393, 39403, 'TYPE')]
===== sent Let $\mathcal{F}$ be a finite type quasi-coherent $\mathcal{O}_S$-module.
 ========

regex_match $\mathcal{F}$
nn candidate a finite type quasi-coherent $\mathcal{O}_S$-module
nn a finite type quasi-coherent $\mathcal{O}_S$-module
new_annotation [(39420, 39433, 'VAR'), (39437, 39488, 'TYPE')]
===== sent \end{proof}

\begin{lemma}
\label{lemma-fitting-ideal-of-finitely-presented}
Let $S$ be a scheme.
 ========

regex_match $S$
nn candidate a scheme
nn a scheme
new_annotation [(39793, 39796, 'VAR'), (39800, 39808, 'TYPE')]
===== sent Let $\mathcal{F}$ be a finitely presented $\mathcal{O}_S$-module.
 ========

regex_match $\mathcal{F}$
new_annotation []
===== sent \end{proof}

\begin{lemma}
\label{lemma-on-subscheme-cut-out-by-Fit-0}
Let $S$ be a scheme.
 ========

regex_match $S$
nn candidate a scheme
nn a scheme
new_annotation [(40157, 40160, 'VAR'), (40164, 40172, 'TYPE')]
===== sent Let $\mathcal{F}$ be a finite type, quasi-coherent $\mathcal{O}_S$-module.
 ========

regex_match $\mathcal{F}$
nn candidate a finite type
nn a finite type
new_annotation [(40178, 40191, 'VAR'), (40195, 40208, 'TYPE')]
===== sent Let $Z_0 \subset S$ be the closed subscheme cut out by
$\text{Fit}_0(\mathcal{F})$.
Let $Z \subset S$ be the scheme theoretic support of $\mathcal{F}$.
Then
\begin{enumerate}
\item $Z \subset Z_0 \subset S$ as closed subschemes,
\item $Z = Z_0 = \text{Supp}(\mathcal{F})$ as closed subsets,
\item there exists a finite type, quasi-coherent $\mathcal{O}_{Z_0}$-module
$\mathcal{G}_0$ with
$$
(Z_0 \to X)_*\mathcal{G}_0 = \mathcal{F}.
$$
\end{enumerate}
\end{lemma}

\begin{proof}
Recall that $Z$ is locally cut out by the annihilator of $\mathcal{F}$, see
Morphisms, Definition \ref{morphisms-definition-scheme-theoretic-support}
(which uses Morphisms, Lemma \ref{morphisms-lemma-scheme-theoretic-support}
to define $Z$). ========

regex_match $Z_0 \subset S$
nn candidate the closed subscheme
nn the closed subscheme
new_annotation [(40253, 40268, 'VAR'), (40272, 40292, 'TYPE')]
===== sent Let $Z_0 \subset S$ be the closed subscheme cut out by
$\text{Fit}_0(\mathcal{F})$.
Let $Z \subset S$ be the scheme theoretic support of $\mathcal{F}$.
Then
\begin{enumerate}
\item $Z \subset Z_0 \subset S$ as closed subschemes,
\item $Z = Z_0 = \text{Supp}(\mathcal{F})$ as closed subsets,
\item there exists a finite type, quasi-coherent $\mathcal{O}_{Z_0}$-module
$\mathcal{G}_0$ with
$$
(Z_0 \to X)_*\mathcal{G}_0 = \mathcal{F}.
$$
\end{enumerate}
\end{lemma}

\begin{proof}
Recall that $Z$ is locally cut out by the annihilator of $\mathcal{F}$, see
Morphisms, Definition \ref{morphisms-definition-scheme-theoretic-support}
(which uses Morphisms, Lemma \ref{morphisms-lemma-scheme-theoretic-support}
to define $Z$). ========

regex_match $Z \subset S$
nn candidate $Z \subset
nn candidate the scheme theoretic support
nn the scheme theoretic support
new_annotation [(40337, 40350, 'VAR'), (40354, 40382, 'TYPE')]
===== sent \end{proof}

\begin{lemma}
\label{lemma-fitting-ideal-generate-locally}
Let $S$ be a scheme. ========

regex_match $S$
nn candidate a scheme
nn a scheme
new_annotation [(41893, 41896, 'VAR'), (41900, 41908, 'TYPE')]
===== sent Let $\mathcal{F}$ be a finite type, quasi-coherent
$\mathcal{O}_S$-module. ========

regex_match $\mathcal{F}$
nn candidate a finite type
nn a finite type
new_annotation [(41914, 41927, 'VAR'), (41931, 41944, 'TYPE')]
===== sent Let $s \in ========

new_annotation []
===== sent Then $\mathcal{F}$ can be
generated by $r$ elements in a neighbourhood of $s$ if and only
if $\text{Fit}_r(\mathcal{F})_s = \mathcal{O}_{S, s}$.
\end{lemma}

\begin{proof}
Follows immediately from
More on Algebra, Lemma \ref{more-algebra-lemma-fitting-ideal-generate-locally}.
\end{proof}

\begin{lemma}
\label{lemma-fitting-ideal-finite-locally-free}
Let $S$ be a scheme. ========

regex_match $S$
nn candidate a scheme
nn a scheme
new_annotation [(42356, 42359, 'VAR'), (42363, 42371, 'TYPE')]
===== sent Let $\mathcal{F}$ be a finite type, quasi-coherent
$\mathcal{O}_S$-module. ========

regex_match $\mathcal{F}$
nn candidate a finite type
nn a finite type
new_annotation [(42377, 42390, 'VAR'), (42394, 42407, 'TYPE')]
===== sent Let $r \geq 0$. The following are equivalent
\begin{enumerate}
\item $\mathcal{F}$ is finite locally free of rank $r$
\item $\text{Fit}_{r - 1}(\mathcal{F}) = 0$ and
$\text{Fit}_r(\mathcal{F}) = \mathcal{O}_S$, and
\item $\text{Fit}_k(\mathcal{F}) = 0$ for $k < r$ and
$\text{Fit}_k(\mathcal{F}) = \mathcal{O}_S$ for $k \geq r$.
\end{enumerate}
\end{lemma}

\begin{proof}
Follows immediately from
More on Algebra, Lemma
\ref{more-algebra-lemma-fitting-ideal-finite-locally-free}.
\end{proof}

\begin{lemma}
\label{lemma-locally-free-rank-r-pullback}
Let $S$ be a scheme. ========

new_annotation []
===== sent Let $r \geq 0$. The following are equivalent
\begin{enumerate}
\item $\mathcal{F}$ is finite locally free of rank $r$
\item $\text{Fit}_{r - 1}(\mathcal{F}) = 0$ and
$\text{Fit}_r(\mathcal{F}) = \mathcal{O}_S$, and
\item $\text{Fit}_k(\mathcal{F}) = 0$ for $k < r$ and
$\text{Fit}_k(\mathcal{F}) = \mathcal{O}_S$ for $k \geq r$.
\end{enumerate}
\end{lemma}

\begin{proof}
Follows immediately from
More on Algebra, Lemma
\ref{more-algebra-lemma-fitting-ideal-finite-locally-free}.
\end{proof}

\begin{lemma}
\label{lemma-locally-free-rank-r-pullback}
Let $S$ be a scheme. ========

regex_match $S$
nn candidate a scheme
nn a scheme
new_annotation [(43002, 43005, 'VAR'), (43009, 43017, 'TYPE')]
===== sent Let $\mathcal{F}$ be a finite type, quasi-coherent
$\mathcal{O}_S$-module. ========

regex_match $\mathcal{F}$
nn candidate a finite type
nn a finite type
new_annotation [(43023, 43036, 'VAR'), (43040, 43053, 'TYPE')]
===== sent \begin{lemma}
\label{lemma-finite-presentation-module}
Let $S$ be a scheme. ========

regex_match $S$
nn candidate a scheme
nn a scheme
new_annotation [(46045, 46048, 'VAR'), (46052, 46060, 'TYPE')]
===== sent Let $\mathcal{F}$ be an $\mathcal{O}_S$-module
of finite presentation. ========

regex_match $\mathcal{F}$
nn candidate finite presentation
nn finite presentation
new_annotation [(46066, 46079, 'VAR'), (46112, 46131, 'TYPE')]
===== sent Let $S = Z_{-1} \subset Z_0 \subset Z_1 \subset \ldots$
be as in Lemma \ref{lemma-locally-free-rank-r-pullback}.
Set $S_r = Z_{r - 1} ========

new_annotation []
===== sent \end{proof}






\section{The singular locus of a morphism}
\label{section-singular-locus-morphism}

\noindent
Let $f : X \to S$ be a finite type morphism of schemes. ========

regex_match $f : X \to S$
nn candidate X \to
nn candidate a finite type morphism
nn a finite type morphism
new_annotation [(47614, 47627, 'VAR'), (47631, 47653, 'TYPE')]
===== sent This can be helpful; it is related to embedding dimensions of
fibres, see Varieties, Section \ref{varieties-section-embedding-dimension}.

\begin{lemma}
\label{lemma-base-change-and-fitting-ideal-omega}
Let $f : X \to S$ be a morphism of schemes which is locally of finite type.
Let $X = Z_{-1} \supset Z_0 \supset Z_1 \supset \ldots$
be the closed subschemes defined by the fitting ideals
of $\Omega_{X/S}$. ========

regex_match $f : X \to S$
nn candidate X \to
nn candidate a morphism
nn a morphism
new_annotation [(49652, 49665, 'VAR'), (49669, 49679, 'TYPE')]
===== sent This can be helpful; it is related to embedding dimensions of
fibres, see Varieties, Section \ref{varieties-section-embedding-dimension}.

\begin{lemma}
\label{lemma-base-change-and-fitting-ideal-omega}
Let $f : X \to S$ be a morphism of schemes which is locally of finite type.
Let $X = Z_{-1} \supset Z_0 \supset Z_1 \supset \ldots$
be the closed subschemes defined by the fitting ideals
of $\Omega_{X/S}$. ========

new_annotation []
===== sent \begin{lemma}
\label{lemma-zero-fitting-ideal-omega-unramified}
Let $f : X \to S$ be a morphism of schemes which is locally of finite type.
 ========

regex_match $f : X \to S$
nn candidate X \to
nn candidate a morphism
nn a morphism
new_annotation [(50764, 50777, 'VAR'), (50781, 50791, 'TYPE')]
===== sent This is exactly
the set of points where $f$ is unramified by
Morphisms, Lemma \ref{morphisms-lemma-unramified-omega-zero}.
\end{proof}

\begin{lemma}
\label{lemma-d-fitting-ideal-omega-smooth}
Let $f : X \to S$ be a morphism of schemes. ========

regex_match $f : X \to S$
nn candidate X \to
nn candidate a morphism
nn a morphism
new_annotation [(51330, 51343, 'VAR'), (51347, 51357, 'TYPE')]
===== sent Let $d \geq 0$ be an integer.
 ========

regex_match $d \geq 0$
nn candidate an integer
nn an integer
new_annotation [(51374, 51384, 'VAR'), (51388, 51398, 'TYPE')]
===== sent Assume
\begin{enumerate}
\item $f$ is flat,
\item $f$ is locally of finite presentation, and
\item every nonempty fibre of $f$ is equidimensional of dimension $d$.
\end{enumerate}
Let $Z \subset X$ be the closed subscheme cut out by the $d$th fitting
ideal of $\Omega_{X/S}$. Then $Z$ is exactly the set of points
where $f$ is not smooth.
 ========

regex_match $Z \subset X$
nn candidate $Z \subset
nn candidate the closed subscheme
nn the closed subscheme
new_annotation [(51584, 51597, 'VAR'), (51601, 51621, 'TYPE')]
===== sent \begin{lemma}
\label{lemma-torsion-sections}
Let $X$ be an integral scheme with generic point $\eta$. Let $\mathcal{F}$
be a quasi-coherent $\mathcal{O}_X$-module. ========

regex_match $X$
nn candidate an integral scheme
nn an integral scheme
new_annotation [(52272, 52275, 'VAR'), (52279, 52297, 'TYPE')]
===== sent \begin{lemma}
\label{lemma-torsion-sections}
Let $X$ be an integral scheme with generic point $\eta$. Let $\mathcal{F}$
be a quasi-coherent $\mathcal{O}_X$-module. ========

new_annotation []
===== sent Let $U \subset X$ be nonempty
open and $s \in \mathcal{F}(U)$. The following are equivalent
\begin{enumerate}
\item for some $x \in U$ the image of $s$ in $\mathcal{F}_x$ is torsion,
\item for all $x \in U$ the image of $s$ in $\mathcal{F}_x$ is torsion,
\item the image of $s$ in $\mathcal{F}_\eta$ is zero,
\item the image of $s$ in $j_*\mathcal{F}_\eta$ is zero, where $j : \eta \to X$
is the inclusion morphism.
 ========

new_annotation []
===== sent \end{proof}

\begin{definition}
\label{definition-torsion}
Let $X$ be an integral scheme. ========

regex_match $X$
nn candidate an integral scheme
nn an integral scheme
new_annotation [(52918, 52921, 'VAR'), (52925, 52943, 'TYPE')]
===== sent Let $\mathcal{F}$ be a quasi-coherent
$\mathcal{O}_X$-module.
 ========

regex_match $\mathcal{F}$
new_annotation []
===== sent \begin{lemma}
\label{lemma-check-torsion-on-affines}
Let $X$ be an integral scheme. ========

regex_match $X$
nn candidate an integral scheme
nn an integral scheme
new_annotation [(53445, 53448, 'VAR'), (53452, 53470, 'TYPE')]
===== sent Let $\mathcal{F}$ be a quasi-coherent
$\mathcal{O}_X$-module. ========

regex_match $\mathcal{F}$
new_annotation []
===== sent \end{proof}

\begin{lemma}
\label{lemma-torsion}
Let $X$ be an integral scheme. ========

regex_match $X$
nn candidate an integral scheme
nn an integral scheme
new_annotation [(53819, 53822, 'VAR'), (53826, 53844, 'TYPE')]
===== sent Let $\mathcal{F}$ be a quasi-coherent
$\mathcal{O}_X$-module. ========

regex_match $\mathcal{F}$
new_annotation []
===== sent \end{proof}

\begin{lemma}
\label{lemma-flat-torsion-free}
Let $X$ be an integral scheme. ========

regex_match $X$
nn candidate an integral scheme
nn an integral scheme
new_annotation [(54293, 54296, 'VAR'), (54300, 54318, 'TYPE')]
===== sent See More on Algebra, Lemma \ref{more-algebra-lemma-flat-torsion-free}.
\end{proof}

\begin{lemma}
\label{lemma-flat-pullback-torsion}
Let $f : X \to Y$ be a flat morphism of integral schemes.
 ========

regex_match $f : X \to Y$
nn candidate X \to
nn candidate a flat morphism
nn a flat morphism
new_annotation [(54558, 54571, 'VAR'), (54575, 54590, 'TYPE')]
===== sent Let $\mathcal{G}$ be a torsion free quasi-coherent $\mathcal{O}_Y$-module.
 ========

regex_match $\mathcal{G}$
nn candidate a torsion
nn a torsion
new_annotation [(54616, 54629, 'VAR'), (54633, 54642, 'TYPE')]
===== sent \end{proof}

\begin{lemma}
\label{lemma-flat-over-integral-integral-fibre}
Let $f : X \to Y$ be a flat morphism of schemes. ========

regex_match $f : X \to Y$
nn candidate X \to
nn candidate a flat morphism
nn a flat morphism
new_annotation [(54968, 54981, 'VAR'), (54985, 55000, 'TYPE')]
===== sent \end{lemma}

\begin{proof}
The algebraic analogue is this: let $A$ be a domain with fraction
field $K$ and let $B$ be a flat $A$-algebra such that $B \otimes_A K$
is a domain. ========

regex_match $A$
nn candidate a domain
nn a domain
new_annotation [(55159, 55162, 'VAR'), (55166, 55174, 'TYPE')]
===== sent \end{lemma}

\begin{proof}
The algebraic analogue is this: let $A$ be a domain with fraction
field $K$ and let $B$ be a flat $A$-algebra such that $B \otimes_A K$
is a domain. ========

regex_match $B$
nn candidate $B$
nn candidate such that $B
nn such that $B
new_annotation [(55207, 55210, 'VAR'), (55233, 55245, 'TYPE')]
===== sent This is true because $B$ is
torsion free by More on Algebra, Lemma
\ref{more-algebra-lemma-flat-torsion-free}
and hence $B \subset B \otimes_A K$.
\end{proof}

\begin{lemma}
\label{lemma-check-torsion}
Let $X$ be an integral scheme. ========

regex_match $X$
nn candidate an integral scheme
nn an integral scheme
new_annotation [(55500, 55503, 'VAR'), (55507, 55525, 'TYPE')]
===== sent Let $\mathcal{F}$ be a quasi-coherent
$\mathcal{O}_X$-module. ========

regex_match $\mathcal{F}$
nn candidate a quasi-coherent
$\mathcal{O}_X$-module
nn a quasi-coherent
$\mathcal{O}_X$-module
new_annotation [(55531, 55544, 'VAR'), (55548, 55587, 'TYPE')]
===== sent See More on Algebra, Lemma
\ref{more-algebra-lemma-check-torsion}.
\end{proof}

\begin{lemma}
\label{lemma-extension-torsion-free}
Let $X$ be an integral scheme. ========

regex_match $X$
nn candidate an integral scheme
nn an integral scheme
new_annotation [(55891, 55894, 'VAR'), (55898, 55916, 'TYPE')]
===== sent \end{proof}

\begin{lemma}
\label{lemma-torsion-free-finite-noetherian-domain}
Let $X$ be a locally Noetherian integral scheme with generic point $\eta$.
Let $\mathcal{F}$ be a nonzero coherent $\mathcal{O}_X$-module.
 ========

regex_match $X$
nn candidate a locally Noetherian integral scheme
nn a locally Noetherian integral scheme
new_annotation [(56366, 56369, 'VAR'), (56373, 56409, 'TYPE')]
===== sent \end{proof}

\begin{lemma}
\label{lemma-torsion-free-finite-noetherian-domain}
Let $X$ be a locally Noetherian integral scheme with generic point $\eta$.
Let $\mathcal{F}$ be a nonzero coherent $\mathcal{O}_X$-module.
 ========

regex_match $\mathcal{F}$
nn candidate a nonzero coherent $\mathcal{O}_X$-module
nn a nonzero coherent $\mathcal{O}_X$-module
new_annotation [(56441, 56454, 'VAR'), (56458, 56499, 'TYPE')]
===== sent \end{proof}

\begin{lemma}
\label{lemma-torsion-free-over-regular-dim-1}
Let $X$ be an integral regular scheme of dimension $\leq 1$.
Let $\mathcal{F}$ be a coherent $\mathcal{O}_X$-module.
 ========

regex_match $X$
nn candidate an integral regular scheme
nn an integral regular scheme
new_annotation [(57123, 57126, 'VAR'), (57130, 57156, 'TYPE')]
===== sent \end{proof}

\begin{lemma}
\label{lemma-torsion-free-over-regular-dim-1}
Let $X$ be an integral regular scheme of dimension $\leq 1$.
Let $\mathcal{F}$ be a coherent $\mathcal{O}_X$-module.
 ========

regex_match $\mathcal{F}$
nn candidate a coherent $\mathcal{O}_X$-module
nn a coherent $\mathcal{O}_X$-module
new_annotation [(57184, 57197, 'VAR'), (57201, 57234, 'TYPE')]
===== sent Hence $\mathcal{F}_x$ is free by More on Algebra, Lemma
\ref{more-algebra-lemma-dedekind-torsion-free-flat}.
\end{proof}

\begin{lemma}
\label{lemma-hom-into-torsion-free}
Let $X$ be an integral scheme. ========

regex_match $X$
nn candidate an integral scheme
nn an integral scheme
new_annotation [(58191, 58194, 'VAR'), (58198, 58216, 'TYPE')]
===== sent Let $\mathcal{F}$, $\mathcal{G}$ be
quasi-coherent $\mathcal{O}_X$-modules.
 ========

new_annotation []
===== sent \end{proof}

\begin{lemma}
\label{lemma-isom-depth-2-torsion-free}
Let $X$ be an integral locally Noetherian scheme. ========

regex_match $X$
nn candidate an integral locally Noetherian scheme
nn an integral locally Noetherian scheme
new_annotation [(58837, 58840, 'VAR'), (58844, 58881, 'TYPE')]
===== sent The reason for
working with coherent modules is that
$\SheafHom_{\mathcal{O}_X}(\mathcal{F}, \mathcal{G})$ is coherent
for every pair of coherent $\mathcal{O}_X$-modules $\mathcal{F}, \mathcal{G}$,
see Modules, Lemma \ref{modules-lemma-internal-hom-locally-kernel-direct-sum}.

\begin{definition}
\label{definition-reflexive}
Let $X$ be an integral locally Noetherian scheme. ========

regex_match $X$
nn candidate an integral locally Noetherian scheme
nn an integral locally Noetherian scheme
new_annotation [(59993, 59996, 'VAR'), (60000, 60037, 'TYPE')]
===== sent Let $\mathcal{F}$
be a coherent $\mathcal{O}_X$-module. ========

new_annotation []
===== sent \begin{lemma}
\label{lemma-check-reflexive-on-affines}
Let $X$ be an integral locally Noetherian scheme. ========

regex_match $X$
nn candidate an integral locally Noetherian scheme
nn an integral locally Noetherian scheme
new_annotation [(60823, 60826, 'VAR'), (60830, 60867, 'TYPE')]
===== sent Let $\mathcal{F}$ be a
coherent $\mathcal{O}_X$-module. ========

new_annotation []
===== sent \end{remark}

\begin{lemma}
\label{lemma-reflexive-torsion-free}
Let $X$ be an integral locally Noetherian scheme. ========

regex_match $X$
nn candidate an integral locally Noetherian scheme
nn an integral locally Noetherian scheme
new_annotation [(61933, 61936, 'VAR'), (61940, 61977, 'TYPE')]
===== sent Let $\mathcal{F}$
be a coherent $\mathcal{O}_X$-module.
 ========

new_annotation []
===== sent See More on Algebra, Lemma
\ref{more-algebra-lemma-reflexive-torsion-free}.
\end{proof}

\begin{lemma}
\label{lemma-check-reflexive}
Let $X$ be an integral locally Noetherian scheme.
 ========

regex_match $X$
nn candidate an integral locally Noetherian scheme
nn an integral locally Noetherian scheme
new_annotation [(62438, 62441, 'VAR'), (62445, 62482, 'TYPE')]
===== sent Let $\mathcal{F}$ be a coherent $\mathcal{O}_X$-module.
 ========

regex_match $\mathcal{F}$
nn candidate a coherent $\mathcal{O}_X$-module
nn a coherent $\mathcal{O}_X$-module
new_annotation [(62488, 62501, 'VAR'), (62505, 62538, 'TYPE')]
===== sent \end{proof}

\begin{lemma}
\label{lemma-dual-reflexive}
Let $X$ be an integral locally Noetherian scheme.
 ========

regex_match $X$
nn candidate an integral locally Noetherian scheme
nn an integral locally Noetherian scheme
new_annotation [(63173, 63176, 'VAR'), (63180, 63217, 'TYPE')]
===== sent Let $\mathcal{F}$, $\mathcal{G}$ be
coherent $\mathcal{O}_X$-modules.
 ========

new_annotation []
===== sent \end{proof}

\begin{lemma}
\label{lemma-reflexive-depth-2}
Let $X$ be an integral locally Noetherian scheme. ========

regex_match $X$
nn candidate an integral locally Noetherian scheme
nn an integral locally Noetherian scheme
new_annotation [(63777, 63780, 'VAR'), (63784, 63821, 'TYPE')]
===== sent Let $\mathcal{F}$
be a coherent $\mathcal{O}_X$-module. ========

new_annotation []
===== sent See More on Algebra, Lemma \ref{more-algebra-lemma-reflexive-depth-2}.
\end{proof}

\noindent
If the scheme is normal, then reflexive is the same thing as
torsion free and $(S_2)$.

\begin{lemma}
\label{lemma-reflexive-over-normal}
Let $X$ be an integral locally Noetherian normal scheme.
 ========

regex_match $X$
nn candidate an integral locally Noetherian normal scheme
nn an integral locally Noetherian normal scheme
new_annotation [(64495, 64498, 'VAR'), (64502, 64546, 'TYPE')]
===== sent Let $\mathcal{F}$ be a coherent $\mathcal{O}_X$-module.
 ========

regex_match $\mathcal{F}$
nn candidate a coherent $\mathcal{O}_X$-module
nn a coherent $\mathcal{O}_X$-module
new_annotation [(64552, 64565, 'VAR'), (64569, 64602, 'TYPE')]
===== sent The following are equivalent
\begin{enumerate}
\item $\mathcal{F}$ is reflexive, and
\item $\mathcal{F}$ is torsion free and has property $(S_2)$.
\end{enumerate}
\end{lemma}

\begin{proof}
This is the scheme theoretic analogue of
More on Algebra, Lemma \ref{more-algebra-lemma-reflexive-over-normal}.
To translate into algebra use Lemma \ref{lemma-check-reflexive-on-affines}.
\end{proof}

\begin{lemma}
\label{lemma-describe-reflexive-hull}
Let $X$ be an integral locally Noetherian normal scheme with
generic point $\eta$. Let $\mathcal{F}$, $\mathcal{G}$ be coherent
$\mathcal{O}_X$-modules. ========

regex_match $X$
nn candidate an integral locally Noetherian normal scheme
nn an integral locally Noetherian normal scheme
new_annotation [(65051, 65054, 'VAR'), (65058, 65102, 'TYPE')]
===== sent The following are equivalent
\begin{enumerate}
\item $\mathcal{F}$ is reflexive, and
\item $\mathcal{F}$ is torsion free and has property $(S_2)$.
\end{enumerate}
\end{lemma}

\begin{proof}
This is the scheme theoretic analogue of
More on Algebra, Lemma \ref{more-algebra-lemma-reflexive-over-normal}.
To translate into algebra use Lemma \ref{lemma-check-reflexive-on-affines}.
\end{proof}

\begin{lemma}
\label{lemma-describe-reflexive-hull}
Let $X$ be an integral locally Noetherian normal scheme with
generic point $\eta$. Let $\mathcal{F}$, $\mathcal{G}$ be coherent
$\mathcal{O}_X$-modules. ========

new_annotation []
===== sent Let $T : \mathcal{G}_\eta \to \mathcal{F}_\eta$
be a linear map. ========

new_annotation []
===== sent In this case
we are asking the following algebra question: Let $R$ be a Noetherian
normal domain with fraction field $K$, let $M$, $N$ be finite $R$-modules,
let $T : M \otimes_R K \to N \otimes_R K$ be a $K$-linear map. ========

regex_match $R$
nn candidate a Noetherian
normal domain
nn a Noetherian
normal domain
new_annotation [(65919, 65922, 'VAR'), (65926, 65952, 'TYPE')]
===== sent In this case
we are asking the following algebra question: Let $R$ be a Noetherian
normal domain with fraction field $K$, let $M$, $N$ be finite $R$-modules,
let $T : M \otimes_R K \to N \otimes_R K$ be a $K$-linear map. ========

new_annotation []
===== sent In this case
we are asking the following algebra question: Let $R$ be a Noetherian
normal domain with fraction field $K$, let $M$, $N$ be finite $R$-modules,
let $T : M \otimes_R K \to N \otimes_R K$ be a $K$-linear map. ========

regex_match $T : M \otimes_R K \to N \otimes_R K$
nn candidate a $K$-linear map
nn a $K$-linear map
new_annotation [(66018, 66055, 'VAR'), (66059, 66075, 'TYPE')]
===== sent \end{proof}

\begin{lemma}
\label{lemma-reflexive-over-regular-dim-2}
Let $X$ be a regular scheme of dimension $\leq 2$.
Let $\mathcal{F}$ be a coherent $\mathcal{O}_X$-module.
 ========

regex_match $X$
nn candidate a regular scheme
nn a regular scheme
new_annotation [(66448, 66451, 'VAR'), (66455, 66471, 'TYPE')]
===== sent \end{proof}

\begin{lemma}
\label{lemma-reflexive-over-regular-dim-2}
Let $X$ be a regular scheme of dimension $\leq 2$.
Let $\mathcal{F}$ be a coherent $\mathcal{O}_X$-module.
 ========

regex_match $\mathcal{F}$
nn candidate a coherent $\mathcal{O}_X$-module
nn a coherent $\mathcal{O}_X$-module
new_annotation [(66499, 66512, 'VAR'), (66516, 66549, 'TYPE')]
===== sent \begin{definition}
\label{definition-effective-Cartier-divisor}
Let $S$ be a scheme.
 ========

regex_match $S$
nn candidate a scheme
nn a scheme
new_annotation [(68004, 68007, 'VAR'), (68011, 68019, 'TYPE')]
===== sent \begin{lemma}
\label{lemma-characterize-effective-Cartier-divisor}
Let $S$ be a scheme.
 ========

regex_match $S$
nn candidate a scheme
nn a scheme
new_annotation [(68813, 68816, 'VAR'), (68820, 68828, 'TYPE')]
===== sent Let $D \subset S$ be a closed subscheme.
 ========

regex_match $D \subset S$
nn candidate $D \subset S$
nn candidate a closed subscheme
nn a closed subscheme
new_annotation [(68834, 68847, 'VAR'), (68851, 68869, 'TYPE')]
===== sent Let $x \in ========

new_annotation []
===== sent \end{proof}

\begin{lemma}
\label{lemma-complement-locally-principal-closed-subscheme}
Let $S$ be a scheme. ========

regex_match $S$
nn candidate a scheme
nn a scheme
new_annotation [(70244, 70247, 'VAR'), (70251, 70259, 'TYPE')]
===== sent Let $Z \subset S$ be a locally principal closed
subscheme. ========

regex_match $Z \subset S$
nn candidate $
nn candidate principal
nn principal
new_annotation [(70265, 70278, 'VAR'), (70292, 70301, 'TYPE')]
===== sent Let $U = S \setminus Z$. ========

new_annotation []
===== sent \end{proof}

\begin{lemma}
\label{lemma-complement-effective-Cartier-divisor}
Let $S$ be a scheme. ========

regex_match $S$
nn candidate a scheme
nn a scheme
new_annotation [(70730, 70733, 'VAR'), (70737, 70745, 'TYPE')]
===== sent Let $D \subset S$ be an effective Cartier divisor.
 ========

regex_match $D \subset S$
nn candidate $D \subset S$
nn candidate an effective Cartier divisor
nn an effective Cartier divisor
new_annotation [(70751, 70764, 'VAR'), (70768, 70796, 'TYPE')]
===== sent Let $U = S \setminus D$. Then $U \to S$ is an affine morphism and $U$
is scheme theoretically dense in $S$.
\end{lemma}

\begin{proof}
Affineness is Lemma \ref{lemma-complement-locally-principal-closed-subscheme}.
The density question is local on $S$, see
Morphisms, Lemma \ref{morphisms-lemma-characterize-scheme-theoretically-dense}.
 ========

new_annotation []
===== sent Thus $A \subset A_f$ which implies that $U \subset S$ is
scheme theoretically dense, see
Morphisms, Example \ref{morphisms-example-scheme-theoretic-closure}.
\end{proof}

\begin{lemma}
\label{lemma-effective-Cartier-makes-dimension-drop}
Let $S$ be a scheme.
 ========

regex_match $S$
nn candidate a scheme
nn a scheme
new_annotation [(71527, 71530, 'VAR'), (71534, 71542, 'TYPE')]
===== sent Let $D \subset S$ be an effective Cartier divisor.
 ========

regex_match $D \subset S$
nn candidate $D \subset S$
nn candidate an effective Cartier divisor
nn an effective Cartier divisor
new_annotation [(71548, 71561, 'VAR'), (71565, 71593, 'TYPE')]
===== sent Let $s \in ========

new_annotation []
===== sent Assume $\dim_s(S) < \infty$.
Let $U = \Spec(A) \subset S$ be an affine open neighbourhood
of $s$ such that $\dim(U) = \dim_s(S)$ and such that $D = V(f)$
for some nonzerodivisor $f \in A$ (see
Lemma \ref{lemma-characterize-effective-Cartier-divisor}).
 ========

regex_match $U = \Spec(A) \subset S$
nn candidate $U
nn candidate an affine open neighbourhood
nn an affine open neighbourhood
new_annotation [(71725, 71749, 'VAR'), (71753, 71781, 'TYPE')]
===== sent \end{proof}

\begin{definition}
\label{definition-sum-effective-Cartier-divisors}
Let $S$ be a scheme. ========

regex_match $S$
nn candidate a scheme
nn a scheme
new_annotation [(72341, 72344, 'VAR'), (72348, 72356, 'TYPE')]
===== sent \end{proof}

\begin{lemma}
\label{lemma-difference-effective-Cartier-divisors}
Let $X$ be a scheme.
 ========

regex_match $X$
nn candidate a scheme
nn a scheme
new_annotation [(73180, 73183, 'VAR'), (73187, 73195, 'TYPE')]
===== sent Let $D, D'$ be two effective Cartier divisors on $X$.
If $D \subset D'$ (as closed subschemes of $X$), then
there exists an effective Cartier divisor $D''$ such
that $D' = D + D''$.
\end{lemma}

\begin{proof}
Omitted.
 ========

new_annotation []
===== sent \end{proof}

\begin{lemma}
\label{lemma-sum-closed-subschemes-effective-Cartier}
Let $X$ be a scheme. ========

regex_match $X$
nn candidate a scheme
nn a scheme
new_annotation [(73500, 73503, 'VAR'), (73507, 73515, 'TYPE')]
===== sent Let $Z, Y$ be two closed subschemes of $X$
with ideal sheaves $\mathcal{I}$ and $\mathcal{J}$. If $\mathcal{I}\mathcal{J}$
defines an effective Cartier divisor $D \subset X$, then $Z$ and $Y$
are effective Cartier divisors and $D = Z + Y$.
\end{lemma}

\begin{proof}
 ========

new_annotation []
===== sent Applying Lemma \ref{lemma-characterize-effective-Cartier-divisor} we obtain
the following algebra situation: $A$ is a ring, $I, J \subset A$
ideals and $f \in A$ a nonzerodivisor such that $IJ = (f)$.
Thus the result follows from
Algebra, Lemma \ref{algebra-lemma-product-ideals-principal}.
\end{proof}

\begin{lemma}
\label{lemma-sum-effective-Cartier-divisors-union}
Let $X$ be a scheme. ========

regex_match $X$
nn candidate a scheme
nn a scheme
new_annotation [(74157, 74160, 'VAR'), (74164, 74172, 'TYPE')]
===== sent Let $D, D' \subset X$ be effective Cartier divisors
such that the scheme theoretic intersection $D \cap D'$ is an effective
Cartier divisor on $D'$. Then $D + D'$ is the scheme theoretic
union of $D$ and $D'$.
\end{lemma}

\begin{proof}
See Morphisms, Definition
\ref{morphisms-definition-scheme-theoretic-intersection-union}
for the definition of scheme theoretic intersection and union.
 ========

new_annotation []
===== sent \end{proof}

\noindent
Recall that we have defined the inverse image of a closed subscheme
under any morphism of schemes in
Schemes, Definition \ref{schemes-definition-inverse-image-closed-subscheme}.

\begin{lemma}
\label{lemma-pullback-locally-principal}
Let $f : S' \to S$ be a morphism of schemes. ========

regex_match $f : S' \to S$
nn candidate S
nn candidate a morphism
nn a morphism
new_annotation [(75149, 75163, 'VAR'), (75167, 75177, 'TYPE')]
===== sent Let $Z \subset S$
be a locally principal closed subscheme. ========

new_annotation []
===== sent \end{proof}

\begin{definition}
\label{definition-pullback-effective-Cartier-divisor}
Let $f : S' \to S$ be a morphism of schemes. ========

regex_match $f : S' \to S$
nn candidate S
nn candidate a morphism
nn a morphism
new_annotation [(75459, 75473, 'VAR'), (75477, 75487, 'TYPE')]
===== sent Let $D \subset S$
be an effective Cartier divisor. ========

new_annotation []
===== sent \begin{lemma}
\label{lemma-pullback-effective-Cartier-defined}
Let $f : X \to Y$ be a morphism of schemes.
 ========

regex_match $f : X \to Y$
nn candidate X \to
nn candidate a morphism
nn a morphism
new_annotation [(76022, 76035, 'VAR'), (76039, 76049, 'TYPE')]
===== sent Let $D \subset Y$ be an effective Cartier divisor.
 ========

regex_match $D \subset Y$
nn candidate $D \subset Y$
nn candidate an effective Cartier divisor
nn an effective Cartier divisor
new_annotation [(76066, 76079, 'VAR'), (76083, 76111, 'TYPE')]
===== sent \end{proof}

\begin{lemma}
\label{lemma-pullback-effective-Cartier-divisors-additive}
Let $f : S' \to S$ be a morphism of schemes.
 ========

regex_match $f : S' \to S$
nn candidate S
nn candidate a morphism
nn a morphism
new_annotation [(77800, 77814, 'VAR'), (77818, 77828, 'TYPE')]
===== sent Let $D_1$, $D_2$ be effective Cartier divisors on $S$.
If the pullbacks of $D_1$ and $D_2$ are defined then the
pullback of $D = D_1 + D_2$ is defined and
$f^*D = f^*D_1 + f^*D_2$.
\end{lemma}

\begin{proof}
Omitted.
 ========

new_annotation []
===== sent \begin{definition}
\label{definition-invertible-sheaf-effective-Cartier-divisor}
Let $S$ be a scheme. ========

regex_match $S$
nn candidate a scheme
nn a scheme
new_annotation [(78436, 78439, 'VAR'), (78443, 78451, 'TYPE')]
===== sent Let $D \subset S$ be an effective Cartier divisor
with ideal sheaf $\mathcal{I}_D$.
\begin{enumerate}
\item The {\it invertible sheaf $\mathcal{O}_S(D)$ associated to $D$}
is defined by
$$
\mathcal{O}_S(D) =
\SheafHom_{\mathcal{O}_S}(\mathcal{I}_D, \mathcal{O}_S) =
\mathcal{I}_D^{\otimes -1}.
$$
\item The {\it canonical section}, usually denoted $1$ or $1_D$, is the
global section of $\mathcal{O}_S(D)$ corresponding to
the inclusion mapping $\mathcal{I}_D \to \mathcal{O}_S$.
\item We write
$\mathcal{O}_S(-D) = \mathcal{O}_S(D)^{\otimes -1} = \mathcal{I}_D$.
\item Given a second effective Cartier divisor $D' \subset S$ we define
$\mathcal{O}_S(D - D') =
\mathcal{O}_S(D) \otimes_{\mathcal{O}_S} \mathcal{O}_S(-D')$.
\end{enumerate}
\end{definition}

\noindent
Some comments. ========

regex_match $D \subset S$
nn candidate $D \subset S$
nn candidate an effective Cartier divisor
nn an effective Cartier divisor
new_annotation [(78457, 78470, 'VAR'), (78474, 78502, 'TYPE')]
===== sent In other words, when we write $\mathcal{O}_S(D - D')$
we may think of $D - D'$ as an element of $\text{EffCart}(S)^{gp}$.

\begin{lemma}
\label{lemma-conormal-effective-Cartier-divisor}
Let $S$ be a scheme and let $D \subset S$ be an effective Cartier divisor.
 ========

regex_match $S$
nn candidate a scheme
nn a scheme
new_annotation [(80197, 80200, 'VAR'), (80204, 80212, 'TYPE')]
===== sent In other words, when we write $\mathcal{O}_S(D - D')$
we may think of $D - D'$ as an element of $\text{EffCart}(S)^{gp}$.

\begin{lemma}
\label{lemma-conormal-effective-Cartier-divisor}
Let $S$ be a scheme and let $D \subset S$ be an effective Cartier divisor.
 ========

regex_match $D \subset S$
nn candidate $D \subset S$
nn candidate an effective Cartier divisor
nn an effective Cartier divisor
new_annotation [(80221, 80234, 'VAR'), (80238, 80266, 'TYPE')]
===== sent \end{proof}

\begin{lemma}
\label{lemma-ses-add-divisor}
Let $X$ be a scheme. ========

regex_match $X$
nn candidate a scheme
nn a scheme
new_annotation [(80520, 80523, 'VAR'), (80527, 80535, 'TYPE')]
===== sent Let $D, C \subset X$ be
effective Cartier divisors with $C \subset D$ and let $D' = D + C$.
 ========

new_annotation []
===== sent Let $D, C \subset X$ be
effective Cartier divisors with $C \subset D$ and let $D' = D + C$.
 ========

new_annotation []
===== sent In the statement of the lemma and in the proof we use the equivalence of
Morphisms, Lemma \ref{morphisms-lemma-i-star-equivalence} to think of
quasi-coherent modules on closed subschemes of $X$
as quasi-coherent modules on $X$. Let $\mathcal{I}$ be the ideal
sheaf of $D$ in $D'$. Then there is a short exact sequence
$$
0 \to \mathcal{I} \to \mathcal{O}_{D'} \to \mathcal{O}_D \to 0
$$
because $D \to D'$ is a closed immersion. ========

regex_match $\mathcal{I}$
nn candidate the ideal
sheaf
nn the ideal
sheaf
new_annotation [(81031, 81044, 'VAR'), (81048, 81063, 'TYPE')]
===== sent \end{proof}

\begin{lemma}
\label{lemma-invertible-sheaf-sum-effective-Cartier-divisors}
Let $S$ be a scheme.
 ========

regex_match $S$
nn candidate a scheme
nn a scheme
new_annotation [(82198, 82201, 'VAR'), (82205, 82213, 'TYPE')]
===== sent Let $D_1$, $D_2$ be effective Cartier divisors on $S$.
Let $D = D_1 + D_2$.
Then there is a unique isomorphism
$$
\mathcal{O}_S(D_1) \otimes_{\mathcal{O}_S} \mathcal{O}_S(D_2)
\longrightarrow
\mathcal{O}_S(D)
$$
which maps $1_{D_1} \otimes 1_{D_2}$ to $1_D$.
\end{lemma}

\begin{proof}
Omitted.
 ========

new_annotation []
===== sent Let $D_1$, $D_2$ be effective Cartier divisors on $S$.
Let $D = D_1 + D_2$.
Then there is a unique isomorphism
$$
\mathcal{O}_S(D_1) \otimes_{\mathcal{O}_S} \mathcal{O}_S(D_2)
\longrightarrow
\mathcal{O}_S(D)
$$
which maps $1_{D_1} \otimes 1_{D_2}$ to $1_D$.
\end{lemma}

\begin{proof}
Omitted.
 ========

new_annotation []
===== sent \end{proof}

\begin{lemma}
\label{lemma-pullback-effective-Cartier-divisors}
Let $f : S' \to S$ be a morphism of schemes.
 ========

regex_match $f : S' \to S$
nn candidate S
nn candidate a morphism
nn a morphism
new_annotation [(82591, 82605, 'VAR'), (82609, 82619, 'TYPE')]
===== sent Let $D$ be a effective Cartier divisors on $S$.
If the pullback of $D$ is defined then
$f^*\mathcal{O}_S(D) = \mathcal{O}_{S'}(f^*D)$
and the canonical section $1_D$ pulls back to
the canonical section $1_{f^*D}$.
\end{lemma}

\begin{proof}
Omitted.
 ========

regex_match $D$
nn candidate a effective Cartier divisors
nn a effective Cartier divisors
new_annotation [(82636, 82639, 'VAR'), (82643, 82671, 'TYPE')]
===== sent \end{proof}

\begin{definition}
\label{definition-regular-section}
Let $(X, \mathcal{O}_X)$ be a locally ringed space.
 ========

regex_match $(X, \mathcal{O}_X)$
nn candidate $(X
nn candidate a locally ringed space
nn a locally ringed space
new_annotation [(82953, 82973, 'VAR'), (82977, 82999, 'TYPE')]
===== sent Let $\mathcal{L}$ be an invertible sheaf on $X$.
A global section $s \in \Gamma(X, \mathcal{L})$ is called a
{\it regular section} if the map $\mathcal{O}_X \to \mathcal{L}$,
$f \mapsto fs$ is injective.
 ========

regex_match $\mathcal{L}$
nn candidate an invertible sheaf
nn an invertible sheaf
new_annotation [(83005, 83018, 'VAR'), (83022, 83041, 'TYPE')]
===== sent \end{definition}

\begin{lemma}
\label{lemma-regular-section-structure-sheaf}
Let $X$ be a locally ringed space. ========

regex_match $X$
nn candidate a locally ringed space
nn a locally ringed space
new_annotation [(83287, 83290, 'VAR'), (83294, 83316, 'TYPE')]
===== sent Let $f \in \Gamma(X, \mathcal{O}_X)$.
The following are equivalent:
\begin{enumerate}
\item $f$ is a regular section, and
\item for any $x \in X$ the image $f \in \mathcal{O}_{X, x}$
is a nonzerodivisor.
 ========

new_annotation []
===== sent \end{proof}

\noindent
Note that a global section $s$ of an invertible $\mathcal{O}_X$-module
$\mathcal{L}$ may be seen as an $\mathcal{O}_X$-module map
$s : \mathcal{O}_X \to \mathcal{L}$. Its dual is therefore a
map $s : \mathcal{L}^{\otimes -1} \to \mathcal{O}_X$.
(See Modules, Definition \ref{modules-definition-powers}
for the definition of the dual invertible sheaf.)

\begin{definition}
\label{definition-zero-scheme-s}
Let $X$ be a scheme. ========

regex_match $X$
nn candidate a scheme
nn a scheme
new_annotation [(84327, 84330, 'VAR'), (84334, 84342, 'TYPE')]
===== sent Let $\mathcal{L}$ be an invertible sheaf.
 ========

regex_match $\mathcal{L}$
nn candidate an invertible sheaf
nn an invertible sheaf
new_annotation [(84348, 84361, 'VAR'), (84365, 84384, 'TYPE')]
===== sent Let $s \in \Gamma(X, \mathcal{L})$ be a global section.
 ========

regex_match $s \in \Gamma(X, \mathcal{L})$
nn candidate $s \in \Gamma(X
nn candidate a global section
nn a global section
new_annotation [(84390, 84420, 'VAR'), (84424, 84440, 'TYPE')]
===== sent The {\it zero scheme} of $s$ is the closed subscheme $Z(s) \subset X$
defined by the quasi-coherent sheaf of ideals
$\mathcal{I} \subset \mathcal{O}_X$ which is the image of the
map $s : \mathcal{L}^{\otimes -1} \to \mathcal{O}_X$.
\end{definition}

\begin{lemma}
\label{lemma-zero-scheme}
Let $X$ be a scheme.
 ========

regex_match $X$
nn candidate a scheme
nn a scheme
new_annotation [(84736, 84739, 'VAR'), (84743, 84751, 'TYPE')]
===== sent Let $\mathcal{L}$ be an invertible sheaf.
 ========

regex_match $\mathcal{L}$
nn candidate an invertible sheaf
nn an invertible sheaf
new_annotation [(84757, 84770, 'VAR'), (84774, 84793, 'TYPE')]
===== sent Let $s \in \Gamma(X, \mathcal{L})$.
\begin{enumerate}
\item Consider closed immersions $i : Z \to X$ such that
$i^*s \in \Gamma(Z, i^*\mathcal{L})$ is zero
ordered by inclusion. ========

new_annotation []
===== sent \end{slogan}
Let $X$ be a scheme.
 ========

regex_match $X$
nn candidate a scheme
nn a scheme
new_annotation [(85620, 85623, 'VAR'), (85627, 85635, 'TYPE')]
===== sent \end{proof}

\begin{remark}
\label{remark-ses-regular-section}
Let $X$ be a scheme, $\mathcal{L}$ an invertible $\mathcal{O}_X$-module,
and $s$ a regular section of $\mathcal{L}$. Then the zero scheme
$D = Z(s)$ is an effective Cartier divisor on $X$ and there are
short exact sequences
$$
0 \to \mathcal{O}_X \to \mathcal{L} \to i_*(\mathcal{L}|_D) \to 0
\quad\text{and}\quad
0 \to \mathcal{L}^{\otimes -1} \to \mathcal{O}_X \to i_*\mathcal{O}_D \to 0.
 ========

regex_match $X$
nn candidate a scheme
nn a scheme
new_annotation [(86533, 86536, 'VAR'), (86540, 86548, 'TYPE')]
===== sent \begin{lemma}
\label{lemma-regular-section-associated-points}
Let $X$ be a locally Noetherian scheme. ========

regex_match $X$
nn candidate a locally Noetherian scheme
nn a locally Noetherian scheme
new_annotation [(87573, 87576, 'VAR'), (87580, 87607, 'TYPE')]
===== sent Let $\mathcal{L}$ be an invertible
$\mathcal{O}_X$-module. ========

regex_match $\mathcal{L}$
nn candidate an invertible
$\mathcal{O}_X$-module
nn an invertible
$\mathcal{O}_X$-module
new_annotation [(87613, 87626, 'VAR'), (87630, 87666, 'TYPE')]
===== sent Let $s \in \Gamma(X, \mathcal{L})$. Then $s$
is a regular section if and only if $s$ does not vanish in the associated
points of $X$.
\end{lemma}

\begin{proof}
Omitted. ========

new_annotation []
===== sent Hint: reduce to the affine case and $\mathcal{L}$ trivial
and then use Lemma \ref{lemma-regular-section-structure-sheaf} and
Algebra, Lemma \ref{algebra-lemma-ass-zero-divisors}.
\end{proof}

\begin{lemma}
\label{lemma-effective-Cartier-in-points}
Let $X$ be a locally Noetherian scheme. ========

regex_match $X$
nn candidate a locally Noetherian scheme
nn a locally Noetherian scheme
new_annotation [(88090, 88093, 'VAR'), (88097, 88124, 'TYPE')]
===== sent Let $D \subset X$ be a closed subscheme
corresponding to the quasi-coherent ideal sheaf
$\mathcal{I} \subset \mathcal{O}_X$.
\begin{enumerate}
\item If for every $x \in D$ the ideal
$\mathcal{I}_x \subset \mathcal{O}_{X, x}$
can be generated by one element, then $D$ is locally principal.
 ========

regex_match $D \subset X$
nn candidate $D \subset X$
nn candidate a closed subscheme
nn a closed subscheme
new_annotation [(88130, 88143, 'VAR'), (88147, 88165, 'TYPE')]
===== sent \end{enumerate}
\end{lemma}

\begin{proof}
Let $\Spec(A)$ be an affine neighbourhood of a point $x \in D$.
Let $\mathfrak p \subset A$ be the prime corresponding to $x$.
Let $I \subset A$ be the ideal defining the trace of $D$ on
$\Spec(A)$. Since $A$ is Noetherian (as $X$ is Noetherian)
the ideal $I$ is generated by finitely many elements, say
$I = (f_1, \ldots, f_r)$. Under the assumption of (1) we have
$I_\mathfrak p = (f)$ for some $f \in A_\mathfrak p$.
Then $f_i = g_i f$ for some $g_i \in A_\mathfrak p$.
 ========

regex_match $\Spec(A)$
nn candidate an affine neighbourhood
nn an affine neighbourhood
new_annotation [(88631, 88641, 'VAR'), (88645, 88668, 'TYPE')]
===== sent \end{enumerate}
\end{lemma}

\begin{proof}
Let $\Spec(A)$ be an affine neighbourhood of a point $x \in D$.
Let $\mathfrak p \subset A$ be the prime corresponding to $x$.
Let $I \subset A$ be the ideal defining the trace of $D$ on
$\Spec(A)$. Since $A$ is Noetherian (as $X$ is Noetherian)
the ideal $I$ is generated by finitely many elements, say
$I = (f_1, \ldots, f_r)$. Under the assumption of (1) we have
$I_\mathfrak p = (f)$ for some $f \in A_\mathfrak p$.
Then $f_i = g_i f$ for some $g_i \in A_\mathfrak p$.
 ========

regex_match $\mathfrak p \subset A$
nn candidate p \subset
nn candidate the prime corresponding
nn the prime corresponding
new_annotation [(88695, 88718, 'VAR'), (88722, 88745, 'TYPE')]
===== sent \end{enumerate}
\end{lemma}

\begin{proof}
Let $\Spec(A)$ be an affine neighbourhood of a point $x \in D$.
Let $\mathfrak p \subset A$ be the prime corresponding to $x$.
Let $I \subset A$ be the ideal defining the trace of $D$ on
$\Spec(A)$. Since $A$ is Noetherian (as $X$ is Noetherian)
the ideal $I$ is generated by finitely many elements, say
$I = (f_1, \ldots, f_r)$. Under the assumption of (1) we have
$I_\mathfrak p = (f)$ for some $f \in A_\mathfrak p$.
Then $f_i = g_i f$ for some $g_i \in A_\mathfrak p$.
 ========

regex_match $I \subset A$
nn candidate I
nn candidate the ideal
nn the ideal
new_annotation [(88758, 88771, 'VAR'), (88775, 88784, 'TYPE')]
===== sent \end{proof}

\begin{lemma}
\label{lemma-effective-Cartier-codimension-1}
Let $X$ be a locally Noetherian scheme.
 ========

regex_match $X$
nn candidate a locally Noetherian scheme
nn a locally Noetherian scheme
new_annotation [(89654, 89657, 'VAR'), (89661, 89688, 'TYPE')]
===== sent \begin{enumerate}
\item Let $D \subset X$ be a locally principal closed subscheme.
 ========

regex_match $D \subset X$
nn candidate $D \subset X$
nn candidate principal
nn principal
new_annotation [(89718, 89731, 'VAR'), (89745, 89754, 'TYPE')]
===== sent Let $\xi \in D$ be a generic point of an irreducible component of $D$.
Then $\dim(\mathcal{O}_{X, \xi}) \leq 1$.
\item ========

regex_match $\xi \in D$
nn candidate a generic point
nn a generic point
new_annotation [(89777, 89788, 'VAR'), (89792, 89807, 'TYPE')]
===== sent Let $D \subset X$ be an effective Cartier divisor.
 ========

regex_match $D \subset X$
nn candidate $D \subset X$
nn candidate an effective Cartier divisor
nn an effective Cartier divisor
new_annotation [(89896, 89909, 'VAR'), (89913, 89941, 'TYPE')]
===== sent Let $\xi \in D$ be a generic point of an irreducible component of $D$.
Then $\dim(\mathcal{O}_{X, \xi}) = 1$.
\end{enumerate}
\end{lemma}

\begin{proof}
Proof of (1). ========

regex_match $\xi \in D$
nn candidate a generic point
nn a generic point
new_annotation [(89947, 89958, 'VAR'), (89962, 89977, 'TYPE')]
===== sent Let $\xi$ correspond to the prime ideal $\mathfrak p \subset A$.
 ========

new_annotation []
===== sent \end{proof}

\begin{lemma}
\label{lemma-integral-effective-Cartier-divisor-dvr}
Let $X$ be a Noetherian scheme. ========

regex_match $X$
nn candidate a Noetherian scheme
nn a Noetherian scheme
new_annotation [(91006, 91009, 'VAR'), (91013, 91032, 'TYPE')]
===== sent Let $D \subset X$ be an
integral closed subscheme which is also an
effective Cartier divisor. ========

new_annotation []
===== sent Thus it is a discrete valuation ring by
Algebra, Lemma \ref{algebra-lemma-characterize-dvr}.
\end{proof}

\begin{lemma}
\label{lemma-effective-Cartier-divisor-Sk}
Let $X$ be a locally Noetherian scheme. ========

regex_match $X$
nn candidate a locally Noetherian scheme
nn a locally Noetherian scheme
new_annotation [(91801, 91804, 'VAR'), (91808, 91835, 'TYPE')]
===== sent Let $D \subset X$ be an
effective Cartier divisor. ========

new_annotation []
===== sent If $X$ is $(S_k)$, then $D$ is $(S_{k - 1})$.
\end{lemma}

\begin{proof}
Let $x \in ========

new_annotation []
===== sent \end{proof}

\begin{lemma}
\label{lemma-normal-effective-Cartier-divisor-S1}
Let $X$ be a locally Noetherian normal scheme. ========

regex_match $X$
nn candidate a locally Noetherian normal scheme
nn a locally Noetherian normal scheme
new_annotation [(92618, 92621, 'VAR'), (92625, 92659, 'TYPE')]
===== sent Let $D \subset X$ be an
effective Cartier divisor. ========

new_annotation []
===== sent Then $D$ is $(S_1)$.
\end{lemma}

\begin{proof}
By Properties, Lemma \ref{properties-lemma-criterion-normal}
we see that $X$ is $(S_2)$. Thus we conclude by
Lemma \ref{lemma-effective-Cartier-divisor-Sk}.
\end{proof}

\begin{lemma}
\label{lemma-weil-divisor-is-cartier-UFD}
Let $X$ be a Noetherian scheme. ========

regex_match $X$
nn candidate a Noetherian scheme
nn a Noetherian scheme
new_annotation [(92990, 92993, 'VAR'), (92997, 93016, 'TYPE')]
===== sent Let $D \subset X$ be a integral
closed subscheme. ========

regex_match $D \subset X$
nn candidate $D \subset X$
nn candidate a integral
closed subscheme
nn a integral
closed subscheme
new_annotation [(93022, 93035, 'VAR'), (93039, 93066, 'TYPE')]
===== sent \end{lemma}

\begin{proof}
Let $x \in D$ and set $A = \mathcal{O}_{X, x}$. Let $\mathfrak p \subset A$
correspond to the generic point of $D$. Then $A_\mathfrak p$ has dimension
$1$ by assumption (1). ========

new_annotation []
===== sent \end{lemma}

\begin{proof}
Let $x \in D$ and set $A = \mathcal{O}_{X, x}$. Let $\mathfrak p \subset A$
correspond to the generic point of $D$. Then $A_\mathfrak p$ has dimension
$1$ by assumption (1). ========

new_annotation []
===== sent Of course $f$ is a nonzerodivisor and we conclude by
Lemma \ref{lemma-effective-Cartier-in-points}.
\end{proof}

\begin{lemma}
\label{lemma-codim-1-part}
Let $X$ be a Noetherian scheme. ========

regex_match $X$
nn candidate a Noetherian scheme
nn a Noetherian scheme
new_annotation [(93740, 93743, 'VAR'), (93747, 93766, 'TYPE')]
===== sent Let $Z \subset X$ be a closed subscheme.
 ========

regex_match $Z \subset X$
nn candidate $
nn candidate a closed subscheme
nn a closed subscheme
new_annotation [(93772, 93785, 'VAR'), (93789, 93807, 'TYPE')]
===== sent \end{lemma}

\begin{proof}
Let $\xi_i \in D_i$ be the generic point and let
$\mathcal{O}_i = \mathcal{O}_{X, \xi_i}$ be the local ring
which is a discrete valuation ring by
Lemma \ref{lemma-integral-effective-Cartier-divisor-dvr}.
Let $a_i \geq 0$ be the minimal valuation of an element of
$\mathcal{I}_{Z, \xi_i} \subset \mathcal{O}_i$.
We claim that the effective Cartier divisor $D = \sum a_i D_i$ works.

 ========

regex_match $\xi_i \in D_i$
nn candidate the generic point
nn the generic point
new_annotation [(94309, 94324, 'VAR'), (94328, 94345, 'TYPE')]
===== sent \end{lemma}

\begin{proof}
Let $\xi_i \in D_i$ be the generic point and let
$\mathcal{O}_i = \mathcal{O}_{X, \xi_i}$ be the local ring
which is a discrete valuation ring by
Lemma \ref{lemma-integral-effective-Cartier-divisor-dvr}.
Let $a_i \geq 0$ be the minimal valuation of an element of
$\mathcal{I}_{Z, \xi_i} \subset \mathcal{O}_i$.
We claim that the effective Cartier divisor $D = \sum a_i D_i$ works.

 ========

regex_match $a_i \geq 0$
nn candidate the minimal valuation
nn the minimal valuation
new_annotation [(94513, 94525, 'VAR'), (94529, 94550, 'TYPE')]
===== sent Namely, suppose that $x \in X$. Let $A = \mathcal{O}_{X, x}$.
Let $f_i \in A$ be a local equation for $D_i$;
we only consider those $i$ such that $x \in ========

new_annotation []
===== sent Namely, suppose that $x \in X$. Let $A = \mathcal{O}_{X, x}$.
Let $f_i \in A$ be a local equation for $D_i$;
we only consider those $i$ such that $x \in ========

regex_match $f_i \in A$
nn candidate a local equation
nn a local equation
new_annotation [(94771, 94782, 'VAR'), (94786, 94802, 'TYPE')]
===== sent \end{proof}

\begin{lemma}
\label{lemma-codimension-1-is-effective-Cartier}
Let $Z \subset X$ be a closed subscheme of a Noetherian scheme. ========

regex_match $Z \subset X$
nn candidate $Z \subset
nn candidate a closed subscheme
nn a closed subscheme
new_annotation [(95788, 95801, 'VAR'), (95805, 95823, 'TYPE')]
===== sent \end{lemma}

\begin{proof}
Let $D = \sum a_i D_i$ be as in Lemma \ref{lemma-codim-1-part}
where $D_i \subset Z$ are the irreducible components of $Z$.
If $D \to Z$ is not an isomorphism, then $\mathcal{O}_Z \to \mathcal{O}_D$
has a nonzero kernel sitting in codimension $\geq 2$. ========

new_annotation []
===== sent \end{proof}

\begin{lemma}
\label{lemma-UFD-one-equation-CM}
Let $R$ be a Noetherian UFD. ========

regex_match $R$
nn candidate a Noetherian UFD
nn a Noetherian UFD
new_annotation [(96576, 96579, 'VAR'), (96583, 96599, 'TYPE')]
===== sent Let $I \subset R$ be an ideal
such that $R/I$ has no embedded primes and such that
every minimal prime over $I$ has height $1$.
 ========

regex_match $I \subset R$
nn candidate $
nn candidate I
nn candidate R/I$
nn R/I$
new_annotation [(96605, 96618, 'VAR'), (96642, 96646, 'TYPE')]
===== sent \end{proof}

\begin{lemma}
\label{lemma-effective-Cartier-divisor-is-a-sum}
Let $X$ be a Noetherian scheme. ========

regex_match $X$
nn candidate a Noetherian scheme
nn a Noetherian scheme
new_annotation [(97087, 97090, 'VAR'), (97094, 97113, 'TYPE')]
===== sent Let $D \subset X$ be an effective
Cartier divisor. ========

regex_match $D \subset X$
nn candidate $D \subset X$
nn candidate an effective
Cartier divisor
nn an effective
Cartier divisor
new_annotation [(97119, 97132, 'VAR'), (97136, 97164, 'TYPE')]
===== sent \end{lemma}

\begin{proof}
Choose $a_i$ as in Lemma \ref{lemma-codim-1-part} and set $D' = \sum a_i D_i$.
Then $D' \to D$ is an inclusion of effective Cartier divisors which
is an isomorphism away from codimension $2$ on $X$. Pick $x \in X$.
Set $A = \mathcal{O}_{X, x}$ and let $f, f' \in A$ be the nonzerodivisor
generating the ideal of $D, D'$ in $A$. ========

regex_match $f, f' \in A$
nn candidate the nonzerodivisor
nn the nonzerodivisor
new_annotation [(97735, 97748, 'VAR'), (97752, 97770, 'TYPE')]
===== sent \end{slogan}
Let $X$ be a Noetherian scheme which has an ample invertible sheaf.
 ========

regex_match $X$
nn candidate a Noetherian scheme
nn a Noetherian scheme
new_annotation [(98302, 98305, 'VAR'), (98309, 98328, 'TYPE')]
===== sent Then every invertible $\mathcal{O}_X$-module is isomorphic to
$$
\mathcal{O}_X(D - D') =
\mathcal{O}_X(D) \otimes_{\mathcal{O}_X} \mathcal{O}_X(D')^{\otimes -1}
$$
for some effective Cartier divisors $D, D'$ in $X$.
\end{lemma}

\begin{proof}
Let $x_1, \ldots, x_n$ be the associated points of $X$
(Lemma \ref{lemma-finite-ass}). ========

regex_match $x_1, \ldots, x_n$
nn candidate the associated points
nn the associated points
new_annotation [(98613, 98631, 'VAR'), (98635, 98656, 'TYPE')]
===== sent Let $\mathcal{L}$ be an ample invertible sheaf.
 ========

regex_match $\mathcal{L}$
nn candidate an ample invertible sheaf
nn an ample invertible sheaf
new_annotation [(98700, 98713, 'VAR'), (98717, 98742, 'TYPE')]
===== sent Let $\mathfrak p_1, \ldots, \mathfrak ========

new_annotation []
===== sent e} \otimes \mathcal{N}$.
Thus both $\mathcal{L}^{\otimes e} \otimes \mathcal{N}$ and
$\mathcal{L}^{\otimes e}$ are invertible sheaves which have global sections
which generate the stalks at the associated points of $X$.
Thus these are regular sections by
Lemma \ref{lemma-regular-section-associated-points}.
Hence $\mathcal{L}^{\otimes e} \otimes \mathcal{N} \cong \mathcal{O}_X(D)$
and $\mathcal{L}^{\otimes e} \cong \mathcal{O}_X(D')$ for some
effective Cartier divisors, see Lemma \ref{lemma-characterize-OD}.
\end{proof}

\begin{lemma}
\label{lemma-wedge-product-ses}
Let $X$ be an integral regular scheme of dimension $2$.
Let $i : D \to X$ be the immersion of an effective Cartier divisor.
 ========

regex_match $X$
nn candidate an integral regular scheme
nn an integral regular scheme
new_annotation [(100523, 100526, 'VAR'), (100530, 100556, 'TYPE')]
===== sent e} \otimes \mathcal{N}$.
Thus both $\mathcal{L}^{\otimes e} \otimes \mathcal{N}$ and
$\mathcal{L}^{\otimes e}$ are invertible sheaves which have global sections
which generate the stalks at the associated points of $X$.
Thus these are regular sections by
Lemma \ref{lemma-regular-section-associated-points}.
Hence $\mathcal{L}^{\otimes e} \otimes \mathcal{N} \cong \mathcal{O}_X(D)$
and $\mathcal{L}^{\otimes e} \cong \mathcal{O}_X(D')$ for some
effective Cartier divisors, see Lemma \ref{lemma-characterize-OD}.
\end{proof}

\begin{lemma}
\label{lemma-wedge-product-ses}
Let $X$ be an integral regular scheme of dimension $2$.
Let $i : D \to X$ be the immersion of an effective Cartier divisor.
 ========

regex_match $i : D \to X$
nn candidate D \to X$
nn candidate the immersion
nn the immersion
new_annotation [(100579, 100592, 'VAR'), (100596, 100609, 'TYPE')]
===== sent Let $\mathcal{F} \to \mathcal{F}' \to i_*\mathcal{G} \to 0$
be an exact sequence of coherent $\mathcal{O}_X$-modules.
 ========

new_annotation []
===== sent To finish the proof, it suffices to see that
$\sigma$ viewed as a global section of
$\mathcal{L}' \otimes \mathcal{L}^{\otimes -1}$ does not
vanish at any codimension point of $X$, except at the generic
point of $D$ and there with vanishing order at most $\min(s, r)$.

\medskip\noindent
Translated into algebra, we arrive at the following problem:
Let $(A, \mathfrak m, \kappa)$ be a discrete valuation ring
with fraction field $K$. Let $M \to M' \to N \to 0$ be an exact sequence
of finite $A$-modules with $\dim_K(M \otimes K) = \dim_K(M' \otimes K) = r$
and with $N \cong \kappa^{\oplus s}$. Show that the induced map
$L = \wedge^r(M)^{**} \to L' = \wedge^r(M')^{**}$ vanishes to
order at most $\min(s, r)$. We will use the structure theorem for
modules over $A$, see
More on Algebra, Lemma
\ref{more-algebra-lemma-generalized-valuation-ring-modules} or
\ref{more-algebra-lemma-modules-PID}.
Dividing out a finite $A$-module by a torsion submodule does not
change the double dual.
 ========

regex_match $(A, \mathfrak m, \kappa)$
nn candidate A
nn candidate a discrete valuation ring
nn a discrete valuation ring
new_annotation [(101959, 101985, 'VAR'), (101989, 102014, 'TYPE')]
===== sent To finish the proof, it suffices to see that
$\sigma$ viewed as a global section of
$\mathcal{L}' \otimes \mathcal{L}^{\otimes -1}$ does not
vanish at any codimension point of $X$, except at the generic
point of $D$ and there with vanishing order at most $\min(s, r)$.

\medskip\noindent
Translated into algebra, we arrive at the following problem:
Let $(A, \mathfrak m, \kappa)$ be a discrete valuation ring
with fraction field $K$. Let $M \to M' \to N \to 0$ be an exact sequence
of finite $A$-modules with $\dim_K(M \otimes K) = \dim_K(M' \otimes K) = r$
and with $N \cong \kappa^{\oplus s}$. Show that the induced map
$L = \wedge^r(M)^{**} \to L' = \wedge^r(M')^{**}$ vanishes to
order at most $\min(s, r)$. We will use the structure theorem for
modules over $A$, see
More on Algebra, Lemma
\ref{more-algebra-lemma-generalized-valuation-ring-modules} or
\ref{more-algebra-lemma-modules-PID}.
Dividing out a finite $A$-module by a torsion submodule does not
change the double dual.
 ========

regex_match $M \to M' \to N \to 0$
nn candidate $M
nn candidate an exact sequence
nn an exact sequence
new_annotation [(102044, 102066, 'VAR'), (102070, 102087, 'TYPE')]
===== sent In this section we discuss the result that the complement of an
affine open in a variety has pure codimension $1$.

\begin{lemma}
\label{lemma-affine-punctured-spec}
Let $(A, \mathfrak m)$ be a Noetherian local ring.
 ========

regex_match $(A, \mathfrak m)$
nn candidate \mathfrak
nn candidate a Noetherian local ring
nn a Noetherian local ring
new_annotation [(103380, 103398, 'VAR'), (103402, 103425, 'TYPE')]
===== sent Moreover, the same holds for any ring $A'$ which is
the target of a local homomorphism of local rings $A \to A'$ such that
$\mathfrak m_{A'} = \sqrt{\mathfrak mA'}$.
\end{remark}

\begin{lemma}
\label{lemma-complement-affine-open-immersion}
\begin{reference}
\cite[EGA IV, Corollaire 21.12.7]{EGA4}
\end{reference}
Let $X$ be a locally Noetherian scheme. ========

regex_match $X$
nn candidate a locally Noetherian scheme
nn a locally Noetherian scheme
new_annotation [(108157, 108160, 'VAR'), (108164, 108191, 'TYPE')]
===== sent Let $U \subset X$ be an open subscheme
such that the inclusion morphism $U \to X$ is affine.
 ========

regex_match $U \subset X$
nn candidate $
nn candidate an open subscheme
nn an open subscheme
new_annotation [(108197, 108210, 'VAR'), (108214, 108231, 'TYPE')]
===== sent Then $\eta \in \Spec(\mathcal{O}_{X, \xi})$ and
we see that the dimension cannot be $0$.
\end{proof}

\begin{lemma}
\label{lemma-complement-affine-open}
Let $X$ be a separated locally Noetherian scheme. ========

regex_match $X$
nn candidate a separated locally Noetherian scheme
nn a separated locally Noetherian scheme
new_annotation [(109565, 109568, 'VAR'), (109572, 109609, 'TYPE')]
===== sent Let $U \subset X$ be an
affine open. ========

new_annotation []
===== sent \begin{lemma}
\label{lemma-complement-open-affine-effective-cartier-divisor}
Let $X$ be a Noetherian separated scheme. ========

regex_match $X$
nn candidate a Noetherian separated scheme
nn a Noetherian separated scheme
new_annotation [(110264, 110267, 'VAR'), (110271, 110300, 'TYPE')]
===== sent Let $U \subset X$ be
a dense affine open. ========

new_annotation []
===== sent Thus $D_i$ is an effective Cartier divisor by
Lemma \ref{lemma-weil-divisor-is-cartier-UFD}.
Hence we can take $D = D_1 + \ldots + D_r$.
\end{proof}






\section{Norms}
\label{section-norms}

\noindent
Let $\pi : X \to Y$ be a finite morphism of schemes and let $d \geq 1$
be an integer. ========

regex_match $\pi : X \to Y$
nn candidate X \to
nn candidate a finite morphism
nn a finite morphism
new_annotation [(111150, 111165, 'VAR'), (111169, 111186, 'TYPE')]
===== sent Thus $D_i$ is an effective Cartier divisor by
Lemma \ref{lemma-weil-divisor-is-cartier-UFD}.
Hence we can take $D = D_1 + \ldots + D_r$.
\end{proof}






\section{Norms}
\label{section-norms}

\noindent
Let $\pi : X \to Y$ be a finite morphism of schemes and let $d \geq 1$
be an integer. ========

new_annotation []
===== sent \begin{lemma}
\label{lemma-finite-trivialize-invertible-upstairs}
Let $\pi : X \to Y$ be a finite morphism of schemes.
 ========

regex_match $\pi : X \to Y$
nn candidate X \to
nn candidate a finite morphism
nn a finite morphism
new_annotation [(112398, 112413, 'VAR'), (112417, 112434, 'TYPE')]
===== sent Let $\mathcal{L}$ be an invertible $\mathcal{O}_X$-module.
 ========

regex_match $\mathcal{L}$
nn candidate an invertible $\mathcal{O}_X$-module
nn an invertible $\mathcal{O}_X$-module
new_annotation [(112451, 112464, 'VAR'), (112468, 112504, 'TYPE')]
===== sent Let $y \in Y$. There exists an open neighbourhood
$V \subset Y$ of $y$ such that $\mathcal{L}|_{\pi^{-1}(V)}$ is trivial.
 ========

new_annotation []
===== sent \end{proof}

\begin{lemma}
\label{lemma-norm-invertible}
Let $\pi : X \to Y$ be a finite morphism of schemes. ========

regex_match $\pi : X \to Y$
nn candidate X \to
nn candidate a finite morphism
nn a finite morphism
new_annotation [(113922, 113937, 'VAR'), (113941, 113958, 'TYPE')]
===== sent We omit the verification that: this is well defined,
additive on Picard groups, and satisfies the property
$\text{Norm}_\pi(\pi^*\mathcal{N}) \cong \mathcal{N}^{\otimes d}$
for all invertible $\mathcal{O}_Y$-modules $\mathcal{N}$.
\end{proof}

\begin{lemma}
\label{lemma-norm-map-invertible}
Let $\pi : X \to Y$ be a finite morphism of schemes. ========

regex_match $\pi : X \to Y$
nn candidate X \to
nn candidate a finite morphism
nn a finite morphism
new_annotation [(115579, 115594, 'VAR'), (115598, 115615, 'TYPE')]
===== sent Let $\pi : X \to Y$ be a finite morphism of schemes. ========

regex_match $\pi : X \to Y$
nn candidate X \to
nn candidate a finite morphism
nn a finite morphism
new_annotation [(117173, 117188, 'VAR'), (117192, 117209, 'TYPE')]
===== sent \end{lemma}

\begin{proof}
Let $\mathcal{L}$ be the ample invertible sheaf on $X$ given to us
by assumption. ========

regex_match $\mathcal{L}$
nn candidate the ample invertible sheaf
nn the ample invertible sheaf
new_annotation [(117383, 117396, 'VAR'), (117400, 117426, 'TYPE')]
===== sent Let $y \in Y$ be a point. ========

regex_match $y \in Y$
nn candidate $y
nn candidate a point
nn a point
new_annotation [(117750, 117759, 'VAR'), (117763, 117770, 'TYPE')]
===== sent Then $Y_t$
is affine by Properties, Lemma \ref{properties-lemma-affine-cap-s-open}.
\end{proof}

\begin{lemma}
\label{lemma-norm-quasi-affine}
Let $\pi : X \to Y$ be a finite morphism of schemes. ========

regex_match $\pi : X \to Y$
nn candidate X \to
nn candidate a finite morphism
nn a finite morphism
new_annotation [(118545, 118560, 'VAR'), (118564, 118581, 'TYPE')]
===== sent \end{proof}

\begin{lemma}
\label{lemma-finite-locally-free-has-norm}
Let $\pi : X \to Y$ be a finite locally free morphism of degree $d \geq 1$.
Then there exists a canonical norm of degree $d$ whose formation commutes
with arbitrary base change.
 ========

regex_match $\pi : X \to Y$
nn candidate X \to
nn candidate a finite locally free morphism
nn a finite locally free morphism
new_annotation [(119176, 119191, 'VAR'), (119195, 119225, 'TYPE')]
===== sent \end{lemma}

\begin{proof}
Let $V \subset Y$ be an affine open such that $(\pi_*\mathcal{O}_X)|_V$
is finite free of rank $d$. Choosing a basis we obtain an isomorphism
$$
\mathcal{O}_V^{\oplus d} \cong (\pi_*\mathcal{O}_X)|_V
$$
For every $f \in \pi_*\mathcal{O}_X(V) = \mathcal{O}_X(\pi^{-1}(V))$
multiplication by $f$ defines a $\mathcal{O}_V$-linear endomorphism
$m_f$ of the displayed free vector bundle. ========

regex_match $V \subset Y$
nn candidate $V
nn candidate an affine
nn an affine
new_annotation [(119381, 119394, 'VAR'), (119398, 119407, 'TYPE')]
===== sent \end{proof}

\begin{lemma}
\label{lemma-norm-in-normal-case}
Let $\pi : X \to Y$ be a finite surjective morphism with $X$ and $Y$
integral and $Y$ normal. ========

regex_match $\pi : X \to Y$
nn candidate X \to
nn candidate a finite surjective morphism
nn a finite surjective morphism
new_annotation [(120776, 120791, 'VAR'), (120795, 120823, 'TYPE')]
===== sent Then there exists a norm of degree
$[R(X) : R(Y)]$ for $\pi$.
\end{lemma}

\begin{proof}
Let $\Spec(B) \subset Y$ be an affine open subset and let
$\Spec(A) \subset X$ be its inverse image. ========

regex_match $\Spec(B) \subset Y$
nn candidate \subset Y$
nn candidate an affine open subset
nn an affine open subset
new_annotation [(120959, 120979, 'VAR'), (120983, 121004, 'TYPE')]
===== sent Then $A$ and $B$
are domains. Let $K$ be the fraction
field of $A$ and $L$ the fraction field of $B$. Picture:
$$
\xymatrix{
L \ar[r] & K \\
 ========

regex_match $K$
nn candidate the fraction
field
nn the fraction
field
new_annotation [(121090, 121093, 'VAR'), (121097, 121115, 'TYPE')]
===== sent To see property (2) let $f \in A$ be contained in the
prime ideal $\mathfrak p \subset A$. Let
$f^m + b_1 f^{m - 1} + \ldots + b_m$ be the minimal
polynomial of $f$ over $L$. By
Algebra, Lemma \ref{algebra-lemma-minimal-polynomial-normal-domain}
we have $b_i \in B$. ========

new_annotation []
===== sent Since $\text{Norm}_{K/L}(f) = b_0^{d/m}$ (see above)
we conclude that the norm vanishes in the image point of $\mathfrak p$.
\end{proof}

\begin{lemma}
\label{lemma-Frobenius-gives-norm-for-reduction}
Let $X$ be a Noetherian scheme. ========

regex_match $X$
nn candidate a Noetherian scheme
nn a Noetherian scheme
new_annotation [(122525, 122528, 'VAR'), (122532, 122551, 'TYPE')]
===== sent Let $p$ be a prime number such that
$p\mathcal{O}_X = 0$. Then for some $e > 0$ there exists a norm
of degree $p^e$ for $X_{red} \to X$ where $X_{red}$ is the reduction
of $X$.
\end{lemma}

\begin{proof}
Let $A$ be a Noetherian ring with $pA = 0$. Let $I \subset A$ be the
ideal of nilpotent elements. ========

regex_match $p$
nn candidate a prime number
nn a prime number
new_annotation [(122557, 122560, 'VAR'), (122564, 122578, 'TYPE')]
===== sent Let $p$ be a prime number such that
$p\mathcal{O}_X = 0$. Then for some $e > 0$ there exists a norm
of degree $p^e$ for $X_{red} \to X$ where $X_{red}$ is the reduction
of $X$.
\end{lemma}

\begin{proof}
Let $A$ be a Noetherian ring with $pA = 0$. Let $I \subset A$ be the
ideal of nilpotent elements. ========

regex_match $A$
nn candidate a Noetherian ring
nn a Noetherian ring
new_annotation [(122761, 122764, 'VAR'), (122768, 122785, 'TYPE')]
===== sent Let $p$ be a prime number such that
$p\mathcal{O}_X = 0$. Then for some $e > 0$ there exists a norm
of degree $p^e$ for $X_{red} \to X$ where $X_{red}$ is the reduction
of $X$.
\end{lemma}

\begin{proof}
Let $A$ be a Noetherian ring with $pA = 0$. Let $I \subset A$ be the
ideal of nilpotent elements. ========

new_annotation []
===== sent \end{proof}

\begin{proposition}
\label{proposition-push-down-ample}
Let $f : X \to Y$ be a finite surjective morphism of schemes.
 ========

regex_match $f : X \to Y$
nn candidate X \to
nn candidate a finite surjective morphism
nn a finite surjective morphism
new_annotation [(123357, 123370, 'VAR'), (123374, 123402, 'TYPE')]
===== sent Then we can apply Lemma \ref{lemma-norm-in-normal-case}.
\end{proof}

\begin{lemma}
\label{lemma-push-down-quasi-affine}
Let $f : X \to Y$ be a finite surjective morphism of schemes.
 ========

regex_match $f : X \to Y$
nn candidate X \to
nn candidate a finite surjective morphism
nn a finite surjective morphism
new_annotation [(124250, 124263, 'VAR'), (124267, 124295, 'TYPE')]
===== sent \begin{lemma}
\label{lemma-relative-Cartier}
Let $f : X \to S$ be a morphism of schemes.
 ========

regex_match $f : X \to S$
nn candidate X \to
nn candidate a morphism
nn a morphism
new_annotation [(125228, 125241, 'VAR'), (125245, 125255, 'TYPE')]
===== sent Let $D \subset X$ be a closed subscheme.
 ========

regex_match $D \subset X$
nn candidate $D \subset X$
nn candidate a closed subscheme
nn a closed subscheme
new_annotation [(125272, 125285, 'VAR'), (125289, 125307, 'TYPE')]
===== sent Let $A \to B$ be a ring
map and $h \in ========

regex_match $A \to B$
nn candidate $
nn candidate a ring
map
nn a ring
map
new_annotation [(125743, 125752, 'VAR'), (125756, 125766, 'TYPE')]
===== sent \begin{definition}
\label{definition-relative-effective-Cartier-divisor}
Let $f : X \to S$ be a morphism of schemes.
 ========

regex_match $f : X \to S$
nn candidate X \to
nn candidate a morphism
nn a morphism
new_annotation [(126287, 126300, 'VAR'), (126304, 126314, 'TYPE')]
===== sent \begin{lemma}
\label{lemma-sum-relative-effective-Cartier-divisor}
Let $f : X \to S$ be a morphism of schemes. ========

regex_match $f : X \to S$
nn candidate X \to
nn candidate a morphism
nn a morphism
new_annotation [(126932, 126945, 'VAR'), (126949, 126959, 'TYPE')]
===== sent This translates into the following algebra fact:
Let $A \to B$ be a ring map and $h_1, h_2 \in B$.
Assume the $h_i$ are nonzerodivisors and that $B/h_iB$ is flat over $A$.
 ========

regex_match $A \to B$
nn candidate a ring map
nn a ring map
new_annotation [(127209, 127218, 'VAR'), (127222, 127232, 'TYPE')]
===== sent The reason is that we have a short exact sequence
$$
0 \to B/h_1B \to B/h_1h_2B \to B/h_2B \to 0
$$
where the first arrow is given by multiplication by $h_2$. Since
the outer two are flat modules over $A$, so is the middle one, see
Algebra, Lemma \ref{algebra-lemma-flat-ses}.
\end{proof}

\begin{lemma}
\label{lemma-difference-relative-effective-Cartier-divisor}
Let $f : X \to S$ be a morphism of schemes. ========

regex_match $f : X \to S$
nn candidate X \to
nn candidate a morphism
nn a morphism
new_annotation [(127764, 127777, 'VAR'), (127781, 127791, 'TYPE')]
===== sent This translates into the following algebra fact:
Let $A \to B$ be a ring map and $h_1, h_2 \in B$.
Assume the $h_i$ are nonzerodivisors, that $B/h_iB$ is flat over $A$, and
that $(h_2) \subset (h_1)$. ========

regex_match $A \to B$
nn candidate a ring map
nn a ring map
new_annotation [(128173, 128182, 'VAR'), (128186, 128196, 'TYPE')]
===== sent Since
the right two are flat modules over $A$, so is the middle one, see
Algebra, Lemma \ref{algebra-lemma-flat-ses}.
\end{proof}

\begin{lemma}
\label{lemma-flat-at-x}
Let $f : X \to S$ be a morphism of schemes.
 ========

regex_match $f : X \to S$
nn candidate X \to
nn candidate a morphism
nn a morphism
new_annotation [(128697, 128710, 'VAR'), (128714, 128724, 'TYPE')]
===== sent Let $D \subset X$ be a relative effective Cartier divisor on $X/S$.
If $x \in D$ and $\mathcal{O}_{X, x}$ is Noetherian, then $f$ is flat at $x$.
\end{lemma}

\begin{proof}
Set $A = \mathcal{O}_{S, f(x)}$ and $B = \mathcal{O}_{X, x}$.
Let $h \in B$ be an element which generates the ideal of $D$.
 ========

regex_match $D \subset X$
nn candidate $D \subset X$
nn candidate a relative effective Cartier divisor
nn a relative effective Cartier divisor
new_annotation [(128741, 128754, 'VAR'), (128758, 128794, 'TYPE')]
===== sent Let $D \subset X$ be a relative effective Cartier divisor on $X/S$.
If $x \in D$ and $\mathcal{O}_{X, x}$ is Noetherian, then $f$ is flat at $x$.
\end{lemma}

\begin{proof}
Set $A = \mathcal{O}_{S, f(x)}$ and $B = \mathcal{O}_{X, x}$.
Let $h \in B$ be an element which generates the ideal of $D$.
 ========

regex_match $h \in B$
nn candidate $h
nn candidate an element
nn an element
new_annotation [(128976, 128985, 'VAR'), (128989, 128999, 'TYPE')]
===== sent Let $I \subset A$ be a finitely generated ideal.
 ========

regex_match $I \subset A$
nn candidate $
nn candidate I
nn candidate a finitely generated ideal
nn a finitely generated ideal
new_annotation [(129120, 129133, 'VAR'), (129137, 129163, 'TYPE')]
===== sent The scheme theoretic version can be found in
More on Morphisms, Section \ref{more-morphisms-section-open-flat}.

\begin{lemma}
\label{lemma-flat-relative-Cartier-divisor}
Let $f : X \to S$ be a morphism of schemes.
 ========

regex_match $f : X \to S$
nn candidate X \to
nn candidate a morphism
nn a morphism
new_annotation [(130139, 130152, 'VAR'), (130156, 130166, 'TYPE')]
===== sent Let $D \subset X$ be a relative effective Cartier divisor.
 ========

regex_match $D \subset X$
nn candidate $D \subset X$
nn candidate a relative effective Cartier divisor
nn a relative effective Cartier divisor
new_annotation [(130183, 130196, 'VAR'), (130200, 130236, 'TYPE')]
===== sent By Algebra, Lemma
\ref{algebra-lemma-flat-finite-presentation-limit-flat}
we may, after enlarging $A_0$, assume that $B_0/h_0B_0$ is flat
over $A_0$. Let $K_0 = \Ker(h_0 : B_0 \to B_0)$.
As $B_0$ is of finite type over $\mathbf{Z}$ we see that $K_0$ is
a finitely generated ideal. ========

new_annotation []
===== sent Let $A_1 \subset A$ be a finite type
$\mathbf{Z}$-subalgebra containing $A_0$ and denote $B_1$, $h_1$, $K_1$
the corresponding objects over $A_1$. By
More on Algebra, Lemma \ref{more-algebra-lemma-base-change-H1-regular}
the map $K_0 \otimes_{A_0} A_1 \to K_1$ is surjective. ========

regex_match $A_1 \subset A$
nn candidate $
nn candidate a finite type
nn a finite type
new_annotation [(131544, 131559, 'VAR'), (131563, 131576, 'TYPE')]
===== sent \begin{lemma}
\label{lemma-michael-artin}
Let $f : X \to S$ be a morphism of schemes.
 ========

regex_match $f : X \to S$
nn candidate X \to
nn candidate a morphism
nn a morphism
new_annotation [(132983, 132996, 'VAR'), (133000, 133010, 'TYPE')]
===== sent Let $D \subset X$ be a relative effective Cartier divisor on $X/S$.
If $f$ is flat at all points of $X \setminus D$, then $f$ is flat.
 ========

regex_match $D \subset X$
nn candidate $D \subset X$
nn candidate a relative effective Cartier divisor
nn a relative effective Cartier divisor
new_annotation [(133027, 133040, 'VAR'), (133044, 133080, 'TYPE')]
===== sent This translates into the following algebra fact:
Let $A \to B$ be a ring map and $h \in B$.
Assume $h$ is a nonzerodivisor, that $B/hB$ is flat over $A$, and
that the localization $B_h$ is flat over $A$. ========

regex_match $A \to B$
nn candidate a ring map
nn a ring map
new_annotation [(133238, 133247, 'VAR'), (133251, 133261, 'TYPE')]
===== sent \begin{lemma}
\label{lemma-fibre-Cartier}
Let $\varphi : X \to S$ be a flat morphism which is locally of finite
presentation. ========

regex_match $\varphi : X \to S$
nn candidate X \to
nn candidate a flat morphism
nn a flat morphism
new_annotation [(134846, 134865, 'VAR'), (134869, 134884, 'TYPE')]
===== sent Let $Z \subset X$ be a closed subscheme.
 ========

regex_match $Z \subset X$
nn candidate $
nn candidate a closed subscheme
nn a closed subscheme
new_annotation [(134930, 134943, 'VAR'), (134947, 134965, 'TYPE')]
===== sent Let $x \in Z$ with image $s \in ========

new_annotation []
===== sent \end{lemma}

\begin{proof}
Choose affine open neighbourhoods $\Spec(A)$ of $s$ and
$\Spec(B)$ of $x$ such that
$\varphi(\Spec(B)) \subset \Spec(A)$.
Let $\mathfrak p \subset A$ be the prime ideal corresponding to $s$.
Let $\mathfrak q \subset B$ be the prime ideal corresponding to $x$.
Let $I \subset B$ be the ideal corresponding to $Z$.
By the initial assumption of the lemma we know that
$A \to B$ is flat and of finite presentation.
 ========

regex_match $\mathfrak p \subset A$
nn candidate p \subset
nn candidate the prime ideal corresponding
nn the prime ideal corresponding
new_annotation [(136212, 136235, 'VAR'), (136239, 136268, 'TYPE')]
===== sent \end{lemma}

\begin{proof}
Choose affine open neighbourhoods $\Spec(A)$ of $s$ and
$\Spec(B)$ of $x$ such that
$\varphi(\Spec(B)) \subset \Spec(A)$.
Let $\mathfrak p \subset A$ be the prime ideal corresponding to $s$.
Let $\mathfrak q \subset B$ be the prime ideal corresponding to $x$.
Let $I \subset B$ be the ideal corresponding to $Z$.
By the initial assumption of the lemma we know that
$A \to B$ is flat and of finite presentation.
 ========

regex_match $\mathfrak q \subset B$
nn candidate the prime ideal corresponding
nn the prime ideal corresponding
new_annotation [(136281, 136304, 'VAR'), (136308, 136337, 'TYPE')]
===== sent \end{lemma}

\begin{proof}
Choose affine open neighbourhoods $\Spec(A)$ of $s$ and
$\Spec(B)$ of $x$ such that
$\varphi(\Spec(B)) \subset \Spec(A)$.
Let $\mathfrak p \subset A$ be the prime ideal corresponding to $s$.
Let $\mathfrak q \subset B$ be the prime ideal corresponding to $x$.
Let $I \subset B$ be the ideal corresponding to $Z$.
By the initial assumption of the lemma we know that
$A \to B$ is flat and of finite presentation.
 ========

regex_match $I \subset B$
nn candidate I
nn candidate the ideal corresponding
nn the ideal corresponding
new_annotation [(136350, 136363, 'VAR'), (136367, 136390, 'TYPE')]
===== sent U = D$ after shrinking $U$.

\medskip\noindent
The final statements of the lemma follow immediately from
parts (2) and (3), combined with the fact that $Z \to S$
is locally of finite presentation if and only if $Z \to X$ is
of finite presentation, see
Morphisms, Lemmas \ref{morphisms-lemma-composition-finite-presentation} and
\ref{morphisms-lemma-finite-presentation-permanence}.
\end{proof}



\section{The normal cone of an immersion}
\label{section-normal-cone}

\noindent
Let $i : Z \to X$ be a closed immersion. ========

regex_match $i : Z \to X$
nn candidate Z \to
nn candidate a closed immersion
nn a closed immersion
new_annotation [(139778, 139791, 'VAR'), (139795, 139813, 'TYPE')]
===== sent Since the sheaves $\mathcal{I}^n/\mathcal{I}^{n + 1}$
are each annihilated by $\mathcal{I}$ this graded algebra
corresponds to a quasi-coherent sheaf of graded $\mathcal{O}_Z$-algebras
by
Morphisms, Lemma \ref{morphisms-lemma-i-star-equivalence}.
This quasi-coherent graded $\mathcal{O}_Z$-algebra is called the
{\it conormal algebra of $Z$ in $X$} and is often simply denoted
$\bigoplus_{n \geq 0} \mathcal{I}^n/\mathcal{I}^{n + 1}$
by the abuse of notation mentioned in
Morphisms, Section \ref{morphisms-section-closed-immersions-quasi-coherent}.

\medskip\noindent
Let $f : Z \to X$ be an immersion. ========

regex_match $f : Z \to X$
nn candidate Z \to X$
nn candidate an immersion
nn an immersion
new_annotation [(140607, 140620, 'VAR'), (140624, 140636, 'TYPE')]
===== sent \partial Z$.

\begin{definition}
\label{definition-conormal-sheaf}
Let $f : Z \to X$ be an immersion. ========

regex_match $f : Z \to X$
nn candidate Z \to X$
nn candidate an immersion
nn an immersion
new_annotation [(141039, 141052, 'VAR'), (141056, 141068, 'TYPE')]
===== sent Finally, note that there is a canonical surjective
map
\begin{equation}
\label{equation-conormal-algebra-quotient}
\text{Sym}^*(\mathcal{C}_{Z/X}) \longrightarrow \mathcal{C}_{Z/X, *}
\end{equation}
of quasi-coherent graded $\mathcal{O}_Z$-algebras which is an isomorphism
in degrees $0$ and $1$.

\begin{lemma}
\label{lemma-affine-conormal-sheaf}
Let $i : Z \to X$ be an immersion. ========

regex_match $i : Z \to X$
nn candidate Z \to X$
nn candidate an immersion
nn an immersion
new_annotation [(142145, 142158, 'VAR'), (142162, 142174, 'TYPE')]
===== sent The conormal algebra
of $i$ has the following properties:
\begin{enumerate}
\item Let $U \subset X$ be any open such that $i(Z)$ is
a closed subset of $U$. Let $\mathcal{I} \subset \mathcal{O}_U$
be the sheaf of ideals corresponding to the closed subscheme
$i(Z) \subset U$. Then
$$
\mathcal{C}_{Z/X, *} =
i^*\left(\bigoplus\nolimits_{n \geq 0} \mathcal{I}^n\right) =
i^{-1}\left(
\bigoplus\nolimits_{n \geq 0} \mathcal{I}^n/\mathcal{I}^{n + 1}
\right)
$$
\item
 ========

new_annotation []
===== sent The conormal algebra
of $i$ has the following properties:
\begin{enumerate}
\item Let $U \subset X$ be any open such that $i(Z)$ is
a closed subset of $U$. Let $\mathcal{I} \subset \mathcal{O}_U$
be the sheaf of ideals corresponding to the closed subscheme
$i(Z) \subset U$. Then
$$
\mathcal{C}_{Z/X, *} =
i^*\left(\bigoplus\nolimits_{n \geq 0} \mathcal{I}^n\right) =
i^{-1}\left(
\bigoplus\nolimits_{n \geq 0} \mathcal{I}^n/\mathcal{I}^{n + 1}
\right)
$$
\item
 ========

new_annotation []
===== sent There is a canonical map
of graded $\mathcal{O}_Z$-algebras
$$
f^*\mathcal{C}_{Z'/X', *}
\longrightarrow
\mathcal{C}_{Z/X, *}
$$
characterized by the following property: For every pair of affine opens
$(\Spec(R) = U \subset X, \Spec(R') = U' \subset X')$ with
$f(U) \subset U'$ such that
$Z \cap U = \Spec(R/I)$ and $Z' \cap U' = \Spec(R'/I')$
the induced map
$$
\Gamma(Z' \cap U', \mathcal{C}_{Z'/X', *}) =
\bigoplus\nolimits (I')^n/(I')^{n + 1}
\longrightarrow
\bigoplus\nolimits_{n \geq 0} I^n/I^{n + 1} =
\Gamma(Z \cap U, \mathcal{C}_{Z/X, *})
$$
is the one induced by the ring map $f^\sharp : R' \to R$ which
has the property $f^\sharp(I') \subset I$.
\end{lemma}

\begin{proof}
Let $\partial Z' = \overline{Z'} \setminus Z'$ and
$\partial Z = \overline{Z} \setminus Z$. ========

new_annotation []
===== sent \end{lemma}

\begin{proof}
Let $R' \to R$ be a ring map, and $I' \subset R'$ an ideal.
 ========

regex_match $R' \to R$
nn candidate R
nn candidate a ring map
nn a ring map
new_annotation [(146004, 146014, 'VAR'), (146018, 146028, 'TYPE')]
===== sent \end{proof}

\begin{definition}
\label{definition-normal-cone}
Let $i : Z \to X$ be an immersion of schemes.
 ========

regex_match $i : Z \to X$
nn candidate Z \to X$
nn candidate an immersion
nn an immersion
new_annotation [(146313, 146326, 'VAR'), (146330, 146342, 'TYPE')]
===== sent Let $X$ be a ringed space and let
$f_1, \ldots, f_r \in \Gamma(X, \mathcal{O}_X)$.
We say that $f_1, \ldots, f_r$ is a {\it regular sequence} if
for each $i = 1, \ldots, r$ the map
\begin{equation}
\label{equation-map-regular}
f_i :
\mathcal{O}_X/(f_1, \ldots, f_{i - 1})
\longrightarrow
\mathcal{O}_X/(f_1, \ldots, f_{i - 1})
\end{equation}
is an injective map of sheaves. ========

regex_match $X$
nn candidate a ringed space
nn a ringed space
new_annotation [(148746, 148749, 'VAR'), (148753, 148767, 'TYPE')]
===== sent \begin{lemma}
\label{lemma-types-regular-sequences-implications}
Let $X$ be a ringed space.
 ========

regex_match $X$
nn candidate a ringed space
nn a ringed space
new_annotation [(150202, 150205, 'VAR'), (150209, 150223, 'TYPE')]
===== sent Let $f_1, \ldots, f_r \in \Gamma(X, \mathcal{O}_X)$.
We have the following implications
$f_1, \ldots, f_r$ is a regular sequence $\Rightarrow$
$f_1, \ldots, f_r$ is a Koszul-regular sequence $\Rightarrow$
$f_1, \ldots, f_r$ is an $H_1$-regular sequence $\Rightarrow$
$f_1, \ldots, f_r$ is a quasi-regular sequence.
 ========

new_annotation []
===== sent Hence the implications follow from
More on Algebra, Lemmas
\ref{more-algebra-lemma-regular-koszul-regular},
\ref{more-algebra-lemma-koszul-regular-H1-regular}, and
\ref{more-algebra-lemma-H1-regular-quasi-regular}.
\end{proof}

\begin{definition}
\label{definition-regular-ideal-sheaf}
Let $X$ be a ringed space. ========

regex_match $X$
nn candidate a ringed space
nn a ringed space
new_annotation [(151339, 151342, 'VAR'), (151346, 151360, 'TYPE')]
===== sent Let $\mathcal{J} \subset \mathcal{O}_X$
be a sheaf of ideals.
 ========

new_annotation []
===== sent \begin{lemma}
\label{lemma-regular-quasi-regular-scheme}
Let $X$ be a ringed space. ========

regex_match $X$
nn candidate a ringed space
nn a ringed space
new_annotation [(152857, 152860, 'VAR'), (152864, 152878, 'TYPE')]
===== sent Let $\mathcal{J}$ be a sheaf of ideals.
 ========

regex_match $\mathcal{J}$
nn candidate a sheaf
nn a sheaf
new_annotation [(152884, 152897, 'VAR'), (152901, 152908, 'TYPE')]
===== sent The lemma immediately reduces to
Lemma \ref{lemma-types-regular-sequences-implications}.
\end{proof}

\begin{lemma}
\label{lemma-quasi-regular-ideal}
Let $X$ be a locally ringed space. ========

regex_match $X$
nn candidate a locally ringed space
nn a locally ringed space
new_annotation [(153299, 153302, 'VAR'), (153306, 153328, 'TYPE')]
===== sent Let $\mathcal{J} \subset \mathcal{O}_X$
be a sheaf of ideals. ========

new_annotation []
===== sent \end{proof}

\begin{lemma}
\label{lemma-generate-regular-ideal}
Let $(X, \mathcal{O}_X)$ be a locally ringed space.
 ========

regex_match $(X, \mathcal{O}_X)$
nn candidate $(X
nn candidate a locally ringed space
nn a locally ringed space
new_annotation [(155574, 155594, 'VAR'), (155598, 155620, 'TYPE')]
===== sent Let $\mathcal{J} \subset \mathcal{O}_X$ be a sheaf of ideals.
 ========

regex_match $\mathcal{J} \subset \mathcal{O}_X$
nn candidate $\mathcal{J
nn candidate a sheaf
nn a sheaf
new_annotation [(155626, 155661, 'VAR'), (155665, 155672, 'TYPE')]
===== sent Let $x \in X$ and $f_1, \ldots, f_r \in \mathcal{J}_x$ whose images
give a basis for the $\kappa(x)$-vector space
$\mathcal{J}_x/\mathfrak m_x\mathcal{J}_x$.
\begin{enumerate}
\item If $\mathcal{J}$ is quasi-regular, then there exists an open
neighbourhood such that $f_1, \ldots, f_r \in \mathcal{O}_X(U)$
form a quasi-regular sequence generating $\mathcal{J}|_U$.
\item If $\mathcal{J}$ is $H_1$-regular, then there exists an open
neighbourhood such that $f_1, \ldots, f_r \in \mathcal{O}_X(U)$
form an $H_1$-regular sequence generating $\mathcal{J}|_U$.
\item If $\mathcal{J}$ is Koszul-regular, then there exists an open
neighbourhood such that $f_1, \ldots, f_r \in \mathcal{O}_X(U)$
form an Koszul-regular sequence generating $\mathcal{J}|_U$.
\end{enumerate}
\end{lemma}

\begin{proof}
First assume that $\mathcal{J}$ is quasi-regular. ========

new_annotation []
===== sent And any sheaf of ideals locally generated
by sections on a scheme is quasi-coherent, see
Schemes, Lemma \ref{schemes-lemma-closed-subspace-scheme}.
\end{proof}

\begin{lemma}
\label{lemma-regular-ideal-sheaf-scheme}
Let $X$ be a scheme. ========

regex_match $X$
nn candidate a scheme
nn a scheme
new_annotation [(158535, 158538, 'VAR'), (158542, 158550, 'TYPE')]
===== sent Let $\mathcal{J}$ be a sheaf of ideals.
 ========

regex_match $\mathcal{J}$
nn candidate a sheaf
nn a sheaf
new_annotation [(158556, 158569, 'VAR'), (158573, 158580, 'TYPE')]
===== sent \end{proof}

\begin{lemma}
\label{lemma-Noetherian-scheme-regular-ideal}
Let $X$ be a locally Noetherian scheme. ========

regex_match $X$
nn candidate a locally Noetherian scheme
nn a locally Noetherian scheme
new_annotation [(160398, 160401, 'VAR'), (160405, 160432, 'TYPE')]
===== sent Let $\mathcal{J} \subset \mathcal{O}_X$
be a quasi-coherent sheaf of ideals. ========

new_annotation []
===== sent Let $x$ be a point of the support of
$\mathcal{O}_X/\mathcal{J}$. The following are equivalent
\begin{enumerate}
\item $\mathcal{J}_x$ is generated by a regular sequence in
$\mathcal{O}_{X, x}$,
\item $\mathcal{J}_x$ is generated by a Koszul-regular sequence in
$\mathcal{O}_{X, x}$,
\item $\mathcal{J}_x$ is generated by an $H_1$-regular sequence in
$\mathcal{O}_{X, x}$,
\item $\mathcal{J}_x$ is generated by a quasi-regular sequence in
$\mathcal{O}_{X, x}$,
\item there exists an affine neighbourhood $U = \Spec(A)$ of $x$ such
that $\mathcal{J}|_U = \widetilde{I}$ and $I$ is generated by a
regular sequence in $A$, and
\item there exists an affine neighbourhood $U = \Spec(A)$ of $x$ such
that $\mathcal{J}|_U = \widetilde{I}$ and $I$ is generated by a
Koszul-regular sequence in $A$, and
\item there exists an affine neighbourhood $U = \Spec(A)$ of $x$ such
that $\mathcal{J}|_U = \widetilde{I}$ and $I$ is generated by an
$H_1$-regular sequence in $A$, and
\item there exists an affine neighbourhood $U = \Spec(A)$ of $x$ such
that $\mathcal{J}|_U = \widetilde{I}$ and $I$ is generated by a
quasi-regular sequence in $A$,
\item there exists a neighbourhood $U$ of $x$ such that $\mathcal{J}|_U$
is regular, and
\item there exists a neighbourhood $U$ of $x$ such that $\mathcal{J}|_U$
is Koszul-regular, and
\item there exists a neighbourhood $U$ of $x$ such that $\mathcal{J}|_U$
is $H_1$-regular, and
\item there exists a neighbourhood $U$ of $x$ such that $\mathcal{J}|_U$
is quasi-regular.
 ========

regex_match $x$
nn candidate a point
nn a point
new_annotation [(160515, 160518, 'VAR'), (160522, 160529, 'TYPE')]
===== sent Let $i : Z \to X$ be an immersion of schemes. ========

regex_match $i : Z \to X$
nn candidate Z \to X$
nn candidate an immersion
nn an immersion
new_annotation [(162946, 162959, 'VAR'), (162963, 162975, 'TYPE')]
===== sent \begin{definition}
\label{definition-regular-immersion}
Let $i : Z \to X$ be an immersion of schemes. ========

regex_match $i : Z \to X$
nn candidate Z \to X$
nn candidate an immersion
nn an immersion
new_annotation [(163871, 163884, 'VAR'), (163888, 163900, 'TYPE')]
===== sent \end{enumerate}
\end{definition}

\noindent
The discussion above shows that this is independent of the choice
of $U$. The conditions are listed in decreasing order of strength, see
Lemma \ref{lemma-regular-quasi-regular-immersion}.
A Koszul-regular closed immersion is smooth locally a regular immersion, see
Lemma \ref{lemma-koszul-regular-smooth-locally-regular}.
In the locally Noetherian case all four notions agree, see
Lemma \ref{lemma-Noetherian-scheme-regular-ideal}.

\begin{lemma}
\label{lemma-regular-quasi-regular-immersion}
Let $i : Z \to X$ be an immersion of schemes.
 ========

regex_match $i : Z \to X$
nn candidate Z \to X$
nn candidate an immersion
nn an immersion
new_annotation [(165006, 165019, 'VAR'), (165023, 165035, 'TYPE')]
===== sent The lemma immediately reduces to
Lemma \ref{lemma-regular-quasi-regular-scheme}.
\end{proof}

\begin{lemma}
\label{lemma-regular-immersion-noetherian}
Let $i : Z \to X$ be an immersion of schemes.
 ========

regex_match $i : Z \to X$
nn candidate Z \to X$
nn candidate an immersion
nn an immersion
new_annotation [(165388, 165401, 'VAR'), (165405, 165417, 'TYPE')]
===== sent \end{lemma}

\begin{proof}
Follows immediately from
Lemma \ref{lemma-regular-quasi-regular-immersion}
and
Lemma \ref{lemma-Noetherian-scheme-regular-ideal}.
\end{proof}

\begin{lemma}
\label{lemma-flat-base-change-regular-immersion}
Let $i : Z \to X$ be a regular (resp.\ Koszul-regular,
$H_1$-regular, quasi-regular) immersion. ========

regex_match $i : Z \to X$
nn candidate Z \to
new_annotation []
===== sent Let $X' \to X$ be a flat
morphism. ========

regex_match $X' \to X$
nn candidate $X
nn candidate a flat
morphism
nn a flat
morphism
new_annotation [(165936, 165946, 'VAR'), (165950, 165965, 'TYPE')]
===== sent \end{lemma}

\begin{proof}
Via
Lemma \ref{lemma-regular-ideal-sheaf-scheme}
this translates into the algebraic statements in
Algebra, Lemmas \ref{algebra-lemma-flat-increases-depth} and
\ref{algebra-lemma-flat-base-change-quasi-regular}
and
More on Algebra,
Lemma \ref{more-algebra-lemma-koszul-regular-flat-base-change}.
\end{proof}

\begin{lemma}
\label{lemma-quasi-regular-immersion}
Let $i : Z \to X$ be an immersion of schemes. ========

regex_match $i : Z \to X$
nn candidate Z \to X$
nn candidate an immersion
nn an immersion
new_annotation [(166485, 166498, 'VAR'), (166502, 166514, 'TYPE')]
===== sent Let $\mathcal{I} \subset \mathcal{O}_X$ be the
corresponding quasi-coherent sheaf of ideals. ========

new_annotation []
===== sent \end{proof}

\begin{lemma}
\label{lemma-transitivity-conormal-quasi-regular}
Let $Z \to Y \to X$ be immersions of schemes. ========

new_annotation []
===== sent \begin{lemma}
\label{lemma-composition-regular-immersion}
Let $i : Z \to Y$ and $j : Y \to X$ be immersions of schemes.
 ========

new_annotation []
===== sent \end{enumerate}
\end{lemma}

\begin{proof}
The algebraic version of (1) is
Algebra, Lemma \ref{algebra-lemma-join-regular-sequences}.
The algebraic version of (2) is
More on Algebra, Lemma \ref{more-algebra-lemma-join-koszul-regular-sequences}.
The algebraic version of (3) is
More on Algebra, Lemma \ref{more-algebra-lemma-join-H1-regular-sequences}.
The algebraic version of (4) is
More on Algebra, Lemma \ref{more-algebra-lemma-join-quasi-regular-H1-regular}.
\end{proof}

\begin{lemma}
\label{lemma-permanence-regular-immersion}
Let $i : Z \to Y$ and $j : Y \to X$ be immersions of schemes. ========

new_annotation []
===== sent Denote $\mathcal{I} \subset \mathcal{O}_X$ the ideal sheaf
of $Y$ and $\mathcal{J} \subset \mathcal{O}_X$ the ideal sheaf of $Z$.
The conormal sequence is $0 \to \mathcal{I}/\mathcal{I}\mathcal{J}
\to \mathcal{J}/\mathcal{J}^2 \to
\mathcal{J}/(\mathcal{I} + \mathcal{J}^2) \to 0$.
Let $z \in Z$ and set $y = i(z)$, $ ========

new_annotation []
===== sent By
Lemma \ref{lemma-generate-regular-ideal}
(applied twice)
we can find an affine open neighbourhood $U$ of $x$ such that
$f_1, \ldots, f_n$ forms a Koszul-regular sequence generating
$\mathcal{I}$ and $f_1, \ldots, f_n, g_1, \ldots, g_m$ forms a
Koszul-regular sequence generating $\mathcal{J}$. Hence by
More on Algebra, Lemma \ref{more-algebra-lemma-truncate-koszul-regular}
we see that $g_1, \ldots, g_m$ induces a Koszul-regular sequence on
$Y \cap U$ cutting out $Z$.
\end{proof}

\begin{lemma}
\label{lemma-extra-permanence-regular-immersion-noetherian}
Let $i : Z \to Y$ and $j : Y \to X$ be immersions of schemes.
 ========

new_annotation []
===== sent An example is $X = \mathbf{A}^1_k = \Spec(k[x])$,
$Y = \Spec(k[x]/(x^2))$ and $Z = \Spec(k[x]/(x))$.
\end{remark}

\begin{lemma}
\label{lemma-koszul-regular-smooth-locally-regular}
Let $i : Z \to X$ be a Koszul regular closed immersion.
 ========

regex_match $i : Z \to X$
nn candidate Z \to
nn candidate a Koszul regular
nn a Koszul regular
new_annotation [(175119, 175132, 'VAR'), (175136, 175152, 'TYPE')]
===== sent The following lemma does not hold for regular immersions
or for Koszul immersions, see
Examples, Lemma \ref{examples-lemma-base-change-regular-sequence}.

\begin{lemma}
\label{lemma-relative-regular-immersion}
Let $f : X \to S$ be a morphism of schemes.
 ========

regex_match $f : X \to S$
nn candidate X \to
nn candidate a morphism
nn a morphism
new_annotation [(176072, 176085, 'VAR'), (176089, 176099, 'TYPE')]
===== sent Let $i : Z \subset X$ be an immersion.
 ========

regex_match $i : Z \subset X$
nn candidate Z \subset
nn candidate an immersion
nn an immersion
new_annotation [(176116, 176133, 'VAR'), (176137, 176149, 'TYPE')]
===== sent \begin{definition}
\label{definition-relative-H1-regular-immersion}
Let $f : X \to S$ be a morphism of schemes.
 ========

regex_match $f : X \to S$
nn candidate X \to
nn candidate a morphism
nn a morphism
new_annotation [(176828, 176841, 'VAR'), (176845, 176855, 'TYPE')]
===== sent Let $i : Z \to X$ be an immersion.
 ========

regex_match $i : Z \to X$
nn candidate Z \to X$
nn candidate an immersion
nn an immersion
new_annotation [(176872, 176885, 'VAR'), (176889, 176901, 'TYPE')]
===== sent \begin{lemma}
\label{lemma-quasi-regular-immersion-flat-at-x}
Let $f : X \to S$ be a morphism of schemes.
 ========

regex_match $f : X \to S$
nn candidate X \to
nn candidate a morphism
nn a morphism
new_annotation [(178013, 178026, 'VAR'), (178030, 178040, 'TYPE')]
===== sent Let $Z \to X$ be a relative quasi-regular immersion.
 ========

regex_match $Z \to X$
nn candidate a relative quasi-regular immersion
nn a relative quasi-regular immersion
new_annotation [(178057, 178066, 'VAR'), (178070, 178104, 'TYPE')]
===== sent If $x \in Z$ and $\mathcal{O}_{X, x}$ is Noetherian, then $f$ is flat at $x$.
\end{lemma}

\begin{proof}
Let $f_1, \ldots, f_r \in \mathcal{O}_{X, x}$ be a quasi-regular
sequence cutting out the ideal of $Z$ at $x$. By
Algebra, Lemma \ref{algebra-lemma-quasi-regular-regular}
we know that $f_1, \ldots, f_r$ is a regular sequence.
 ========

regex_match $f_1, \ldots, f_r \in \mathcal{O}_{X, x}$
nn candidate a quasi-regular
sequence
nn a quasi-regular
sequence
new_annotation [(178215, 178256, 'VAR'), (178260, 178284, 'TYPE')]
===== sent \end{proof}

\begin{lemma}
\label{lemma-relative-regular-immersion-flat-in-neighbourhood}
Let $X \to S$ be a morphism of schemes.
 ========

regex_match $X \to S$
nn candidate $X
nn candidate a morphism
nn a morphism
new_annotation [(178910, 178919, 'VAR'), (178923, 178933, 'TYPE')]
===== sent Let $Z \to X$ be an immersion.
 ========

regex_match $Z \to X$
nn candidate $Z
nn candidate an immersion
nn an immersion
new_annotation [(178950, 178959, 'VAR'), (178963, 178975, 'TYPE')]
===== sent \end{proof}

\begin{lemma}
\label{lemma-flat-relative-H1-regular}
Let $X \to S$ be a morphism of schemes.
 ========

regex_match $X \to S$
nn candidate $X
nn candidate a morphism
nn a morphism
new_annotation [(180657, 180666, 'VAR'), (180670, 180680, 'TYPE')]
===== sent Let $Z \to X$ be a relative $H_1$-regular immersion.
 ========

regex_match $Z \to X$
nn candidate a relative $H_1$-regular immersion
nn a relative $H_1$-regular immersion
new_annotation [(180697, 180706, 'VAR'), (180710, 180744, 'TYPE')]
===== sent Let $A_0 \subset A_1 \subset A$ be a larger finite type
$\mathbf{Z}$-subalgebra of $A$. ========

regex_match $A_0 \subset A_1 \subset A$
nn candidate $
nn candidate A_1 \subset
nn candidate a larger finite type
nn a larger finite type
new_annotation [(182509, 182536, 'VAR'), (182540, 182560, 'TYPE')]
===== sent Let $\varphi : X \to S$ be a flat morphism which is locally of finite
presentation. ========

regex_match $\varphi : X \to S$
nn candidate X \to
nn candidate a flat morphism
nn a flat morphism
new_annotation [(184695, 184714, 'VAR'), (184718, 184733, 'TYPE')]
===== sent Let $T \subset X$ be a closed subscheme.
 ========

regex_match $T \subset X$
nn candidate $T \subset
nn candidate a closed subscheme
nn a closed subscheme
new_annotation [(184779, 184792, 'VAR'), (184796, 184814, 'TYPE')]
===== sent Let $x \in T$ with image $s \in ========

new_annotation []
===== sent \end{lemma}

\begin{proof}
Choose affine open neighbourhoods $\Spec(A)$ of $s$ and
$\Spec(B)$ of $x$ such that
$\varphi(\Spec(B)) \subset \Spec(A)$.
Let $\mathfrak p \subset A$ be the prime ideal corresponding to $s$.
Let $\mathfrak q \subset B$ be the prime ideal corresponding to $x$.
Let $I \subset B$ be the ideal corresponding to $T$.
By the initial assumption of the lemma we know that
$A \to B$ is flat and of finite presentation.
 ========

regex_match $\mathfrak p \subset A$
nn candidate p \subset
nn candidate the prime ideal corresponding
nn the prime ideal corresponding
new_annotation [(186059, 186082, 'VAR'), (186086, 186115, 'TYPE')]
===== sent \end{lemma}

\begin{proof}
Choose affine open neighbourhoods $\Spec(A)$ of $s$ and
$\Spec(B)$ of $x$ such that
$\varphi(\Spec(B)) \subset \Spec(A)$.
Let $\mathfrak p \subset A$ be the prime ideal corresponding to $s$.
Let $\mathfrak q \subset B$ be the prime ideal corresponding to $x$.
Let $I \subset B$ be the ideal corresponding to $T$.
By the initial assumption of the lemma we know that
$A \to B$ is flat and of finite presentation.
 ========

regex_match $\mathfrak q \subset B$
nn candidate the prime ideal corresponding
nn the prime ideal corresponding
new_annotation [(186128, 186151, 'VAR'), (186155, 186184, 'TYPE')]
===== sent \end{lemma}

\begin{proof}
Choose affine open neighbourhoods $\Spec(A)$ of $s$ and
$\Spec(B)$ of $x$ such that
$\varphi(\Spec(B)) \subset \Spec(A)$.
Let $\mathfrak p \subset A$ be the prime ideal corresponding to $s$.
Let $\mathfrak q \subset B$ be the prime ideal corresponding to $x$.
Let $I \subset B$ be the ideal corresponding to $T$.
By the initial assumption of the lemma we know that
$A \to B$ is flat and of finite presentation.
 ========

regex_match $I \subset B$
nn candidate I
nn candidate the ideal corresponding
nn the ideal corresponding
new_annotation [(186197, 186210, 'VAR'), (186214, 186237, 'TYPE')]
===== sent In other words,
$T \cap U = Z$ after shrinking $U$.

\medskip\noindent
The final statements of the lemma follow immediately from
part (2), combined with the fact that $Z \to S$
is locally of finite presentation if and only if $Z \to X$ is
of finite presentation, see
Morphisms, Lemmas \ref{morphisms-lemma-composition-finite-presentation} and
\ref{morphisms-lemma-finite-presentation-permanence}.
\end{proof}

\noindent
The following lemma is an enhancement of
Morphisms, Lemma \ref{morphisms-lemma-section-smooth-morphism}.

\begin{lemma}
\label{lemma-section-smooth-regular-immersion}
Let $f : X \to S$ be a smooth morphism of schemes.
 ========

regex_match $f : X \to S$
nn candidate X \to
nn candidate a smooth morphism
nn a smooth morphism
new_annotation [(189719, 189732, 'VAR'), (189736, 189753, 'TYPE')]
===== sent Let $\sigma : S \to X$ be a section of $f$.
Then $\sigma$ is a regular immersion.
 ========

regex_match $\sigma : S \to X$
nn candidate $\sigma
nn candidate a section
nn a section
new_annotation [(189770, 189788, 'VAR'), (189792, 189801, 'TYPE')]
===== sent Let $T = \sigma(S)$ be the corresponding closed subscheme of $X$.
 ========

regex_match $T = \sigma(S)$
nn candidate $T
nn candidate the corresponding closed subscheme
nn the corresponding closed subscheme
new_annotation [(190081, 190096, 'VAR'), (190100, 190134, 'TYPE')]
===== sent Let $k$ be a field and let $A$ be a smooth $k$-algebra.
 ========

regex_match $k$
nn candidate a field
nn a field
new_annotation [(190680, 190683, 'VAR'), (190687, 190694, 'TYPE')]
===== sent Let $k$ be a field and let $A$ be a smooth $k$-algebra.
 ========

regex_match $A$
nn candidate a smooth $k$-algebra
nn a smooth $k$-algebra
new_annotation [(190703, 190706, 'VAR'), (190710, 190730, 'TYPE')]
===== sent Let $\mathfrak m \subset A$ be a maximal ideal whose residue field is $k$.
Then $\mathfrak m$ is generated by a quasi-regular sequence, possibly
after replacing $A$ by $A_g$ for some $g \in A$, $g \not \in \mathfrak ========

regex_match $\mathfrak m \subset A$
nn candidate a maximal ideal
nn a maximal ideal
new_annotation [(190736, 190759, 'VAR'), (190763, 190778, 'TYPE')]
===== sent Let $T = i(Y)$ be the corresponding closed subscheme of $X$. ========

regex_match $T = i(Y)$
nn candidate $T
nn candidate the corresponding closed subscheme
nn the corresponding closed subscheme
new_annotation [(192736, 192746, 'VAR'), (192750, 192784, 'TYPE')]
===== sent If $i$ is a Koszul-regular (resp.\ $H_1$-regular, quasi-regular)
immersion, then so is $j$.
\end{lemma}

\begin{proof}
Let $y \in Y$ be any point. ========

new_annotation []
===== sent Will Robinson!}.

\medskip\noindent
Let $(X, \mathcal{O}_X)$ be a locally ringed space.
 ========

regex_match $(X, \mathcal{O}_X)$
nn candidate $(X
nn candidate a locally ringed space
nn a locally ringed space
new_annotation [(197755, 197775, 'VAR'), (197779, 197801, 'TYPE')]
===== sent Moreover, $\mathcal{S}(U)$
is a multiplicative subset of the ring $\mathcal{O}_X(U)$ for
each $U$. Hence we may consider
the presheaf of rings
$$
U \longmapsto \mathcal{S}(U)^{-1} \mathcal{O}_X(U),
$$
see Modules, Lemma \ref{modules-lemma-simple-invert}.

\begin{definition}
\label{definition-sheaf-meromorphic-functions}
Let $(X, \mathcal{O}_X)$ be a locally ringed space.
 ========

regex_match $(X, \mathcal{O}_X)$
nn candidate $(X
nn candidate a locally ringed space
nn a locally ringed space
new_annotation [(198574, 198594, 'VAR'), (198598, 198620, 'TYPE')]
===== sent \begin{example}
\label{example-no-change}
Let $A = \mathbf{C}[x, \{y_\alpha\}_{\alpha \in \mathbf{C}}]/
((x - \alpha)y_\alpha, y_\alpha y_\beta)$. ========

new_annotation []
===== sent Let $X = \Spec(A)$.
In this case $\mathcal{O}_X = \mathcal{K}_X$, since on
any affine open $D(f)$ the ring $A_f$ any nonzerodivisor is
a unit (proof omitted).
 ========

new_annotation []
===== sent \end{example}

\begin{definition}
\label{definition-pullback-meromorphic-sections}
Let $f : (X, \mathcal{O}_X) \to (Y, \mathcal{O}_Y)$ be a morphism
of locally ringed spaces. ========

regex_match $f : (X, \mathcal{O}_X) \to (Y, \mathcal{O}_Y)$
nn candidate (X
nn candidate a morphism
nn a morphism
new_annotation [(199580, 199627, 'VAR'), (199631, 199641, 'TYPE')]
===== sent We say that {\it pullbacks of meromorphic
functions are defined for $f$} if for every pair of open
$U \subset X$, $V \subset Y$ such that $f(U) \subset V$, and any
section $s \in \Gamma(V, \mathcal{S}_Y)$ the pullback
$f^\sharp(s) \in \Gamma(U, \mathcal{O}_X)$ is an element
of $\Gamma(U, \mathcal{S}_X)$.
\end{definition}

\noindent
In this case there is an induced map
$f^\sharp : f^{-1}\mathcal{K}_Y \to \mathcal{K}_X$,
in other words we obtain a commutative diagram of morphisms
of ringed spaces
$$
\xymatrix{
(X, \mathcal{K}_X) \ar[r] \ar[d]^f &
(X, \mathcal{O}_X) \ar[d]^f \\
(Y, \mathcal{K}_Y) \ar[r] &
(Y, \mathcal{O}_X)
}
$$
We sometimes denote $f^*(s) = f^\sharp(s)$ for a
section $s \in \Gamma(Y, \mathcal{K}_Y)$.

\begin{lemma}
\label{lemma-pullback-meromorphic-sections-defined}
Let $f : X \to Y$ be a morphism of schemes.
 ========

regex_match $f : X \to Y$
nn candidate X \to
nn candidate a morphism
nn a morphism
new_annotation [(200464, 200477, 'VAR'), (200481, 200491, 'TYPE')]
===== sent p$.
See Algebra, Lemma \ref{algebra-lemma-minimal-prime-reduced-ring}.
\end{proof}

\noindent
Let $(X, \mathcal{O}_X)$ be a locally ringed space.
 ========

regex_match $(X, \mathcal{O}_X)$
nn candidate $(X
nn candidate a locally ringed space
nn a locally ringed space
new_annotation [(201617, 201637, 'VAR'), (201641, 201663, 'TYPE')]
===== sent Let $\mathcal{F}$ be a sheaf of $\mathcal{O}_X$-modules.
 ========

regex_match $\mathcal{F}$
nn candidate a sheaf
nn a sheaf
new_annotation [(201669, 201682, 'VAR'), (201686, 201693, 'TYPE')]
===== sent \mathcal{S}(U)^{-1}\mathcal{F}(U)$.
Its sheafification is the sheaf
$\mathcal{F} \otimes_{\mathcal{O}_X} \mathcal{K}_X$, see
Modules, Lemma \ref{modules-lemma-simple-invert-module}.

\begin{definition}
\label{definition-meromorphic-section}
Let $X$ be a locally ringed space.
 ========

regex_match $X$
nn candidate a locally ringed space
nn a locally ringed space
new_annotation [(202000, 202003, 'VAR'), (202007, 202029, 'TYPE')]
===== sent Let $\mathcal{F}$ be a sheaf of $\mathcal{O}_X$-modules.
 ========

regex_match $\mathcal{F}$
nn candidate a sheaf
nn a sheaf
new_annotation [(202035, 202048, 'VAR'), (202052, 202059, 'TYPE')]
===== sent \begin{lemma}
\label{lemma-meromorphic-section-restricts-to-zero}
Let $X$ be a quasi-compact scheme. ========

regex_match $X$
nn candidate a quasi-compact scheme
nn a quasi-compact scheme
new_annotation [(203326, 203329, 'VAR'), (203333, 203355, 'TYPE')]
===== sent Let $h \in \Gamma(X, \mathcal{O}_X)$ and
$f \in \Gamma(X, \mathcal{K}_X)$ such that $f$ restricts
to zero on $X_h$. ========

new_annotation []
===== sent \end{proof}

\begin{lemma}
\label{lemma-locally-Noetherian-K}
Let $X$ be a locally Noetherian scheme.
 ========

regex_match $X$
nn candidate a locally Noetherian scheme
nn a locally Noetherian scheme
new_annotation [(204168, 204171, 'VAR'), (204175, 204202, 'TYPE')]
===== sent It is clear that $\mathcal{S}_x$ is contained
in the set of nonzerodivisors of $\mathcal{O}_{X, x} = A_\mathfrak p$.
For the converse, let $f, g \in A$, $g \not \in \mathfrak p$ and
assume $f/g$ is a nonzerodivisor in $A_{\mathfrak p}$. Let
$I = \{a \in A \mid af = 0\}$. Then we see that $I_{\mathfrak p} = 0$ by
exactness of localization. ========

new_annotation []
===== sent Let $f \in \Gamma(X, \mathcal{K}_X)$ be a meromorphic function.
 ========

regex_match $f \in \Gamma(X, \mathcal{K}_X)$
nn candidate $
nn candidate \in \Gamma(X
nn candidate a meromorphic function
nn a meromorphic function
new_annotation [(205400, 205432, 'VAR'), (205436, 205458, 'TYPE')]
===== sent \end{proof}

\begin{lemma}
\label{lemma-reduced-finite-irreducible}
Let $X$ be a scheme.
 ========

regex_match $X$
nn candidate a scheme
nn a scheme
new_annotation [(206771, 206774, 'VAR'), (206778, 206786, 'TYPE')]
===== sent In particular $\mathcal{K}_{X, x}$
is the total quotient ring of $\mathcal{O}_{X, x}$.
\item For any affine open $\Spec(A) = U \subset X$ we have
that $\mathcal{K}_X(U)$ equals the total quotient ring of $A$.
\end{enumerate}
\end{lemma}

\begin{proof}
Let $X$ be as in the lemma. ========

new_annotation []
===== sent Let $X^{(0)} \subset X$ be the
set of generic points of irreducible components of $X$. Let
$$
f :
Y = \coprod\nolimits_{\eta \in X^{(0)}} \Spec(\kappa(\eta))
\longrightarrow
X
$$
be the inclusion of the generic points into $X$ using the
canonical maps of Schemes, Section \ref{schemes-section-points}.
(This morphism was used in
Morphisms, Definition \ref{morphisms-definition-normalization}
to define the normalization of $X$.)
We claim that $\mathcal{K}_X = f_*\mathcal{O}_Y$.
First note that $\mathcal{K}_Y = \mathcal{O}_Y$ as $Y$ is a disjoint
union of spectra of field. ========

new_annotation []
===== sent $$
Let $\Spec(A) = U \subset X$ be an affine open.
 ========

regex_match $\Spec(A) = U \subset X$
nn candidate U
nn candidate an affine
nn an affine
new_annotation [(208171, 208195, 'VAR'), (208199, 208208, 'TYPE')]
===== sent Let $x \in ========

new_annotation []
===== sent Part (3) we saw during the course of the proof that
$\mathcal{K}_X = f_*\mathcal{O}_Y$.
\end{proof}

\begin{lemma}
\label{lemma-reduced-normalization}
Let $X$ be a scheme.
 ========

regex_match $X$
nn candidate a scheme
nn a scheme
new_annotation [(209838, 209841, 'VAR'), (209845, 209853, 'TYPE')]
===== sent \end{proof}

\begin{lemma}
\label{lemma-meromorphic-functions-integral-scheme}
Let $X$ be an integral scheme with generic point $\eta$. We have
\begin{enumerate}
\item the sheaf of meromorphic functions is
isomorphic to the constant sheaf with value the
function field (see
Morphisms, Definition \ref{morphisms-definition-function-field})
of $X$.
\item for any quasi-coherent sheaf $\mathcal{F}$ on $X$ the
sheaf $\mathcal{K}_X(\mathcal{F})$ is isomorphic to the
constant sheaf with value $\mathcal{F}_\eta$.
\end{enumerate}
\end{lemma}

\begin{proof}
Omitted.
 ========

regex_match $X$
nn candidate an integral scheme
nn an integral scheme
new_annotation [(210596, 210599, 'VAR'), (210603, 210621, 'TYPE')]
===== sent \end{proof}

\begin{definition}
\label{definition-regular-meromorphic-section}
Let $X$ be a locally ringed space.
 ========

regex_match $X$
nn candidate a locally ringed space
nn a locally ringed space
new_annotation [(211157, 211160, 'VAR'), (211164, 211186, 'TYPE')]
===== sent Let $\mathcal{L}$ be an invertible $\mathcal{O}_X$-module.
 ========

regex_match $\mathcal{L}$
nn candidate an invertible $\mathcal{O}_X$-module
nn an invertible $\mathcal{O}_X$-module
new_annotation [(211192, 211205, 'VAR'), (211209, 211245, 'TYPE')]
===== sent \begin{lemma}
\label{lemma-meromorphic-sections-pullback}
Let $f : X \to Y$ be a morphism of locally ringed spaces.
 ========

regex_match $f : X \to Y$
nn candidate X \to
nn candidate a morphism
nn a morphism
new_annotation [(211829, 211842, 'VAR'), (211846, 211856, 'TYPE')]
===== sent \begin{enumerate}
\item Let $\mathcal{F}$ be a sheaf of $\mathcal{O}_Y$-modules.
 ========

regex_match $\mathcal{F}$
nn candidate a sheaf
nn a sheaf
new_annotation [(212043, 212056, 'VAR'), (212060, 212067, 'TYPE')]
===== sent There is a canonical pullback map
$f^* : \Gamma(Y, \mathcal{K}_Y(\mathcal{F})) \to
\Gamma(X, \mathcal{K}_X(f^*\mathcal{F}))$
for meromorphic sections of $\mathcal{F}$.
\item Let $\mathcal{L}$ be an invertible $\mathcal{O}_X$-module.
 ========

regex_match $\mathcal{L}$
nn candidate an invertible $\mathcal{O}_X$-module
nn an invertible $\mathcal{O}_X$-module
new_annotation [(212274, 212287, 'VAR'), (212291, 212327, 'TYPE')]
===== sent \begin{lemma}
\label{lemma-regular-meromorphic-section-exists}
Let $X$ be a scheme.
 ========

regex_match $X$
nn candidate a scheme
nn a scheme
new_annotation [(212657, 212660, 'VAR'), (212664, 212672, 'TYPE')]
===== sent Let $\mathcal{L}$ be an invertible $\mathcal{O}_X$-module.
 ========

regex_match $\mathcal{L}$
nn candidate an invertible $\mathcal{O}_X$-module
nn an invertible $\mathcal{O}_X$-module
new_annotation [(212678, 212691, 'VAR'), (212695, 212731, 'TYPE')]
===== sent Let $G \subset X$ be the set of
generic points of irreducible components of $X$. For each $\eta \in G$
denote $j_\eta : \eta \to X$ the canonical morphism
of $\eta = \Spec(\kappa(\eta))$ into $X$
(see Schemes, Lemma \ref{schemes-lemma-characterize-points}). ========

regex_match $G \subset X$
nn candidate $G \subset X$
nn candidate the set
nn the set
new_annotation [(213304, 213317, 'VAR'), (213321, 213328, 'TYPE')]
===== sent Hence the result follows from
Algebra, Lemma \ref{algebra-lemma-total-ring-fractions-no-embedded-points}.
\end{proof}

\begin{lemma}
\label{lemma-regular-meromorphic-ideal-denominators}
Let $X$ be a scheme.
 ========

regex_match $X$
nn candidate a scheme
nn a scheme
new_annotation [(215539, 215542, 'VAR'), (215546, 215554, 'TYPE')]
===== sent Let $\mathcal{L}$ be an invertible $\mathcal{O}_X$-module.
 ========

regex_match $\mathcal{L}$
nn candidate an invertible $\mathcal{O}_X$-module
nn an invertible $\mathcal{O}_X$-module
new_annotation [(215560, 215573, 'VAR'), (215577, 215613, 'TYPE')]
===== sent Let $s$ be a regular meromorphic section of $\mathcal{L}$.
Let us denote $\mathcal{I} \subset \mathcal{O}_X$ the
sheaf of ideals defined by the rule
$$
\mathcal{I}(V)
=
\{f \in \mathcal{O}_X(V) \mid fs \in \mathcal{L}(V)\}.
$$
The formula makes sense since
$\mathcal{L}(V) \subset \mathcal{K}_X(\mathcal{L})(V)$.
Then $\mathcal{I}$ is a quasi-coherent sheaf of ideals and
we have injective maps
$$
1 : \mathcal{I} \longrightarrow \mathcal{O}_X,
\quad
s : \mathcal{I} \longrightarrow \mathcal{L}
$$
whose cokernels are supported on closed nowhere dense subsets of $X$.
\end{lemma}

\begin{proof}
The question is local on $X$.
Hence we may assume that $X = \Spec(A)$,
and $\mathcal{L} = \mathcal{O}_X$. ========

regex_match $s$
nn candidate a regular meromorphic section
nn a regular meromorphic section
new_annotation [(215619, 215622, 'VAR'), (215626, 215655, 'TYPE')]
===== sent \end{proof}

\begin{definition}
\label{definition-regular-meromorphic-ideal-denominators}
Let $X$ be a scheme.
 ========

regex_match $X$
nn candidate a scheme
nn a scheme
new_annotation [(217268, 217271, 'VAR'), (217275, 217283, 'TYPE')]
===== sent Let $\mathcal{L}$ be an invertible $\mathcal{O}_X$-module.
 ========

regex_match $\mathcal{L}$
nn candidate an invertible $\mathcal{O}_X$-module
nn an invertible $\mathcal{O}_X$-module
new_annotation [(217289, 217302, 'VAR'), (217306, 217342, 'TYPE')]
===== sent Let $s$ be a regular meromorphic section of $\mathcal{L}$.
The sheaf of ideals $\mathcal{I}$ constructed in
Lemma \ref{lemma-regular-meromorphic-ideal-denominators}
is called the {\it ideal sheaf of denominators of $s$}.
\end{definition}

\noindent
Here is a lemma which will be used later.

 ========

regex_match $s$
nn candidate a regular meromorphic section
nn a regular meromorphic section
new_annotation [(217348, 217351, 'VAR'), (217355, 217384, 'TYPE')]
===== sent \begin{lemma}
\label{lemma-make-maps-regular-section}
Suppose given
\begin{enumerate}
\item $X$ a locally Noetherian scheme,
\item $\mathcal{L}$ an invertible $\mathcal{O}_X$-module,
\item $s$ a regular meromorphic section of $\mathcal{L}$, and
\item $\mathcal{F}$ coherent on $X$
without embedded associated points and $\text{Supp}(\mathcal{F}) = X$.
\end{enumerate}
Let $\mathcal{I} \subset \mathcal{O}_X$ be the ideal of
denominators of $s$. Let $T \subset X$ be the union
of the supports of $\mathcal{O}_X/\mathcal{I}$ and
$\mathcal{L}/s(\mathcal{I})$ which is a nowhere dense closed
subset $T \subset X$ according to
Lemma \ref{lemma-regular-meromorphic-ideal-denominators}.
 ========

regex_match $\mathcal{I} \subset \mathcal{O}_X$
nn candidate $\mathcal{I
nn candidate the ideal
nn the ideal
new_annotation [(218008, 218043, 'VAR'), (218047, 218056, 'TYPE')]
===== sent \begin{lemma}
\label{lemma-make-maps-regular-section}
Suppose given
\begin{enumerate}
\item $X$ a locally Noetherian scheme,
\item $\mathcal{L}$ an invertible $\mathcal{O}_X$-module,
\item $s$ a regular meromorphic section of $\mathcal{L}$, and
\item $\mathcal{F}$ coherent on $X$
without embedded associated points and $\text{Supp}(\mathcal{F}) = X$.
\end{enumerate}
Let $\mathcal{I} \subset \mathcal{O}_X$ be the ideal of
denominators of $s$. Let $T \subset X$ be the union
of the supports of $\mathcal{O}_X/\mathcal{I}$ and
$\mathcal{L}/s(\mathcal{I})$ which is a nowhere dense closed
subset $T \subset X$ according to
Lemma \ref{lemma-regular-meromorphic-ideal-denominators}.
 ========

regex_match $T \subset X$
nn candidate $T \subset
nn candidate the union
nn the union
new_annotation [(218085, 218098, 'VAR'), (218102, 218111, 'TYPE')]
===== sent Write $\mathcal{F} = \widetilde{M}$.
Let $I = \{x \in A \mid x(a/b) \in A\}$
so that $\mathcal{I} = \widetilde{I}$ (see
proof of Lemma \ref{lemma-regular-meromorphic-ideal-denominators}).
 ========

new_annotation []
===== sent \begin{lemma}
\label{lemma-components-locally-finite}
Let $X$ be a locally Noetherian scheme. ========

regex_match $X$
nn candidate a locally Noetherian scheme
nn a locally Noetherian scheme
new_annotation [(220360, 220363, 'VAR'), (220367, 220394, 'TYPE')]
===== sent Let $Z \subset X$ be a closed
subscheme. ========

regex_match $Z \subset X$
nn candidate $
nn candidate a closed
subscheme
nn a closed
subscheme
new_annotation [(220400, 220413, 'VAR'), (220417, 220435, 'TYPE')]
===== sent The collection of irreducible components of $Z$
is locally finite in $X$.
\end{lemma}

\begin{proof}
Let $U \subset X$ be a quasi-compact open subscheme. ========

regex_match $U \subset X$
nn candidate $U
nn candidate a quasi-compact open subscheme
nn a quasi-compact open subscheme
new_annotation [(220542, 220555, 'VAR'), (220559, 220589, 'TYPE')]
===== sent See
Properties, Lemma \ref{properties-lemma-codimension-local-ring}.

\begin{definition}
\label{definition-Weil-divisor}
Let $X$ be a locally Noetherian integral scheme.
 ========

regex_match $X$
nn candidate a locally Noetherian integral scheme
nn a locally Noetherian integral scheme
new_annotation [(221238, 221241, 'VAR'), (221245, 221281, 'TYPE')]
===== sent \begin{definition}
\label{definition-order-vanishing}
Let $X$ be a locally Noetherian integral scheme. ========

regex_match $X$
nn candidate a locally Noetherian integral scheme
nn a locally Noetherian integral scheme
new_annotation [(221999, 222002, 'VAR'), (222006, 222042, 'TYPE')]
===== sent Let $f \in R(X)^*$.
For every prime divisor $Z \subset X$ we define the
{\it order of vanishing of $f$ along $Z$} as the integer
$$
\text{ord}_Z(f) = \text{ord}_{\mathcal{O}_{X, \xi}}(f)
$$
where the right hand side is the notion of
Algebra, Definition \ref{algebra-definition-ord}
and $\xi$ is the generic point of $Z$.
\end{definition}

\noindent
Note that for $f, g \in R(X)^*$ we have
$$
\text{ord}_Z(fg) = \text{ord}_Z(f) + \text{ord}_Z(g).
 ========

new_annotation []
===== sent \begin{lemma}
\label{lemma-divisor-locally-finite}
Let $X$ be a locally Noetherian integral scheme. ========

regex_match $X$
nn candidate a locally Noetherian integral scheme
nn a locally Noetherian integral scheme
new_annotation [(222947, 222950, 'VAR'), (222954, 222990, 'TYPE')]
===== sent Let $f \in R(X)^*$.
 ========

new_annotation []
===== sent \begin{definition}
\label{definition-principal-divisor}
Let $X$ be a locally Noetherian integral scheme. ========

regex_match $X$
nn candidate a locally Noetherian integral scheme
nn a locally Noetherian integral scheme
new_annotation [(223754, 223757, 'VAR'), (223761, 223797, 'TYPE')]
===== sent Let $f \in R(X)^*$.
The {\it principal Weil divisor associated to $f$} is the Weil divisor
$$
\text{div}(f) = \text{div}_X(f) = \sum \text{ord}_Z(f) [Z]
$$
where the sum is over prime divisors and $\text{ord}_Z(f)$ is as in
Definition \ref{definition-order-vanishing}. This makes sense
by Lemma \ref{lemma-divisor-locally-finite}.
\end{definition}

\begin{lemma}
\label{lemma-div-additive}
Let $X$ be a locally Noetherian integral scheme. ========

new_annotation []
===== sent Let $f \in R(X)^*$.
The {\it principal Weil divisor associated to $f$} is the Weil divisor
$$
\text{div}(f) = \text{div}_X(f) = \sum \text{ord}_Z(f) [Z]
$$
where the sum is over prime divisors and $\text{ord}_Z(f)$ is as in
Definition \ref{definition-order-vanishing}. This makes sense
by Lemma \ref{lemma-divisor-locally-finite}.
\end{definition}

\begin{lemma}
\label{lemma-div-additive}
Let $X$ be a locally Noetherian integral scheme. ========

regex_match $X$
nn candidate a locally Noetherian integral scheme
nn a locally Noetherian integral scheme
new_annotation [(224193, 224196, 'VAR'), (224200, 224236, 'TYPE')]
===== sent Let $f, g \in R(X)^*$. Then
$$
\text{div}_X(fg) = \text{div}_X(f) + \text{div}_X(g)
 ========

new_annotation []
===== sent \begin{definition}
\label{definition-class-group}
Let $X$ be a locally Noetherian integral scheme. ========

regex_match $X$
nn candidate a locally Noetherian integral scheme
nn a locally Noetherian integral scheme
new_annotation [(224686, 224689, 'VAR'), (224693, 224729, 'TYPE')]
===== sent \medskip\noindent
Let $X$ be a scheme. ========

regex_match $X$
nn candidate a scheme
nn a scheme
new_annotation [(225554, 225557, 'VAR'), (225561, 225569, 'TYPE')]
===== sent Let $\mathcal{L}$ be an invertible $\mathcal{O}_X$-module.
 ========

regex_match $\mathcal{L}$
nn candidate an invertible $\mathcal{O}_X$-module
nn an invertible $\mathcal{O}_X$-module
new_annotation [(225575, 225588, 'VAR'), (225592, 225628, 'TYPE')]
===== sent Let $\xi \in X$ be a point. ========

regex_match $\xi \in X$
nn candidate a point
nn a point
new_annotation [(225634, 225645, 'VAR'), (225649, 225656, 'TYPE')]
===== sent \begin{definition}
\label{definition-order-vanishing-meromorphic}
Let $X$ be a locally Noetherian integral scheme.
 ========

regex_match $X$
nn candidate a locally Noetherian integral scheme
nn a locally Noetherian integral scheme
new_annotation [(226813, 226816, 'VAR'), (226820, 226856, 'TYPE')]
===== sent Let $\mathcal{L}$ be an invertible $\mathcal{O}_X$-module.
 ========

regex_match $\mathcal{L}$
nn candidate an invertible $\mathcal{O}_X$-module
nn an invertible $\mathcal{O}_X$-module
new_annotation [(226862, 226875, 'VAR'), (226879, 226915, 'TYPE')]
===== sent Let $s \in \Gamma(X, \mathcal{K}_X(\mathcal{L}))$
be a regular meromorphic section of $\mathcal{L}$.
For every prime divisor $Z \subset X$ we define the
{\it order of vanishing of $s$ along $Z$} as the integer
$$
\text{ord}_{Z, \mathcal{L}}(s)
= \text{ord}_{\mathcal{O}_{X, \xi}}(s/s_\xi)
$$
where the right hand side is the notion of
Algebra, Definition \ref{algebra-definition-ord},
$\xi \in Z$ is the generic point,
and $s_\xi \in \mathcal{L}_\xi$ is a generator.
 ========

new_annotation []
===== sent \begin{lemma}
\label{lemma-divisor-meromorphic-locally-finite}
Let $X$ be a locally Noetherian integral scheme. ========

regex_match $X$
nn candidate a locally Noetherian integral scheme
nn a locally Noetherian integral scheme
new_annotation [(227546, 227549, 'VAR'), (227553, 227589, 'TYPE')]
===== sent Let $\mathcal{L}$ be an
invertible $\mathcal{O}_X$-module. ========

new_annotation []
===== sent Let $s \in \mathcal{K}_X(\mathcal{L})$ be a
regular (i.e., nonzero) meromorphic section of $\mathcal{L}$. Then the sets
$$
\{Z \subset X \mid Z \text{ a prime divisor with generic point }\xi
\text{ and }s\text{ not in }\mathcal{L}_\xi\}
$$
and
$$
\{Z \subset X \mid Z \text{ is a prime divisor and }
\text{ord}_{Z, \mathcal{L}}(s) \not = 0\}
$$
are locally finite in $X$.
\end{lemma}

\begin{proof}
There exists a nonempty open subscheme $U \subset X$ such that $s$
corresponds to a section of $\Gamma(U, \mathcal{L})$ which generates
$\mathcal{L}$ over $U$. ========

new_annotation []
===== sent \end{proof}

\begin{lemma}
\label{lemma-divisor-meromorphic-well-defined}
Let $X$ be a locally Noetherian integral scheme.
 ========

regex_match $X$
nn candidate a locally Noetherian integral scheme
nn a locally Noetherian integral scheme
new_annotation [(228481, 228484, 'VAR'), (228488, 228524, 'TYPE')]
===== sent Let $\mathcal{L}$ be an invertible $\mathcal{O}_X$-module.
 ========

regex_match $\mathcal{L}$
nn candidate an invertible $\mathcal{O}_X$-module
nn an invertible $\mathcal{O}_X$-module
new_annotation [(228530, 228543, 'VAR'), (228547, 228583, 'TYPE')]
===== sent Let $s, s' \in \mathcal{K}_X(\mathcal{L})$ be nonzero
meromorphic sections of $\mathcal{L}$. Then $f = s/s'$
is an element of $R(X)^*$ and we have
$$
\sum \text{ord}_{Z, \mathcal{L}}(s)[Z]
=
\sum \text{ord}_{Z, \mathcal{L}}(s')[Z]
+
\text{div}(f)
$$
as Weil divisors.
 ========

new_annotation []
===== sent \end{proof}

\begin{definition}
\label{definition-divisor-invertible-sheaf}
Let $X$ be a locally Noetherian integral scheme.
 ========

regex_match $X$
nn candidate a locally Noetherian integral scheme
nn a locally Noetherian integral scheme
new_annotation [(229110, 229113, 'VAR'), (229117, 229153, 'TYPE')]
===== sent Let $\mathcal{L}$ be an invertible $\mathcal{O}_X$-module.
 ========

regex_match $\mathcal{L}$
nn candidate an invertible $\mathcal{O}_X$-module
nn an invertible $\mathcal{O}_X$-module
new_annotation [(229159, 229172, 'VAR'), (229176, 229212, 'TYPE')]
===== sent \begin{lemma}
\label{lemma-c1-additive}
Let $X$ be a locally Noetherian integral scheme.
 ========

regex_match $X$
nn candidate a locally Noetherian integral scheme
nn a locally Noetherian integral scheme
new_annotation [(229915, 229918, 'VAR'), (229922, 229958, 'TYPE')]
===== sent Let $\mathcal{L}$, $\mathcal{N}$ be invertible $\mathcal{O}_X$-modules.
 ========

new_annotation []
===== sent Let $s$, resp.\ $t$ be a nonzero meromorphic section
of $\mathcal{L}$, resp.\ $\mathcal{N}$. Then $st$ is a nonzero
meromorphic section of $\mathcal{L} \otimes \mathcal{N}$, and
$$
\text{div}_{\mathcal{L} \otimes \mathcal{N}}(st)
=
\text{div}_\mathcal{L}(s) + \text{div}_\mathcal{N}(t)
$$
in $\text{Div}(X)$. ========

new_annotation []
===== sent In particular, the Weil divisor class of
$\mathcal{L} \otimes_{\mathcal{O}_X} \mathcal{N}$ is the sum
of the Weil divisor classes of $\mathcal{L}$ and $\mathcal{N}$.
\end{lemma}

\begin{proof}
Let $s$, resp.\ $t$ be a nonzero meromorphic section
of $\mathcal{L}$, resp.\ $\mathcal{N}$. Then $st$ is a nonzero
meromorphic section of $\mathcal{L} \otimes \mathcal{N}$.
Let $Z \subset X$ be a prime divisor. ========

new_annotation []
===== sent In particular, the Weil divisor class of
$\mathcal{L} \otimes_{\mathcal{O}_X} \mathcal{N}$ is the sum
of the Weil divisor classes of $\mathcal{L}$ and $\mathcal{N}$.
\end{lemma}

\begin{proof}
Let $s$, resp.\ $t$ be a nonzero meromorphic section
of $\mathcal{L}$, resp.\ $\mathcal{N}$. Then $st$ is a nonzero
meromorphic section of $\mathcal{L} \otimes \mathcal{N}$.
Let $Z \subset X$ be a prime divisor. ========

regex_match $Z \subset X$
nn candidate $Z \subset
nn candidate a prime divisor
nn a prime divisor
new_annotation [(230712, 230725, 'VAR'), (230729, 230744, 'TYPE')]
===== sent Let $\xi \in Z$ be its generic
point. ========

new_annotation []
===== sent \begin{lemma}
\label{lemma-normal-c1-injective}
Let $X$ be a locally Noetherian integral scheme. If $X$ is normal,
then the map (\ref{equation-c1}) $\Pic(X) \to \text{Cl}(X)$
is injective.
 ========

regex_match $X$
nn candidate a locally Noetherian integral scheme
nn a locally Noetherian integral scheme
new_annotation [(231461, 231464, 'VAR'), (231468, 231504, 'TYPE')]
===== sent \end{lemma}

\begin{proof}
Let $\mathcal{L}$ be an invertible $\mathcal{O}_X$-module whose
associated Weil divisor class is trivial. ========

regex_match $\mathcal{L}$
nn candidate whose
associated Weil divisor class
nn whose
associated Weil divisor class
new_annotation [(231629, 231642, 'VAR'), (231683, 231718, 'TYPE')]
===== sent Let $s$ be a regular
meromorphic section of $\mathcal{L}$. The assumption means that
$\text{div}_\mathcal{L}(s) = \text{div}(f)$ for some
$f \in ========

regex_match $s$
nn candidate a regular
meromorphic section
nn a regular
meromorphic section
new_annotation [(231735, 231738, 'VAR'), (231742, 231771, 'TYPE')]
===== sent \end{proof}

\begin{lemma}
\label{lemma-local-rings-UFD-c1-bijective}
Let $X$ be a locally Noetherian integral scheme. ========

regex_match $X$
nn candidate a locally Noetherian integral scheme
nn a locally Noetherian integral scheme
new_annotation [(232986, 232989, 'VAR'), (232993, 233029, 'TYPE')]
===== sent In other words, $\text{div}_\mathcal{L}(fs) = [D]$.
Let $x \in X$ and let $ ========

new_annotation []
===== sent In other words, $\text{div}_\mathcal{L}(fs) = [D]$.
Let $x \in X$ and let $ ========

new_annotation []
===== sent \begin{lemma}
\label{lemma-in-image-pullback}
Let $\varphi : X \to Y$ be a morphism of schemes. ========

regex_match $\varphi : X \to Y$
nn candidate X \to
nn candidate a morphism
nn a morphism
new_annotation [(235405, 235424, 'VAR'), (235428, 235438, 'TYPE')]
===== sent Let $\mathcal{L}$
be an invertible $\mathcal{O}_X$-module. ========

new_annotation []
===== sent Assume that
\begin{enumerate}
\item $X$ is locally Noetherian,
\item $Y$ is locally Noetherian, integral, and normal,
\item $\varphi$ is flat with integral (hence nonempty) fibres,
\item $\varphi$ is either quasi-compact or locally of finite type,
\item $\mathcal{L}$ is trivial when restricted to the generic fibre of
$\varphi$.
\end{enumerate}
Then $\mathcal{L} \cong \varphi^*\mathcal{N}$ for some invertible
$\mathcal{O}_Y$-module $\mathcal{N}$.
\end{lemma}

\begin{proof}
Let $\xi \in Y$ be the generic point. ========

regex_match $\xi \in Y$
nn candidate the generic point
nn the generic point
new_annotation [(235991, 236002, 'VAR'), (236006, 236023, 'TYPE')]
===== sent Let $X_\xi$ be the scheme theoretic
fibre of $\varphi$ over $\xi$. Denote $\mathcal{L}_\xi$ the pullback of
$\mathcal{L}$ to $X_\xi$. Assumption (5) means that $\mathcal{L}_\xi$
is trivial. ========

regex_match $X_\xi$
nn candidate the scheme
nn the scheme
new_annotation [(236029, 236036, 'VAR'), (236040, 236050, 'TYPE')]
===== sent Let $\mathcal{N} \subset \mathcal{K}_Y$ be the $\mathcal{O}_Y$-module
whose sections over an open $V \subset Y$ are those meromorphic functions
$g \in \mathcal{K}_Y(V)$ such that
$\varphi^*(g)s \in \mathcal{L}(\varphi^{-1}V)$.
A priori $\varphi^*(g)s$ is a section of $\mathcal{K}_X(\mathcal{L})$ over
$\varphi^{-1}V$. We claim that $\mathcal{N}$ is an invertible
$\mathcal{O}_Y$-module and that the map
$$
\varphi^*\mathcal{N} \longrightarrow \mathcal{L},\quad
g ========

regex_match $\mathcal{N} \subset \mathcal{K}_Y$
nn candidate $\mathcal{N
nn candidate whose sections
nn whose sections
new_annotation [(236572, 236607, 'VAR'), (236638, 236652, 'TYPE')]
===== sent Of course
$R_{\mathfrak p_i} \to A_{\mathfrak q_i}$ has ramification index $1$.
Let $e_i, e'_i \geq 0$ be the  valuation of $a, a'$ in $A_{\mathfrak q_i}$.
Then $e_i + e'_i$ is the valuation of $rr'$ in $R_{\mathfrak p_i}$. Note that
$$
\mathfrak p_1^{(e_1 + e'_1)} \cap \ldots \cap \mathfrak p_i^{(e_n + e'_n)} =
(rr')
$$
in $R$ by Algebra, Lemma
\ref{algebra-lemma-normal-domain-intersection-localizations-height-1}.
Set
$$
I = \mathfrak p_1^{(e_1)} \cap \ldots ========

new_annotation []
===== sent \medskip\noindent
Let $y \in Y$ be a point.
 ========

regex_match $y \in Y$
nn candidate $y
nn candidate a point
nn a point
new_annotation [(239610, 239619, 'VAR'), (239623, 239630, 'TYPE')]
===== sent By the uniqueness for
the ring map $\mathcal{O}_{Y, y} \to \mathcal{O}_{X, x''}$
we find that $g$ and $g'$ differ (multiplicatively)
by an element in $\mathcal{O}_{Y, y}^*$. Hence we see that $\varphi^*(g)s$
is a generator for $\mathcal{L}$ on an open neighbourhood
of $\varphi^{-1}(y)$. Let $Z \subset X$ be the set of points
$z \in X$ such that $\varphi^*(g)s$ does not generate $\mathcal{L}_z$.
The arguments above show that $Z$ is closed and that $Z = \varphi^{-1}(T)$
for some subset $T \subset Y$ with $y \not \in T$. If we can show that
$T$ is closed, then $g$ will be a generator for $\mathcal{N}$ as an
$\mathcal{O}_Y$-module in the open neighbourhood $Y \setminus T$ of $y$
thereby finishing the proof (some details omitted).

 ========

regex_match $Z \subset X$
nn candidate $Z \subset
nn candidate the set
nn the set
new_annotation [(240588, 240601, 'VAR'), (240605, 240612, 'TYPE')]
===== sent Z$.
\end{proof}

\begin{lemma}
\label{lemma-closure-effective-cartier-divisor}
Let $X$ be a locally Noetherian scheme. ========

regex_match $X$
nn candidate a locally Noetherian scheme
nn a locally Noetherian scheme
new_annotation [(241433, 241436, 'VAR'), (241440, 241467, 'TYPE')]
===== sent Let $U \subset X$ be
an open and let $D \subset U$ be an effective Cartier divisor.
 ========

new_annotation []
===== sent Let $U \subset X$ be
an open and let $D \subset U$ be an effective Cartier divisor.
 ========

regex_match $D \subset U$
nn candidate $D \subset U$
nn candidate an effective Cartier divisor
nn an effective Cartier divisor
new_annotation [(241506, 241519, 'VAR'), (241523, 241551, 'TYPE')]
===== sent If $\mathcal{O}_{X, x}$ is a UFD for all $x \in X \setminus U$,
then there exists an effective Cartier divisor $D' \subset X$
with $D = U \cap D'$.
\end{lemma}

\begin{proof}
Let $D' \subset X$ be the scheme theoretic image of the morphism $D \to X$.
Since $X$ is locally Noetherian the morphism $D \to X$ is quasi-compact, see
Properties, Lemma \ref{properties-lemma-immersion-into-noetherian}.
Hence the formation of $D'$ commutes with passing to opens in $X$ by
Morphisms, Lemma \ref{morphisms-lemma-quasi-compact-scheme-theoretic-image}.
 ========

regex_match $D' \subset X$
nn candidate $D
nn candidate the scheme theoretic image
nn the scheme theoretic image
new_annotation [(241732, 241746, 'VAR'), (241750, 241776, 'TYPE')]
===== sent Let $I \subset A$ be the ideal corresponding to $D'$.
Let $\mathfrak p \subset A$ be a prime ideal corresponding to a point
of $X \setminus ========

regex_match $I \subset A$
nn candidate $I
nn candidate the ideal corresponding
nn the ideal corresponding
new_annotation [(242144, 242157, 'VAR'), (242161, 242184, 'TYPE')]
===== sent Let $I \subset A$ be the ideal corresponding to $D'$.
Let $\mathfrak p \subset A$ be a prime ideal corresponding to a point
of $X \setminus ========

regex_match $\mathfrak p \subset A$
nn candidate p \subset
nn candidate a prime ideal
nn a prime ideal
new_annotation [(242198, 242221, 'VAR'), (242225, 242238, 'TYPE')]
===== sent \end{proof}

\begin{lemma}
\label{lemma-extend-invertible-module}
Let $X$ be a locally Noetherian scheme. ========

regex_match $X$
nn candidate a locally Noetherian scheme
nn a locally Noetherian scheme
new_annotation [(243191, 243194, 'VAR'), (243198, 243225, 'TYPE')]
===== sent Let $U \subset X$ be
an open and let $\mathcal{L}$ be an invertible $\mathcal{O}_U$-module.
 ========

new_annotation []
===== sent Let $U \subset X$ be
an open and let $\mathcal{L}$ be an invertible $\mathcal{O}_U$-module.
 ========

regex_match $\mathcal{L}$
nn candidate $\mathcal{L}$
nn candidate an invertible $\mathcal{O}_U$-module
nn an invertible $\mathcal{O}_U$-module
new_annotation [(243264, 243277, 'VAR'), (243281, 243317, 'TYPE')]
===== sent W$.
Let $W = \Spec(A)$ be an affine open neighbourhood.
 ========

regex_match $W = \Spec(A)$
nn candidate $W
nn candidate an affine open neighbourhood
nn an affine open neighbourhood
new_annotation [(243870, 243884, 'VAR'), (243888, 243916, 'TYPE')]
===== sent \end{proof}

\begin{lemma}
\label{lemma-open-subscheme-UFD}
Let $R$ be a UFD. ========

regex_match $R$
nn candidate a UFD
nn a UFD
new_annotation [(246357, 246360, 'VAR'), (246364, 246369, 'TYPE')]
===== sent Thus if $U \subset \mathbf{A}^n_R$ is open, then the map
$\Pic(\mathbf{A}^n_R) \to \Pic(U)$ is surjective
by Lemma \ref{lemma-extend-invertible-module}.
The vanishing of $\Pic(\mathbf{A}^n_R)$ is equivalent to
the vanishing of the picard group of the UFD $R[x_1, \ldots, x_n]$
which is proved in
More on Algebra, Lemma \ref{more-algebra-lemma-UFD-Pic-trivial}.
\end{proof}

\begin{lemma}
\label{lemma-Pic-projective-space-UFD}
Let $R$ be a UFD. ========

regex_match $R$
nn candidate a UFD
nn a UFD
new_annotation [(247298, 247301, 'VAR'), (247305, 247310, 'TYPE')]
===== sent Hence $X$ is an integral Noetherian scheme all of whose
local rings are UFDs and we see that $\Pic(X) = \text{Cl}(X)$
by Lemma \ref{lemma-local-rings-UFD-c1-bijective}.

\medskip\noindent
The displayed map is a group homomorphism by
Constructions, Lemma \ref{constructions-lemma-apply-modules}.
The map is injective because $H^0$ of
$\mathcal{O}_X$ and $\mathcal{O}_X(m)$ are non-isomorphic $R$-modules
if $m > 0$, see Cohomology of Schemes, Lemma
\ref{coherent-lemma-cohomology-projective-space-over-ring}.
Let $\mathcal{L}$ be an invertible module on $X$.
Consider the open $U = D_+(T_0) \cong \mathbf{A}^n_R$.
The complement $H = X \setminus U$ is a prime divisor because it is
isomorphic to $\text{Proj}(R[T_1, \ldots, T_n])$ which is
integral by the discussion in the previous paragraph.
 ========

regex_match $\mathcal{L}$
nn candidate an invertible module
nn an invertible module
new_annotation [(248383, 248396, 'VAR'), (248400, 248420, 'TYPE')]
===== sent Let $s \in \mathcal{L}(U)$ be a trivializing section.
 ========

regex_match $s \in \mathcal{L}(U)$
nn candidate $s
nn candidate a trivializing section
nn a trivializing section
new_annotation [(248916, 248938, 'VAR'), (248942, 248964, 'TYPE')]
===== sent Recall that a relative Proj is always
separated over the base, see
Constructions, Lemma \ref{constructions-lemma-relative-proj-separated}.

\begin{lemma}
\label{lemma-relative-proj-quasi-compact}
Let $S$ be a scheme. ========

regex_match $S$
nn candidate a scheme
nn a scheme
new_annotation [(249699, 249702, 'VAR'), (249706, 249714, 'TYPE')]
===== sent Let $\mathcal{A}$ be a quasi-coherent graded
$\mathcal{O}_S$-algebra. ========

regex_match $\mathcal{A}$
nn candidate a quasi-coherent
nn a quasi-coherent
new_annotation [(249720, 249733, 'VAR'), (249737, 249753, 'TYPE')]
===== sent \end{proof}

\begin{lemma}
\label{lemma-relative-proj-finite-type}
Let $S$ be a scheme. ========

regex_match $S$
nn candidate a scheme
nn a scheme
new_annotation [(251534, 251537, 'VAR'), (251541, 251549, 'TYPE')]
===== sent Let $\mathcal{A}$ be a quasi-coherent graded
$\mathcal{O}_S$-algebra. ========

regex_match $\mathcal{A}$
nn candidate a quasi-coherent
nn a quasi-coherent
new_annotation [(251555, 251568, 'VAR'), (251572, 251588, 'TYPE')]
===== sent Thus $\text{Proj}(A)$ is of finite type over $R$.
To see the statement on $\mathcal{O}_X(d)$ use part (2) of
Algebra, Lemma \ref{algebra-lemma-dehomogenize-finite-type}.
\end{proof}

\begin{lemma}
\label{lemma-relative-proj-universally-closed}
Let $S$ be a scheme. ========

regex_match $S$
nn candidate a scheme
nn a scheme
new_annotation [(252841, 252844, 'VAR'), (252848, 252856, 'TYPE')]
===== sent Let $\mathcal{A}$ be a quasi-coherent graded
$\mathcal{O}_S$-algebra. ========

regex_match $\mathcal{A}$
nn candidate a quasi-coherent
nn a quasi-coherent
new_annotation [(252862, 252875, 'VAR'), (252879, 252895, 'TYPE')]
===== sent Let $\mathcal{A}$ be the quasi-coherent $\mathcal{O}_X$-algebra
associated to the graded $R$-algebra $A$. ========

regex_match $\mathcal{A}$
nn candidate the quasi-coherent $\mathcal{O}_X$-algebra
nn the quasi-coherent $\mathcal{O}_X$-algebra
new_annotation [(253486, 253499, 'VAR'), (253503, 253545, 'TYPE')]
===== sent \end{proof}

\begin{lemma}
\label{lemma-relative-proj-proper}
Let $S$ be a scheme. ========

regex_match $S$
nn candidate a scheme
nn a scheme
new_annotation [(254472, 254475, 'VAR'), (254479, 254487, 'TYPE')]
===== sent Let $\mathcal{A}$ be a quasi-coherent graded
$\mathcal{O}_S$-algebra. ========

regex_match $\mathcal{A}$
nn candidate a quasi-coherent
nn a quasi-coherent
new_annotation [(254493, 254506, 'VAR'), (254510, 254526, 'TYPE')]
===== sent This follows from Lemma \ref{lemma-relative-proj-universally-closed}.
\end{proof}

\begin{lemma}
\label{lemma-relative-proj-projective}
Let $S$ be a scheme. ========

regex_match $S$
nn candidate a scheme
nn a scheme
new_annotation [(257041, 257044, 'VAR'), (257048, 257056, 'TYPE')]
===== sent Let $\mathcal{A}$ be a quasi-coherent graded
$\mathcal{O}_S$-algebra. ========

regex_match $\mathcal{A}$
nn candidate a quasi-coherent
nn a quasi-coherent
new_annotation [(257062, 257075, 'VAR'), (257079, 257095, 'TYPE')]
===== sent \end{lemma}

\begin{proof}
Namely, the morphism associated to the graded $\mathcal{O}_S$-algebra map
$$
\text{Sym}_{\mathcal{O}_X}^*(\mathcal{A}_1)
\longrightarrow
\mathcal{A}
$$
is a closed immersion $X \to \mathbf{P}(\mathcal{A}_1)$, see
Constructions, Lemma
\ref{constructions-lemma-surjective-generated-degree-1-map-relative-proj}.
\end{proof}

\begin{lemma}
\label{lemma-relative-proj-flat}
Let $S$ be a scheme. ========

regex_match $S$
nn candidate a scheme
nn a scheme
new_annotation [(257781, 257784, 'VAR'), (257788, 257796, 'TYPE')]
===== sent Let $\mathcal{A}$ be a quasi-coherent graded
$\mathcal{O}_S$-algebra. ========

regex_match $\mathcal{A}$
nn candidate a quasi-coherent
nn a quasi-coherent
new_annotation [(257802, 257815, 'VAR'), (257819, 257835, 'TYPE')]
===== sent $p$ is flat and $\mathcal{O}_X(d)$ is
flat over $S$.
\end{lemma}

\begin{proof}
Affine locally flatness of $X$ over $S$ reduces to the following statement:
Let $R$ be a ring, let $A$ be a graded $R$-algebra with
$A_d$ flat over $R$ for $d \gg 0$, let $f \in A_d$
for some $d > 0$, then $A_{(f)}$ is flat over $R$.
 ========

regex_match $R$
nn candidate a ring
nn a ring
new_annotation [(258198, 258201, 'VAR'), (258205, 258211, 'TYPE')]
===== sent $p$ is flat and $\mathcal{O}_X(d)$ is
flat over $S$.
\end{lemma}

\begin{proof}
Affine locally flatness of $X$ over $S$ reduces to the following statement:
Let $R$ be a ring, let $A$ be a graded $R$-algebra with
$A_d$ flat over $R$ for $d \gg 0$, let $f \in A_d$
for some $d > 0$, then $A_{(f)}$ is flat over $R$.
 ========

regex_match $A$
nn candidate a graded $R$-algebra
nn a graded $R$-algebra
new_annotation [(258217, 258220, 'VAR'), (258224, 258244, 'TYPE')]
===== sent $p$ is flat and $\mathcal{O}_X(d)$ is
flat over $S$.
\end{lemma}

\begin{proof}
Affine locally flatness of $X$ over $S$ reduces to the following statement:
Let $R$ be a ring, let $A$ be a graded $R$-algebra with
$A_d$ flat over $R$ for $d \gg 0$, let $f \in A_d$
for some $d > 0$, then $A_{(f)}$ is flat over $R$.
 ========

new_annotation []
===== sent Since $A_{(f)} = \colim A_{nd}$ where the transition maps
are given by multiplication by $f$, this follows from
Algebra, Lemma \ref{algebra-lemma-colimit-flat}.
Argue similarly to get flatness of $\mathcal{O}_X(d)$ over $S$.
\end{proof}

\begin{lemma}
\label{lemma-relative-proj-finite-presentation}
Let $S$ be a scheme. ========

regex_match $S$
nn candidate a scheme
nn a scheme
new_annotation [(258656, 258659, 'VAR'), (258663, 258671, 'TYPE')]
===== sent Let $\mathcal{A}$ be a quasi-coherent graded
$\mathcal{O}_S$-algebra. ========

regex_match $\mathcal{A}$
nn candidate a quasi-coherent
nn a quasi-coherent
new_annotation [(258677, 258690, 'VAR'), (258694, 258710, 'TYPE')]
===== sent \end{lemma}

\begin{proof}
Affine locally this reduces to the following statement:
Let $R$ be a ring and let $A$ be a finitely presented graded $R$-algebra.
 ========

regex_match $R$
nn candidate a ring
nn a ring
new_annotation [(259105, 259108, 'VAR'), (259112, 259118, 'TYPE')]
===== sent \end{lemma}

\begin{proof}
Affine locally this reduces to the following statement:
Let $R$ be a ring and let $A$ be a finitely presented graded $R$-algebra.
 ========

regex_match $A$
new_annotation []
===== sent The finite presentation condition implies we can choose
a presentation
$$
A = R[X_1, \ldots, X_n]/(F_1, \ldots, F_m)
$$
where $R[X_1, \ldots, X_n]$ is a polynomial ring graded by giving
weights $d_i$ to $X_i$ and $F_1, \ldots, F_m$ are homogeneous polynomials
of degree $e_j$. Let $R_0 \subset R$ be the subring generated by
the coefficients of the polynomials $F_1, \ldots, F_m$.
 ========

regex_match $R_0 \subset R$
nn candidate the subring
nn the subring
new_annotation [(259622, 259637, 'VAR'), (259641, 259652, 'TYPE')]
===== sent \begin{lemma}
\label{lemma-closed-subscheme-proj}
Let $S$ be a scheme. ========

regex_match $S$
nn candidate a scheme
nn a scheme
new_annotation [(260757, 260760, 'VAR'), (260764, 260772, 'TYPE')]
===== sent Let $\mathcal{A}$ be a quasi-coherent graded
$\mathcal{O}_S$-algebra. ========

regex_match $\mathcal{A}$
nn candidate a quasi-coherent
nn a quasi-coherent
new_annotation [(260778, 260791, 'VAR'), (260795, 260811, 'TYPE')]
===== sent Let
$p : X = \underline{\text{Proj}}_S(\mathcal{A}) \to S$ be the relative
Proj of $\mathcal{A}$. Let $i : Z \to X$ be a closed subscheme. Denote
$\mathcal{I} \subset \mathcal{A}$ the kernel of the canonical map
$$
\mathcal{A}
\longrightarrow
\bigoplus\nolimits_{d \geq 0} p_*\left((i_*\mathcal{O}_Z)(d)\right)
$$
If $p$ is quasi-compact, then there is an isomorphism
$Z = \underline{\text{Proj}}_S(\mathcal{A}/\mathcal{I})$.
\end{lemma}

\begin{proof}
The morphism $p$ is separated by
Constructions, Lemma \ref{constructions-lemma-relative-proj-separated}.
As $p$ is quasi-compact, $p_*$ transforms quasi-coherent modules
into quasi-coherent modules, see
Schemes, Lemma \ref{schemes-lemma-push-forward-quasi-coherent}.
Hence $\mathcal{I}$ is a quasi-coherent $\mathcal{O}_S$-module.
 ========

regex_match $i : Z \to X$
nn candidate a closed subscheme
nn a closed subscheme
new_annotation [(260946, 260959, 'VAR'), (260963, 260981, 'TYPE')]
===== sent Let $d$ be the least common multiple of $\deg(f_0), \ldots, \deg(f_n)$.
Note that each of the twists $\mathcal{O}_X(nd)$ is invertible, trivialized
by $f_i^{nd/\deg(f_i)}$ over $D_{+}(f_i)$, and that for any quasi-coherent
module $\mathcal{F}$ on $X$ the multiplication maps
$\mathcal{O}_X(nd) \otimes_{\mathcal{O}_X} \mathcal{F}(m)
\to \mathcal{F}(nd + m)$ are isomorphisms, see
Constructions, Lemma \ref{constructions-lemma-when-invertible}.
Observe that $J'$ is the ideal generated by the elements $g/f_0^e$ where
$g \in I$ is homogeneous of degree $e\deg(f_0)$ (see proof of
Constructions, Lemma
\ref{constructions-lemma-surjective-graded-rings-map-proj}).
 ========

regex_match $d$
nn candidate the least common multiple
nn the least common multiple
new_annotation [(262795, 262798, 'VAR'), (262802, 262827, 'TYPE')]
===== sent \end{proof}

\begin{example}
\label{example-closed-subscheme-of-proj}
Let $A$ be a graded ring. ========

regex_match $A$
nn candidate a graded ring
nn a graded ring
new_annotation [(264478, 264481, 'VAR'), (264485, 264498, 'TYPE')]
===== sent Let $X = \text{Proj}(A)$ and $S = \Spec(A_0)$.
Given a graded ideal $I \subset A$ we obtain a closed subscheme
$V_+(I) = \text{Proj}(A/I) \to X$ by Constructions, Lemma
\ref{constructions-lemma-surjective-graded-rings-map-proj}.
 ========

new_annotation []
===== sent For example, suppose that $A = \mathbf{Q}[x, y]$
with $\deg(x) = 2$ and $\deg(y) = 3$. Then $d = 6$.
Let $I = (y^2)$. Then we see $y \in I(V_+(I))$ because
for any homogeneous $f \in A$ such that $6 | \deg(fy)$
we have $y | f$, hence $fy \in I$. It follows that
$I(V_+(I)) = (y)$ but $x^n y \not \in I$ for all $n$
hence $I(V_+(I))$ is not equal to the saturation.
 ========

new_annotation []
===== sent \end{example}

\begin{lemma}
\label{lemma-equation-codim-1-in-projective-space}
Let $R$ be a UFD. ========

regex_match $R$
nn candidate a UFD
nn a UFD
new_annotation [(266775, 266778, 'VAR'), (266782, 266787, 'TYPE')]
===== sent Let $Z \subset \mathbf{P}^n_R$ be a closed subscheme
which has no embedded points such that every irreducible component
of $Z$ has codimension $1$ in $\mathbf{P}^n_R$.
Then the ideal $I(Z) \subset R[T_0, \ldots, T_n]$ corresponding
to $Z$ is principal.
 ========

regex_match $Z \subset \mathbf{P}^n_R$
nn candidate $
nn candidate a closed subscheme
nn a closed subscheme
new_annotation [(266793, 266819, 'VAR'), (266823, 266841, 'TYPE')]
===== sent Thus $Z$ is an effective Cartier divisor by
Lemma \ref{lemma-codimension-1-is-effective-Cartier}.
Let $\mathcal{I} \subset \mathcal{O}_X$ be the quasi-coherent
sheaf of ideals corresponding to $Z$.
Choose an isomorphism $\mathcal{O}(m) \to \mathcal{I}$
for some $m \in \mathbf{Z}$, see
Lemma \ref{lemma-Pic-projective-space-UFD}.
 ========

regex_match $\mathcal{I} \subset \mathcal{O}_X$
nn candidate $\mathcal{I
nn candidate the quasi-coherent
sheaf
nn the quasi-coherent
sheaf
new_annotation [(267397, 267432, 'VAR'), (267436, 267460, 'TYPE')]
===== sent In case the closed subscheme is locally cut out by finitely many
equations we can define it by a finite type ideal sheaf of
$\mathcal{A}$.

\begin{lemma}
\label{lemma-closed-subscheme-proj-finite}
Let $S$ be a quasi-compact and quasi-separated scheme.
 ========

regex_match $S$
nn candidate a quasi-compact and quasi-separated scheme
nn a quasi-compact and quasi-separated scheme
new_annotation [(268953, 268956, 'VAR'), (268960, 269002, 'TYPE')]
===== sent Let $\mathcal{A}$ be a quasi-coherent graded $\mathcal{O}_S$-algebra. ========

regex_match $\mathcal{A}$
new_annotation []
===== sent Let
$p : X = \underline{\text{Proj}}_S(\mathcal{A}) \to S$ be the relative
Proj of $\mathcal{A}$. Let $i : Z \to X$ be a closed subscheme.
 ========

regex_match $i : Z \to X$
nn candidate a closed subscheme
nn a closed subscheme
new_annotation [(269176, 269189, 'VAR'), (269193, 269211, 'TYPE')]
===== sent \end{proof}

\noindent
The following version of Lemma \ref{lemma-closed-subscheme-proj-finite}
will be used in the proof of
Lemma \ref{lemma-composition-admissible-blowups}.

\begin{lemma}
\label{lemma-closed-subscheme-proj-finite-type}
Let $S$ be a quasi-compact and quasi-separated scheme.
 ========

regex_match $S$
nn candidate a quasi-compact and quasi-separated scheme
nn a quasi-compact and quasi-separated scheme
new_annotation [(271225, 271228, 'VAR'), (271232, 271274, 'TYPE')]
===== sent Let $\mathcal{A}$ be a quasi-coherent graded $\mathcal{O}_S$-algebra.
 ========

regex_match $\mathcal{A}$
new_annotation []
===== sent Let $p : X = \underline{\text{Proj}}_S(\mathcal{A}) \to S$ be the relative
Proj of $\mathcal{A}$. Let $i : Z \to X$ be a closed subscheme.
 ========

regex_match $p : X = \underline{\text{Proj}}_S(\mathcal{A}) \to S$
nn candidate $p
nn candidate the relative
Proj
nn the relative
Proj
new_annotation [(271350, 271404, 'VAR'), (271408, 271425, 'TYPE')]
===== sent Let $p : X = \underline{\text{Proj}}_S(\mathcal{A}) \to S$ be the relative
Proj of $\mathcal{A}$. Let $i : Z \to X$ be a closed subscheme.
 ========

regex_match $i : Z \to X$
nn candidate a closed subscheme
nn a closed subscheme
new_annotation [(271448, 271461, 'VAR'), (271465, 271483, 'TYPE')]
===== sent Let $U \subset X$ be an open. ========

regex_match $U \subset X$
nn candidate $U
new_annotation []
===== sent Assume that
\begin{enumerate}
\item $p$ is quasi-compact,
\item $i$ of finite presentation,
\item $U \cap p(i(Z)) = \emptyset$,
\item $U$ is quasi-compact,
\item $\mathcal{A}_n$ is a finite type $\mathcal{O}_S$-module for all $n$.
\end{enumerate}
Then there exists a $d > 0$ and a quasi-coherent finite type
$\mathcal{O}_S$-submodule $\mathcal{F} \subset \mathcal{A}_d$ with (a)
$Z = \underline{\text{Proj}}_S(\mathcal{A}/\mathcal{F}\mathcal{A})$
and (b) the support of $\mathcal{A}_d/\mathcal{F}$ is disjoint from $U$.
\end{lemma}

\begin{proof}
Let $\mathcal{I} \subset \mathcal{A}$ be the sheaf of quasi-coherent
graded ideals constructed in Lemma \ref{lemma-closed-subscheme-proj}.
Let $U_i$, $R_i$, $A_i$, $I_i$, $f_{i, j}$, $h_{i, j, k}$, and $d$
be as constructed in the proof of
Lemma \ref{lemma-closed-subscheme-proj-finite}.
Since $U \cap p(i(Z)) = \emptyset$ we see that
$\mathcal{I}_d|_U = \mathcal{A}_d|_U$ (by our construction of
$\mathcal{I}$ as a kernel). ========

regex_match $\mathcal{I} \subset \mathcal{A}$
nn candidate $\mathcal{I
nn candidate the sheaf
nn the sheaf
new_annotation [(272066, 272099, 'VAR'), (272103, 272112, 'TYPE')]
===== sent Assume that
\begin{enumerate}
\item $p$ is quasi-compact,
\item $i$ of finite presentation,
\item $U \cap p(i(Z)) = \emptyset$,
\item $U$ is quasi-compact,
\item $\mathcal{A}_n$ is a finite type $\mathcal{O}_S$-module for all $n$.
\end{enumerate}
Then there exists a $d > 0$ and a quasi-coherent finite type
$\mathcal{O}_S$-submodule $\mathcal{F} \subset \mathcal{A}_d$ with (a)
$Z = \underline{\text{Proj}}_S(\mathcal{A}/\mathcal{F}\mathcal{A})$
and (b) the support of $\mathcal{A}_d/\mathcal{F}$ is disjoint from $U$.
\end{lemma}

\begin{proof}
Let $\mathcal{I} \subset \mathcal{A}$ be the sheaf of quasi-coherent
graded ideals constructed in Lemma \ref{lemma-closed-subscheme-proj}.
Let $U_i$, $R_i$, $A_i$, $I_i$, $f_{i, j}$, $h_{i, j, k}$, and $d$
be as constructed in the proof of
Lemma \ref{lemma-closed-subscheme-proj-finite}.
Since $U \cap p(i(Z)) = \emptyset$ we see that
$\mathcal{I}_d|_U = \mathcal{A}_d|_U$ (by our construction of
$\mathcal{I}$ as a kernel). ========

new_annotation []
===== sent \end{proof}

\begin{lemma}
\label{lemma-conormal-sheaf-section-projective-bundle}
Let $X$ be a scheme. ========

regex_match $X$
nn candidate a scheme
nn a scheme
new_annotation [(273253, 273256, 'VAR'), (273260, 273268, 'TYPE')]
===== sent Let $\mathcal{E}$ be a quasi-coherent
$\mathcal{O}_X$-module. ========

regex_match $\mathcal{E}$
new_annotation []
===== sent Let $U \subset X$ be an affine open and $k \in \mathcal{E}(U)$ and
$s \in \mathcal{E}(U)$ be local sections such that $k$ maps to
zero in $\mathcal{L}$ and $s$ maps to a generator $\overline{s}$
of $\mathcal{L}$.
 ========

regex_match $U \subset X$
nn candidate $U
nn candidate an affine
nn an affine
new_annotation [(274603, 274616, 'VAR'), (274620, 274629, 'TYPE')]
===== sent \begin{definition}
\label{definition-blow-up}
Let $X$ be a scheme. ========

regex_match $X$
nn candidate a scheme
nn a scheme
new_annotation [(276903, 276906, 'VAR'), (276910, 276918, 'TYPE')]
===== sent Let $\mathcal{I} \subset \mathcal{O}_X$ be a
quasi-coherent sheaf of ideals, and let $Z \subset X$ be the closed subscheme
corresponding to $\mathcal{I}$, see
Schemes, Definition \ref{schemes-definition-immersion}.
The {\it blowing up of $X$ along $Z$}, or the
{\it blowing up of $X$ in the ideal sheaf $\mathcal{I}$} is
the morphism
 ========

new_annotation []
===== sent Let $\mathcal{I} \subset \mathcal{O}_X$ be a
quasi-coherent sheaf of ideals, and let $Z \subset X$ be the closed subscheme
corresponding to $\mathcal{I}$, see
Schemes, Definition \ref{schemes-definition-immersion}.
The {\it blowing up of $X$ along $Z$}, or the
{\it blowing up of $X$ in the ideal sheaf $\mathcal{I}$} is
the morphism
 ========

regex_match $Z \subset X$
nn candidate $Z \subset
nn candidate the closed subscheme
nn the closed subscheme
new_annotation [(277005, 277018, 'VAR'), (277022, 277042, 'TYPE')]
===== sent Note that $\mathcal{O}_{X'}(1)$ is $b$-relatively very ample, even though
$b$ need not be of finite type or even quasi-compact, because
$X'$ comes equipped with a closed immersion into $\mathbf{P}(\mathcal{I})$,
see Morphisms, Example \ref{morphisms-example-very-ample}.

\begin{lemma}
\label{lemma-blowing-up-affine}
Let $X$ be a scheme. ========

regex_match $X$
nn candidate a scheme
nn a scheme
new_annotation [(278530, 278533, 'VAR'), (278537, 278545, 'TYPE')]
===== sent Let $\mathcal{I} \subset \mathcal{O}_X$ be a
quasi-coherent sheaf of ideals. ========

new_annotation []
===== sent Let $U = \Spec(A)$ be an affine open
subscheme of $X$ and let $I \subset A$ be the ideal corresponding to
$\mathcal{I}|_U$. If $b : X' \to X$ is the blow up of $X$ in $\mathcal{I}$,
then there is a canonical isomorphism
$$
b^{-1}(U) = \text{Proj}(\bigoplus\nolimits_{d \geq 0} I^d)
$$
of $b^{-1}(U)$ with the homogeneous spectrum of the Rees algebra
of $I$ in $A$. ========

regex_match $U = \Spec(A)$
nn candidate $
nn candidate an affine open
subscheme
nn an affine open
subscheme
new_annotation [(278628, 278642, 'VAR'), (278646, 278670, 'TYPE')]
===== sent Let $U = \Spec(A)$ be an affine open
subscheme of $X$ and let $I \subset A$ be the ideal corresponding to
$\mathcal{I}|_U$. If $b : X' \to X$ is the blow up of $X$ in $\mathcal{I}$,
then there is a canonical isomorphism
$$
b^{-1}(U) = \text{Proj}(\bigoplus\nolimits_{d \geq 0} I^d)
$$
of $b^{-1}(U)$ with the homogeneous spectrum of the Rees algebra
of $I$ in $A$. ========

regex_match $I \subset A$
nn candidate I
nn candidate X$
nn X$
new_annotation [(278686, 278699, 'VAR'), (278763, 278765, 'TYPE')]
===== sent \end{proof}

\begin{lemma}
\label{lemma-flat-base-change-blowing-up}
Let $X_1 \to X_2$ be a flat morphism of schemes. ========

regex_match $X_1 \to X_2$
nn candidate a flat morphism
nn a flat morphism
new_annotation [(279837, 279850, 'VAR'), (279854, 279869, 'TYPE')]
===== sent Let $Z_2 \subset X_2$ be a
closed subscheme. ========

new_annotation []
===== sent Let $Z_1$ be the inverse image of $Z_2$ in $X_1$.
Let $X'_i$ be the blow up of $Z_i$ in $X_i$. Then there exists a cartesian
diagram
$$
\xymatrix{
X_1' \ar[r] \ar[d] & X_2' \ar[d] \\
 ========

regex_match $Z_1$
nn candidate $
nn candidate the inverse image
nn the inverse image
new_annotation [(279931, 279936, 'VAR'), (279940, 279957, 'TYPE')]
===== sent Let $Z_1$ be the inverse image of $Z_2$ in $X_1$.
Let $X'_i$ be the blow up of $Z_i$ in $X_i$. Then there exists a cartesian
diagram
$$
\xymatrix{
X_1' \ar[r] \ar[d] & X_2' \ar[d] \\
 ========

regex_match $X'_i$
nn candidate the blow
nn the blow
new_annotation [(279981, 279987, 'VAR'), (279991, 279999, 'TYPE')]
===== sent \end{lemma}

\begin{proof}
Let $\mathcal{I}_2$ be the ideal sheaf of $Z_2$ in $X_2$.
Denote $g : X_1 \to X_2$ the given morphism. ========

regex_match $\mathcal{I}_2$
nn candidate the ideal sheaf
nn the ideal sheaf
new_annotation [(280175, 280190, 'VAR'), (280194, 280209, 'TYPE')]
===== sent Thus we see that $X_1 \times_{X_2} X_2' = X_1'$.
\end{proof}

\begin{lemma}
\label{lemma-blowing-up-gives-effective-Cartier-divisor}
Let $X$ be a scheme. ========

regex_match $X$
nn candidate a scheme
nn a scheme
new_annotation [(280926, 280929, 'VAR'), (280933, 280941, 'TYPE')]
===== sent Let $Z \subset X$ be a closed subscheme.
 ========

regex_match $Z \subset X$
nn candidate $
nn candidate a closed subscheme
nn a closed subscheme
new_annotation [(280947, 280960, 'VAR'), (280964, 280982, 'TYPE')]
===== sent Namely, on the affine open corresponding to the blowup algebra
$A' = A[\frac{I}{a}]$ mentioned above $\psi_{univ, 1}$ corresponds to
the $A'$-module map
$$
I \otimes_A A'
\longrightarrow
\left(\Big(\bigoplus\nolimits_{d \geq 0} I^d\Big)_{a^{(1)}}\right)_1
$$
where $a^{(1)}$ is as in Algebra, Definition \ref{algebra-definition-blow-up}.
We omit the verification that this is the map
$I \otimes_A A' \to IA' = aA'$.
\end{proof}

\begin{lemma}[Universal property blowing up]
\label{lemma-universal-property-blowing-up}
Let $X$ be a scheme. ========

regex_match $X$
nn candidate a scheme
nn a scheme
new_annotation [(283083, 283086, 'VAR'), (283090, 283098, 'TYPE')]
===== sent Let $Z \subset X$ be a closed subscheme.
 ========

regex_match $Z \subset X$
nn candidate $
nn candidate a closed subscheme
nn a closed subscheme
new_annotation [(283104, 283117, 'VAR'), (283121, 283139, 'TYPE')]
===== sent Let $\mathcal{C}$ be the full subcategory of $(\Sch/X)$ consisting
of $Y \to X$ such that the inverse image of $Z$ is an effective
Cartier divisor on $Y$. Then the blowing up $b : X' \to X$ of $Z$ in $X$
is a final object of $\mathcal{C}$.
\end{lemma}

\begin{proof}
We see that $b : X' \to X$ is an object of $\mathcal{C}$ according to
Lemma \ref{lemma-blowing-up-gives-effective-Cartier-divisor}.
Let $f : Y \to X$ be an object of $\mathcal{C}$. We have to show there exists
a unique morphism $Y \to X'$ over $X$. Let $D ========

regex_match $\mathcal{C}$
nn candidate the full subcategory
nn the full subcategory
new_annotation [(283145, 283158, 'VAR'), (283162, 283182, 'TYPE')]
===== sent Let $\mathcal{C}$ be the full subcategory of $(\Sch/X)$ consisting
of $Y \to X$ such that the inverse image of $Z$ is an effective
Cartier divisor on $Y$. Then the blowing up $b : X' \to X$ of $Z$ in $X$
is a final object of $\mathcal{C}$.
\end{lemma}

\begin{proof}
We see that $b : X' \to X$ is an object of $\mathcal{C}$ according to
Lemma \ref{lemma-blowing-up-gives-effective-Cartier-divisor}.
Let $f : Y \to X$ be an object of $\mathcal{C}$. We have to show there exists
a unique morphism $Y \to X'$ over $X$. Let $D ========

regex_match $f : Y \to X$
nn candidate an object
nn an object
new_annotation [(283544, 283557, 'VAR'), (283561, 283570, 'TYPE')]
===== sent Let $\mathcal{C}$ be the full subcategory of $(\Sch/X)$ consisting
of $Y \to X$ such that the inverse image of $Z$ is an effective
Cartier divisor on $Y$. Then the blowing up $b : X' \to X$ of $Z$ in $X$
is a final object of $\mathcal{C}$.
\end{lemma}

\begin{proof}
We see that $b : X' \to X$ is an object of $\mathcal{C}$ according to
Lemma \ref{lemma-blowing-up-gives-effective-Cartier-divisor}.
Let $f : Y \to X$ be an object of $\mathcal{C}$. We have to show there exists
a unique morphism $Y \to X'$ over $X$. Let $D ========

new_annotation []
===== sent = f^{-1}(Z)$.
Let $\mathcal{I} \subset \mathcal{O}_X$ be the ideal sheaf of $Z$
and let $\mathcal{I}_D$ be the ideal sheaf of $D$. Then
$f^*\mathcal{I} \to \mathcal{I}_D$ is a surjection
to an invertible $\mathcal{O}_Y$-module. ========

regex_match $\mathcal{I} \subset \mathcal{O}_X$
nn candidate $\mathcal{I
nn candidate the ideal sheaf
nn the ideal sheaf
new_annotation [(283682, 283717, 'VAR'), (283721, 283736, 'TYPE')]
===== sent = f^{-1}(Z)$.
Let $\mathcal{I} \subset \mathcal{O}_X$ be the ideal sheaf of $Z$
and let $\mathcal{I}_D$ be the ideal sheaf of $D$. Then
$f^*\mathcal{I} \to \mathcal{I}_D$ is a surjection
to an invertible $\mathcal{O}_Y$-module. ========

regex_match $\mathcal{I}_D$
nn candidate the ideal sheaf
nn the ideal sheaf
new_annotation [(283752, 283767, 'VAR'), (283771, 283786, 'TYPE')]
===== sent \end{proof}

\begin{lemma}
\label{lemma-characterize-affine-blowup}
Let $b : X' \to X$ be the blowing up of the scheme $X$ along a closed
subscheme $Z$. Let $U = \Spec(A)$ be an affine open of $X$ and let
$I \subset A$ be the ideal corresponding to $Z \cap U$.
Let $a \in I$ and let $x' \in X'$ be a point mapping to a point of $U$.
Then $x'$ is a point of the affine open $U' = \Spec(A[\frac{I}{a}])$
if and only if the image of $a$ in $\mathcal{O}_{X', x'}$ cuts
out the exceptional divisor.
 ========

regex_match $b : X' \to X$
nn candidate X
nn candidate the scheme
nn the scheme
new_annotation [(284788, 284802, 'VAR'), (284824, 284834, 'TYPE')]
===== sent \end{proof}

\begin{lemma}
\label{lemma-characterize-affine-blowup}
Let $b : X' \to X$ be the blowing up of the scheme $X$ along a closed
subscheme $Z$. Let $U = \Spec(A)$ be an affine open of $X$ and let
$I \subset A$ be the ideal corresponding to $Z \cap U$.
Let $a \in I$ and let $x' \in X'$ be a point mapping to a point of $U$.
Then $x'$ is a point of the affine open $U' = \Spec(A[\frac{I}{a}])$
if and only if the image of $a$ in $\mathcal{O}_{X', x'}$ cuts
out the exceptional divisor.
 ========

regex_match $U = \Spec(A)$
nn candidate $U
nn candidate an affine
nn an affine
new_annotation [(284873, 284887, 'VAR'), (284891, 284900, 'TYPE')]
===== sent \end{proof}

\begin{lemma}
\label{lemma-characterize-affine-blowup}
Let $b : X' \to X$ be the blowing up of the scheme $X$ along a closed
subscheme $Z$. Let $U = \Spec(A)$ be an affine open of $X$ and let
$I \subset A$ be the ideal corresponding to $Z \cap U$.
Let $a \in I$ and let $x' \in X'$ be a point mapping to a point of $U$.
Then $x'$ is a point of the affine open $U' = \Spec(A[\frac{I}{a}])$
if and only if the image of $a$ in $\mathcal{O}_{X', x'}$ cuts
out the exceptional divisor.
 ========

new_annotation []
===== sent \end{proof}

\begin{lemma}
\label{lemma-characterize-affine-blowup}
Let $b : X' \to X$ be the blowing up of the scheme $X$ along a closed
subscheme $Z$. Let $U = \Spec(A)$ be an affine open of $X$ and let
$I \subset A$ be the ideal corresponding to $Z \cap U$.
Let $a \in I$ and let $x' \in X'$ be a point mapping to a point of $U$.
Then $x'$ is a point of the affine open $U' = \Spec(A[\frac{I}{a}])$
if and only if the image of $a$ in $\mathcal{O}_{X', x'}$ cuts
out the exceptional divisor.
 ========

regex_match $x' \in X'$
nn candidate a point mapping
nn a point mapping
new_annotation [(284999, 285010, 'VAR'), (285014, 285029, 'TYPE')]
===== sent This gives a morphism $\Spec(\mathcal{O}_{X', x"}) \to \Spec(A') = U'$.
By the universal property of blowing up
(Lemma \ref{lemma-universal-property-blowing-up})
this is a morphism over
$X'$, which of course implies that $x' \in U'$.
\end{proof}

\begin{lemma}
\label{lemma-blow-up-effective-Cartier-divisor}
Let $X$ be a scheme. ========

regex_match $X$
nn candidate a scheme
nn a scheme
new_annotation [(286428, 286431, 'VAR'), (286435, 286443, 'TYPE')]
===== sent Let $Z \subset X$ be an effective Cartier divisor.
 ========

regex_match $Z \subset X$
nn candidate $
nn candidate an effective Cartier divisor
nn an effective Cartier divisor
new_annotation [(286449, 286462, 'VAR'), (286466, 286494, 'TYPE')]
===== sent \end{proof}

\begin{lemma}
\label{lemma-blow-up-reduced-scheme}
Let $X$ be a scheme. ========

regex_match $X$
nn candidate a scheme
nn a scheme
new_annotation [(286749, 286752, 'VAR'), (286756, 286764, 'TYPE')]
===== sent Let $\mathcal{I} \subset \mathcal{O}_X$ be a
quasi-coherent sheaf of ideals. ========

new_annotation []
===== sent \end{lemma}

\begin{proof}
Combine Lemma \ref{lemma-blowing-up-affine}
with Algebra, Lemma \ref{algebra-lemma-blowup-reduced}.
\end{proof}

\begin{lemma}
\label{lemma-blow-up-integral-scheme}
Let $X$ be a scheme. ========

regex_match $X$
nn candidate a scheme
nn a scheme
new_annotation [(287116, 287119, 'VAR'), (287123, 287131, 'TYPE')]
===== sent Let $\mathcal{I} \subset \mathcal{O}_X$ be a
nonzero quasi-coherent sheaf of ideals. ========

new_annotation []
===== sent \end{lemma}

\begin{proof}
Combine Lemma \ref{lemma-blowing-up-affine}
with Algebra, Lemma \ref{algebra-lemma-blowup-domain}.
\end{proof}

\begin{lemma}
\label{lemma-blow-up-and-irreducible-components}
Let $X$ be a scheme. ========

regex_match $X$
nn candidate a scheme
nn a scheme
new_annotation [(287503, 287506, 'VAR'), (287510, 287518, 'TYPE')]
===== sent Let $Z \subset X$ be a closed subscheme.
 ========

regex_match $Z \subset X$
nn candidate $
nn candidate a closed subscheme
nn a closed subscheme
new_annotation [(287524, 287537, 'VAR'), (287541, 287559, 'TYPE')]
===== sent Let $b : X' \to X$ be the blowing up of $X$ along $Z$. ========

regex_match $b : X' \to X$
nn candidate X
new_annotation []
===== sent \end{proof}

\begin{lemma}
\label{lemma-blow-up-pullback-effective-Cartier}
Let $X$ be a scheme. ========

regex_match $X$
nn candidate a scheme
nn a scheme
new_annotation [(288221, 288224, 'VAR'), (288228, 288236, 'TYPE')]
===== sent Let $b : X' \to X$ be a blow up of $X$ in a closed
subscheme. ========

regex_match $b : X' \to X$
nn candidate X
nn candidate a blow
nn a blow
new_annotation [(288242, 288256, 'VAR'), (288260, 288266, 'TYPE')]
===== sent By Lemmas \ref{lemma-blowing-up-affine} and
\ref{lemma-characterize-effective-Cartier-divisor}
this reduces to the following algebra fact:
Let $A$ be a ring, $I \subset A$ an ideal, $a \in I$, and $x \in A$
a nonzerodivisor. ========

regex_match $A$
nn candidate a ring
nn a ring
new_annotation [(288619, 288622, 'VAR'), (288626, 288632, 'TYPE')]
===== sent \end{proof}

\begin{lemma}
\label{lemma-blowing-up-two-ideals}
Let $X$ be a scheme. ========

regex_match $X$
nn candidate a scheme
nn a scheme
new_annotation [(289027, 289030, 'VAR'), (289034, 289042, 'TYPE')]
===== sent Let $\mathcal{I} \subset \mathcal{O}_X$ and
$\mathcal{J}$ be quasi-coherent sheaves of ideals. ========

new_annotation []
===== sent Let $b : X' \to X$
be the blowing up of $X$ in $\mathcal{I}$. Let $b' : X'' \to X'$ be the
blowing up of $X'$ in $b^{-1}\mathcal{J} \mathcal{O}_{X'}$. Then $X'' \to X$
is canonically isomorphic to the blowing up of $X$ in $\mathcal{I}\mathcal{J}$.
\end{lemma}

\begin{proof}
Let $E \subset X'$ be the exceptional divisor of $b$ which is an effective
Cartier divisor by
Lemma \ref{lemma-blowing-up-gives-effective-Cartier-divisor}.
Then $(b')^{-1}E$ is an effective Cartier divisor on $X''$ by
Lemma \ref{lemma-blow-up-pullback-effective-Cartier}.
Let $E' \subset X''$ be the exceptional divisor of $b'$ (also an effective
Cartier divisor). ========

new_annotation []
===== sent Let $b : X' \to X$
be the blowing up of $X$ in $\mathcal{I}$. Let $b' : X'' \to X'$ be the
blowing up of $X'$ in $b^{-1}\mathcal{J} \mathcal{O}_{X'}$. Then $X'' \to X$
is canonically isomorphic to the blowing up of $X$ in $\mathcal{I}\mathcal{J}$.
\end{lemma}

\begin{proof}
Let $E \subset X'$ be the exceptional divisor of $b$ which is an effective
Cartier divisor by
Lemma \ref{lemma-blowing-up-gives-effective-Cartier-divisor}.
Then $(b')^{-1}E$ is an effective Cartier divisor on $X''$ by
Lemma \ref{lemma-blow-up-pullback-effective-Cartier}.
Let $E' \subset X''$ be the exceptional divisor of $b'$ (also an effective
Cartier divisor). ========

new_annotation []
===== sent Let $b : X' \to X$
be the blowing up of $X$ in $\mathcal{I}$. Let $b' : X'' \to X'$ be the
blowing up of $X'$ in $b^{-1}\mathcal{J} \mathcal{O}_{X'}$. Then $X'' \to X$
is canonically isomorphic to the blowing up of $X$ in $\mathcal{I}\mathcal{J}$.
\end{lemma}

\begin{proof}
Let $E \subset X'$ be the exceptional divisor of $b$ which is an effective
Cartier divisor by
Lemma \ref{lemma-blowing-up-gives-effective-Cartier-divisor}.
Then $(b')^{-1}E$ is an effective Cartier divisor on $X''$ by
Lemma \ref{lemma-blow-up-pullback-effective-Cartier}.
Let $E' \subset X''$ be the exceptional divisor of $b'$ (also an effective
Cartier divisor). ========

regex_match $E \subset X'$
nn candidate $E \subset X'$
nn candidate the exceptional divisor
nn the exceptional divisor
new_annotation [(289418, 289432, 'VAR'), (289436, 289459, 'TYPE')]
===== sent Let $b : X' \to X$
be the blowing up of $X$ in $\mathcal{I}$. Let $b' : X'' \to X'$ be the
blowing up of $X'$ in $b^{-1}\mathcal{J} \mathcal{O}_{X'}$. Then $X'' \to X$
is canonically isomorphic to the blowing up of $X$ in $\mathcal{I}\mathcal{J}$.
\end{lemma}

\begin{proof}
Let $E \subset X'$ be the exceptional divisor of $b$ which is an effective
Cartier divisor by
Lemma \ref{lemma-blowing-up-gives-effective-Cartier-divisor}.
Then $(b')^{-1}E$ is an effective Cartier divisor on $X''$ by
Lemma \ref{lemma-blow-up-pullback-effective-Cartier}.
Let $E' \subset X''$ be the exceptional divisor of $b'$ (also an effective
Cartier divisor). ========

regex_match $E' \subset X''$
nn candidate E
nn candidate the exceptional divisor
nn the exceptional divisor
new_annotation [(289690, 289706, 'VAR'), (289710, 289733, 'TYPE')]
===== sent \end{proof}

\begin{lemma}
\label{lemma-blowing-up-projective}
Let $X$ be a scheme. ========

regex_match $X$
nn candidate a scheme
nn a scheme
new_annotation [(290790, 290793, 'VAR'), (290797, 290805, 'TYPE')]
===== sent Let $\mathcal{I} \subset \mathcal{O}_X$ be a
quasi-coherent sheaf of ideals. ========

new_annotation []
===== sent Let $b : X' \to X$ be the blowing up of $X$
in the ideal sheaf $\mathcal{I}$. If $\mathcal{I}$ is of finite type, then
\begin{enumerate}
\item $b : X' \to X$ is a projective morphism, and
\item $\mathcal{O}_{X'}(1)$ is a $b$-relatively ample invertible sheaf.
 ========

regex_match $b : X' \to X$
nn candidate X
nn candidate the ideal sheaf
nn the ideal sheaf
new_annotation [(290888, 290902, 'VAR'), (290931, 290946, 'TYPE')]
===== sent \end{proof}

\begin{lemma}
\label{lemma-composition-finite-type-blowups}
Let $X$ be a quasi-compact and quasi-separated scheme.
 ========

regex_match $X$
nn candidate a quasi-compact and quasi-separated scheme
nn a quasi-compact and quasi-separated scheme
new_annotation [(291963, 291966, 'VAR'), (291970, 292012, 'TYPE')]
===== sent Let $Z \subset X$ be a closed subscheme of finite presentation.
 ========

regex_match $Z \subset X$
nn candidate $Z \subset
nn candidate a closed subscheme
nn a closed subscheme
new_annotation [(292018, 292031, 'VAR'), (292035, 292053, 'TYPE')]
===== sent Let $b : X' \to X$ be the blowing up with center $Z$. Let $Z' \subset X'$ be
a closed subscheme of finite presentation.
 ========

regex_match $b : X' \to X$
nn candidate X
nn candidate $Z
nn $Z
new_annotation [(292082, 292096, 'VAR'), (292136, 292138, 'TYPE')]
===== sent Let $b : X' \to X$ be the blowing up with center $Z$. Let $Z' \subset X'$ be
a closed subscheme of finite presentation.
 ========

new_annotation []
===== sent Let $X'' \to X'$ be the blowing up with center $Z'$.
There exists a closed subscheme $Y \subset X$ of finite presentation,
such that
\begin{enumerate}
\item $Y = Z \cup b(Z')$ set theoretically, and
\item the composition $X'' \to X$ is isomorphic to the blowing up
of $X$ in $Y$.
\end{enumerate}
\end{lemma}

\begin{proof}
 ========

regex_match $X'' \to X'$
nn candidate $X
nn candidate a closed subscheme $Y \subset X$
nn a closed subscheme $Y \subset X$
new_annotation [(292202, 292214, 'VAR'), (292264, 292296, 'TYPE')]
===== sent Let $S$ be a scheme and let $Z \subset S$ be a closed subscheme.
 ========

regex_match $S$
nn candidate a scheme
nn a scheme
new_annotation [(296196, 296199, 'VAR'), (296203, 296211, 'TYPE')]
===== sent Let $S$ be a scheme and let $Z \subset S$ be a closed subscheme.
 ========

regex_match $Z \subset S$
nn candidate $Z \subset
nn candidate a closed subscheme
nn a closed subscheme
new_annotation [(296220, 296233, 'VAR'), (296237, 296255, 'TYPE')]
===== sent Let $b : S' \to S$ be the blowing up of $S$ in $Z$ and denote $E \subset S'$
the exceptional divisor $E = b^{-1}Z$. In the following we will often
consider a scheme $X$ over $S$ and form the cartesian diagram
$$
\xymatrix{
\text{pr}_{S'}^{-1}E \ar[r] \ar[d] &
X \times_S S' \ar[r]_-{\text{pr}_X} \ar[d]_{\text{pr}_{S'}} &
X \ar[d]^f \\
E \ar[r] & S' \ar[r] & S
}
$$
Since $E$ is an effective Cartier divisor
(Lemma \ref{lemma-blowing-up-gives-effective-Cartier-divisor})
we see that $\text{pr}_{S'}^{-1}E \subset X \times_S ========

regex_match $b : S' \to S$
nn candidate S
nn candidate Z$
nn Z$
new_annotation [(296261, 296275, 'VAR'), (296305, 296307, 'TYPE')]
===== sent In the situation of Definition \ref{definition-strict-transform}.
\begin{enumerate}
\item The strict transform $X'$ of $X$ is the blowup of $X$ in the closed
subscheme $f^{-1}Z$ of $X$.
\item For a quasi-coherent $\mathcal{O}_X$-module $\mathcal{F}$ the
strict transform $\mathcal{F}'$ is canonically isomorphic to
the pushforward along $X' \to X \times_S S'$ of the strict transform of
$\mathcal{F}$ relative to the blowing up $X' \to X$.
\end{enumerate}
\end{lemma}

\begin{proof}
Let $X'' \to X$ be the blowup of $X$ in $f^{-1}Z$. By the universal
property of blowing up (Lemma \ref{lemma-universal-property-blowing-up})
there exists a commutative diagram
$$
\xymatrix{
X'' \ar[r] \ar[d] & X \ar[d] \\
 ========

regex_match $X'' \to X$
nn candidate $X
nn candidate the blowup
nn the blowup
new_annotation [(298873, 298884, 'VAR'), (298888, 298898, 'TYPE')]
===== sent \medskip\noindent
In the notation above, let $\mathcal{F}$ correspond to the $B$-module $N$.
The strict transform of $\mathcal{F}$ corresponds to the
$B \otimes_A A[\frac{I}{a}]$-module
$$
N' = (N \otimes_A A[\textstyle{\frac{I}{a}}])/a\text{-power-torsion}
$$
see Properties, Lemma
\ref{properties-lemma-sections-supported-on-closed-subset}.
The strict transform of $\mathcal{F}$ relative to the blowup of
$X$ in $f^{-1}Z$ corresponds to the $B[\frac{J}{b}]$-module
$N \otimes_B B[\frac{J}{b}]/b\text{-power-torsion}$. In exactly the same
way as above one proves that these two modules are isomorphic.
 ========

new_annotation []
===== sent S'$.
\item Let $\mathcal{F}$ be a quasi-coherent $\mathcal{O}_X$-module.
 ========

regex_match $\mathcal{F}$
nn candidate a quasi-coherent $\mathcal{O}_X$-module
nn a quasi-coherent $\mathcal{O}_X$-module
new_annotation [(301484, 301497, 'VAR'), (301501, 301540, 'TYPE')]
===== sent Let $y \in N \otimes_A A[\frac{I}{a}]$
with $a^n y = 0$. If $\mathfrak q \subset B$
is a prime and $a \not \in \mathfrak q$, then $y$ maps to
zero in $(N \otimes_A A[\frac{I}{a}])_\mathfrak q$. on the other hand,
if $a \in \mathfrak q$, then $N_\mathfrak q$ is a flat $A$-module
and we see that
$N_\mathfrak q \otimes_A A[\frac{I}{a}]
=(N \otimes_A A[\frac{I}{a}])_\mathfrak q$
has no $a$-power torsion (as $A[\frac{I}{a}]$ doesn't).
 ========

new_annotation []
===== sent We conclude that
$y$ is zero by
Algebra, Lemma \ref{algebra-lemma-characterize-zero-local}.
\end{proof}

\begin{lemma}
\label{lemma-strict-transform-affine}
Let $S$ be a scheme. ========

regex_match $S$
nn candidate a scheme
nn a scheme
new_annotation [(303023, 303026, 'VAR'), (303030, 303038, 'TYPE')]
===== sent Let $Z \subset S$ be a closed subscheme.
 ========

regex_match $Z \subset S$
nn candidate $
nn candidate a closed subscheme
nn a closed subscheme
new_annotation [(303044, 303057, 'VAR'), (303061, 303079, 'TYPE')]
===== sent Let $b : S' \to S$ be the blowing up of $Z$ in $S$. Let
$g : X \to Y$ be an affine morphism of schemes over $S$.
Let $\mathcal{F}$ be a quasi-coherent sheaf on $X$.
Let $g' : X \times_S S' \to Y \times_S S'$ be the base change
of $g$. Let $\mathcal{F}'$ be the strict transform of $\mathcal{F}$
relative to $b$. Then $g'_*\mathcal{F}'$ is the strict transform
of $g_*\mathcal{F}$.
\end{lemma}

\begin{proof}
 ========

regex_match $b : S' \to S$
nn candidate S
nn candidate X \to
nn X \to
new_annotation [(303085, 303099, 'VAR'), (303142, 303147, 'TYPE')]
===== sent Let $b : S' \to S$ be the blowing up of $Z$ in $S$. Let
$g : X \to Y$ be an affine morphism of schemes over $S$.
Let $\mathcal{F}$ be a quasi-coherent sheaf on $X$.
Let $g' : X \times_S S' \to Y \times_S S'$ be the base change
of $g$. Let $\mathcal{F}'$ be the strict transform of $\mathcal{F}$
relative to $b$. Then $g'_*\mathcal{F}'$ is the strict transform
of $g_*\mathcal{F}$.
\end{lemma}

\begin{proof}
 ========

regex_match $\mathcal{F}$
nn candidate a quasi-coherent sheaf
nn a quasi-coherent sheaf
new_annotation [(303198, 303211, 'VAR'), (303215, 303237, 'TYPE')]
===== sent Let $b : S' \to S$ be the blowing up of $Z$ in $S$. Let
$g : X \to Y$ be an affine morphism of schemes over $S$.
Let $\mathcal{F}$ be a quasi-coherent sheaf on $X$.
Let $g' : X \times_S S' \to Y \times_S S'$ be the base change
of $g$. Let $\mathcal{F}'$ be the strict transform of $\mathcal{F}$
relative to $b$. Then $g'_*\mathcal{F}'$ is the strict transform
of $g_*\mathcal{F}$.
\end{lemma}

\begin{proof}
 ========

regex_match $g' : X \times_S S' \to Y \times_S S'$
nn candidate : X \times_S S
nn candidate the base change
nn the base change
new_annotation [(303250, 303288, 'VAR'), (303292, 303307, 'TYPE')]
===== sent Let $b : S' \to S$ be the blowing up of $Z$ in $S$. Let
$g : X \to Y$ be an affine morphism of schemes over $S$.
Let $\mathcal{F}$ be a quasi-coherent sheaf on $X$.
Let $g' : X \times_S S' \to Y \times_S S'$ be the base change
of $g$. Let $\mathcal{F}'$ be the strict transform of $\mathcal{F}$
relative to $b$. Then $g'_*\mathcal{F}'$ is the strict transform
of $g_*\mathcal{F}$.
\end{lemma}

\begin{proof}
 ========

regex_match $\mathcal{F}'$
nn candidate the strict transform
nn the strict transform
new_annotation [(303320, 303334, 'VAR'), (303338, 303358, 'TYPE')]
===== sent Observe that $g'_*\text{pr}_X^*\mathcal{F} = \text{pr}_Y^*g_*\mathcal{F}$
by Cohomology of Schemes, Lemma \ref{coherent-lemma-affine-base-change}.
Let $\mathcal{K} \subset \text{pr}_X^*\mathcal{F}$ be the subsheaf
of sections supported in the inverse image of $Z$ in $X \times_S S'$.
 ========

regex_match $\mathcal{K} \subset \text{pr}_X^*\mathcal{F}$
nn candidate the subsheaf
nn the subsheaf
new_annotation [(303640, 303686, 'VAR'), (303690, 303702, 'TYPE')]
===== sent \end{proof}

\begin{lemma}
\label{lemma-strict-transform-different-centers}
Let $S$ be a scheme. ========

regex_match $S$
nn candidate a scheme
nn a scheme
new_annotation [(304217, 304220, 'VAR'), (304224, 304232, 'TYPE')]
===== sent Let $Z \subset S$ be a closed subscheme.
 ========

regex_match $Z \subset S$
nn candidate $
nn candidate a closed subscheme
nn a closed subscheme
new_annotation [(304238, 304251, 'VAR'), (304255, 304273, 'TYPE')]
===== sent Let $D \subset S$ be an effective Cartier divisor.
 ========

regex_match $D \subset S$
nn candidate $D \subset S$
nn candidate an effective Cartier divisor
nn an effective Cartier divisor
new_annotation [(304279, 304292, 'VAR'), (304296, 304324, 'TYPE')]
===== sent Let $Z' \subset S$ be the closed subscheme cut out by the product
of the ideal sheaves of $Z$ and $D$.
Let $S' \to S$ be the blowup of $S$ in $Z$.
\begin{enumerate}
\item ========

regex_match $Z' \subset S$
nn candidate Z
nn candidate the closed subscheme
nn the closed subscheme
new_annotation [(304330, 304344, 'VAR'), (304348, 304368, 'TYPE')]
===== sent Let $Z' \subset S$ be the closed subscheme cut out by the product
of the ideal sheaves of $Z$ and $D$.
Let $S' \to S$ be the blowup of $S$ in $Z$.
\begin{enumerate}
\item ========

regex_match $S' \to S$
nn candidate S
nn candidate the blowup
nn the blowup
new_annotation [(304433, 304443, 'VAR'), (304447, 304457, 'TYPE')]
===== sent Let $f : X \to S$ be a morphism of schemes and let $\mathcal{F}$
be a quasi-coherent $\mathcal{O}_X$-module. ========

regex_match $f : X \to S$
nn candidate X \to
nn candidate a morphism
nn a morphism
new_annotation [(304562, 304575, 'VAR'), (304579, 304589, 'TYPE')]
===== sent Let $f : X \to S$ be a morphism of schemes and let $\mathcal{F}$
be a quasi-coherent $\mathcal{O}_X$-module. ========

new_annotation []
===== sent Let $A$ be a ring, $I \subset A$ an ideal,
$x \in A$ a nonzerodivisor, and $a \in I$. ========

regex_match $A$
nn candidate a ring
nn a ring
new_annotation [(305198, 305201, 'VAR'), (305205, 305211, 'TYPE')]
===== sent Let $M$ be an $A$-module
whose $x$-torsion is zero. ========

regex_match $M$
nn candidate whose $x$-torsion
nn whose $x$-torsion
new_annotation [(305284, 305287, 'VAR'), (305305, 305322, 'TYPE')]
===== sent \end{proof}

\begin{lemma}
\label{lemma-strict-transform-composition-blowups}
Let $S$ be a scheme. ========

regex_match $S$
nn candidate a scheme
nn a scheme
new_annotation [(305797, 305800, 'VAR'), (305804, 305812, 'TYPE')]
===== sent Let $Z \subset S$ be a closed subscheme.
 ========

regex_match $Z \subset S$
nn candidate $
nn candidate a closed subscheme
nn a closed subscheme
new_annotation [(305818, 305831, 'VAR'), (305835, 305853, 'TYPE')]
===== sent Let $b : S' \to S$ be the blowing up with center $Z$. Let $Z' \subset S'$ be
a closed subscheme. ========

regex_match $b : S' \to S$
nn candidate S
nn candidate $Z
nn $Z
new_annotation [(305859, 305873, 'VAR'), (305913, 305915, 'TYPE')]
===== sent Let $b : S' \to S$ be the blowing up with center $Z$. Let $Z' \subset S'$ be
a closed subscheme. ========

new_annotation []
===== sent Let $S'' \to S'$ be the blowing up with center $Z'$.
Let $Y \subset S$ be a closed subscheme such that
$Y = Z \cup b(Z')$ set theoretically and the composition $S'' \to S$
is isomorphic to the blowing up of $S$ in $Y$.
 ========

regex_match $S'' \to S'$
nn candidate $S
nn candidate a closed subscheme
nn a closed subscheme
new_annotation [(305956, 305968, 'VAR'), (306026, 306044, 'TYPE')]
===== sent Let $S'' \to S'$ be the blowing up with center $Z'$.
Let $Y \subset S$ be a closed subscheme such that
$Y = Z \cup b(Z')$ set theoretically and the composition $S'' \to S$
is isomorphic to the blowing up of $S$ in $Y$.
 ========

regex_match $Y \subset S$
nn candidate a closed subscheme
nn a closed subscheme
new_annotation [(306009, 306022, 'VAR'), (306026, 306044, 'TYPE')]
===== sent In this situation, given any scheme $X$ over $S$ and
$\mathcal{F} \in \QCoh(\mathcal{O}_X)$ we have
\begin{enumerate}
\item the strict transform of $\mathcal{F}$ with respect to the blowing
up of $S$ in $Y$ is equal to the strict transform with respect to the
blowup $S'' \to S'$ in $Z'$ of the strict transform of $\mathcal{F}$
with respect to the blowup $S' \to S$ of $S$ in $Z$, and
\item the strict transform of $X$ with respect to the blowing
up of $S$ in $Y$ is equal to the strict transform with respect to the
blowup $S'' \to S'$ in $Z'$ of the strict transform of $X$
with respect to the blowup $S' \to S$ of $S$ in $Z$.
\end{enumerate}
\end{lemma}

\begin{proof}
Let $\mathcal{F}'$ be the strict transform of $\mathcal{F}$ with respect
to the blowup $S' \to S$ of $S$ in $Z$.
Let $\mathcal{F}''$ be the strict transform of $\mathcal{F}'$ with respect
to the blowup $S'' \to S'$ of $S'$ in $Z'$.
Let $\mathcal{G}$ be the strict transform of $\mathcal{F}$ with respect
to the blowup $S'' \to S$ of $S$ in $Y$.
We also label the morphisms
$$
\xymatrix{
X \times_S S'' \ar[r]_q ========

regex_match $\mathcal{F}'$
nn candidate the strict transform
nn the strict transform
new_annotation [(306848, 306862, 'VAR'), (306866, 306886, 'TYPE')]
===== sent In this situation, given any scheme $X$ over $S$ and
$\mathcal{F} \in \QCoh(\mathcal{O}_X)$ we have
\begin{enumerate}
\item the strict transform of $\mathcal{F}$ with respect to the blowing
up of $S$ in $Y$ is equal to the strict transform with respect to the
blowup $S'' \to S'$ in $Z'$ of the strict transform of $\mathcal{F}$
with respect to the blowup $S' \to S$ of $S$ in $Z$, and
\item the strict transform of $X$ with respect to the blowing
up of $S$ in $Y$ is equal to the strict transform with respect to the
blowup $S'' \to S'$ in $Z'$ of the strict transform of $X$
with respect to the blowup $S' \to S$ of $S$ in $Z$.
\end{enumerate}
\end{lemma}

\begin{proof}
Let $\mathcal{F}'$ be the strict transform of $\mathcal{F}$ with respect
to the blowup $S' \to S$ of $S$ in $Z$.
Let $\mathcal{F}''$ be the strict transform of $\mathcal{F}'$ with respect
to the blowup $S'' \to S'$ of $S'$ in $Z'$.
Let $\mathcal{G}$ be the strict transform of $\mathcal{F}$ with respect
to the blowup $S'' \to S$ of $S$ in $Y$.
We also label the morphisms
$$
\xymatrix{
X \times_S S'' \ar[r]_q ========

regex_match $\mathcal{F}''$
nn candidate the strict transform
nn the strict transform
new_annotation [(306961, 306976, 'VAR'), (306980, 307000, 'TYPE')]
===== sent In this situation, given any scheme $X$ over $S$ and
$\mathcal{F} \in \QCoh(\mathcal{O}_X)$ we have
\begin{enumerate}
\item the strict transform of $\mathcal{F}$ with respect to the blowing
up of $S$ in $Y$ is equal to the strict transform with respect to the
blowup $S'' \to S'$ in $Z'$ of the strict transform of $\mathcal{F}$
with respect to the blowup $S' \to S$ of $S$ in $Z$, and
\item the strict transform of $X$ with respect to the blowing
up of $S$ in $Y$ is equal to the strict transform with respect to the
blowup $S'' \to S'$ in $Z'$ of the strict transform of $X$
with respect to the blowup $S' \to S$ of $S$ in $Z$.
\end{enumerate}
\end{lemma}

\begin{proof}
Let $\mathcal{F}'$ be the strict transform of $\mathcal{F}$ with respect
to the blowup $S' \to S$ of $S$ in $Z$.
Let $\mathcal{F}''$ be the strict transform of $\mathcal{F}'$ with respect
to the blowup $S'' \to S'$ of $S'$ in $Z'$.
Let $\mathcal{G}$ be the strict transform of $\mathcal{F}$ with respect
to the blowup $S'' \to S$ of $S$ in $Y$.
We also label the morphisms
$$
\xymatrix{
X \times_S S'' \ar[r]_q ========

regex_match $\mathcal{G}$
nn candidate the strict transform
nn the strict transform
new_annotation [(307080, 307093, 'VAR'), (307097, 307117, 'TYPE')]
===== sent Let $a\in A$. ========

new_annotation []
===== sent Thus $a^n(x - y) = 0$. Hence $y$ maps to the class of $x$ in
$M_2/a\text{-power torsion}$ as desired.
\end{proof}

\begin{lemma}
\label{lemma-strict-transform-blowup-fitting-ideal}
Let $S$ be a scheme. ========

regex_match $S$
nn candidate a scheme
nn a scheme
new_annotation [(310512, 310515, 'VAR'), (310519, 310527, 'TYPE')]
===== sent Let $\mathcal{F}$ be a finite type
quasi-coherent $\mathcal{O}_S$-module. ========

regex_match $\mathcal{F}$
nn candidate a finite type
nn a finite type
new_annotation [(310533, 310546, 'VAR'), (310550, 310563, 'TYPE')]
===== sent Let $Z_k \subset S$ be the closed
subscheme cut out by $\text{Fit}_k(\mathcal{F})$, see
Section \ref{section-fitting-ideals}.
Let $S' \to S$ be the blowup of $S$ in $Z_k$ and let
$\mathcal{F}'$ be the strict transform of $\mathcal{F}$.
Then $\mathcal{F}'$ can locally be generated by $\leq k$
sections.
 ========

regex_match $Z_k \subset S$
nn candidate $
nn candidate the closed
subscheme
nn the closed
subscheme
new_annotation [(310607, 310622, 'VAR'), (310626, 310646, 'TYPE')]
===== sent Let $Z_k \subset S$ be the closed
subscheme cut out by $\text{Fit}_k(\mathcal{F})$, see
Section \ref{section-fitting-ideals}.
Let $S' \to S$ be the blowup of $S$ in $Z_k$ and let
$\mathcal{F}'$ be the strict transform of $\mathcal{F}$.
Then $\mathcal{F}'$ can locally be generated by $\leq k$
sections.
 ========

regex_match $S' \to S$
nn candidate S
nn candidate the blowup
nn the blowup
new_annotation [(310733, 310743, 'VAR'), (310747, 310757, 'TYPE')]
===== sent \end{lemma}

\begin{proof}
Recall that $\mathcal{F}'$ can locally be generated by $\leq k$
sections if and only if $\text{Fit}_k(\mathcal{F}') = \mathcal{O}_{S'}$, see
Lemma \ref{lemma-fitting-ideal-generate-locally}.
Hence this lemma is a translation of
More on Algebra, Lemma \ref{more-algebra-lemma-blowup-fitting-ideal}.
\end{proof}

\begin{lemma}
\label{lemma-strict-transform-blowup-fitting-ideal-locally-free}
Let $S$ be a scheme. ========

regex_match $S$
nn candidate a scheme
nn a scheme
new_annotation [(311327, 311330, 'VAR'), (311334, 311342, 'TYPE')]
===== sent Let $\mathcal{F}$ be a finite type
quasi-coherent $\mathcal{O}_S$-module. ========

regex_match $\mathcal{F}$
nn candidate a finite type
nn a finite type
new_annotation [(311348, 311361, 'VAR'), (311365, 311378, 'TYPE')]
===== sent Let $Z_k \subset S$ be the closed
subscheme cut out by $\text{Fit}_k(\mathcal{F})$, see
Section \ref{section-fitting-ideals}.
 ========

regex_match $Z_k \subset S$
nn candidate $
nn candidate the closed
subscheme
nn the closed
subscheme
new_annotation [(311422, 311437, 'VAR'), (311441, 311461, 'TYPE')]
===== sent Assume that $\mathcal{F}$ is locally free of rank $k$ on $S \setminus Z_k$.
Let $S' \to S$ be the blowup of $S$ in $Z_k$ and let
$\mathcal{F}'$ be the strict transform of $\mathcal{F}$.
Then $\mathcal{F}'$ is locally free of rank $k$.
\end{lemma}

\begin{proof}
Translation of More on Algebra, Lemma
\ref{more-algebra-lemma-blowup-fitting-ideal-locally-free}.
\end{proof}







\section{Admissible blowups}
\label{section-admissible-blowups}

\noindent
To have a bit more control over our blowups we introduce the following
standard terminology.

 ========

regex_match $S' \to S$
nn candidate S
nn candidate the blowup
nn the blowup
new_annotation [(311624, 311634, 'VAR'), (311638, 311648, 'TYPE')]
===== sent \begin{definition}
\label{definition-admissible-blowup}
Let $X$ be a scheme. ========

regex_match $X$
nn candidate a scheme
nn a scheme
new_annotation [(312152, 312155, 'VAR'), (312159, 312167, 'TYPE')]
===== sent Let $U \subset X$ be an open subscheme. ========

regex_match $U \subset X$
nn candidate $
nn candidate an open subscheme
nn an open subscheme
new_annotation [(312173, 312186, 'VAR'), (312190, 312207, 'TYPE')]
===== sent \end{slogan}
Let $X$ be a quasi-compact and quasi-separated scheme.
 ========

regex_match $X$
nn candidate a quasi-compact and quasi-separated scheme
nn a quasi-compact and quasi-separated scheme
new_annotation [(313318, 313321, 'VAR'), (313325, 313367, 'TYPE')]
===== sent Let $U \subset X$ be a quasi-compact open subscheme.
 ========

regex_match $U \subset X$
nn candidate $U
nn candidate a quasi-compact open subscheme
nn a quasi-compact open subscheme
new_annotation [(313373, 313386, 'VAR'), (313390, 313420, 'TYPE')]
===== sent Let $b : X' \to X$ be a $U$-admissible blowup.
 ========

regex_match $b : X' \to X$
nn candidate X
nn candidate a $U$-admissible blowup
nn a $U$-admissible blowup
new_annotation [(313426, 313440, 'VAR'), (313444, 313467, 'TYPE')]
===== sent Let $X'' \to X'$ be a $U$-admissible blowup.
 ========

regex_match $X'' \to X'$
nn candidate $X
nn candidate a $U$-admissible blowup
nn a $U$-admissible blowup
new_annotation [(313473, 313485, 'VAR'), (313489, 313512, 'TYPE')]
===== sent \end{lemma}

\begin{proof}
Immediate from the more precise
Lemma \ref{lemma-composition-finite-type-blowups}.
\end{proof}

\begin{lemma}
\label{lemma-extend-admissible-blowups}
Let $X$ be a quasi-compact and quasi-separated scheme.
 ========

regex_match $X$
nn candidate a quasi-compact and quasi-separated scheme
nn a quasi-compact and quasi-separated scheme
new_annotation [(313756, 313759, 'VAR'), (313763, 313805, 'TYPE')]
===== sent Let $U, V \subset X$ be quasi-compact open subschemes.
 ========

new_annotation []
===== sent Let $b : V' \to V$ be a $U \cap V$-admissible blowup.
 ========

regex_match $b : V' \to V$
nn candidate V
nn candidate a $U
nn a $U
new_annotation [(313866, 313880, 'VAR'), (313884, 313888, 'TYPE')]
===== sent Then there exists a $U$-admissible blowup $X' \to X$
whose restriction to $V$ is $V'$.
\end{lemma}

\begin{proof}
Let $\mathcal{I} \subset \mathcal{O}_V$ be the finite type
quasi-coherent sheaf of ideals such that $V(\mathcal{I})$ is
disjoint from $U \cap V$ and such that $V'$ is isomorphic to the
blow up of $V$ in $\mathcal{I}$. Let
$\mathcal{I}' \subset ========

regex_match $\mathcal{I} \subset \mathcal{O}_V$
nn candidate $\mathcal{I
nn candidate the finite type
nn the finite type
new_annotation [(314034, 314069, 'VAR'), (314073, 314088, 'TYPE')]
===== sent \end{proof}

\begin{lemma}
\label{lemma-dominate-admissible-blowups}
Let $X$ be a quasi-compact and quasi-separated scheme.
 ========

regex_match $X$
nn candidate a quasi-compact and quasi-separated scheme
nn a quasi-compact and quasi-separated scheme
new_annotation [(314776, 314779, 'VAR'), (314783, 314825, 'TYPE')]
===== sent Let $U \subset X$ be a quasi-compact open subscheme.
 ========

regex_match $U \subset X$
nn candidate $U
nn candidate a quasi-compact open subscheme
nn a quasi-compact open subscheme
new_annotation [(314831, 314844, 'VAR'), (314848, 314878, 'TYPE')]
===== sent Let $b_i : X_i \to X$, $i = 1, \ldots, n$ be $U$-admissible blowups.
 ========

new_annotation []
===== sent \end{lemma}

\begin{proof}
Let $\mathcal{I}_i \subset \mathcal{O}_X$ be the finite type
quasi-coherent sheaf of ideals such that $V(\mathcal{I}_i)$ is
disjoint from $U$ and such that $X_i$ is isomorphic to the
blow up of $X$ in $\mathcal{I}_i$. Set
$\mathcal{I} = \mathcal{I}_1 \cdot \ldots \cdot \mathcal{I}_n$
and let $X'$ be the blowup of $X$ in $\mathcal{I}$. Then
$X' \to X$ factors through $b_i$ by Lemma \ref{lemma-blowing-up-two-ideals}.
\end{proof}

\begin{lemma}
\label{lemma-separate-disjoint-opens-by-blowing-up}
\begin{slogan}
Separate irreducible components by blowing up.
 ========

regex_match $\mathcal{I}_i \subset \mathcal{O}_X$
nn candidate the finite type
nn the finite type
new_annotation [(315174, 315211, 'VAR'), (315215, 315230, 'TYPE')]
===== sent \end{lemma}

\begin{proof}
Let $\mathcal{I}_i \subset \mathcal{O}_X$ be the finite type
quasi-coherent sheaf of ideals such that $V(\mathcal{I}_i)$ is
disjoint from $U$ and such that $X_i$ is isomorphic to the
blow up of $X$ in $\mathcal{I}_i$. Set
$\mathcal{I} = \mathcal{I}_1 \cdot \ldots \cdot \mathcal{I}_n$
and let $X'$ be the blowup of $X$ in $\mathcal{I}$. Then
$X' \to X$ factors through $b_i$ by Lemma \ref{lemma-blowing-up-two-ideals}.
\end{proof}

\begin{lemma}
\label{lemma-separate-disjoint-opens-by-blowing-up}
\begin{slogan}
Separate irreducible components by blowing up.
 ========

regex_match $X'$
nn candidate the blowup
nn the blowup
new_annotation [(315463, 315467, 'VAR'), (315471, 315481, 'TYPE')]
===== sent \end{slogan}
Let $X$ be a quasi-compact and quasi-separated scheme.
 ========

regex_match $X$
nn candidate a quasi-compact and quasi-separated scheme
nn a quasi-compact and quasi-separated scheme
new_annotation [(315747, 315750, 'VAR'), (315754, 315796, 'TYPE')]
===== sent Let $U, V$ be quasi-compact disjoint open subschemes of $X$.
Then there exist a $U \cup ========

new_annotation []
===== sent Since $\mathcal{I}$ and $\mathcal{J}$ are of finite type and $X$
is quasi-compact there exists an $n > 0$ such that
$\mathcal{I}^n \mathcal{J}^n = 0$. We may and do replace $\mathcal{I}$
by $\mathcal{I}^n$ and $\mathcal{J}$ by $\mathcal{J}^n$. Whence
$\mathcal{I} \mathcal{J} = 0$. Let $b : X' \to X$ be the blowing
up in $\mathcal{I} + \mathcal{J}$. This is $U \cup ========

regex_match $b : X' \to X$
nn candidate X
nn candidate \mathcal{I
nn \mathcal{I
new_annotation [(316753, 316767, 'VAR'), (316790, 316800, 'TYPE')]
===== sent \begin{lemma}
\label{lemma-filter-after-modification}
Let $X$ be an integral scheme. ========

regex_match $X$
nn candidate an integral scheme
nn an integral scheme
new_annotation [(318918, 318921, 'VAR'), (318925, 318943, 'TYPE')]
===== sent Let $\mathcal{E}$ be a finite locally free
$\mathcal{O}_X$-module. ========

regex_match $\mathcal{E}$
new_annotation []
===== sent Assume $r > 1$.
Let $P = \mathbf{P}(\mathcal{E})$ with structure morphism $\pi : P \to X$,
see Constructions, Section \ref{constructions-section-projective-bundle}.
Then $\pi$ is proper (Lemma \ref{lemma-relative-proj-proper}).
 ========

new_annotation []
===== sent In particular, there exists a section $s : U \to P_U$ of $\pi$.
Let $X' \subset P$ be the scheme theoretic image of the
morphism $U \to P_U ========

regex_match $X' \subset P$
nn candidate $X
nn candidate the scheme theoretic image
nn the scheme theoretic image
new_annotation [(319898, 319912, 'VAR'), (319916, 319942, 'TYPE')]
===== sent \end{proof}

\begin{lemma}
\label{lemma-extend-rational-map-after-modification}
Let $S$ be a scheme. ========

regex_match $S$
nn candidate a scheme
nn a scheme
new_annotation [(320859, 320862, 'VAR'), (320866, 320874, 'TYPE')]
===== sent Let $X$, $Y$ be schemes over $S$.
Assume $X$ is Noetherian and $Y$ is proper over $S$.
Given an $S$-rational map $f : U \to Y$ from $X$ to $Y$
there exists a morphism $p : X' \to X$ and an
$S$-morphism $f' : X' \to Y$ such that
\begin{enumerate}
\item $p$ is proper and $p^{-1}(U) \to U$ is an isomorphism,
\item $f'|_{p^{-1}(U)}$ is equal to $f \circ p|_{p^{-1}(U)}$.
\end{enumerate}
\end{lemma}

\begin{proof}
Denote $j : U \to X$ the inclusion morphism. ========

new_annotation []
===== sent Let $X' \subset Y \times_S X$
be the scheme theoretic image of $(f, j) : U \to Y \times_S X$
(Morphisms, Definition \ref{morphisms-definition-scheme-theoretic-image}).
 ========

new_annotation []
file:  more-morphisms.tex
===== sent \item Let $S$ be a scheme. ========

regex_match $S$
nn candidate a scheme
nn a scheme
new_annotation [(1304, 1307, 'VAR'), (1311, 1319, 'TYPE')]
===== sent Let $i_X : X \to X'$ be a thickening.
 ========

regex_match $i_X : X \to X'$
nn candidate a thickening
nn a thickening
new_annotation [(1650, 1666, 'VAR'), (1670, 1682, 'TYPE')]
===== sent \begin{lemma}
\label{lemma-first-order-thickening}
Let $X$ be a scheme over a base $S$. Consider a short exact sequence
$$
0 \to \mathcal{I} \to \mathcal{A} \to \mathcal{O}_X \to 0
$$
of sheaves on $X$ where $\mathcal{A}$ is a sheaf of
$f^{-1}\mathcal{O}_S$-algebras,
$\mathcal{A} \to \mathcal{O}_X$ is a surjection
of sheaves of $f^{-1}\mathcal{O}_S$-algebras, and $\mathcal{I}$ is its kernel.
 ========

regex_match $X$
nn candidate a scheme
nn a scheme
new_annotation [(2797, 2800, 'VAR'), (2804, 2812, 'TYPE')]
===== sent Let $U = \Spec(B)$
be an affine open of $X$. Set $A = \Gamma(U, \mathcal{A})$. Note that
since $H^1(U, \mathcal{I}) = 0$ (see Cohomology of Schemes, Lemma
\ref{coherent-lemma-quasi-coherent-affine-cohomology-zero})
the map $A \to B$ is surjective. ========

new_annotation []
===== sent More precisely, we will
use Cohomology of Schemes, Lemma
\ref{coherent-lemma-quasi-compact-h1-zero-covering}. Let $\mathcal{F}$ be a
quasi-coherent $\mathcal{O}_{X'}$-module. ========

new_annotation []
===== sent Thus $\mathcal{F}$ has a finite filtration whose successive quotients
have vanishing first cohomology and it follows by a simple
induction argument that $H^1(X', \mathcal{F}) = 0$.
\end{proof}

\begin{lemma}
\label{lemma-base-change-thickening}
Let $S \subset S'$ be a thickening of schemes. ========

regex_match $S \subset S'$
nn candidate $S
nn candidate a thickening
nn a thickening
new_annotation [(7048, 7062, 'VAR'), (7066, 7078, 'TYPE')]
===== sent Let $X' \to S'$ be a morphism
and set $X = S \times_{S'} X'$. ========

regex_match $X' \to S'$
nn candidate $X
nn candidate a morphism
nn a morphism
new_annotation [(7095, 7106, 'VAR'), (7110, 7120, 'TYPE')]
===== sent \end{lemma}

\begin{proof}
The statement means the following: Let $X \to X'$ be a morphism
of schemes and let $\{g_i : X'_i \to X'\}$
be an fpqc covering such that the base change $X_i \to X'_i$
is a thickening for all $i$. ========

regex_match $X \to X'$
nn candidate $X
nn candidate a morphism
nn a morphism
new_annotation [(7799, 7809, 'VAR'), (7813, 7823, 'TYPE')]
===== sent \end{lemma}

\begin{proof}
The statement means the following: Let $X \to X'$ be a morphism
of schemes and let $\{g_i : X'_i \to X'\}$
be an fpqc covering such that the base change $X_i \to X'_i$
is a thickening for all $i$. ========

new_annotation []
===== sent \begin{lemma}
\label{lemma-thicken-property-morphisms}
Let $(f, f') : (X \subset X') \to (S \subset S')$ be a morphism
of thickenings. ========

regex_match $(f, f') : (X \subset X') \to (S \subset S')$
nn candidate S \subset
nn candidate a morphism
nn a morphism
new_annotation [(8647, 8692, 'VAR'), (8696, 8706, 'TYPE')]
===== sent \end{proof}

\begin{lemma}
\label{lemma-thicken-property-relatively-ample}
Let $(f, f') : (X \subset X') \to (S \subset S')$ be a morphism
of thickenings. ========

regex_match $(f, f') : (X \subset X') \to (S \subset S')$
nn candidate S \subset
nn candidate a morphism
nn a morphism
new_annotation [(10500, 10545, 'VAR'), (10549, 10559, 'TYPE')]
===== sent Let $\mathcal{L}'$ be an invertible sheaf on $X'$
and denote $\mathcal{L}$ the restriction to $X$.
 ========

regex_match $\mathcal{L}'$
nn candidate an invertible sheaf
nn an invertible sheaf
new_annotation [(10580, 10594, 'VAR'), (10598, 10617, 'TYPE')]
===== sent This is Limits, Lemma \ref{limits-lemma-ample-on-reduction}.
\end{proof}

\begin{lemma}
\label{lemma-thicken-property-morphisms-cartesian}
Let $(f, f') : (X \subset X') \to (S \subset S')$ be a morphism
of thickenings such that $X = S \times_{S'} X'$. ========

regex_match $(f, f') : (X \subset X') \to (S \subset S')$
nn candidate S \subset
nn candidate a morphism
nn a morphism
new_annotation [(11330, 11375, 'VAR'), (11379, 11389, 'TYPE')]
===== sent Let
$U' \subset S'$ be an affine open and let $V' \subset X'$ be an affine open
lying over $U'$. Let $U' = \Spec(A')$ and denote $I \subset A'$ be the ideal
defining the closed subscheme $U' \cap S$. Say $V' = \Spec(B')$.
Then $V' \cap X = \Spec(B'/IB')$. Setting $A = A'/I$ and
$B = B'/IB'$ we get a commutative diagram
$$
\xymatrix{
0 \ar[r] &
IB' \ar[r] &
B' \ar[r] &
B \ar[r] & 0 \\
0 \ar[r] &
IA' \ar[r] \ar[u] &
A' \ar[r] \ar[u] &
A \ar[r] \ar[u] & 0
}
$$
with exact rows and $I^2 = 0$.

\medskip\noindent
 ========

regex_match $V' \subset X'$
nn candidate $V
nn candidate an affine
nn an affine
new_annotation [(13362, 13377, 'VAR'), (13381, 13390, 'TYPE')]
===== sent Let
$U' \subset S'$ be an affine open and let $V' \subset X'$ be an affine open
lying over $U'$. Let $U' = \Spec(A')$ and denote $I \subset A'$ be the ideal
defining the closed subscheme $U' \cap S$. Say $V' = \Spec(B')$.
Then $V' \cap X = \Spec(B'/IB')$. Setting $A = A'/I$ and
$B = B'/IB'$ we get a commutative diagram
$$
\xymatrix{
0 \ar[r] &
IB' \ar[r] &
B' \ar[r] &
B \ar[r] & 0 \\
0 \ar[r] &
IA' \ar[r] \ar[u] &
A' \ar[r] \ar[u] &
A \ar[r] \ar[u] & 0
}
$$
with exact rows and $I^2 = 0$.

\medskip\noindent
 ========

new_annotation []
===== sent Let $U' \subset S'$ be the open subscheme whose
underlying topological space is the same as $U$. Then $X' \to S'$
factors through $U'$ and we conclude that $X' \to U'$ is a closed
immersion by part (1). ========

regex_match $U' \subset S'$
nn candidate U
nn candidate the open subscheme
nn the open subscheme
new_annotation [(16880, 16895, 'VAR'), (16899, 16917, 'TYPE')]
===== sent \begin{lemma}
\label{lemma-properties-that-extend-over-thickenings}
Let $(f, f') : (X \subset X') \to (Y \to Y')$ be a morphism
of thickenings. ========

regex_match $(f, f') : (X \subset X') \to (Y \to Y')$
nn candidate (X \subset X
nn candidate a morphism
nn a morphism
new_annotation [(17492, 17533, 'VAR'), (17537, 17547, 'TYPE')]
===== sent This follows immediately from
Morphisms, Lemma \ref{morphisms-lemma-check-closed-infinitesimally}.

\medskip\noindent
Case (4) follows from the following algebra statement: Let $A$ be a ring and
let $I \subset A$ be a locally nilpotent ideal. ========

regex_match $A$
nn candidate a ring
nn a ring
new_annotation [(19750, 19753, 'VAR'), (19757, 19763, 'TYPE')]
===== sent This follows immediately from
Morphisms, Lemma \ref{morphisms-lemma-check-closed-infinitesimally}.

\medskip\noindent
Case (4) follows from the following algebra statement: Let $A$ be a ring and
let $I \subset A$ be a locally nilpotent ideal. ========

regex_match $I \subset A$
nn candidate I
nn candidate a locally nilpotent ideal
nn a locally nilpotent ideal
new_annotation [(19772, 19785, 'VAR'), (19789, 19814, 'TYPE')]
===== sent Let $B$ be a finite type
$A$-algebra. ========

regex_match $B$
nn candidate a finite type
nn a finite type
new_annotation [(19820, 19823, 'VAR'), (19827, 19840, 'TYPE')]
===== sent Let $V' \subset Y'$ be the open subscheme whose
underlying topological space is the same as $V$. Then $X' \to V'$
factors through $V'$ and we conclude that $X' \to V'$ is a closed
immersion by part (3).

\medskip\noindent
Case (8) follows from Lemma \ref{lemma-thicken-property-morphisms}
and the definition of proper morphisms as being the quasi-compact,
universally closed, and separated morphisms that are locally of finite type.
 ========

regex_match $V' \subset Y'$
nn candidate $V
nn candidate the open subscheme
nn the open subscheme
new_annotation [(21685, 21700, 'VAR'), (21704, 21722, 'TYPE')]
===== sent \begin{lemma}
\label{lemma-picard-group-first-order-thickening}
Let $X \subset X'$ be a first order thickening
with ideal sheaf $\mathcal{I}$. Then there is a canonical
exact sequence
$$
\xymatrix{
0 \ar[r] &
H^0(X, \mathcal{I}) \ar[r] &
H^0(X', \mathcal{O}_{X'}^*) \ar[r] &
H^0(X, \mathcal{O}^*_X) \ar ` ========

regex_match $X \subset X'$
nn candidate $X
nn candidate a first order thickening
nn a first order thickening
new_annotation [(22340, 22354, 'VAR'), (22358, 22382, 'TYPE')]
===== sent \end{lemma}

\begin{proof}
This is the long exact cohomology sequence associated to the
short exact sequence of sheaves of abelian groups
$$
0 \to \mathcal{I} \to \mathcal{O}_{X'}^* \to \mathcal{O}_X^* \to 0
$$
where the first map sends a local section $f$ of $\mathcal{I}$
to the invertible section $1 + f$ of $\mathcal{O}_{X'}$.
We also use the identification of the Picard group of a
ringed space with the first cohomology group of the sheaf
of invertible functions, see
Cohomology, Lemma \ref{cohomology-lemma-h1-invertible}.
\end{proof}

\begin{lemma}
\label{lemma-torsion-pic-thickening}
Let $X \subset X'$ be a thickening. ========

regex_match $X \subset X'$
nn candidate $X
nn candidate a thickening
nn a thickening
new_annotation [(23388, 23402, 'VAR'), (23406, 23418, 'TYPE')]
===== sent Let $n$ be an integer
invertible in $\mathcal{O}_X$. Then the map
$\Pic(X')[n] \to \Pic(X)[n]$ is bijective ========

regex_match $n$
nn candidate an integer
invertible
nn an integer
invertible
new_annotation [(23424, 23427, 'VAR'), (23431, 23452, 'TYPE')]
===== sent \end{proof}

\begin{proof}[Proof in general]
Let $\mathcal{I} \subset \mathcal{O}_{X'}$ be the quasi-coherent ideal
sheaf cutting out $X$. ========

regex_match $\mathcal{I} \subset \mathcal{O}_{X'}$
nn candidate $\mathcal{I
nn candidate the quasi-coherent ideal
sheaf
nn the quasi-coherent ideal
sheaf
new_annotation [(24083, 24121, 'VAR'), (24125, 24155, 'TYPE')]
===== sent Choose an
open subscheme $U \subset X$ such that $i$ identifies $Z$ with a closed
subscheme $Z \subset U$. Let $\mathcal{I} \subset \mathcal{O}_U$ be the
quasi-coherent sheaf of ideals defining $Z$ in $U$. Then we can consider
the closed subscheme $Z' \subset U$ defined by the quasi-coherent sheaf
of ideals $\mathcal{I}^2$.

\begin{definition}
\label{definition-first-order-infinitesimal-neighbourhood}
Let $i : Z \to X$ be an immersion of schemes. ========

new_annotation []
===== sent Choose an
open subscheme $U \subset X$ such that $i$ identifies $Z$ with a closed
subscheme $Z \subset U$. Let $\mathcal{I} \subset \mathcal{O}_U$ be the
quasi-coherent sheaf of ideals defining $Z$ in $U$. Then we can consider
the closed subscheme $Z' \subset U$ defined by the quasi-coherent sheaf
of ideals $\mathcal{I}^2$.

\begin{definition}
\label{definition-first-order-infinitesimal-neighbourhood}
Let $i : Z \to X$ be an immersion of schemes. ========

regex_match $i : Z \to X$
nn candidate an immersion
nn an immersion
new_annotation [(25389, 25402, 'VAR'), (25406, 25418, 'TYPE')]
===== sent \begin{lemma}
\label{lemma-first-order-infinitesimal-neighbourhood}
Let $i : Z \to X$ be an immersion of schemes. ========

regex_match $i : Z \to X$
nn candidate Z \to X$
nn candidate an immersion
nn an immersion
new_annotation [(25818, 25831, 'VAR'), (25835, 25847, 'TYPE')]
===== sent The first order infinitesimal
neighbourhood $Z'$ of $Z$ in $X$ has the following universal property:
Given any commutative diagram
$$
\xymatrix{
Z \ar[d]_i & T \ar[l]^a \ar[d] \\
X & T' \ar[l]_b
}
$$
where $T \subset T'$ is a first order thickening over $X$, there exists
a unique morphism $(a', a) : (T \subset T') \to (Z \subset Z')$ of
thickenings over $X$.
\end{lemma}

\begin{proof}
Let $U \subset X$ be the open used in the construction of $Z'$, i.e., an
open such that $Z$ is identified with a closed subscheme of $U$ cut out by
the quasi-coherent sheaf of ideals $\mathcal{I}$.
 ========

regex_match $U \subset X$
nn candidate $U
nn candidate the construction
nn the construction
new_annotation [(26252, 26265, 'VAR'), (26286, 26302, 'TYPE')]
===== sent Since $|T| = |T'|$ we see that $b(T') \subset U$. Hence we can
think of $b$ as a morphism into $U$. Let $\mathcal{J} \subset \mathcal{O}_{T'}$
be the ideal cutting out $T$. ========

new_annotation []
===== sent \end{proof}

\begin{lemma}
\label{lemma-infinitesimal-neighbourhood-conormal}
Let $i : Z \to X$ be an immersion of schemes. ========

regex_match $i : Z \to X$
nn candidate Z \to X$
nn candidate an immersion
nn an immersion
new_annotation [(27107, 27120, 'VAR'), (27124, 27136, 'TYPE')]
===== sent Let $Z \subset Z'$ be
the first order infinitesimal neighbourhood of $Z$ in $X$.
Then the diagram
$$
\xymatrix{
Z \ar[r] \ar[d] & Z' \ar[d] \\
Z \ar[r] & X
}
$$
induces a map of conormal sheaves $\mathcal{C}_{Z/X} \to \mathcal{C}_{Z/Z'}$ by
Morphisms, Lemma \ref{morphisms-lemma-conormal-functorial}.
This map is an isomorphism.
 ========

new_annotation []
===== sent \begin{definition}
\label{definition-formally-unramified}
Let $f : X \to S$ be a morphism of schemes.
 ========

regex_match $f : X \to S$
nn candidate X \to
nn candidate a morphism
nn a morphism
new_annotation [(28168, 28181, 'VAR'), (28185, 28195, 'TYPE')]
===== sent \end{proof}

\begin{lemma}
\label{lemma-formally-unramified-on-opens}
Let $f : X \to S$ be a morphism of schemes.
 ========

regex_match $f : X \to S$
nn candidate X \to
nn candidate a morphism
nn a morphism
new_annotation [(29689, 29702, 'VAR'), (29706, 29716, 'TYPE')]
===== sent Let $U \subset X$ and $V \subset S$ be open such that
$f(U) \subset V$. ========

new_annotation []
===== sent If $f$ is formally unramified, so is $f|_U : U \to V$.
\end{lemma}

\begin{proof}
Consider a solid diagram
$$
\xymatrix{
U \ar[d]_{f|_U} & T \ar[d]^i \ar[l]^a \\
V & T' \ar[l] \ar@{-->}[lu]
}
$$
as in Definition \ref{definition-formally-unramified}. If $f$ is formally
ramified, then there exists at most one
$S$-morphism $a' : T' \to X$ such that $a'|_T = a$.
Hence clearly there exists at most one such morphism into $U$.
\end{proof}

\begin{lemma}
\label{lemma-affine-formally-unramified}
Let $f : X \to S$ be a morphism of schemes.
 ========

regex_match $f : X \to S$
nn candidate X \to
nn candidate a morphism
nn a morphism
new_annotation [(30297, 30310, 'VAR'), (30314, 30324, 'TYPE')]
===== sent \begin{lemma}
\label{lemma-formally-unramified-differentials}
Let $f : X \to S$ be a morphism of schemes.
 ========

regex_match $f : X \to S$
nn candidate X \to
nn candidate a morphism
nn a morphism
new_annotation [(30958, 30971, 'VAR'), (30975, 30985, 'TYPE')]
===== sent We recall some of the arguments of the proof of
Morphisms, Lemma \ref{morphisms-lemma-differentials-affine}.
Let $W \subset X \times_S X$ be an open such that
$\Delta : X \to X \times_S X$ induces a closed immersion into $W$.
Let $\mathcal{J} \subset \mathcal{O}_W$ be the ideal sheaf of this
closed immersion. Let $X' \subset W$ be the closed subscheme
defined by the quasi-coherent sheaf of ideals $\mathcal{J}^2$.
 ========

regex_match $W \subset X \times_S X$
nn candidate $W \subset X \times_S X$
nn candidate such that
$\Delta
nn such that
$\Delta
new_annotation [(32072, 32096, 'VAR'), (32108, 32125, 'TYPE')]
===== sent We recall some of the arguments of the proof of
Morphisms, Lemma \ref{morphisms-lemma-differentials-affine}.
Let $W \subset X \times_S X$ be an open such that
$\Delta : X \to X \times_S X$ induces a closed immersion into $W$.
Let $\mathcal{J} \subset \mathcal{O}_W$ be the ideal sheaf of this
closed immersion. Let $X' \subset W$ be the closed subscheme
defined by the quasi-coherent sheaf of ideals $\mathcal{J}^2$.
 ========

regex_match $\mathcal{J} \subset \mathcal{O}_W$
nn candidate $\mathcal{J
nn candidate the ideal sheaf
nn the ideal sheaf
new_annotation [(32189, 32224, 'VAR'), (32228, 32243, 'TYPE')]
===== sent We recall some of the arguments of the proof of
Morphisms, Lemma \ref{morphisms-lemma-differentials-affine}.
Let $W \subset X \times_S X$ be an open such that
$\Delta : X \to X \times_S X$ induces a closed immersion into $W$.
Let $\mathcal{J} \subset \mathcal{O}_W$ be the ideal sheaf of this
closed immersion. Let $X' \subset W$ be the closed subscheme
defined by the quasi-coherent sheaf of ideals $\mathcal{J}^2$.
 ========

regex_match $X' \subset W$
nn candidate the closed subscheme
nn the closed subscheme
new_annotation [(32274, 32288, 'VAR'), (32292, 32312, 'TYPE')]
===== sent \end{slogan}
Let $f : X \to S$ be a morphism of schemes.
 ========

regex_match $f : X \to S$
nn candidate X \to
nn candidate a morphism
nn a morphism
new_annotation [(34049, 34062, 'VAR'), (34066, 34076, 'TYPE')]
===== sent \end{enumerate}
\end{lemma}

\begin{proof}
Use Lemma \ref{lemma-formally-unramified-differentials} and
Morphisms, Lemma \ref{morphisms-lemma-unramified-omega-zero}.
\end{proof}









\section{Universal first order thickenings}
\label{section-universal-thickening}

\noindent
Let $h : Z \to X$ be a morphism of schemes. ========

regex_match $h : Z \to X$
nn candidate Z \to
nn candidate a morphism
nn a morphism
new_annotation [(34597, 34610, 'VAR'), (34614, 34624, 'TYPE')]
===== sent \begin{lemma}
\label{lemma-universal-thickening}
Let $h : Z \to X$ be a formally unramified morphism of schemes.
 ========

regex_match $h : Z \to X$
nn candidate Z \to
nn candidate a formally unramified morphism
nn a formally unramified morphism
new_annotation [(35403, 35416, 'VAR'), (35420, 35450, 'TYPE')]
===== sent Let $V' \subset Z'$ be the open subscheme such that $V = Z \cap V'$.
Then we claim that $V \subset V'$ is the universal first order thickening of
$V$ over $U$. Namely, suppose given any diagram
$$
\xymatrix{
V \ar[d]_h & T \ar[l]^a \ar[d] \\
 ========

regex_match $V' \subset Z'$
nn candidate $V
nn candidate the open subscheme
nn the open subscheme
new_annotation [(36242, 36257, 'VAR'), (36261, 36279, 'TYPE')]
===== sent By the general remarks above
$Z_i \subset Z_i'$ is also a universal first order thickening of
$Z_i$ over $X$. Let $Z'_{i, j} \subset Z'_i$ be the open subscheme
such that $Z_i \cap Z_j = Z'_{i, j} \cap Z_i$. By the general remarks
we see that both $Z'_{i, j}$ and $Z'_{j, i}$ are universal first
order thickenings of $Z_i \cap Z_j$ over $X$. ========

regex_match $Z'_{i, j} \subset Z'_i$
nn candidate j} \subset
nn candidate the open subscheme
nn the open subscheme
new_annotation [(38370, 38394, 'VAR'), (38398, 38416, 'TYPE')]
===== sent \end{proof}

\begin{definition}
\label{definition-universal-thickening}
Let $h : Z \to X$ be a formally unramified morphism of schemes.
 ========

regex_match $h : Z \to X$
nn candidate Z \to
nn candidate a formally unramified morphism
nn a formally unramified morphism
new_annotation [(40309, 40322, 'VAR'), (40326, 40356, 'TYPE')]
===== sent \begin{lemma}
\label{lemma-immersion-universal-thickening}
Let $i : Z \to X$ be an immersion of schemes. ========

regex_match $i : Z \to X$
nn candidate Z \to X$
nn candidate an immersion
nn an immersion
new_annotation [(41127, 41140, 'VAR'), (41144, 41156, 'TYPE')]
===== sent \end{proof}

\begin{lemma}
\label{lemma-universal-thickening-unramified}
Let $Z \to X$ be a formally unramified morphism of schemes.
 ========

regex_match $Z \to X$
nn candidate a formally unramified morphism
nn a formally unramified morphism
new_annotation [(42162, 42171, 'VAR'), (42175, 42205, 'TYPE')]
===== sent \medskip\noindent
Let $T \subset T'$ be a first order thickening. ========

regex_match $T \subset T'$
nn candidate $T \subset
nn candidate a first order thickening
nn a first order thickening
new_annotation [(42649, 42663, 'VAR'), (42667, 42691, 'TYPE')]
===== sent Let $Z \subset Z'$ be the universal
first order thickening of $Z$ over $X$. Let $W \subset W'$ be the universal
first order thickening of $W$ over $Y$. ========

regex_match $Z \subset Z'$
nn candidate $
nn candidate the universal
first order thickening
nn the universal
first order thickening
new_annotation [(43991, 44005, 'VAR'), (44009, 44045, 'TYPE')]
===== sent Let $Z \subset Z'$ be the universal
first order thickening of $Z$ over $X$. Let $W \subset W'$ be the universal
first order thickening of $W$ over $Y$. ========

regex_match $W \subset W'$
nn candidate $W
nn candidate the universal
first order thickening
nn the universal
first order thickening
new_annotation [(44067, 44081, 'VAR'), (44085, 44121, 'TYPE')]
===== sent More precisely, let $h : Z \to X$ be a formally unramified morphism of schemes.
 ========

regex_match $h : Z \to X$
nn candidate a formally unramified morphism
nn a formally unramified morphism
new_annotation [(47069, 47082, 'VAR'), (47086, 47116, 'TYPE')]
===== sent Let $V \subset Z$, $U \subset X$ be opens such that $h(V) \subset U$.
 ========

new_annotation []
===== sent Let $Z'$ be the universal first order thickening of $Z$ over $X$.
Then $h|_V : V \to U$ is formally unramified and the universal first
order thickening of $V$ over $U$ is the open subscheme $V' \subset ========

regex_match $Z'$
nn candidate the universal first order thickening
nn the universal first order thickening
new_annotation [(47203, 47207, 'VAR'), (47211, 47247, 'TYPE')]
===== sent The compatibility of universal thickenings can be deduced from the proof of
Lemma \ref{lemma-universal-thickening},
or from
Algebra, Lemma \ref{algebra-lemma-universal-thickening-localize}
or deduced from
Lemma \ref{lemma-universal-thickening-fibre-product-flat}.
\end{proof}

\begin{lemma}
\label{lemma-differentials-universally-unramified}
Let $h : Z \to X$ be a formally unramified morphism of schemes over $S$.
Let $Z \subset Z'$ be the universal first order thickening of $Z$
over $X$ with structure morphism $h' : Z' \to X$. The canonical map
$$
c_{h'} : (h')^*\Omega_{X/S} \longrightarrow \Omega_{Z'/S}
$$
induces an isomorphism
$h^*\Omega_{X/S} \to \Omega_{Z'/S} \otimes \mathcal{O}_Z$.
\end{lemma}

\begin{proof}
The map $c_{h'}$ is the map defined in
Morphisms, Lemma \ref{morphisms-lemma-functoriality-differentials}.
If $i : Z \to Z'$ is the given closed immersion, then
$i^*c_{h'}$ is a map
$h^*\Omega_{X/S} \to \Omega_{Z'/S} \otimes \mathcal{O}_Z$.
Checking that it is an isomorphism reduces to the affine case
by localization, see
Lemma \ref{lemma-universal-thickening-localize}
and
Morphisms, Lemma \ref{morphisms-lemma-differentials-restrict-open}.
 ========

regex_match $h : Z \to X$
nn candidate $h
nn candidate a formally unramified morphism
nn a formally unramified morphism
new_annotation [(47935, 47948, 'VAR'), (47952, 47982, 'TYPE')]
===== sent The compatibility of universal thickenings can be deduced from the proof of
Lemma \ref{lemma-universal-thickening},
or from
Algebra, Lemma \ref{algebra-lemma-universal-thickening-localize}
or deduced from
Lemma \ref{lemma-universal-thickening-fibre-product-flat}.
\end{proof}

\begin{lemma}
\label{lemma-differentials-universally-unramified}
Let $h : Z \to X$ be a formally unramified morphism of schemes over $S$.
Let $Z \subset Z'$ be the universal first order thickening of $Z$
over $X$ with structure morphism $h' : Z' \to X$. The canonical map
$$
c_{h'} : (h')^*\Omega_{X/S} \longrightarrow \Omega_{Z'/S}
$$
induces an isomorphism
$h^*\Omega_{X/S} \to \Omega_{Z'/S} \otimes \mathcal{O}_Z$.
\end{lemma}

\begin{proof}
The map $c_{h'}$ is the map defined in
Morphisms, Lemma \ref{morphisms-lemma-functoriality-differentials}.
If $i : Z \to Z'$ is the given closed immersion, then
$i^*c_{h'}$ is a map
$h^*\Omega_{X/S} \to \Omega_{Z'/S} \otimes \mathcal{O}_Z$.
Checking that it is an isomorphism reduces to the affine case
by localization, see
Lemma \ref{lemma-universal-thickening-localize}
and
Morphisms, Lemma \ref{morphisms-lemma-differentials-restrict-open}.
 ========

regex_match $Z \subset Z'$
nn candidate $Z \subset
nn candidate the universal first order thickening
nn the universal first order thickening
new_annotation [(48008, 48022, 'VAR'), (48026, 48062, 'TYPE')]
===== sent In this case the result is
Algebra, Lemma \ref{algebra-lemma-differentials-universal-thickening}.
\end{proof}

\begin{lemma}
\label{lemma-universally-unramified-differentials-sequence}
Let $h : Z \to X$ be a formally unramified morphism of schemes over $S$.
There is a canonical exact sequence
$$
\mathcal{C}_{Z/X} \to h^*\Omega_{X/S} \to \Omega_{Z/S} \to 0.
 ========

regex_match $h : Z \to X$
nn candidate Z \to
nn candidate a formally unramified morphism
nn a formally unramified morphism
new_annotation [(48944, 48957, 'VAR'), (48961, 48991, 'TYPE')]
===== sent Then we have $(i')^*\Omega_{Z'/Y} = i^*\Omega_{X/Y}$ by
Lemma \ref{lemma-differentials-universally-unramified}.
\end{proof}

\begin{lemma}
\label{lemma-transitivity-conormal}
Let $Z \to Y \to X$ be formally unramified morphisms of schemes.
 ========

new_annotation []
===== sent \begin{definition}
\label{definition-formally-etale}
Let $f : X \to S$ be a morphism of schemes.
 ========

regex_match $f : X \to S$
nn candidate X \to
nn candidate a morphism
nn a morphism
new_annotation [(53190, 53203, 'VAR'), (53207, 53217, 'TYPE')]
===== sent \end{lemma}

\begin{proof}
Let $T' = \bigcup T'_i$ be an affine open covering, and let
$T_i = T \cap T'_i$. ========

regex_match $T' = \bigcup T'_i$
nn candidate $T
nn candidate affine
nn affine
new_annotation [(54259, 54278, 'VAR'), (54285, 54291, 'TYPE')]
===== sent \end{proof}

\begin{lemma}
\label{lemma-formally-etale-on-opens}
Let $f : X \to S$ be a morphism of schemes.
 ========

regex_match $f : X \to S$
nn candidate X \to
nn candidate a morphism
nn a morphism
new_annotation [(55147, 55160, 'VAR'), (55164, 55174, 'TYPE')]
===== sent Let $U \subset X$ and $V \subset S$ be open subschemes such that
$f(U) \subset V$. If $f$ is formally \'etale, so is $f|_U : U \to V$.
\end{lemma}

\begin{proof}
Consider a solid diagram
$$
\xymatrix{
U \ar[d]_{f|_U} & T \ar[d]^i \ar[l]^a \\
V & T' \ar[l] \ar@{-->}[lu]
}
$$
as in Definition \ref{definition-formally-etale}. If $f$ is formally
ramified, then there exists exactly one $S$-morphism $a' : T' \to X$
such that $a'|_T = a$. ========

new_annotation []
===== sent Since $|T'| = |T|$ we conclude that $a'(T') \subset U$
which gives our unique morphism from $T'$ into $U$.
\end{proof}

\begin{lemma}
\label{lemma-characterize-formally-etale}
Let $f : X \to S$ be a morphism of schemes.
 ========

regex_match $f : X \to S$
nn candidate X \to
nn candidate a morphism
nn a morphism
new_annotation [(55803, 55816, 'VAR'), (55820, 55830, 'TYPE')]
===== sent In other words, the pullback of $\mathcal{C}_{X/S}$ to $X$ via
the identity morphism is zero, i.e., $\mathcal{C}_{X/S} = 0$.
\end{proof}

\begin{lemma}
\label{lemma-affine-formally-etale}
Let $f : X \to S$ be a morphism of schemes.
 ========

regex_match $f : X \to S$
nn candidate X \to
nn candidate a morphism
nn a morphism
new_annotation [(58246, 58259, 'VAR'), (58263, 58273, 'TYPE')]
===== sent This is immediate from the definitions
(Definition \ref{definition-formally-etale} and
Algebra, Definition \ref{algebra-definition-formally-etale})
by the equivalence of categories of rings and affine schemes,
see
Schemes, Lemma \ref{schemes-lemma-category-affine-schemes}.
\end{proof}

\begin{lemma}
\label{lemma-etale-formally-etale}
Let $f : X \to S$ be a morphism of schemes.
 ========

regex_match $f : X \to S$
nn candidate X \to
nn candidate a morphism
nn a morphism
new_annotation [(58800, 58813, 'VAR'), (58817, 58827, 'TYPE')]
===== sent Throughout this section we use that
a sheaf on a thickening $X'$ of $X$ can be seen as a sheaf on $X$.

\begin{lemma}
\label{lemma-difference-derivation}
Let $S$ be a scheme.
Let $X \subset X'$ and $Y \subset Y'$ be two first order thickenings
over $S$. Let $(a, a'), (b, b') : (X \subset X') \to (Y \subset Y')$
be two morphisms of thickenings over $S$. Assume that
\begin{enumerate}
\item $a = b$, and
\item the two maps $a^*\mathcal{C}_{Y/Y'} \to \mathcal{C}_{X/X'}$
(Morphisms, Lemma \ref{morphisms-lemma-conormal-functorial})
are equal.
 ========

regex_match $S$
nn candidate a scheme
nn a scheme
new_annotation [(60500, 60503, 'VAR'), (60507, 60515, 'TYPE')]
===== sent Throughout this section we use that
a sheaf on a thickening $X'$ of $X$ can be seen as a sheaf on $X$.

\begin{lemma}
\label{lemma-difference-derivation}
Let $S$ be a scheme.
Let $X \subset X'$ and $Y \subset Y'$ be two first order thickenings
over $S$. Let $(a, a'), (b, b') : (X \subset X') \to (Y \subset Y')$
be two morphisms of thickenings over $S$. Assume that
\begin{enumerate}
\item $a = b$, and
\item the two maps $a^*\mathcal{C}_{Y/Y'} \to \mathcal{C}_{X/X'}$
(Morphisms, Lemma \ref{morphisms-lemma-conormal-functorial})
are equal.
 ========

new_annotation []
===== sent Throughout this section we use that
a sheaf on a thickening $X'$ of $X$ can be seen as a sheaf on $X$.

\begin{lemma}
\label{lemma-difference-derivation}
Let $S$ be a scheme.
Let $X \subset X'$ and $Y \subset Y'$ be two first order thickenings
over $S$. Let $(a, a'), (b, b') : (X \subset X') \to (Y \subset Y')$
be two morphisms of thickenings over $S$. Assume that
\begin{enumerate}
\item $a = b$, and
\item the two maps $a^*\mathcal{C}_{Y/Y'} \to \mathcal{C}_{X/X'}$
(Morphisms, Lemma \ref{morphisms-lemma-conormal-functorial})
are equal.
 ========

new_annotation []
===== sent \end{proof}

\noindent
Note that in the situation of the lemma above we may write
$D$ as
\begin{equation}
\label{equation-D}
D = \text{d}_{Y/S} \circ \theta
\end{equation}
where $\theta$ is an $\mathcal{O}_Y$-linear map
$\theta : \Omega_{Y/S} \to a_*\mathcal{C}_{X/X'}$.
Of course, then by adjunction again we may view $\theta$ as an
$\mathcal{O}_X$-linear map
$\theta : a^*\Omega_{Y/S} \to \mathcal{C}_{X/X'}$.

\begin{lemma}
\label{lemma-action-by-derivations}
Let $S$ be a scheme.
 ========

regex_match $S$
nn candidate a scheme
nn a scheme
new_annotation [(62449, 62452, 'VAR'), (62456, 62464, 'TYPE')]
===== sent Let $(a, a') : (X \subset X') \to (Y \subset Y')$
be a morphism of first order thickenings over $S$.
Let
$$
\theta : a^*\Omega_{Y/S} \to \mathcal{C}_{X/X'}
$$
be an $\mathcal{O}_X$-linear map. ========

new_annotation []
===== sent \end{remark}

\begin{lemma}
\label{lemma-sheaf}
Let $S$ be a scheme.
Let $X \subset X'$ and $Y \subset Y'$ be first order thickenings
over $S$. Assume given a morphism $a : X \to Y$ and a map
$A : a^*\mathcal{C}_{Y/Y'} \to \mathcal{C}_{X/X'}$ of
$\mathcal{O}_X$-modules. ========

regex_match $S$
nn candidate a scheme
nn a scheme
new_annotation [(63787, 63790, 'VAR'), (63794, 63802, 'TYPE')]
===== sent \end{remark}

\begin{lemma}
\label{lemma-sheaf}
Let $S$ be a scheme.
Let $X \subset X'$ and $Y \subset Y'$ be first order thickenings
over $S$. Assume given a morphism $a : X \to Y$ and a map
$A : a^*\mathcal{C}_{Y/Y'} \to \mathcal{C}_{X/X'}$ of
$\mathcal{O}_X$-modules. ========

new_annotation []
===== sent $$
In this case the map $A : a^*\mathcal{C}_{Y/Y'} \to \mathcal{C}_{X/X'}$
is determined by $a$: the map
$h^*\mathcal{C}_{Z/Z'} \to \mathcal{C}_{Y/Y'}$ is surjective (because
we assumed $Y = Z \times_{Z'} Y'$),
hence the pullback $g^*\mathcal{C}_{Z/Z'} = a^*h^*\mathcal{C}_{Z/Z'} \to
a^*\mathcal{C}_{Y/Y'}$ is surjective, and the composition
$g^*\mathcal{C}_{Z/Z'} \to a^*\mathcal{C}_{Y/Y'} \to \mathcal{C}_{X/X'}$
has to be the canonical map induced by $g'$. Thus the sheaf of
Lemma \ref{lemma-sheaf}
is just given by the rule
$$
U' \mapsto
\{a' : U' \to Y'\text{ over }Z'\text{ with } a'|_U = a|_U\}
$$
and we act on this by the sheaf
$\SheafHom_{\mathcal{O}_X}(a^*\Omega_{Y/Z}, \mathcal{C}_{X/X'})$.
\end{remark}

\begin{lemma}
\label{lemma-omega-deformation}
Let $S$ be a scheme. ========

regex_match $S$
nn candidate a scheme
nn a scheme
new_annotation [(67147, 67150, 'VAR'), (67154, 67162, 'TYPE')]
===== sent Let $X \subset X'$ be a first order thickening over
$S$. Let $Y$ be a scheme over $S$. Let
$a', b' : X' \to Y$ be two morphisms over $S$ with
$a = a'|_X = b'|_X$. ========

regex_match $X \subset X'$
nn candidate $
nn candidate a scheme
nn a scheme
new_annotation [(67168, 67182, 'VAR'), (67232, 67240, 'TYPE')]
===== sent Let $X \subset X'$ be a first order thickening over
$S$. Let $Y$ be a scheme over $S$. Let
$a', b' : X' \to Y$ be two morphisms over $S$ with
$a = a'|_X = b'|_X$. ========

regex_match $Y$
nn candidate a scheme
nn a scheme
new_annotation [(67225, 67228, 'VAR'), (67232, 67240, 'TYPE')]
===== sent \begin{lemma}
\label{lemma-deform}
Let $(f, f') : (X \subset X') \to (S \subset S')$ be a morphism
of first order thickenings. ========

regex_match $(f, f') : (X \subset X') \to (S \subset S')$
nn candidate S \subset
nn candidate a morphism
nn a morphism
new_annotation [(74016, 74061, 'VAR'), (74065, 74075, 'TYPE')]
===== sent (I) Let $W' \subset S'$ be an affine open and let $U' \subset X'$
and $V' \subset Y'$ be affine opens lying over $W'$ with $f'(U') \subset V'$.
Let $W' = \Spec(R')$ and denote $I \subset R'$ be the ideal
defining the closed subscheme $W' \cap S$. Say $U' = \Spec(B')$
and $V' = \Spec(A')$. Then we get a commutative diagram
$$
\xymatrix{
0 \ar[r] &
IB' \ar[r] &
B' \ar[r] &
B \ar[r] & 0 \\
0 \ar[r] &
IA' \ar[r] \ar[u] &
A' \ar[r] \ar[u] &
A \ar[r] \ar[u] & 0
}
$$
with exact rows. ========

regex_match $W' \subset S'$
nn candidate $W
nn candidate an affine
nn an affine
new_annotation [(80659, 80674, 'VAR'), (80678, 80687, 'TYPE')]
===== sent (I) Let $W' \subset S'$ be an affine open and let $U' \subset X'$
and $V' \subset Y'$ be affine opens lying over $W'$ with $f'(U') \subset V'$.
Let $W' = \Spec(R')$ and denote $I \subset R'$ be the ideal
defining the closed subscheme $W' \cap S$. Say $U' = \Spec(B')$
and $V' = \Spec(A')$. Then we get a commutative diagram
$$
\xymatrix{
0 \ar[r] &
IB' \ar[r] &
B' \ar[r] &
B \ar[r] & 0 \\
0 \ar[r] &
IA' \ar[r] \ar[u] &
A' \ar[r] \ar[u] &
A \ar[r] \ar[u] & 0
}
$$
with exact rows. ========

new_annotation []
===== sent (I) Let $W' \subset S'$ be an affine open and let $U' \subset X'$
and $V' \subset Y'$ be affine opens lying over $W'$ with $f'(U') \subset V'$.
Let $W' = \Spec(R')$ and denote $I \subset R'$ be the ideal
defining the closed subscheme $W' \cap S$. Say $U' = \Spec(B')$
and $V' = \Spec(A')$. Then we get a commutative diagram
$$
\xymatrix{
0 \ar[r] &
IB' \ar[r] &
B' \ar[r] &
B \ar[r] & 0 \\
0 \ar[r] &
IA' \ar[r] \ar[u] &
A' \ar[r] \ar[u] &
A \ar[r] \ar[u] & 0
}
$$
with exact rows. ========

new_annotation []
===== sent Then $f$ is an isomorphism of $X$ with an open subscheme $V \subset Y$.
Let $V' \subset Y'$ be the open subscheme whose underlying topological
space is $V$. Then $f'$ is a map from $X'$ to $V'$ which is an isomorphism by
(\ref{item-isomorphism}). ========

regex_match $V' \subset Y'$
nn candidate $V
nn candidate the open subscheme
nn the open subscheme
new_annotation [(82440, 82455, 'VAR'), (82459, 82477, 'TYPE')]
===== sent (I) Let $W' \subset S'$ be an affine open and let $U' \subset X'$
and $V' \subset Y'$ be affine opens lying over $W'$ with $f'(U') \subset V'$.
Let $W' = \Spec(R')$ and denote $I \subset R'$ be the ideal
defining the closed subscheme $W' \cap S$. Say $U' = \Spec(B')$
and $V' = \Spec(A')$. Then we get a commutative diagram
$$
\xymatrix{
0 \ar[r] &
IB' \ar[r] &
B' \ar[r] &
B \ar[r] & 0 \\
0 \ar[r] &
IA' \ar[r] \ar[u] &
A' \ar[r] \ar[u] &
A \ar[r] \ar[u] & 0
}
$$
with exact rows.
 ========

regex_match $W' \subset S'$
nn candidate $W
nn candidate an affine
nn an affine
new_annotation [(93710, 93725, 'VAR'), (93729, 93738, 'TYPE')]
===== sent (I) Let $W' \subset S'$ be an affine open and let $U' \subset X'$
and $V' \subset Y'$ be affine opens lying over $W'$ with $f'(U') \subset V'$.
Let $W' = \Spec(R')$ and denote $I \subset R'$ be the ideal
defining the closed subscheme $W' \cap S$. Say $U' = \Spec(B')$
and $V' = \Spec(A')$. Then we get a commutative diagram
$$
\xymatrix{
0 \ar[r] &
IB' \ar[r] &
B' \ar[r] &
B \ar[r] & 0 \\
0 \ar[r] &
IA' \ar[r] \ar[u] &
A' \ar[r] \ar[u] &
A \ar[r] \ar[u] & 0
}
$$
with exact rows.
 ========

new_annotation []
===== sent (I) Let $W' \subset S'$ be an affine open and let $U' \subset X'$
and $V' \subset Y'$ be affine opens lying over $W'$ with $f'(U') \subset V'$.
Let $W' = \Spec(R')$ and denote $I \subset R'$ be the ideal
defining the closed subscheme $W' \cap S$. Say $U' = \Spec(B')$
and $V' = \Spec(A')$. Then we get a commutative diagram
$$
\xymatrix{
0 \ar[r] &
IB' \ar[r] &
B' \ar[r] &
B \ar[r] & 0 \\
0 \ar[r] &
IA' \ar[r] \ar[u] &
A' \ar[r] \ar[u] &
A \ar[r] \ar[u] & 0
}
$$
with exact rows.
 ========

new_annotation []
===== sent Then $f$ is an isomorphism of $X$ with an open subscheme $V \subset Y$.
Let $V' \subset Y'$ be the open subscheme whose underlying topological
space is $V$. Then $f'$ is a map from $X'$ to $V'$ which is an isomorphism by
(\ref{item-isomorphism-fp-over-ft}). ========

regex_match $V' \subset Y'$
nn candidate $V
nn candidate the open subscheme
nn the open subscheme
new_annotation [(95304, 95319, 'VAR'), (95323, 95341, 'TYPE')]
===== sent Hence $f'$ is finite locally free by
Morphisms, Lemma \ref{morphisms-lemma-finite-flat}.
\end{proof}

\begin{lemma}[Deformations of projective schemes]
\label{lemma-deform-projective}
Let $f : X \to S$ be a morphism of schemes which is proper, flat, and
of finite presentation. ========

regex_match $f : X \to S$
nn candidate X \to
nn candidate a morphism
nn a morphism
new_annotation [(100726, 100739, 'VAR'), (100743, 100753, 'TYPE')]
===== sent Let $\mathcal{L}$ be $f$-ample. ========

new_annotation []
===== sent Let $U' = \Spec(R') \subset S'$ be an affine open.
 ========

regex_match $U' = \Spec(R') \subset S'$
nn candidate U
nn candidate an affine
nn an affine
new_annotation [(105866, 105893, 'VAR'), (105897, 105906, 'TYPE')]
===== sent \begin{definition}
\label{definition-formally-smooth}
Let $f : X \to S$ be a morphism of schemes.
 ========

regex_match $f : X \to S$
nn candidate X \to
nn candidate a morphism
nn a morphism
new_annotation [(110287, 110300, 'VAR'), (110304, 110314, 'TYPE')]
===== sent \end{proof}

\begin{lemma}
\label{lemma-formally-etale-unramified-smooth}
Let $f : X \to S$ be a morphism of schemes.
 ========

regex_match $f : X \to S$
nn candidate X \to
nn candidate a morphism
nn a morphism
new_annotation [(111833, 111846, 'VAR'), (111850, 111860, 'TYPE')]
===== sent \end{proof}

\begin{lemma}
\label{lemma-formally-smooth-on-opens}
Let $f : X \to S$ be a morphism of schemes.
 ========

regex_match $f : X \to S$
nn candidate X \to
nn candidate a morphism
nn a morphism
new_annotation [(112071, 112084, 'VAR'), (112088, 112098, 'TYPE')]
===== sent Let $U \subset X$ and $V \subset S$ be open subschemes such that
$f(U) \subset V$. If $f$ is formally smooth, so is $f|_U : ========

new_annotation []
===== sent \end{proof}

\begin{lemma}
\label{lemma-affine-formally-smooth}
Let $f : X \to S$ be a morphism of schemes.
 ========

regex_match $f : X \to S$
nn candidate X \to
nn candidate a morphism
nn a morphism
new_annotation [(112843, 112856, 'VAR'), (112860, 112870, 'TYPE')]
===== sent It is a victory of the
functorial point of view in that it implies (combined with
Limits,
Proposition \ref{limits-proposition-characterize-locally-finite-presentation})
that we can recognize whether a morphism $f : X \to S$ is smooth in terms of
``simple'' properties of the functor $h_X : \Sch/S \to \textit{Sets}$.

\begin{lemma}[Infinitesimal lifting criterion]
\label{lemma-smooth-formally-smooth}
Let $f : X \to S$ be a morphism of schemes.
 ========

regex_match $f : X \to S$
nn candidate X \to
nn candidate a morphism
nn a morphism
new_annotation [(113816, 113829, 'VAR'), (113833, 113843, 'TYPE')]
===== sent S & T' \ar[l] \ar@{-->}[lu]
}
$$
as in Definition \ref{definition-formally-smooth}.
We will show the dotted arrow exists thereby
proving that $f$ is formally smooth.

\medskip\noindent
Let $\mathcal{F}$ be the sheaf of sets on $T'$ of Lemma \ref{lemma-sheaf}
in the special case discussed in Remark \ref{remark-special-case}.
Let
$$
\mathcal{H} =
\SheafHom_{\mathcal{O}_T}(a^*\Omega_{X/S}, \mathcal{C}_{T/T'})
$$
be the sheaf of $\mathcal{O}_T$-modules with action
$\mathcal{H} \times \mathcal{F} \to \mathcal{F}$ as in
Lemma \ref{lemma-action-sheaf}. Our goal is simply
to show that $\mathcal{F}(T) \not = \emptyset$. ========

regex_match $\mathcal{F}$
nn candidate the sheaf
nn the sheaf
new_annotation [(115087, 115100, 'VAR'), (115104, 115113, 'TYPE')]
===== sent \end{proof}

\noindent
Locally projective quasi-coherent modules are defined in
Properties, Section \ref{properties-section-locally-projective}.

\begin{lemma}
\label{lemma-formally-smooth-sheaf-differentials}
Let $f : X \to Y$ be a formally smooth morphism of schemes.
 ========

regex_match $f : X \to Y$
nn candidate X \to
nn candidate a formally smooth morphism
nn a formally smooth morphism
new_annotation [(117852, 117865, 'VAR'), (117869, 117895, 'TYPE')]
===== sent Hence $\Omega_{U/V}$ is locally projective, see
Properties, Section \ref{properties-section-locally-projective}.
\end{proof}

\begin{lemma}
\label{lemma-h1-is-zero}
Let $T$ be an affine scheme. ========

regex_match $T$
nn candidate an affine scheme
nn an affine scheme
new_annotation [(118648, 118651, 'VAR'), (118655, 118671, 'TYPE')]
===== sent Let $\mathcal{F}$, $\mathcal{G}$ be quasi-coherent
$\mathcal{O}_T$-modules. ========

new_annotation []
===== sent By
Cohomology, Lemma \ref{cohomology-lemma-cohomology-products}
we conclude that $H^1 = 0$ because the cohomology of a quasi-coherent sheaf
on an affine scheme is zero, see Cohomology of Schemes, Lemma
\ref{coherent-lemma-quasi-coherent-affine-cohomology-zero}.
\end{proof}

\begin{lemma}
\label{lemma-formally-smooth}
Let $f : X \to Y$ be a morphism of schemes. ========

regex_match $f : X \to Y$
nn candidate X \to
nn candidate a morphism
nn a morphism
new_annotation [(119655, 119668, 'VAR'), (119672, 119682, 'TYPE')]
===== sent Let $\mathcal{F}$ be the sheaf of sets on $T'$ of
Lemma \ref{lemma-sheaf} as in the special case discussed in
Remark \ref{remark-special-case}.
Let
$$
\mathcal{H} =
\SheafHom_{\mathcal{O}_T}(a^*\Omega_{X/Y}, \mathcal{C}_{T/T'})
$$
be the sheaf of $\mathcal{O}_T$-modules on $T$
with action $\mathcal{H} \times \mathcal{F} \to \mathcal{F}$ as in
Lemma \ref{lemma-action-sheaf}.
 ========

regex_match $\mathcal{F}$
nn candidate the sheaf
nn the sheaf
new_annotation [(120990, 121003, 'VAR'), (121007, 121016, 'TYPE')]
===== sent To see this, for every $t \in T$ we can
choose an affine open $W \subset T$ neighbourhood of $t$
such that $a(W)$ is contained in $U_{ji}$ for some $i, j$.
Let $W' \subset T'$ be the corresponding open subscheme.
 ========

regex_match $W' \subset T'$
nn candidate $W
nn candidate the corresponding open subscheme
nn the corresponding open subscheme
new_annotation [(122100, 122115, 'VAR'), (122119, 122151, 'TYPE')]
===== sent Hence $\Omega_{X/Y}$ is locally projective, see
Properties, Lemma \ref{properties-lemma-locally-projective}.
Hence $a^*\Omega_{X/Y}$ is locally projective, see
Properties, Lemma \ref{properties-lemma-locally-projective-pullback}.
Hence
$$
H^1(T, \mathcal{H}) =
H^1(T, \SheafHom_{\mathcal{O}_T}(a^*\Omega_{X/Y}, \mathcal{C}_{T/T'}) = 0
$$
by
Lemma \ref{lemma-h1-is-zero}
as desired.
\end{proof}

\begin{lemma}
\label{lemma-triangle-differentials-formally-smooth}
Let $f : X \to Y$, $g : Y \to S$ be morphisms of schemes.
 ========

new_annotation []
===== sent This is
Algebra, Lemma \ref{algebra-lemma-ses-formally-smooth}.
\end{proof}

\begin{lemma}
\label{lemma-differentials-formally-unramified-formally-smooth}
Let $h : Z \to X$ be a formally unramified morphism of schemes over $S$.
Assume that $Z$ is formally smooth over $S$. ========

regex_match $h : Z \to X$
nn candidate Z \to
nn candidate a formally unramified morphism
nn a formally unramified morphism
new_annotation [(123627, 123640, 'VAR'), (123644, 123674, 'TYPE')]
===== sent \end{lemma}

\begin{proof}
Let $Z \to Z'$ be the universal first order thickening of $Z$ over $X$.
 ========

regex_match $Z \to Z'$
nn candidate the universal first order thickening
nn the universal first order thickening
new_annotation [(123962, 123972, 'VAR'), (123976, 124012, 'TYPE')]
===== sent In the proof of
Lemma \ref{lemma-two-unramified-morphisms}
we identified the sequence above with the sequence
$$
\mathcal{C}_{Z/Z''} \to
\mathcal{C}_{Z/Z'} \to
(i')^*\Omega_{Z'/Z''} \to 0
$$
Let $U'' \subset Z''$ be an affine open. Denote $U \subset Z$ and
$U' \subset Z'$ the corresponding affine open subschemes.
 ========

regex_match $U'' \subset Z''$
nn candidate $U'' \subset Z''$
nn candidate an affine
nn an affine
new_annotation [(125566, 125583, 'VAR'), (125587, 125596, 'TYPE')]
===== sent \begin{lemma}
\label{lemma-lifting-along-artinian-at-point}
Let $f : X \to S$ be a morphism of schemes.
 ========

regex_match $f : X \to S$
nn candidate X \to
nn candidate a morphism
nn a morphism
new_annotation [(126722, 126735, 'VAR'), (126739, 126749, 'TYPE')]
===== sent Let $x \in X$.
Assume that $S$ is locally Noetherian and $f$ locally of finite type.
 ========

new_annotation []
===== sent In this affine case the lemma is identical to
Algebra, Lemma \ref{algebra-lemma-smooth-test-artinian}.
\end{proof}

\noindent
Sometimes it is useful to know that one only needs to check the
lifting criterion for small extensions ``centered'' at points
of finite type (see
Morphisms, Section \ref{morphisms-section-points-finite-type}).

\begin{lemma}
\label{lemma-lifting-along-artinian}
Let $f : X \to S$ be a morphism of schemes.
 ========

regex_match $f : X \to S$
nn candidate X \to
nn candidate a morphism
nn a morphism
new_annotation [(128658, 128671, 'VAR'), (128675, 128685, 'TYPE')]
===== sent \begin{lemma}
\label{lemma-check-smoothness-on-infinitesimal-nbhds}
Let $f : X \to S$ be a finite type morphism of locally Noetherian schemes.
 ========

regex_match $f : X \to S$
nn candidate X \to
nn candidate a finite type morphism
nn a finite type morphism
new_annotation [(130264, 130277, 'VAR'), (130281, 130303, 'TYPE')]
===== sent Let $Z \subset S$ be a closed subscheme with $n$th infinitesimal
neighbourhood $Z_n \subset S$. Set $X_n = Z_n \times_S X$.
\begin{enumerate}
\item If $X_n \to Z_n$ is smooth for all $n$, then $f$
is smooth at every point of $f^{-1}(Z)$.
\item ========

regex_match $Z \subset S$
nn candidate $Z \subset
nn candidate a closed subscheme
nn a closed subscheme
new_annotation [(130339, 130352, 'VAR'), (130356, 130374, 'TYPE')]
===== sent If $X_n \to Z_n$ is \'etale for all $n$, then $f$
is \'etale at every point of $f^{-1}(Z)$.
\end{enumerate}
\end{lemma}

\begin{proof}
Assume $X_n \to Z_n$ is smooth for all $n$.
Let $x \in X$ be a point lying over a point of $Z$.
Given a small extension $B' \to B$ and morphisms $\alpha$, $\beta$ as in
Lemma \ref{lemma-lifting-along-artinian-at-point} part (3)
the maximal ideal of $B'$ is nilpotent (as $B'$ is Artinian)
and hence the morphism $\beta$ factors through $Z_n$ and $\alpha$ factors
through $X_n$ for a suitable $n$. ========

regex_match $x \in X$
nn candidate a point
nn a point
new_annotation [(130762, 130771, 'VAR'), (130775, 130782, 'TYPE')]
===== sent Part (2) follows from (1) and the fact that a morphism
is \'etale if and only if it is smooth of relative dimension $0$.
\end{proof}

\begin{lemma}
\label{lemma-check-flatness-on-infinitesimal-nbhds}
Let $f : X \to S$ be a morphism of locally Noetherian schemes.
 ========

regex_match $f : X \to S$
nn candidate X \to
nn candidate a morphism
nn a morphism
new_annotation [(131433, 131446, 'VAR'), (131450, 131460, 'TYPE')]
===== sent Let $Z \subset S$ be a closed subscheme with $n$th infinitesimal
neighbourhood $Z_n \subset S$. Set $X_n = Z_n \times_S X$.
If $X_n \to Z_n$ is flat for all $n$, then $f$
is flat at every point of $f^{-1}(Z)$.
\end{lemma}

\begin{proof}
This is a translation of Algebra, Lemma \ref{algebra-lemma-flat-module-powers}
into the language of schemes.
 ========

regex_match $Z \subset S$
nn candidate $Z \subset
nn candidate a closed subscheme
nn a closed subscheme
new_annotation [(131496, 131509, 'VAR'), (131513, 131531, 'TYPE')]
===== sent \end{proof}








\section{The naive cotangent complex}
\label{section-netherlander}

\noindent
This section is the continuation of
Modules, Section \ref{modules-section-netherlander}
which in turn continues the discussion in
Algebra, Section \ref{algebra-section-netherlander}.

\begin{definition}
\label{definition-netherlander}
Let $f : X \to Y$ be a morphism of schemes.
 ========

regex_match $f : X \to Y$
nn candidate X \to
nn candidate a morphism
nn a morphism
new_annotation [(132175, 132188, 'VAR'), (132192, 132202, 'TYPE')]
===== sent The {\it naive cotangent complex of $f$}
is the complex defined in Modules, Definition
\ref{modules-definition-cotangent-complex-morphism-ringed-topoi}.
Notation: $\NL_f$ or $\NL_{X/Y}$.
\end{definition}

\begin{lemma}
\label{lemma-NL-affine}
Let $f : X \to Y$ be a morphism of schemes. ========

regex_match $f : X \to Y$
nn candidate X \to
nn candidate a morphism
nn a morphism
new_annotation [(132462, 132475, 'VAR'), (132479, 132489, 'TYPE')]
===== sent \end{proof}

\begin{lemma}
\label{lemma-netherlander-quasi-coherent}
Let $f : X \to Y$ be a morphism of schemes. ========

regex_match $f : X \to Y$
nn candidate X \to
nn candidate a morphism
nn a morphism
new_annotation [(134059, 134072, 'VAR'), (134076, 134086, 'TYPE')]
===== sent This follows by checking over affines
using Lemma \ref{lemma-NL-affine}.
\end{proof}

\begin{lemma}
\label{lemma-netherlander-fp}
Let $f : X \to Y$ be a morphism of schemes. ========

regex_match $f : X \to Y$
nn candidate X \to
nn candidate a morphism
nn a morphism
new_annotation [(134827, 134840, 'VAR'), (134844, 134854, 'TYPE')]
===== sent I/I^2 \to
\bigoplus\nolimits_{i = 1, \ldots, n} A\text{d}x_i \to 0 \to \ldots
$$
by Algebra, Section \ref{algebra-section-netherlander}
and in particular
Algebra, Lemma \ref{algebra-lemma-NL-homotopy}.
\end{proof}

\begin{lemma}
\label{lemma-NL-formally-smooth}
Let $f : X \to Y$ be a morphism of schemes. ========

regex_match $f : X \to Y$
nn candidate X \to
nn candidate a morphism
nn a morphism
new_annotation [(135755, 135768, 'VAR'), (135772, 135782, 'TYPE')]
===== sent This follows from Algebra, Proposition
\ref{algebra-proposition-characterize-formally-smooth}
and Lemma \ref{lemma-formally-smooth}.
\end{proof}

\begin{lemma}
\label{lemma-NL-formally-etale}
Let $f : X \to Y$ be a morphism of schemes. ========

regex_match $f : X \to Y$
nn candidate X \to
nn candidate a morphism
nn a morphism
new_annotation [(136200, 136213, 'VAR'), (136217, 136227, 'TYPE')]
===== sent Conversely, if (2) holds, then $f$ is formally smooth by
Lemma \ref{lemma-NL-formally-smooth}
and formally unramified by
Lemma \ref{lemma-formally-unramified-differentials}
and hence formally \'etale by
Lemmas \ref{lemma-formally-etale-unramified-smooth}.
\end{proof}

\begin{lemma}
\label{lemma-NL-smooth}
Let $f : X \to Y$ be a morphism of schemes. ========

regex_match $f : X \to Y$
nn candidate X \to
nn candidate a morphism
nn a morphism
new_annotation [(136946, 136959, 'VAR'), (136963, 136973, 'TYPE')]
===== sent \end{proof}

\begin{lemma}
\label{lemma-exact-sequence-NL}
Let $f : X \to Y$ and $g : Y \to Z$ be morphisms of schemes.
 ========

new_annotation []
===== sent Namely, for a scheme $T$ a morphism $W \to T$ is the same thing as
a collection of morphism $W_i \to T$ which agree on overlaps, see
Descent, Lemma \ref{descent-lemma-fpqc-universal-effective-epimorphisms}.
\end{proof}

\noindent
A suitably general case is where one morphism is
a closed immersion and the second is integral, however
a further condition is needed to ensure the pushout exists.

\begin{proposition}
\label{proposition-pushout-along-closed-immersion-and-integral}
\begin{reference}
\cite[Theorem 7.1 part iii]{Ferrand-Conducteur}
\end{reference}
Let $i : Z \to X$ and $j : Z \to Y$ be morphisms of schemes. ========

new_annotation []
===== sent Let $f : X \to T$ and $g : Y \to T$ be morphisms of schemes
with $f \circ i = g \circ j$. ========

new_annotation []
===== sent Thus we get a morphism
of ringed spaces
$$
h : Y \amalg_Z X \longrightarrow T
$$
by our choice of the structure sheaf of $Y \amalg_Z X$
as the fibre product of $a_*\mathcal{O}_X$ and $b_*\mathcal{O}_Y$
over $c_*\mathcal{O}_Z$. To show that $h$ is a morphism of
locally ringed spaces (and hence a morphism of schemes),
let $s \in Y \amalg_Z X$ be a point mapping to $t \in T$.
Then either $s$ is the image of a $y \in Y$ or the image of
an $x \in X$. Consider
$$
\mathcal{O}_{T, t} \to \mathcal{O}_{Y \amalg_Z X, s}
\to \mathcal{O}_{Y, y}
\quad\text{or}\quad
\mathcal{O}_{T, t} \to \mathcal{O}_{Y \amalg_Z X, s}
\to \mathcal{O}_{X, x}
$$
Since the composition and the second map are local ring homomorphisms,
we conclude.
 ========

regex_match $s \in Y \amalg_Z X$
nn candidate a point mapping
nn a point mapping
new_annotation [(145725, 145745, 'VAR'), (145749, 145764, 'TYPE')]
===== sent \end{proof}

\begin{lemma}
\label{lemma-pushout-separated}
Let $S$ be a scheme. ========

regex_match $S$
nn candidate a scheme
nn a scheme
new_annotation [(146187, 146190, 'VAR'), (146194, 146202, 'TYPE')]
===== sent Let $i : Z \to X$ and $j : Z \to Y$
be morphisms of schemes over $S$. ========

new_annotation []
===== sent Thus we may apply
Morphisms, Lemma \ref{morphisms-lemma-image-universally-closed-separated}.
\end{proof}

\begin{lemma}
\label{lemma-pushout-finite-type}
Let $S$ be a locally Noetherian scheme. ========

regex_match $S$
nn candidate a locally Noetherian scheme
nn a locally Noetherian scheme
new_annotation [(146991, 146994, 'VAR'), (146998, 147025, 'TYPE')]
===== sent Let $i : Z \to X$ and $j : Z \to Y$
be morphisms of schemes locally of finite type over $S$. ========

new_annotation []
===== sent \begin{lemma}
\label{lemma-pushout-along-closed-immersions}
Let $i : Z \to X$ and $j : Z \to Y$ be closed immersions of schemes.
 ========

new_annotation []
===== sent \end{proof}

\begin{lemma}
\label{lemma-pushout-along-closed-immersions-properties-above}
Let $i : Z \to X$ and $j : Z \to Y$ be closed immersions of schemes.
 ========

new_annotation []
===== sent Let $f : X' \to X$ and $g : Y' \to Y$ be morphisms of schemes and let
$\varphi : X' \times_{X, i} Z \to Y' \times_{Y, j} Z$
be an isomorphism of schemes over $Z$. Consider the morphism
$$
h :
X' \amalg_{X' \times_{X, i} Z, \varphi} Y'
\longrightarrow
X \amalg_Z Y
$$
Then we have
\begin{enumerate}
\item $h$ is locally of finite type if and only if $f$ and $g$ are
locally of finite type,
\item $h$ is flat if and only if $f$ and $g$ are flat,
\item $h$ is flat and locally of finite presentation if and only if
$f$ and $g$ are flat and locally of finite presentation,
\item $h$ is smooth if and only if $f$ and $g$ are smooth,
\item $h$ is \'etale if and only if $f$ and $g$ are \'etale, and
\item add more here as needed.
 ========

new_annotation []
===== sent \begin{lemma}
\label{lemma-pushout-along-thickening}
Let $X \to X'$ be a thickening of schemes and let $X \to Y$ be an affine
morphism of schemes. ========

regex_match $X \to X'$
nn candidate $X
nn candidate a thickening
nn a thickening
new_annotation [(150100, 150110, 'VAR'), (150114, 150126, 'TYPE')]
===== sent \begin{lemma}
\label{lemma-pushout-along-thickening}
Let $X \to X'$ be a thickening of schemes and let $X \to Y$ be an affine
morphism of schemes. ========

regex_match $X \to Y$
nn candidate an affine
nn an affine
new_annotation [(150146, 150155, 'VAR'), (150159, 150168, 'TYPE')]
===== sent Namely, let $S$ be
a scheme and let $b : Y \to S$ and $a' : X' \to S$ be morphisms such that
$$
\xymatrix{
X \ar[r] \ar[d]
&
X' \ar[d]^{a'}
\\
Y \ar[r]^b
&
S
}
$$
commutes. ========

new_annotation []
===== sent Namely, let $S$ be
a scheme and let $b : Y \to S$ and $a' : X' \to S$ be morphisms such that
$$
\xymatrix{
X \ar[r] \ar[d]
&
X' \ar[d]^{a'}
\\
Y \ar[r]^b
&
S
}
$$
commutes. ========

new_annotation []
===== sent In the following lemma we use the fibre product of categories as
defined in
Categories, Example \ref{categories-example-2-fibre-product-categories}.

\begin{lemma}
\label{lemma-equivalence-categories-schemes-over-pushout}
Let $X \to X'$ be a thickening of schemes and let $X \to Y$ be an
affine morphism of schemes. ========

regex_match $X \to X'$
nn candidate $X
nn candidate a thickening
nn a thickening
new_annotation [(153149, 153159, 'VAR'), (153163, 153175, 'TYPE')]
===== sent In the following lemma we use the fibre product of categories as
defined in
Categories, Example \ref{categories-example-2-fibre-product-categories}.

\begin{lemma}
\label{lemma-equivalence-categories-schemes-over-pushout}
Let $X \to X'$ be a thickening of schemes and let $X \to Y$ be an
affine morphism of schemes. ========

new_annotation []
===== sent Let $Y' = Y \amalg_X X'$ be the pushout
(see Lemma \ref{lemma-pushout-along-thickening}). ========

regex_match $Y' = Y \amalg_X X'$
nn candidate $Y
nn candidate the pushout
nn the pushout
new_annotation [(153243, 153263, 'VAR'), (153267, 153278, 'TYPE')]
===== sent \end{lemma}

\begin{proof}
Let $(V, U', \varphi)$ be an object of the fibre product category.
 ========

regex_match $(V, U', \varphi)$
nn candidate $(V
nn candidate an object
nn an object
new_annotation [(153777, 153795, 'VAR'), (153799, 153808, 'TYPE')]
===== sent In particular $U \to V$ is affine, see
Morphisms, Lemma \ref{morphisms-lemma-base-change-affine}.
Hence we can apply Lemma \ref{lemma-pushout-along-thickening}
to get a pushout $V' = V \amalg_U U'$. Denote $V' \to Y'$ the morphism
we obtain in virtue of the fact that $V'$ is a pushout and because
we are given morphisms $V \to Y$ and $U' \to X'$ agreeing on $U$
as morphisms into $Y'$. Setting $G(V, U', \varphi) = V'$
gives the functor $G$.

\medskip\noindent
Let us prove that $G$ is a left adjoint to $F$. Let $Z$ be a scheme
over $Y'$. We have to show that
$$
\Mor(V', Z) = \Mor((V, U', \varphi), F(Z))
$$
where the morphism sets are taking in their respective categories.
 ========

regex_match $Z$
nn candidate a scheme
nn a scheme
new_annotation [(154598, 154601, 'VAR'), (154605, 154613, 'TYPE')]
===== sent Let $g' : V' \to Z$ be a morphism. ========

regex_match $g' : V' \to Z$
nn candidate a morphism
nn a morphism
new_annotation [(154766, 154781, 'VAR'), (154785, 154795, 'TYPE')]
===== sent In this case the statement we have to
prove is that $D' \otimes_{B'} B \cong D$ and $D' \otimes_{B'} A' \cong C'$.
This is a special case of More on Algebra, Lemma
\ref{more-algebra-lemma-module-over-fibre-product}.
\end{proof}

\begin{lemma}
\label{lemma-scheme-over-pushout-flat-modules}
Let $X \to X'$ be a thickening of schemes and let $X \to Y$ be an
affine morphism of schemes. ========

regex_match $X \to X'$
nn candidate $X
nn candidate a thickening
nn a thickening
new_annotation [(156616, 156626, 'VAR'), (156630, 156642, 'TYPE')]
===== sent In this case the statement we have to
prove is that $D' \otimes_{B'} B \cong D$ and $D' \otimes_{B'} A' \cong C'$.
This is a special case of More on Algebra, Lemma
\ref{more-algebra-lemma-module-over-fibre-product}.
\end{proof}

\begin{lemma}
\label{lemma-scheme-over-pushout-flat-modules}
Let $X \to X'$ be a thickening of schemes and let $X \to Y$ be an
affine morphism of schemes. ========

new_annotation []
===== sent Let $Y' = Y \amalg_X X'$ be the pushout
(see Lemma \ref{lemma-pushout-along-thickening}). ========

regex_match $Y' = Y \amalg_X X'$
nn candidate $Y
nn candidate the pushout
nn the pushout
new_annotation [(156710, 156730, 'VAR'), (156734, 156745, 'TYPE')]
===== sent Let $V' \to Y'$
be a morphism of schemes. ========

new_annotation []
===== sent \begin{theorem}
\label{theorem-openness-flatness}
\begin{reference}
\cite[IV Theorem 11.3.1]{EGA}
\end{reference}
Let $S$ be a scheme.
 ========

regex_match $S$
nn candidate a scheme
nn a scheme
new_annotation [(160404, 160407, 'VAR'), (160411, 160419, 'TYPE')]
===== sent Let $f : X \to S$ be a morphism which is locally of finite presentation.
 ========

regex_match $f : X \to S$
nn candidate X \to
nn candidate a morphism
nn a morphism
new_annotation [(160425, 160438, 'VAR'), (160442, 160452, 'TYPE')]
===== sent Let $\mathcal{F}$ be a quasi-coherent $\mathcal{O}_X$-module which is
locally of finite presentation. ========

regex_match $\mathcal{F}$
nn candidate a quasi-coherent $\mathcal{O}_X$-module
nn a quasi-coherent $\mathcal{O}_X$-module
new_annotation [(160498, 160511, 'VAR'), (160515, 160554, 'TYPE')]
===== sent In this case the
theorem is exactly
Algebra, Theorem \ref{algebra-theorem-openness-flatness}.
\end{proof}

\begin{lemma}
\label{lemma-flat-locus-base-change}
Let $S$ be a scheme.
Let
$$
\xymatrix{
X' \ar[r]_{g'} \ar[d]_{f'} & X \ar[d]^f \\
 ========

regex_match $S$
nn candidate a scheme
nn a scheme
new_annotation [(160983, 160986, 'VAR'), (160990, 160998, 'TYPE')]
===== sent Let $\mathcal{F}$ be a quasi-coherent $\mathcal{O}_X$-module.
 ========

regex_match $\mathcal{F}$
nn candidate a quasi-coherent $\mathcal{O}_X$-module
nn a quasi-coherent $\mathcal{O}_X$-module
new_annotation [(161121, 161134, 'VAR'), (161138, 161177, 'TYPE')]
===== sent Let $x' \in X'$ with images
$x = g'(x')$ and $s' = g'(x')$.
\begin{enumerate}
\item If $\mathcal{F}$ is flat over $S$ at $x$, then
$(g')^*\mathcal{F}$ is flat over $S'$ at $x'$.
\item If $g$ is flat at $s'$ and $(g')^*\mathcal{F}$ is flat over $S'$ at
$x'$, then $\mathcal{F}$ is flat over $S$ at $x$.
\end{enumerate}
In particular, if $g$ is flat, $f$ is locally of finite presentation,
and $\mathcal{F}$ is locally of finite presentation,
then formation of the open subset of
Theorem \ref{theorem-openness-flatness}
commutes with base change.
 ========

new_annotation []
===== sent If $\mathcal{F}$ is flat over $S$ at $x$, then
the theorem states this question is intimately related to the
question of whether the restriction of $\mathcal{F}$ to the fibre
$$
\mathcal{F}_s = (X_s \to X)^*\mathcal{F}
$$
is flat over $Y_s$ at $x$. Below you will find a ``Noetherian'' version,
a ``finitely presented'' version, and earlier we treated a ``nilpotent''
version, see
Lemma \ref{lemma-flatness-morphism-thickenings}.

\begin{theorem}
\label{theorem-criterion-flatness-fibre-Noetherian}
Let $S$ be a scheme.
 ========

regex_match $S$
nn candidate a scheme
nn a scheme
new_annotation [(163274, 163277, 'VAR'), (163281, 163289, 'TYPE')]
===== sent Let $f : X \to Y$ be a morphism of schemes over $S$.
Let $\mathcal{F}$ be a quasi-coherent $\mathcal{O}_X$-module.
 ========

regex_match $f : X \to Y$
nn candidate X \to
nn candidate a morphism
nn a morphism
new_annotation [(163295, 163308, 'VAR'), (163312, 163322, 'TYPE')]
===== sent Let $f : X \to Y$ be a morphism of schemes over $S$.
Let $\mathcal{F}$ be a quasi-coherent $\mathcal{O}_X$-module.
 ========

regex_match $\mathcal{F}$
nn candidate a quasi-coherent $\mathcal{O}_X$-module
nn a quasi-coherent $\mathcal{O}_X$-module
new_annotation [(163348, 163361, 'VAR'), (163365, 163404, 'TYPE')]
===== sent Let $x \in X$. Set $y = f(x)$ and $s \in S$ the image of $x$ in $S$.
Assume $S$, $X$, $Y$ locally Noetherian,
$\mathcal{F}$ coherent, and $\mathcal{F}_x \not = 0$.
 ========

new_annotation []
===== sent \begin{theorem}
\label{theorem-criterion-flatness-fibre}
Let $S$ be a scheme.
 ========

regex_match $S$
nn candidate a scheme
nn a scheme
new_annotation [(164905, 164908, 'VAR'), (164912, 164920, 'TYPE')]
===== sent Let $f : X \to Y$ be a morphism of schemes over $S$.
Let $\mathcal{F}$ be a quasi-coherent $\mathcal{O}_X$-module.
 ========

regex_match $f : X \to Y$
nn candidate X \to
nn candidate a morphism
nn a morphism
new_annotation [(164926, 164939, 'VAR'), (164943, 164953, 'TYPE')]
===== sent Let $f : X \to Y$ be a morphism of schemes over $S$.
Let $\mathcal{F}$ be a quasi-coherent $\mathcal{O}_X$-module.
 ========

regex_match $\mathcal{F}$
nn candidate a quasi-coherent $\mathcal{O}_X$-module
nn a quasi-coherent $\mathcal{O}_X$-module
new_annotation [(164979, 164992, 'VAR'), (164996, 165035, 'TYPE')]
===== sent Assume
\begin{enumerate}
\item $X$ is locally of finite presentation over $S$,
\item $\mathcal{F}$ an $\mathcal{O}_X$-module of finite presentation, and
\item $Y$ is locally of finite type over $S$.
\end{enumerate}
Let $x \in X$. Set $y = f(x)$ and let $s \in S$ be the image of $x$ in $S$.
If $\mathcal{F}_x \not = 0$, then the following are equivalent:
\begin{enumerate}
\item $\mathcal{F}$ is flat over $S$ at $x$, and
$\mathcal{F}_s$ is flat over $Y_s$ at $x$, and
\item $Y$ is flat over $S$ at $y$ and $\mathcal{F}$ is
flat over $Y$ at $x$.
\end{enumerate}
Moreover, the set of points $x$ where (1) and (2) hold is open in
$\text{Supp}(\mathcal{F})$.
\end{theorem}

\begin{proof}
Consider the ring maps
$$
\mathcal{O}_{S, s} \longrightarrow
 ========

new_annotation []
===== sent Assume
\begin{enumerate}
\item $X$ is locally of finite presentation over $S$,
\item $\mathcal{F}$ an $\mathcal{O}_X$-module of finite presentation, and
\item $Y$ is locally of finite type over $S$.
\end{enumerate}
Let $x \in X$. Set $y = f(x)$ and let $s \in S$ be the image of $x$ in $S$.
If $\mathcal{F}_x \not = 0$, then the following are equivalent:
\begin{enumerate}
\item $\mathcal{F}$ is flat over $S$ at $x$, and
$\mathcal{F}_s$ is flat over $Y_s$ at $x$, and
\item $Y$ is flat over $S$ at $y$ and $\mathcal{F}$ is
flat over $Y$ at $x$.
\end{enumerate}
Moreover, the set of points $x$ where (1) and (2) hold is open in
$\text{Supp}(\mathcal{F})$.
\end{theorem}

\begin{proof}
Consider the ring maps
$$
\mathcal{O}_{S, s} \longrightarrow
 ========

regex_match $s \in S$
nn candidate the image
nn the image
new_annotation [(165290, 165299, 'VAR'), (165303, 165312, 'TYPE')]
===== sent \begin{lemma}
\label{lemma-morphism-between-flat-Noetherian}
Let $S$ be a scheme.
 ========

regex_match $S$
nn candidate a scheme
nn a scheme
new_annotation [(167747, 167750, 'VAR'), (167754, 167762, 'TYPE')]
===== sent Let $f : X \to Y$ be a morphism of schemes over $S$.
Assume
\begin{enumerate}
\item $S$, $X$, $Y$ are locally Noetherian,
\item $X$ is flat over $S$,
\item for every $s \in S$ the morphism
$f_s : X_s \to Y_s$ is flat.
 ========

regex_match $f : X \to Y$
nn candidate X \to
nn candidate a morphism
nn a morphism
new_annotation [(167768, 167781, 'VAR'), (167785, 167795, 'TYPE')]
===== sent This is a special case of
Theorem \ref{theorem-criterion-flatness-fibre-Noetherian}.
\end{proof}

\begin{lemma}
\label{lemma-morphism-between-flat}
Let $S$ be a scheme.
 ========

regex_match $S$
nn candidate a scheme
nn a scheme
new_annotation [(168249, 168252, 'VAR'), (168256, 168264, 'TYPE')]
===== sent Let $f : X \to Y$ be a morphism of schemes over $S$.
Assume
\begin{enumerate}
\item $X$ is locally of finite presentation over $S$,
\item $X$ is flat over $S$,
\item for every $s \in S$ the morphism
$f_s : X_s \to Y_s$ is flat, and
\item $Y$ is locally of finite type over $S$.
\end{enumerate}
Then $f$ is flat. If $f$ is also surjective, then $Y$ is flat over $S$.
\end{lemma}

\begin{proof}
 ========

regex_match $f : X \to Y$
nn candidate X \to
nn candidate a morphism
nn a morphism
new_annotation [(168270, 168283, 'VAR'), (168287, 168297, 'TYPE')]
===== sent This is a special case of
Theorem \ref{theorem-criterion-flatness-fibre}.
\end{proof}

\begin{lemma}
\label{lemma-base-change-criterion-flatness-fibre}
Let $S$ be a scheme. ========

regex_match $S$
nn candidate a scheme
nn a scheme
new_annotation [(168815, 168818, 'VAR'), (168822, 168830, 'TYPE')]
===== sent Let $f : X \to Y$ be a morphism of schemes over $S$.
Let $\mathcal{F}$ be a quasi-coherent $\mathcal{O}_X$-module.
 ========

regex_match $f : X \to Y$
nn candidate $
nn candidate X \to
nn candidate a morphism
nn a morphism
new_annotation [(168836, 168849, 'VAR'), (168853, 168863, 'TYPE')]
===== sent Let $f : X \to Y$ be a morphism of schemes over $S$.
Let $\mathcal{F}$ be a quasi-coherent $\mathcal{O}_X$-module.
 ========

regex_match $\mathcal{F}$
nn candidate a quasi-coherent $\mathcal{O}_X$-module
nn a quasi-coherent $\mathcal{O}_X$-module
new_annotation [(168889, 168902, 'VAR'), (168906, 168945, 'TYPE')]
===== sent Y'_{s'} \ar[r] & Y_s
}
$$
the horizontal morphisms are flat as they are base changes by the flat
morphism $\Spec(\kappa(s')) \to \Spec(\kappa(s))$.
\end{proof}

\begin{lemma}
\label{lemma-base-change-flatness-fibres}
Let $S$ be a scheme. ========

regex_match $S$
nn candidate a scheme
nn a scheme
new_annotation [(170848, 170851, 'VAR'), (170855, 170863, 'TYPE')]
===== sent Let $f : X \to Y$ be a morphism of schemes over $S$.
Assume
\begin{enumerate}
\item $X$ is locally of finite presentation over $S$,
\item $X$ is flat over $S$, and
\item $Y$ is locally of finite type over $S$.
\end{enumerate}
Then the set
$$
U = \{x \in X \mid X\text{ flat at }x \text{ over }Y\}.
$$
is open in $X$ and its formation commutes with arbitrary base change.
 ========

regex_match $f : X \to Y$
nn candidate $
nn candidate X \to
nn candidate a morphism
nn a morphism
new_annotation [(170869, 170882, 'VAR'), (170886, 170896, 'TYPE')]
===== sent \begin{lemma}
\label{lemma-flat-and-free-at-point-fibre}
Let $f : X \to S$ be a morphism of schemes of finite presentation.
 ========

regex_match $f : X \to S$
nn candidate X \to
nn candidate a morphism
nn a morphism
new_annotation [(171926, 171939, 'VAR'), (171943, 171953, 'TYPE')]
===== sent Let $\mathcal{F}$ be a finitely presented $\mathcal{O}_X$-module.
 ========

regex_match $\mathcal{F}$
new_annotation []
===== sent Let $x \in X$ with image $s \in ========

new_annotation []
===== sent \end{proof}

\begin{lemma}
\label{lemma-finite-free-open}
Let $f : X \to S$ be a morphism of schemes which is
locally of finite presentation.
 ========

regex_match $f : X \to S$
nn candidate X \to
nn candidate a morphism
nn a morphism
new_annotation [(173485, 173498, 'VAR'), (173502, 173512, 'TYPE')]
===== sent Let $\mathcal{F}$ be a finitely presented $\mathcal{O}_X$-module
flat over $S$. ========

regex_match $\mathcal{F}$
nn candidate $S$.
nn $S$.
new_annotation [(173569, 173582, 'VAR'), (173640, 173644, 'TYPE')]
===== sent Let $x \in X$ mapping to $s \in S$.
 ========

new_annotation []
===== sent \begin{lemma}
\label{lemma-integral-closure-smooth-pullback}
Let $f : Y \to X$ be a smooth morphism of schemes. ========

regex_match $f : Y \to X$
nn candidate : Y \to X$
nn candidate a smooth morphism
nn a smooth morphism
new_annotation [(174595, 174608, 'VAR'), (174612, 174629, 'TYPE')]
===== sent Let $\mathcal{A}$ be a
quasi-coherent sheaf of $\mathcal{O}_X$-algebras. ========

new_annotation []
===== sent Let $Y_i \to X_i' \to X_i$ be the normalization of $X_i$ in $Y_i$.
Then $X_2' \cong X_2 \times_{X_1} X_1'$.
\end{lemma}

\begin{proof}
 ========

regex_match $Y_i \to X_i' \to X_i$
nn candidate $Y_i \to X_i
nn candidate the normalization
nn the normalization
new_annotation [(175417, 175439, 'VAR'), (175443, 175460, 'TYPE')]
===== sent Observe that $X_2'$ is the relative spectrum of the integral closure
of $\mathcal{O}_{X_2}$ in $f_{2, *}\mathcal{O}_{Y_2}$.
If $\mathcal{A}' \subset f_{1, *}\mathcal{O}_{Y_1}$ denotes the integral
closure of $\mathcal{O}_{X_2}$, then $X_2 \times_{X_1} X_1'$ is the
relative spectrum of $\varphi^*\mathcal{A}'$, see
Constructions, Lemma \ref{constructions-lemma-spec-properties}.
By
Cohomology of Schemes, Lemma \ref{coherent-lemma-flat-base-change-cohomology}
we know that $f_{2, *}\mathcal{O}_{Y_2} = \varphi^*f_{1, *}\mathcal{O}_{Y_1}$.
Hence the result follows from
Lemma \ref{lemma-integral-closure-smooth-pullback}.
\end{proof}

\begin{lemma}[Normalization and smooth morphisms]
\label{lemma-normalization-and-smooth}
Let $X \to Y$ be a smooth morphism of schemes. ========

regex_match $X \to Y$
nn candidate $X
nn candidate a smooth morphism
nn a smooth morphism
new_annotation [(176669, 176678, 'VAR'), (176682, 176699, 'TYPE')]
===== sent We conclude that there exists a factorization
$X^\nu \to X' \to X$, see
Morphisms, Lemma \ref{morphisms-lemma-normalization-normal}
which is an isomorphism as $X'$ is normal and integral over $X$.
\end{proof}

\begin{lemma}[Normalization and henselization]
\label{lemma-normalization-and-henselization}
Let $X$ be a locally Noetherian scheme. ========

regex_match $X$
nn candidate a locally Noetherian scheme
nn a locally Noetherian scheme
new_annotation [(178125, 178128, 'VAR'), (178132, 178159, 'TYPE')]
===== sent Let $\nu : X^\nu \to X$
be the normalization morphism. ========

new_annotation []
===== sent \Spec(\mathcal{O}_{X, x}^{sh}) \to \Spec(\mathcal{O}_{X, x}^{sh})
$$
is the normalization of $\Spec(\mathcal{O}_{X, x}^h)$,
resp.\ $\Spec(\mathcal{O}_{X, x}^{sh})$.
\end{lemma}

\begin{proof}
Let $\eta_1, \ldots, \eta_r$ be the generic points of the irreducible
components of $X$ passing through $x$. The base change of the normalization to
$\Spec(\mathcal{O}_{X, x})$ is the spectrum of the
integral closure of $\mathcal{O}_{X, x}$ in $\prod ========

regex_match $\eta_1, \ldots, \eta_r$
nn candidate $\eta_1
nn candidate \ldots
nn candidate the generic points
nn the generic points
new_annotation [(178574, 178598, 'VAR'), (178602, 178620, 'TYPE')]
===== sent Let $A$ be a Noetherian local ring; recall that this implies
the henselization $A^h$ and strict henselization $A^{sh}$
are Noetherian too (More on Algebra, Lemma
\ref{more-algebra-lemma-henselization-noetherian}).
 ========

regex_match $A$
nn candidate a Noetherian local ring
nn a Noetherian local ring
new_annotation [(179204, 179207, 'VAR'), (179211, 179234, 'TYPE')]
===== sent Let $\mathfrak p_1, \ldots, \mathfrak p_r$
be its minimal primes. ========

new_annotation []
===== sent Let $A'$ be the integral closure
of $A$ in $\prod \kappa(\mathfrak p_i)$.
Problem: show that
$A' \otimes_A A^h$, resp.\ $A' \otimes_A A^{sh}$ is constructed
from the Noetherian local ring $A^h$, resp.\ $A^{sh}$ in the same
manner.

 ========

regex_match $A'$
nn candidate the integral closure
nn the integral closure
new_annotation [(179484, 179488, 'VAR'), (179492, 179512, 'TYPE')]
===== sent \begin{definition}
\label{definition-normal}
Let $f : X \to Y$ be a morphism of schemes.
 ========

regex_match $f : X \to Y$
nn candidate X \to
nn candidate a morphism
nn a morphism
new_annotation [(181368, 181381, 'VAR'), (181385, 181395, 'TYPE')]
===== sent \begin{enumerate}
\item Let $x \in X$, and $y = f(x)$. We say that $f$ is {\it normal at $x$}
if $f$ is flat at $x$, and the scheme $X_y$ is geometrically
normal at $x$ over $\kappa(y)$ (see
Varieties, Definition \ref{varieties-definition-geometrically-normal}).
 ========

new_annotation []
===== sent \begin{lemma}
\label{lemma-normal}
Let $f : X \to Y$ be a morphism of schemes.
 ========

regex_match $f : X \to Y$
nn candidate X \to
nn candidate a morphism
nn a morphism
new_annotation [(182047, 182060, 'VAR'), (182064, 182074, 'TYPE')]
===== sent \end{lemma}

\begin{proof}
Let $f : X \to Y$ be a smooth morphism.
 ========

regex_match $f : X \to Y$
nn candidate X \to
nn candidate a smooth morphism
nn a smooth morphism
new_annotation [(182477, 182490, 'VAR'), (182494, 182511, 'TYPE')]
===== sent \end{lemma}

\begin{proof}
Let $f : X \to Y$ be a morphism of schemes.
 ========

regex_match $f : X \to Y$
nn candidate X \to
nn candidate a morphism
nn a morphism
new_annotation [(183493, 183506, 'VAR'), (183510, 183520, 'TYPE')]
===== sent Let $\{\varphi_i : Y_i \to Y\}_{i \in I}$ be an fppf covering of $Y$.
Denote $f_i : X_i \to Y_i$ the base change of $f$ by $\varphi_i$.
Let $i \in I$ and let $y_i \in Y_i$ be a point.
 ========

regex_match $\{\varphi_i : Y_i \to Y\}_{i \in I}$
nn candidate $\{\varphi_i
nn candidate an fppf
nn an fppf
new_annotation [(183537, 183574, 'VAR'), (183578, 183585, 'TYPE')]
===== sent Let $\{\varphi_i : Y_i \to Y\}_{i \in I}$ be an fppf covering of $Y$.
Denote $f_i : X_i \to Y_i$ the base change of $f$ by $\varphi_i$.
Let $i \in I$ and let $y_i \in Y_i$ be a point.
 ========

new_annotation []
===== sent Let $\{\varphi_i : Y_i \to Y\}_{i \in I}$ be an fppf covering of $Y$.
Denote $f_i : X_i \to Y_i$ the base change of $f$ by $\varphi_i$.
Let $i \in I$ and let $y_i \in Y_i$ be a point.
 ========

regex_match $y_i \in Y_i$
nn candidate a point
nn a point
new_annotation [(183691, 183704, 'VAR'), (183708, 183715, 'TYPE')]
===== sent \medskip\noindent
Let $\{X_i \to X\}$ be an fppf covering of $X$.
Let $y \in Y$. In this case $\{X_{i, y} \to X_y\}$ is an
fppf covering of the fibre. ========

regex_match $\{X_i \to X\}$
nn candidate an fppf
nn an fppf
new_annotation [(184204, 184219, 'VAR'), (184223, 184230, 'TYPE')]
===== sent \medskip\noindent
Let $\{X_i \to X\}$ be an fppf covering of $X$.
Let $y \in Y$. In this case $\{X_{i, y} \to X_y\}$ is an
fppf covering of the fibre. ========

new_annotation []
===== sent Thus we have to deal
with $\mathcal{P}_3$.

\medskip\noindent
Let $f : X \to Y$ be a morphism of schemes.
 ========

regex_match $f : X \to Y$
nn candidate X \to
nn candidate a morphism
nn a morphism
new_annotation [(185376, 185389, 'VAR'), (185393, 185403, 'TYPE')]
===== sent Let $\{\varphi_i : Y_i \to Y\}_{i \in I}$ be an fpqc covering of $Y$.
Denote $f_i : X_i \to Y_i$ the base change of $f$ by $\varphi_i$.
Let $i \in I$ and let $y_i \in Y_i$ be a point.
 ========

regex_match $\{\varphi_i : Y_i \to Y\}_{i \in I}$
nn candidate $\{\varphi_i
nn candidate an fpqc
nn an fpqc
new_annotation [(185420, 185457, 'VAR'), (185461, 185468, 'TYPE')]
===== sent Let $\{\varphi_i : Y_i \to Y\}_{i \in I}$ be an fpqc covering of $Y$.
Denote $f_i : X_i \to Y_i$ the base change of $f$ by $\varphi_i$.
Let $i \in I$ and let $y_i \in Y_i$ be a point.
 ========

new_annotation []
===== sent Let $\{\varphi_i : Y_i \to Y\}_{i \in I}$ be an fpqc covering of $Y$.
Denote $f_i : X_i \to Y_i$ the base change of $f$ by $\varphi_i$.
Let $i \in I$ and let $y_i \in Y_i$ be a point.
 ========

regex_match $y_i \in Y_i$
nn candidate a point
nn a point
new_annotation [(185574, 185587, 'VAR'), (185591, 185598, 'TYPE')]
===== sent \medskip\noindent
Let $\{X_i \to X\}$ be a smooth covering of $X$.
Let $y \in Y$. In this case $\{X_{i, y} \to X_y\}$ is a
smooth covering of the fibre. ========

regex_match $\{X_i \to X\}$
nn candidate $y
nn $y
new_annotation [(185985, 186000, 'VAR'), (186034, 186036, 'TYPE')]
===== sent \medskip\noindent
Let $\{X_i \to X\}$ be a smooth covering of $X$.
Let $y \in Y$. In this case $\{X_{i, y} \to X_y\}$ is a
smooth covering of the fibre. ========

new_annotation []
===== sent \end{proof}



\section{Regular morphisms}
\label{section-regular}

\noindent
Compare with Section \ref{section-normal}. The algebraic version of
this notion is discussed in
More on Algebra, Section \ref{more-algebra-section-regular}.

\begin{definition}
\label{definition-regular}
Let $f : X \to Y$ be a morphism of schemes.
 ========

regex_match $f : X \to Y$
nn candidate X \to
nn candidate a morphism
nn a morphism
new_annotation [(186586, 186599, 'VAR'), (186603, 186613, 'TYPE')]
===== sent \begin{enumerate}
\item Let $x \in X$, and $y = f(x)$. We say that $f$ is {\it regular at $x$}
if $f$ is flat at $x$, and the scheme $X_y$ is geometrically
regular at $x$ over $\kappa(y)$ (see
Varieties, Definition \ref{varieties-definition-geometrically-regular}).
 ========

new_annotation []
===== sent \begin{lemma}
\label{lemma-regular}
Let $f : X \to Y$ be a morphism of schemes.
 ========

regex_match $f : X \to Y$
nn candidate X \to
nn candidate a morphism
nn a morphism
new_annotation [(187270, 187283, 'VAR'), (187287, 187297, 'TYPE')]
===== sent Let $x \in X$ with $y = f(x)$. By definition $f$ is flat at $x$
if and only if $\mathcal{O}_{Y, y} \to \mathcal{O}_{X, x}$ is a
flat ring map, and $X_y$ is geometrically regular at $x$ over
$\kappa(y)$ if and only if
$\mathcal{O}_{X_y, x} = \mathcal{O}_{X, x}/\mathfrak m_y\mathcal{O}_{X, x}$
is a geometrically regular algebra over $\kappa(y)$. Hence
Whether or not $f$ is regular at $x$
depends only on the local homomorphism of local rings
$\mathcal{O}_{Y, y} \to \mathcal{O}_{X, x}$.
 ========

new_annotation []
===== sent \end{lemma}

\begin{proof}
Let $f : X \to Y$ be a smooth morphism.
 ========

regex_match $f : X \to Y$
nn candidate X \to
nn candidate a smooth morphism
nn a smooth morphism
new_annotation [(189886, 189899, 'VAR'), (189903, 189920, 'TYPE')]
===== sent Thus we have to deal
with $\mathcal{P}_3$.

\medskip\noindent
Let $f : X \to Y$ be a morphism of schemes.
 ========

regex_match $f : X \to Y$
nn candidate X \to
nn candidate a morphism
nn a morphism
new_annotation [(191413, 191426, 'VAR'), (191430, 191440, 'TYPE')]
===== sent Let $\{\varphi_i : Y_i \to Y\}_{i \in I}$ be an fpqc covering of $Y$.
Denote $f_i : X_i \to Y_i$ the base change of $f$ by $\varphi_i$.
Let $i \in I$ and let $y_i \in Y_i$ be a point.
 ========

regex_match $\{\varphi_i : Y_i \to Y\}_{i \in I}$
nn candidate $\{\varphi_i
nn candidate an fpqc
nn an fpqc
new_annotation [(191457, 191494, 'VAR'), (191498, 191505, 'TYPE')]
===== sent Let $\{\varphi_i : Y_i \to Y\}_{i \in I}$ be an fpqc covering of $Y$.
Denote $f_i : X_i \to Y_i$ the base change of $f$ by $\varphi_i$.
Let $i \in I$ and let $y_i \in Y_i$ be a point.
 ========

new_annotation []
===== sent Let $\{\varphi_i : Y_i \to Y\}_{i \in I}$ be an fpqc covering of $Y$.
Denote $f_i : X_i \to Y_i$ the base change of $f$ by $\varphi_i$.
Let $i \in I$ and let $y_i \in Y_i$ be a point.
 ========

regex_match $y_i \in Y_i$
nn candidate a point
nn a point
new_annotation [(191611, 191624, 'VAR'), (191628, 191635, 'TYPE')]
===== sent $$
Hence in this situation we have that $X_y$ is geometrically regular if and
only if $X_{i, y_i}$ is geometrically regular, see
Varieties, Lemma \ref{varieties-lemma-geometrically-regular-upstairs}.
This fact implies $\mathcal{P}_3$ is fpqc local on the target.

\medskip\noindent
Let $\{X_i \to X\}$ be a smooth covering of $X$.
Let $y \in Y$. In this case $\{X_{i, y} \to X_y\}$ is a
smooth covering of the fibre. ========

regex_match $\{X_i \to X\}$
nn candidate $y
nn $y
new_annotation [(192025, 192040, 'VAR'), (192074, 192076, 'TYPE')]
===== sent $$
Hence in this situation we have that $X_y$ is geometrically regular if and
only if $X_{i, y_i}$ is geometrically regular, see
Varieties, Lemma \ref{varieties-lemma-geometrically-regular-upstairs}.
This fact implies $\mathcal{P}_3$ is fpqc local on the target.

\medskip\noindent
Let $\{X_i \to X\}$ be a smooth covering of $X$.
Let $y \in Y$. In this case $\{X_{i, y} \to X_y\}$ is a
smooth covering of the fibre. ========

new_annotation []
===== sent \begin{definition}
\label{definition-CM}
Let $f : X \to Y$ be a morphism of schemes.
 ========

regex_match $f : X \to Y$
nn candidate X \to
nn candidate a morphism
nn a morphism
new_annotation [(192716, 192729, 'VAR'), (192733, 192743, 'TYPE')]
===== sent \begin{enumerate}
\item Let $x \in X$, and $y = f(x)$. We say that $f$ is
{\it Cohen-Macaulay at $x$} if $f$ is flat at $x$, and the
local ring of the scheme $X_y$ at $x$ is Cohen-Macaulay.
 ========

new_annotation []
===== sent \begin{lemma}
\label{lemma-CM}
Let $f : X \to Y$ be a morphism of schemes.
 ========

regex_match $f : X \to Y$
nn candidate X \to
nn candidate a morphism
nn a morphism
new_annotation [(193214, 193227, 'VAR'), (193231, 193241, 'TYPE')]
===== sent \end{proof}

\begin{lemma}
\label{lemma-CM-dimension}
Let $f : X \to Y$ be a morphism of locally Noetherian schemes
which is locally of finite type and Cohen-Macaulay.
 ========

regex_match $f : X \to Y$
nn candidate X \to
nn candidate a morphism
nn a morphism
new_annotation [(193589, 193602, 'VAR'), (193606, 193616, 'TYPE')]
===== sent Hence the result follows from
Morphisms, Lemma \ref{morphisms-lemma-rel-dimension-dimension}.
\end{proof}

\begin{lemma}
\label{lemma-composition-CM}
Let $f : X \to Y$ and $g : Y \to Z$ be morphisms of schemes. ========

new_annotation []
===== sent Let $x \in X$ with images $y \in Y$ and $z \in Z$.
\begin{enumerate}
\item If $f$ is Cohen-Macaulay at $x$ and $g$ is Cohen-Macaulay
at $f(x)$, then $g \circ f$ is Cohen-Macaulay at $x$.
\item If $f$ and $g$ are Cohen-Macaulay, then $g \circ f$ is Cohen-Macaulay.
 ========

new_annotation []
===== sent If $f \circ g$ is Cohen-Macaulay and $f$ is flat, then
$f$ is Cohen-Macaulay and $g$ is Cohen-Macaulay at every point in
the image of $f$.
\end{enumerate}
\end{lemma}

\begin{proof}
Consider the map of Noetherian local rings
$$
\mathcal{O}_{Y_z, y} \to \mathcal{O}_{X_z, x}
$$
and observe that its fibre is
$$
\mathcal{O}_{X_z, x}/\mathfrak m_{Y_z, y}\mathcal{O}_{X_z, x} =
\mathcal{O}_{X_y, x}
$$
Thus the lemma this follows from
Algebra, Lemma \ref{algebra-lemma-CM-goes-up}.
\end{proof}

\begin{lemma}
\label{lemma-flat-morphism-from-CM-scheme}
Let $f : X \to Y$ be a flat morphism of locally Noetherian schemes.
 ========

regex_match $f : X \to Y$
nn candidate X \to
nn candidate a flat morphism
nn a flat morphism
new_annotation [(195430, 195443, 'VAR'), (195447, 195462, 'TYPE')]
===== sent After translating into algebra this follows from
Algebra, Lemma \ref{algebra-lemma-CM-goes-up}.
\end{proof}

\begin{lemma}
\label{lemma-base-change-CM}
Let $f : X \to Y$ be a morphism of schemes.
 ========

regex_match $f : X \to Y$
nn candidate X \to
nn candidate a morphism
nn a morphism
new_annotation [(195795, 195808, 'VAR'), (195812, 195822, 'TYPE')]
===== sent Let $Y' \to Y$ be locally of finite type. ========

new_annotation []
===== sent Let $f' : X' \to Y'$
be the base change of $f$.
Let $x' \in X'$ be a point with image $x \in X$.
\begin{enumerate}
\item If $f$ is Cohen-Macaulay at $x$, then
$f' : X' \to Y'$ is Cohen-Macaulay at $x'$.
\item If $f$ is flat at $x$ and $f'$ is Cohen-Macaulay at $x'$, then $f$
is Cohen-Macaulay at $x$.
\item ========

new_annotation []
===== sent Let $f' : X' \to Y'$
be the base change of $f$.
Let $x' \in X'$ be a point with image $x \in X$.
\begin{enumerate}
\item If $f$ is Cohen-Macaulay at $x$, then
$f' : X' \to Y'$ is Cohen-Macaulay at $x'$.
\item If $f$ is flat at $x$ and $f'$ is Cohen-Macaulay at $x'$, then $f$
is Cohen-Macaulay at $x$.
\item ========

regex_match $x' \in X'$
nn candidate x
nn candidate a point
nn a point
new_annotation [(195994, 196005, 'VAR'), (196009, 196016, 'TYPE')]
===== sent Hence we see that $f$
is Cohen-Macaulay at $x$.
\end{proof}

\begin{lemma}
\label{lemma-flat-finite-presentation-CM-open}
\begin{reference}
\cite[IV Corollary 12.1.7(iii)]{EGA}
\end{reference}
Let $f : X \to S$ be a morphism of schemes which is flat and locally
of finite presentation. ========

regex_match $f : X \to S$
nn candidate X \to
nn candidate a morphism
nn a morphism
new_annotation [(198789, 198802, 'VAR'), (198806, 198816, 'TYPE')]
===== sent Parts (2) and (3) follow from
Algebra, Lemma \ref{algebra-lemma-generic-CM-flat-finite-presentation}.
Part (4) follows either from
Algebra, Lemma \ref{algebra-lemma-CM-locus-commutes-base-change}
or
Varieties, Lemma \ref{varieties-lemma-CM-base-change}.
\end{proof}

\begin{lemma}
\label{lemma-flat-finite-presentation-characterize-CM}
Let $f : X \to S$ be a morphism of schemes which is flat and locally
of finite presentation. ========

regex_match $f : X \to S$
nn candidate X \to
nn candidate a morphism
nn a morphism
new_annotation [(199862, 199875, 'VAR'), (199879, 199889, 'TYPE')]
===== sent Let $x \in X$ with image $s \in ========

new_annotation []
===== sent Let $\mathfrak p \subset A$
be the prime ideal corresponding to ========

new_annotation []
===== sent \end{proof}

\begin{lemma}
\label{lemma-flat-finite-presentation-CM-pieces}
Let $f : X \to S$ be a morphism of schemes which is flat and locally
of finite presentation. ========

regex_match $f : X \to S$
nn candidate X \to
nn candidate a morphism
nn a morphism
new_annotation [(202237, 202250, 'VAR'), (202254, 202264, 'TYPE')]
===== sent This follows by combining
Lemma \ref{lemma-flat-finite-presentation-CM-open}
with
Morphisms, Lemma
\ref{morphisms-lemma-flat-finite-presentation-CM-fibres-relative-dimension}.
\end{proof}

\begin{lemma}
\label{lemma-flat-finite-presentation-specialization-dimension}
Let $f : X \to S$ be a morphism of schemes which is flat and locally
of finite presentation.
 ========

regex_match $f : X \to S$
nn candidate X \to
nn candidate a morphism
nn a morphism
new_annotation [(202932, 202945, 'VAR'), (202949, 202959, 'TYPE')]
===== sent Thus we have to deal
with $\mathcal{P}_2$.

\medskip\noindent
Let $f : X \to Y$ be a morphism of schemes.
 ========

regex_match $f : X \to Y$
nn candidate X \to
nn candidate a morphism
nn a morphism
new_annotation [(204125, 204138, 'VAR'), (204142, 204152, 'TYPE')]
===== sent Let $\{\varphi_i : Y_i \to Y\}_{i \in I}$ be an fppf covering of $Y$.
Denote $f_i : X_i \to Y_i$ the base change of $f$ by $\varphi_i$.
Let $i \in I$ and let $y_i \in Y_i$ be a point.
 ========

regex_match $\{\varphi_i : Y_i \to Y\}_{i \in I}$
nn candidate $\{\varphi_i
nn candidate an fppf
nn an fppf
new_annotation [(204169, 204206, 'VAR'), (204210, 204217, 'TYPE')]
===== sent Let $\{\varphi_i : Y_i \to Y\}_{i \in I}$ be an fppf covering of $Y$.
Denote $f_i : X_i \to Y_i$ the base change of $f$ by $\varphi_i$.
Let $i \in I$ and let $y_i \in Y_i$ be a point.
 ========

new_annotation []
===== sent Let $\{\varphi_i : Y_i \to Y\}_{i \in I}$ be an fppf covering of $Y$.
Denote $f_i : X_i \to Y_i$ the base change of $f$ by $\varphi_i$.
Let $i \in I$ and let $y_i \in Y_i$ be a point.
 ========

regex_match $y_i \in Y_i$
nn candidate a point
nn a point
new_annotation [(204323, 204336, 'VAR'), (204340, 204347, 'TYPE')]
===== sent \medskip\noindent
Let $\{X_i \to X\}$ be a syntomic covering of $X$.
Let $y \in Y$. In this case $\{X_{i, y} \to X_y\}$ is a
syntomic covering of the fibre. ========

regex_match $\{X_i \to X\}$
nn candidate $y
nn $y
new_annotation [(204903, 204918, 'VAR'), (204954, 204956, 'TYPE')]
===== sent \medskip\noindent
Let $\{X_i \to X\}$ be a syntomic covering of $X$.
Let $y \in Y$. In this case $\{X_{i, y} \to X_y\}$ is a
syntomic covering of the fibre. ========

new_annotation []
===== sent The following lemma is very closely related to
Divisors, Lemma \ref{divisors-lemma-fibre-Cartier}.

\begin{lemma}
\label{lemma-slice-given-element}
Let $f : X \to S$ be a morphism of schemes.
 ========

regex_match $f : X \to S$
nn candidate X \to
nn candidate a morphism
nn a morphism
new_annotation [(205629, 205642, 'VAR'), (205646, 205656, 'TYPE')]
===== sent Let $x \in X$ be a point with image $s \in ========

regex_match $x \in X$
nn candidate a point
nn a point
new_annotation [(205673, 205682, 'VAR'), (205686, 205693, 'TYPE')]
===== sent S$.
Let $h \in \mathfrak m_x \subset \mathcal{O}_{X, x}$.
Assume
\begin{enumerate}
\item $f$ is locally of finite presentation,
\item $f$ is flat at $x$, and
\item the image $\overline{h}$ of $h$ in
$\mathcal{O}_{X_s, x} = \mathcal{O}_{X, x}/\mathfrak m_s\mathcal{O}_{X, x}$
is a nonzerodivisor.
 ========

new_annotation []
===== sent Let $x_0 \in X_0$, resp.\ $s_0 \in S_0$
be the image of $x$, resp.\ $s$. We may also assume there exists an element
$h_0 \in \Gamma(X_0, \mathcal{O}_{X_0})$ which restricts to $h$ on $X$.
(If you used the algebra reference above then this is clear; if you used
the references to the chapter on limits then this follows from
Limits, Lemma \ref{limits-lemma-descend-finite-presentation}
by thinking of $h$ as a morphism $X \to \mathbf{A}^1_S$.)
Note that $\mathcal{O}_{X_s, x}$ is a localization of
$\mathcal{O}_{(X_0)_{s_0}, x_0} \otimes_{\kappa(s_0)} \kappa(s)$, so that
$\mathcal{O}_{(X_0)_{s_0}, x_0} \to \mathcal{O}_{X_s, x}$ is a flat
local ring map, in particular faithfully flat. ========

new_annotation []
===== sent The corresponding algebra statement
is the following (we drop the subscript ${}_0$ here):
Let $A \to B$ be a flat, finite type ring map of Noetherian rings.
 ========

regex_match $A \to B$
nn candidate a flat, finite type ring map
nn a flat, finite type ring map
new_annotation [(208763, 208772, 'VAR'), (208776, 208804, 'TYPE')]
===== sent Let $\mathfrak q \subset B$ be a prime lying over $\mathfrak p \subset A$.
 ========

regex_match $\mathfrak q \subset B$
nn candidate q
nn candidate a prime lying
nn a prime lying
new_annotation [(208830, 208853, 'VAR'), (208857, 208870, 'TYPE')]
===== sent \end{proof}

\begin{lemma}
\label{lemma-slice-given-elements}
Let $f : X \to S$ be a morphism of schemes.
 ========

regex_match $f : X \to S$
nn candidate X \to
nn candidate a morphism
nn a morphism
new_annotation [(209998, 210011, 'VAR'), (210015, 210025, 'TYPE')]
===== sent Let $x \in X$ be a point with image $s \in ========

regex_match $x \in X$
nn candidate a point
nn a point
new_annotation [(210042, 210051, 'VAR'), (210055, 210062, 'TYPE')]
===== sent S$.
Let $h_1, \ldots, h_r \in \mathcal{O}_{X, x}$.
Assume
\begin{enumerate}
\item $f$ is locally of finite presentation,
\item $f$ is flat at $x$, and
\item the images of $h_1, \ldots, h_r$ in
$\mathcal{O}_{X_s, x} = \mathcal{O}_{X, x}/\mathfrak m_s\mathcal{O}_{X, x}$
form a regular sequence.
 ========

new_annotation []
===== sent \end{proof}

\begin{lemma}
\label{lemma-slice-once}
Let $f : X \to S$ be a morphism of schemes.
 ========

regex_match $f : X \to S$
nn candidate X \to
nn candidate a morphism
nn a morphism
new_annotation [(213165, 213178, 'VAR'), (213182, 213192, 'TYPE')]
===== sent Let $x \in X$ be a point with image $s \in ========

regex_match $x \in X$
nn candidate a point
nn a point
new_annotation [(213209, 213218, 'VAR'), (213222, 213229, 'TYPE')]
===== sent \end{lemma}

\begin{proof}
Pick any $h \in \mathfrak m_x \subset \mathcal{O}_{X, x}$ which
maps to a nonzerodivisor in $\mathcal{O}_{X_s, x}$ and apply
Lemma \ref{lemma-slice-given-element}.
\end{proof}

\begin{lemma}
\label{lemma-slice-CM}
\begin{reference}
\cite[IV Proposition 17.16.1]{EGA}
\end{reference}
Let $f : X \to S$ be a morphism of schemes.
 ========

regex_match $f : X \to S$
nn candidate X \to
nn candidate a morphism
nn a morphism
new_annotation [(213924, 213937, 'VAR'), (213941, 213951, 'TYPE')]
===== sent Let $x \in X$ be a point with image $s \in ========

regex_match $x \in X$
nn candidate a point
nn a point
new_annotation [(213968, 213977, 'VAR'), (213981, 213988, 'TYPE')]
===== sent \end{proof}

\begin{lemma}
\label{lemma-qf-fp-flat-neighbourhood-dominates-fppf}
Let $f : X \to S$ be a flat morphism of schemes which is
locally of finite presentation Let $s \in S$ be a point in the image of $f$.
Then there exists a commutative diagram
$$
\xymatrix{
S' \ar[rr] \ar[rd]_g & & X \ar[ld]^f \\
& S
}
$$
where $g : S' \to S$ is flat, locally of finite presentation,
locally quasi-finite, and $s \in g(S')$.
\end{lemma}

\begin{proof}
The fibre $X_s$ is not empty by assumption. ========

regex_match $f : X \to S$
nn candidate X \to
nn candidate a flat morphism
nn a flat morphism
new_annotation [(216978, 216991, 'VAR'), (216995, 217010, 'TYPE')]
===== sent \end{proof}

\begin{lemma}
\label{lemma-qf-fp-flat-neighbourhood-dominates-fppf}
Let $f : X \to S$ be a flat morphism of schemes which is
locally of finite presentation Let $s \in S$ be a point in the image of $f$.
Then there exists a commutative diagram
$$
\xymatrix{
S' \ar[rr] \ar[rd]_g & & X \ar[ld]^f \\
& S
}
$$
where $g : S' \to S$ is flat, locally of finite presentation,
locally quasi-finite, and $s \in g(S')$.
\end{lemma}

\begin{proof}
The fibre $X_s$ is not empty by assumption. ========

regex_match $s \in S$
nn candidate $s
nn candidate a point
nn a point
new_annotation [(217066, 217075, 'VAR'), (217079, 217086, 'TYPE')]
===== sent \begin{lemma}
\label{lemma-qf-fp-flat-dominates-fppf}
Let $S$ be a scheme. ========

regex_match $S$
nn candidate a scheme
nn a scheme
new_annotation [(217802, 217805, 'VAR'), (217809, 217817, 'TYPE')]
===== sent Let $\mathcal{U} = \{S_i \to S\}_{i \in I}$ be an fppf
covering of $S$, see
Topologies, Definition \ref{topologies-definition-fppf-covering}.
 ========

regex_match $\mathcal{U} = \{S_i \to S\}_{i \in I}$
nn candidate $\mathcal{U
nn candidate an fppf
nn an fppf
new_annotation [(217823, 217862, 'VAR'), (217866, 217873, 'TYPE')]
===== sent \begin{lemma}
\label{lemma-empty-generic-fibre}
Let $f : X \to Y$ be a finite type morphism of schemes. ========

regex_match $f : X \to Y$
nn candidate X \to
nn candidate a finite type morphism
nn a finite type morphism
new_annotation [(218834, 218847, 'VAR'), (218851, 218873, 'TYPE')]
===== sent Assume
$Y$ irreducible with generic point $\eta$. If $X_\eta = \emptyset$
then there exists a nonempty open $V \subset Y$ such that
$X_V = V \times_Y X = \emptyset$.
\end{lemma}

\begin{proof}
Follows immediately from the more general
Morphisms,
Lemma \ref{morphisms-lemma-quasi-compact-generic-point-not-in-image}.
\end{proof}

\begin{lemma}
\label{lemma-nonempty-generic-fibre}
Let $f : X \to Y$ be a finite type morphism of schemes. ========

regex_match $f : X \to Y$
nn candidate X \to
nn candidate a finite type morphism
nn a finite type morphism
new_annotation [(219270, 219283, 'VAR'), (219287, 219309, 'TYPE')]
===== sent \end{lemma}

\begin{proof}
This follows, upon taking affine opens, from
Algebra, Lemma \ref{algebra-lemma-characterize-image-finite-type}.
(Of course it also follows from generic flatness.)
\end{proof}

\begin{lemma}
\label{lemma-nowhere-dense-generic-fibre}
Let $f : X \to Y$ be a finite type morphism of schemes. ========

regex_match $f : X \to Y$
nn candidate X \to
nn candidate a finite type morphism
nn a finite type morphism
new_annotation [(219764, 219777, 'VAR'), (219781, 219803, 'TYPE')]
===== sent Assume
$Y$ irreducible with generic point $\eta$.
If $Z \subset X$ is a closed subset with $Z_\eta$ nowhere dense
in $X_\eta$, then there exists a nonempty open $V \subset Y$ such
that $Z_y$ is nowhere dense in $X_y$ for all $y \in V$.
\end{lemma}

\begin{proof}
Let $Y' \subset Y$ be the reduction of $Y$.
Set $X' = Y' \times_Y X$ and $Z' = Y' \times_Y Z$.
As $Y' ========

regex_match $Y' \subset Y$
nn candidate $Y
nn candidate the reduction
nn the reduction
new_annotation [(220083, 220097, 'VAR'), (220101, 220114, 'TYPE')]
===== sent \end{proof}

\begin{lemma}
\label{lemma-scheme-theoretically-dense-generic-fibre}
Let $f : X \to Y$ be a finite type morphism of schemes.
 ========

regex_match $f : X \to Y$
nn candidate X \to
nn candidate a finite type morphism
nn a finite type morphism
new_annotation [(221488, 221501, 'VAR'), (221505, 221527, 'TYPE')]
===== sent Assume $Y$ irreducible with generic point $\eta$.
Let $U \subset X$ be an open subscheme such that $U_\eta$ is
scheme theoretically dense in $X_\eta$.
Then there exists a nonempty open $V \subset Y$ such
that $U_y$ is scheme theoretically dense in $X_y$ for all $y \in V$.
\end{lemma}

\begin{proof}
Let $Y' \subset Y$ be the reduction of $Y$.
Let $X' = Y' \times_Y X$ and $U' = Y' \times_Y U$.
As $Y' \to Y$ induces a bijection on points, and as
$U' \to U$ and $X' \to X$ induce isomorphisms of scheme theoretic fibres,
we may replace $Y$ by $Y'$ and $X$ by $X'$.
 ========

regex_match $U \subset X$
nn candidate $U \subset X$
nn candidate an open subscheme
nn an open subscheme
new_annotation [(221594, 221607, 'VAR'), (221611, 221628, 'TYPE')]
===== sent Assume $Y$ irreducible with generic point $\eta$.
Let $U \subset X$ be an open subscheme such that $U_\eta$ is
scheme theoretically dense in $X_\eta$.
Then there exists a nonempty open $V \subset Y$ such
that $U_y$ is scheme theoretically dense in $X_y$ for all $y \in V$.
\end{lemma}

\begin{proof}
Let $Y' \subset Y$ be the reduction of $Y$.
Let $X' = Y' \times_Y X$ and $U' = Y' \times_Y U$.
As $Y' \to Y$ induces a bijection on points, and as
$U' \to U$ and $X' \to X$ induce isomorphisms of scheme theoretic fibres,
we may replace $Y$ by $Y'$ and $X$ by $X'$.
 ========

regex_match $Y' \subset Y$
nn candidate $Y
nn candidate the reduction
nn the reduction
new_annotation [(221844, 221858, 'VAR'), (221862, 221875, 'TYPE')]
===== sent Assume $Y$ irreducible with generic point $\eta$.
Let $U \subset X$ be an open subscheme such that $U_\eta$ is
scheme theoretically dense in $X_\eta$.
Then there exists a nonempty open $V \subset Y$ such
that $U_y$ is scheme theoretically dense in $X_y$ for all $y \in V$.
\end{lemma}

\begin{proof}
Let $Y' \subset Y$ be the reduction of $Y$.
Let $X' = Y' \times_Y X$ and $U' = Y' \times_Y U$.
As $Y' \to Y$ induces a bijection on points, and as
$U' \to U$ and $X' \to X$ induce isomorphisms of scheme theoretic fibres,
we may replace $Y$ by $Y'$ and $X$ by $X'$.
 ========

new_annotation []
===== sent Thus we can
find finitely many $g_1, \ldots, g_m \in B$ such that $D(g_j) \subset U$
and such that $U_\eta = D(g_1)_\eta \cup \ldots \cup D(g_m)_\eta$.
The fact that $U_\eta$ is scheme theoretically dense in
$X_\eta$ means that $B_K \to \bigoplus_j (B_K)_{g_j}$
is injective, see
Morphisms, Example \ref{morphisms-example-scheme-theoretic-closure}.
By
Algebra, Lemma \ref{algebra-lemma-when-injective-covering}
this is equivalent to the injectivity of
$B_K \to \bigoplus\nolimits_{j = 1, \ldots, m} B_K$,
$b \mapsto (g_1b, \ldots, g_mb)$. Let $M$ be the cokernel of this
map over $A$, i.e., such that we have an exact sequence
$$
0 \to I \to B \xrightarrow{(g_1, \ldots, g_m)}
\bigoplus\nolimits_{j = 1, \ldots, m} B \to M \to 0
$$
After replacing $A$ by $A_h$ for some nonzero $h$ we may assume that $B$
is a flat, finitely presented $A$-algebra, and that $M$
is flat over $A$, see
Algebra, Lemma \ref{algebra-lemma-generic-flatness}.
The flatness of $B$ over $A$ implies that $B$ is torsion free as an
$A$-module, see
More on Algebra, Lemma \ref{more-algebra-lemma-flat-torsion-free}.
 ========

regex_match $M$
nn candidate the cokernel
nn the cokernel
new_annotation [(223514, 223517, 'VAR'), (223521, 223533, 'TYPE')]
===== sent Let $T$ be the closure of $T(y)$ in $X$.
Endow $T$ with the induced reduced scheme structure.
 ========

regex_match $T$
nn candidate the closure
nn the closure
new_annotation [(225299, 225302, 'VAR'), (225306, 225317, 'TYPE')]
===== sent In the following lemma we apply this to the generic fibre of $f$.

\begin{lemma}
\label{lemma-cover-generic-fibre-neighbourhood}
Let $f : X \to Y$ be a finite type morphism of schemes. ========

regex_match $f : X \to Y$
nn candidate X \to
nn candidate a finite type morphism
nn a finite type morphism
new_annotation [(225772, 225785, 'VAR'), (225789, 225811, 'TYPE')]
===== sent $\eta$. Let
$X_\eta = Z_{1, \eta} \cup \ldots \cup Z_{n, \eta}$ be a covering of
the generic fibre by closed subsets of $X_\eta$.
Let $Z_i$ be the closure of $Z_{i, \eta}$ in $X$ (see discussion above).
 ========

regex_match $Z_i$
nn candidate the closure
nn the closure
new_annotation [(226000, 226005, 'VAR'), (226009, 226020, 'TYPE')]
===== sent \begin{lemma}
\label{lemma-reduction-generic-fibre}
Let $f : X \to Y$ be a morphism of schemes. ========

regex_match $f : X \to Y$
nn candidate X \to
nn candidate a morphism
nn a morphism
new_annotation [(227221, 227234, 'VAR'), (227238, 227248, 'TYPE')]
===== sent Let $\eta \in Y$ be a generic
point of an irreducible component of $Y$. Then
$(X_\eta)_{red} = (X_{red})_\eta$.
\end{lemma}

\begin{proof}
Choose an affine neighbourhood $\Spec(A) \subset Y$ of $\eta$.
Choose an affine open $\Spec(B) \subset X$ mapping into $\Spec(A)$
via the morphism $f$. ========

regex_match $\eta \in Y$
nn candidate $
nn candidate a generic
point
nn a generic
point
new_annotation [(227265, 227277, 'VAR'), (227281, 227296, 'TYPE')]
===== sent Let $\mathfrak p \subset A$ be the minimal prime
corresponding to $\eta$. Let $B_{red}$ be the quotient of $B$ by
$\sqrt{(0)}$. The algebraic content of the lemma is that
$B_{red} \otimes_A \kappa(\mathfrak p)$ is reduced. ========

regex_match $\mathfrak p \subset A$
nn candidate p \subset
nn candidate the quotient
nn the quotient
new_annotation [(227556, 227579, 'VAR'), (227643, 227655, 'TYPE')]
===== sent Let $\mathfrak p \subset A$ be the minimal prime
corresponding to $\eta$. Let $B_{red}$ be the quotient of $B$ by
$\sqrt{(0)}$. The algebraic content of the lemma is that
$B_{red} \otimes_A \kappa(\mathfrak p)$ is reduced. ========

regex_match $B_{red}$
nn candidate the quotient
nn the quotient
new_annotation [(227630, 227639, 'VAR'), (227643, 227655, 'TYPE')]
===== sent \end{proof}

\begin{lemma}
\label{lemma-make-generic-fibre-geometrically-reduced}
Let $f : X \to Y$ be a morphism of schemes.
 ========

regex_match $f : X \to Y$
nn candidate X \to
nn candidate a morphism
nn a morphism
new_annotation [(228712, 228725, 'VAR'), (228729, 228739, 'TYPE')]
===== sent \end{enumerate}
\end{lemma}

\begin{proof}
Let $V = \Spec(A)$ be a nonempty affine open of $Y$.
 ========

regex_match $V = \Spec(A)$
nn candidate $V
nn candidate a nonempty affine
nn a nonempty affine
new_annotation [(229396, 229410, 'VAR'), (229414, 229431, 'TYPE')]
===== sent By assumption the radical of $A$ is a prime ideal $\mathfrak p$.
Let $K = f.f(A/\mathfrak p)$ be the fraction field.
 ========

regex_match $K = f.f(A/\mathfrak p)$
nn candidate the fraction field
nn the fraction field
new_annotation [(229514, 229538, 'VAR'), (229542, 229560, 'TYPE')]
===== sent Let $p$ be the characteristic of $K$ if positive and $1$
if the characteristic is zero. ========

regex_match $p$
nn candidate the characteristic
nn the characteristic
new_annotation [(229566, 229569, 'VAR'), (229573, 229591, 'TYPE')]
===== sent By
Varieties, Lemma \ref{varieties-lemma-finite-extension-geometrically-reduced}
there exists a finite purely inseparable field extension
$K \subset K'$ such that $X_{K'}$ is geometrically reduced over $K'$.
Choose elements $x_1, \ldots, x_n \in K'$ which generate $K'$ over
$K$ and such that some $p$-power of $x_i$ is in $A/\mathfrak p$.
Let $A' \subset K'$ be the finite $A$-subalgebra of $K'$ generated by
$x_1, \ldots, x_n$. Note that $A'$ is a domain with fraction field ========

regex_match $A' \subset K'$
nn candidate A' \subset
nn candidate the finite $A$-subalgebra
nn the finite $A$-subalgebra
new_annotation [(229994, 230009, 'VAR'), (230013, 230038, 'TYPE')]
===== sent This can be achieved by shrinking $Y'$ and hence $V$, see
Morphisms, Proposition \ref{morphisms-proposition-generic-flatness}.
\end{proof}

\begin{lemma}
\label{lemma-make-components-generic-fibre-geometrically-irreducible}
Let $f : X \to Y$ be a morphism of schemes.
 ========

regex_match $f : X \to Y$
nn candidate X \to
nn candidate a morphism
nn a morphism
new_annotation [(230975, 230988, 'VAR'), (230992, 231002, 'TYPE')]
===== sent \end{enumerate}
\end{lemma}

\begin{proof}
Let $V = \Spec(A)$ be a nonempty affine open of $Y$.
 ========

regex_match $V = \Spec(A)$
nn candidate $V
nn candidate a nonempty affine
nn a nonempty affine
new_annotation [(231620, 231634, 'VAR'), (231638, 231655, 'TYPE')]
===== sent By assumption the radical of $A$ is a prime ideal $\mathfrak p$.
Let $K = f.f(A/\mathfrak p)$ be the fraction field. ========

regex_match $K = f.f(A/\mathfrak p)$
nn candidate the fraction field
nn the fraction field
new_annotation [(231738, 231762, 'VAR'), (231766, 231784, 'TYPE')]
===== sent By
Varieties, Lemma
\ref{varieties-lemma-finite-extension-geometrically-irreducible-components}
there exists a finite separable field extension
$K \subset K'$ such that all irreducible components of $X_{K'}$ are
geometrically irreducible over $K'$.
Choose an element $\alpha \in K'$ which generates $K'$ over
$K$, see
Fields, Lemma \ref{fields-lemma-primitive-element}.
Let $P(T) \in K[T]$ be the minimal polynomial for $\alpha$ over $K$.
After replacing $\alpha$ by $f \alpha$ for some
$f \in A$, $f \not \in ========

regex_match $P(T) \in K[T]$
nn candidate the minimal polynomial
nn the minimal polynomial
new_annotation [(232160, 232175, 'VAR'), (232179, 232201, 'TYPE')]
===== sent \end{proof}







\section{Relative assassins}
\label{section-assassin}

\begin{lemma}
\label{lemma-relative-assassin-in-neighbourhood}
Let $f : X \to S$ be a morphism of schemes.
 ========

regex_match $f : X \to S$
nn candidate X \to
nn candidate a morphism
nn a morphism
new_annotation [(233464, 233477, 'VAR'), (233481, 233491, 'TYPE')]
===== sent Let $\mathcal{F}$ be a quasi-coherent $\mathcal{O}_X$-module.
 ========

regex_match $\mathcal{F}$
nn candidate a quasi-coherent $\mathcal{O}_X$-module
nn a quasi-coherent $\mathcal{O}_X$-module
new_annotation [(233508, 233521, 'VAR'), (233525, 233564, 'TYPE')]
===== sent Let $\xi \in \text{Ass}_{X/S}(\mathcal{F})$ and set
$Z = \overline{\{\xi\}} \subset X$.
If $f$ is locally of finite type and $\mathcal{F}$ is a
finite type $\mathcal{O}_X$-module, then there exists a nonempty
open $V \subset Z$ such that for every $s \in f(V)$ the generic
points of $V_s$ are elements of $\text{Ass}_{X/S}(\mathcal{F})$.
\end{lemma}

\begin{proof}
We may replace $S$ by an affine open neighbourhood of $f(\xi)$
and $X$ by an affine open neighbourhood of $\xi$. ========

new_annotation []
===== sent Moreover, we may write $\mathcal{F} = \widetilde{M}$ for some finite
$B$-module $M$, see
Properties, Lemma \ref{properties-lemma-finite-type-module}.
Let $\mathfrak q \subset B$ be the prime corresponding to $\xi$ and
let $\mathfrak p \subset A$ be the corresponding prime of $A$.
 ========

regex_match $\mathfrak q \subset B$
nn candidate q
nn candidate the prime corresponding
nn the prime corresponding
new_annotation [(234387, 234410, 'VAR'), (234414, 234437, 'TYPE')]
===== sent Moreover, we may write $\mathcal{F} = \widetilde{M}$ for some finite
$B$-module $M$, see
Properties, Lemma \ref{properties-lemma-finite-type-module}.
Let $\mathfrak q \subset B$ be the prime corresponding to $\xi$ and
let $\mathfrak p \subset A$ be the corresponding prime of $A$.
 ========

regex_match $\mathfrak p \subset A$
nn candidate p \subset
nn candidate the corresponding prime
nn the corresponding prime
new_annotation [(234455, 234478, 'VAR'), (234482, 234505, 'TYPE')]
===== sent Let $Q$ be the cokernel so that we obtain a short exact sequence
$$
0 \to B/\mathfrak q \to M \to Q \to 0
$$
of finite $B$-modules. ========

regex_match $Q$
nn candidate the cokernel
nn the cokernel
new_annotation [(235594, 235597, 'VAR'), (235601, 235613, 'TYPE')]
===== sent \kappa(\mathfrak p')$
is an associated prime of $(B/\mathfrak q) \otimes_A \kappa(\mathfrak p')$ by
Divisors, Lemma \ref{divisors-lemma-minimal-support-in-ass}.
\end{proof}

\begin{lemma}
\label{lemma-bad-case}
Let $f : X \to Y$ be a morphism of schemes. ========

regex_match $f : X \to Y$
nn candidate X \to
nn candidate a morphism
nn a morphism
new_annotation [(236491, 236504, 'VAR'), (236508, 236518, 'TYPE')]
===== sent Let $\mathcal{F}$ be a
quasi-coherent $\mathcal{O}_X$-module. ========

new_annotation []
===== sent Let $U \subset X$ be an open
subscheme. ========

regex_match $U \subset X$
nn candidate $
nn candidate an open
subscheme
nn an open
subscheme
new_annotation [(236597, 236610, 'VAR'), (236614, 236631, 'TYPE')]
===== sent Assume
\begin{enumerate}
\item $f$ is of finite type,
\item $\mathcal{F}$ is of finite type,
\item $Y$ is irreducible with generic point $\eta$, and
\item $\text{Ass}_{X_\eta}(\mathcal{F}_\eta)$ is not contained in $U_\eta$.
\end{enumerate}
Then there exists a nonempty open subscheme $V \subset Y$ such that
for all $y \in V$ the set $\text{Ass}_{X_y}(\mathcal{F}_y)$ is not
contained in $U_y$.
\end{lemma}

\begin{proof}
Let $Z \subset X$ be the scheme theoretic support of $\mathcal{F}$, see
Morphisms, Definition \ref{morphisms-definition-scheme-theoretic-support}.
Then $Z_\eta$ is the scheme theoretic support of $\mathcal{F}_\eta$
(Morphisms, Lemma \ref{morphisms-lemma-flat-pullback-support}).
 ========

regex_match $Z \subset X$
nn candidate $Z \subset
nn candidate the scheme theoretic support
nn the scheme theoretic support
new_annotation [(237060, 237073, 'VAR'), (237077, 237105, 'TYPE')]
===== sent \end{proof}

\begin{lemma}
\label{lemma-good-case}
Let $f : X \to Y$ be a morphism of schemes. ========

regex_match $f : X \to Y$
nn candidate X \to
nn candidate a morphism
nn a morphism
new_annotation [(237927, 237940, 'VAR'), (237944, 237954, 'TYPE')]
===== sent Let $\mathcal{F}$ be a
quasi-coherent $\mathcal{O}_X$-module. ========

new_annotation []
===== sent Let $U \subset X$ be an open
subscheme. ========

regex_match $U \subset X$
nn candidate $
nn candidate an open
subscheme
nn an open
subscheme
new_annotation [(238033, 238046, 'VAR'), (238050, 238067, 'TYPE')]
===== sent \medskip\noindent
Write $X = \Spec(B)$. Let $N$ be a finite $B$-module such that
$\mathcal{F} = \widetilde{N}$.
Note that $B_K$ is Noetherian as it is a
finite type $K$-algebra. ========

regex_match $N$
nn candidate a finite $B$-module
nn a finite $B$-module
new_annotation [(239385, 239388, 'VAR'), (239392, 239411, 'TYPE')]
===== sent Let $I$ and $M$ be the kernel and
cokernel of this map over $A$, i.e., such that we have an exact sequence
$$
0 \to I \to N \xrightarrow{(g_1, \ldots, g_m)}
\bigoplus\nolimits_{j = 1, \ldots, m} N \to M \to 0
$$
After replacing $A$ by $A_h$ for some nonzero $h$ we may assume that
$B$ is a flat, finitely presented $A$-algebra and that
both $M$ and $N$ are flat over $A$, see
Algebra, Lemma \ref{algebra-lemma-generic-flatness}.
The flatness of $N$ over $A$ implies that $N$ is torsion free as an
$A$-module, see
More on Algebra, Lemma \ref{more-algebra-lemma-flat-torsion-free}.
 ========

new_annotation []
===== sent \end{proof}

\begin{lemma}
\label{lemma-base-change-assassin-in-U}
Let $f : X \to S$ be a morphism which is locally of finite type.
 ========

regex_match $f : X \to S$
nn candidate X \to
nn candidate a morphism
nn a morphism
new_annotation [(241624, 241637, 'VAR'), (241641, 241651, 'TYPE')]
===== sent Let $\mathcal{F}$ be a quasi-coherent $\mathcal{O}_X$-module
of finite type. ========

regex_match $\mathcal{F}$
nn candidate finite type
nn finite type
new_annotation [(241689, 241702, 'VAR'), (241749, 241760, 'TYPE')]
===== sent Let $U \subset X$ be an open subscheme.
 ========

regex_match $U \subset X$
nn candidate $
nn candidate an open subscheme
nn an open subscheme
new_annotation [(241766, 241779, 'VAR'), (241783, 241800, 'TYPE')]
===== sent Let $g : S' \to S$ be a morphism of schemes, let
$f' : X' = X_{S'} \to S'$ be the base change of $f$,
let $g' : X' \to X$ be the projection, set
$\mathcal{F}' = (g')^*\mathcal{F}$, and set
$U' = (g')^{-1}(U)$. ========

regex_match $g : S' \to S$
nn candidate $g
nn candidate a morphism
nn a morphism
new_annotation [(241806, 241820, 'VAR'), (241824, 241834, 'TYPE')]
===== sent Let $g : S' \to S$ be a morphism of schemes, let
$f' : X' = X_{S'} \to S'$ be the base change of $f$,
let $g' : X' \to X$ be the projection, set
$\mathcal{F}' = (g')^*\mathcal{F}$, and set
$U' = (g')^{-1}(U)$. ========

regex_match $g' : X' \to X$
nn candidate the projection
nn the projection
new_annotation [(241908, 241923, 'VAR'), (241927, 241941, 'TYPE')]
===== sent Finally, let $s' \in S'$ with image $s = g(s')$.
In this case
$$
\text{Ass}_{X_s}(\mathcal{F}_s) \subset U_s
\Leftrightarrow
\text{Ass}_{X'_{s'}}(\mathcal{F}'_{s'}) \subset U'_{s'}.
$$
\end{lemma}

\begin{proof}
This follows immediately from
Divisors, Lemma \ref{divisors-lemma-base-change-relative-assassin}.
See also
Divisors, Remark \ref{divisors-remark-base-change-relative-assassin}.
\end{proof}

\begin{lemma}
\label{lemma-relative-assassin-constructible}
Let $f : X \to Y$ be a morphism of finite presentation.
 ========

new_annotation []
===== sent Finally, let $s' \in S'$ with image $s = g(s')$.
In this case
$$
\text{Ass}_{X_s}(\mathcal{F}_s) \subset U_s
\Leftrightarrow
\text{Ass}_{X'_{s'}}(\mathcal{F}'_{s'}) \subset U'_{s'}.
$$
\end{lemma}

\begin{proof}
This follows immediately from
Divisors, Lemma \ref{divisors-lemma-base-change-relative-assassin}.
See also
Divisors, Remark \ref{divisors-remark-base-change-relative-assassin}.
\end{proof}

\begin{lemma}
\label{lemma-relative-assassin-constructible}
Let $f : X \to Y$ be a morphism of finite presentation.
 ========

regex_match $f : X \to Y$
nn candidate X \to
nn candidate a morphism
nn a morphism
new_annotation [(242478, 242491, 'VAR'), (242495, 242505, 'TYPE')]
===== sent Let $\mathcal{F}$ be a quasi-coherent $\mathcal{O}_X$-module
of finite presentation. ========

regex_match $\mathcal{F}$
nn candidate finite presentation
nn finite presentation
new_annotation [(242534, 242547, 'VAR'), (242594, 242613, 'TYPE')]
===== sent Let $U \subset X$ be an open subscheme
such that $U \to Y$ is quasi-compact. ========

regex_match $U \subset X$
nn candidate $
nn candidate an open subscheme
nn an open subscheme
new_annotation [(242619, 242632, 'VAR'), (242636, 242653, 'TYPE')]
===== sent Then the set
$$
E = \{y \in Y \mid \text{Ass}_{X_y}(\mathcal{F}_y) \subset U_y\}
$$
is locally constructible in $Y$.
\end{lemma}

\begin{proof}
Let $y \in Y$. We have to show that there exists an open neighbourhood
$V$ of $y$ in $Y$ such that $E \cap V$ is constructible in $V$. ========

new_annotation []
===== sent To see this let $Z \subset Y$ be an irreducible closed subscheme.
 ========

regex_match $Z \subset Y$
nn candidate $Z \subset
nn candidate an irreducible closed subscheme
nn an irreducible closed subscheme
new_annotation [(244133, 244146, 'VAR'), (244150, 244181, 'TYPE')]
===== sent This follows from
Lemmas \ref{lemma-bad-case} and
\ref{lemma-good-case}
applied to the base change $(X, \mathcal{F}, U) \times_Y Z$ over $Z$.
\end{proof}











\section{Reduced fibres}
\label{section-reduced}

\begin{lemma}
\label{lemma-nonreduced-in-neighbourhood}
Let $f : X \to Y$ be a morphism of schemes. ========

regex_match $f : X \to Y$
nn candidate X \to
nn candidate a morphism
nn a morphism
new_annotation [(244553, 244566, 'VAR'), (244570, 244580, 'TYPE')]
===== sent \end{lemma}

\begin{proof}
Let $Y' \subset Y$ be the reduction of $Y$. Let $X' \to Y'$
be the base change of $f$. Note that $Y' \to Y$
induces a bijection on points and that $X' \to X$ identifies fibres.
 ========

regex_match $Y' \subset Y$
nn candidate $Y
nn candidate the reduction
nn the reduction
new_annotation [(244831, 244845, 'VAR'), (244849, 244862, 'TYPE')]
===== sent \end{lemma}

\begin{proof}
Let $Y' \subset Y$ be the reduction of $Y$. Let $X' \to Y'$
be the base change of $f$. Note that $Y' \to Y$
induces a bijection on points and that $X' \to X$ identifies fibres.
 ========

new_annotation []
===== sent \label{lemma-base-change-fibres-geometrically-reduced}
Let $f : X \to Y$ be a morphism of schemes.
 ========

regex_match $f : X \to Y$
nn candidate X \to
nn candidate a morphism
nn a morphism
new_annotation [(246863, 246876, 'VAR'), (246880, 246890, 'TYPE')]
===== sent Let $g : Y' \to Y$ be any morphism, and denote
$f' : X' \to Y'$ the base change of $f$.
Then
\begin{align*}
\{y' \in Y' \mid X'_{y'}\text{ is geometrically reduced}\} \\
 ========

new_annotation []
===== sent This comes down to the statement that for $y' \in Y'$ with image
$y \in Y$ the fibre $X'_{y'} = X_y \times_y y'$ is geometrically
reduced over $\kappa(y')$ if and only if $X_y$ is geometrically
reduced over $\kappa(y)$. This follows from
Varieties, Lemma \ref{varieties-lemma-geometrically-reduced-upstairs}.
\end{proof}

\begin{lemma}
\label{lemma-not-geometrically-reduced-in-neighbourhood}
Let $f : X \to Y$ be a morphism of schemes. ========

regex_match $f : X \to Y$
nn candidate X \to
nn candidate a morphism
nn a morphism
new_annotation [(247574, 247587, 'VAR'), (247591, 247601, 'TYPE')]
===== sent Let $\eta'$ be the generic point of $Y'$.
Consider the morphism $X' \to X_{Y'}$ (which is the reduction
morphism) and the resulting morphism of generic fibres
$X'_{\eta'} \to X_{\eta'}$.
Since $X'_{\eta'}$ is geometrically reduced, and $X_\eta$
is not this cannot be an isomorphism, see
Varieties, Lemma \ref{varieties-lemma-geometrically-reduced-upstairs}.
Hence $X_{\eta'}$ is nonreduced. ========

regex_match $\eta'$
nn candidate the generic point
nn the generic point
new_annotation [(248112, 248119, 'VAR'), (248123, 248140, 'TYPE')]
===== sent \end{proof}

\begin{lemma}
\label{lemma-geometrically-reduced-generic-fibre}
Let $f : X \to Y$ be a morphism of schemes.
 ========

regex_match $f : X \to Y$
nn candidate X \to
nn candidate a morphism
nn a morphism
new_annotation [(248893, 248906, 'VAR'), (248910, 248920, 'TYPE')]
===== sent \end{lemma}

\begin{proof}
Let $Y' \subset Y$ be the reduction of $Y$. Let $X' \to Y'$
be the base change of $f$. Note that $Y' \to Y$
induces a bijection on points and that $X' \to X$ identifies fibres.
 ========

regex_match $Y' \subset Y$
nn candidate $Y
nn candidate the reduction
nn the reduction
new_annotation [(249245, 249259, 'VAR'), (249263, 249276, 'TYPE')]
===== sent \end{lemma}

\begin{proof}
Let $Y' \subset Y$ be the reduction of $Y$. Let $X' \to Y'$
be the base change of $f$. Note that $Y' \to Y$
induces a bijection on points and that $X' \to X$ identifies fibres.
 ========

new_annotation []
===== sent After shrinking $Y$ a bit we may also assume that
$X \to Y$ is flat and of finite presentation, see
Morphisms, Proposition \ref{morphisms-proposition-generic-flatness}.

\medskip\noindent
As $X_\eta$ is geometrically reduced there exists an open dense
subset $V \subset X_\eta$ such that $V \to \Spec(K)$ is smooth, see
Varieties, Lemma \ref{varieties-lemma-geometrically-reduced-dense-smooth-open}.
Let $U \subset X$ be the set of points where $f$ is smooth. ========

regex_match $U \subset X$
nn candidate $U
nn candidate the set
nn the set
new_annotation [(250098, 250111, 'VAR'), (250115, 250122, 'TYPE')]
===== sent Thus it suffices to show that, after
shrinking $Y$, for all $y \in Y$ the scheme $U_y$ is scheme theoretically
dense in $X_y$, see
Morphisms, Lemma \ref{morphisms-lemma-reduced-subscheme-closure}.
This follows from
Lemma \ref{lemma-scheme-theoretically-dense-generic-fibre}.
\end{proof}

\begin{lemma}
\label{lemma-geometrically-reduced-constructible}
Let $f : X \to Y$ be a morphism which is quasi-compact and
locally of finite presentation. ========

regex_match $f : X \to Y$
nn candidate X \to
nn candidate a morphism
nn a morphism
new_annotation [(250948, 250961, 'VAR'), (250965, 250975, 'TYPE')]
===== sent Then the set
$$
E = \{y \in Y \mid X_y\text{ is geometrically reduced}\}
$$
is locally constructible in $Y$.
\end{lemma}

\begin{proof}
Let $y \in Y$. We have to show that there exists an open neighbourhood
$V$ of $y$ in $Y$ such that $E \cap V$ is constructible in $V$. ========

new_annotation []
===== sent To see this let $Z \subset Y$ be an irreducible closed subscheme.
 ========

regex_match $Z \subset Y$
nn candidate $Z \subset
nn candidate an irreducible closed subscheme
nn an irreducible closed subscheme
new_annotation [(252265, 252278, 'VAR'), (252282, 252313, 'TYPE')]
===== sent \end{proof}

\begin{lemma}
\label{lemma-proper-flat-over-dvr-reduced-fibre}
Let $X \to \Spec(R)$ be a proper flat morphism where $R$ is a
discrete valuation ring. ========

regex_match $X \to \Spec(R)$
nn candidate $X
nn candidate a proper flat morphism
nn a proper flat morphism
new_annotation [(252964, 252980, 'VAR'), (252984, 253006, 'TYPE')]
===== sent \end{lemma}

\begin{proof}
Let $x \in X$ be a point in the generic fibre $X_\eta$
such that $\mathcal{O}_{X_\eta}$ is nonreduced.
 ========

regex_match $x \in X$
nn candidate a point
nn a point
new_annotation [(253169, 253178, 'VAR'), (253182, 253189, 'TYPE')]
===== sent Let $x \leadsto x'$ be a specialization with $x'$
in the special fibre; such a specialization exists
as a proper morphism is closed. ========

regex_match $x \leadsto x'$
nn candidate a specialization
nn a specialization
new_annotation [(253313, 253328, 'VAR'), (253332, 253348, 'TYPE')]
===== sent Consider the local
ring $A = \mathcal{O}_{X, x'}$. Let $\pi \in R$ be a uniformizer.
 ========

regex_match $\pi \in R$
nn candidate a uniformizer
nn a uniformizer
new_annotation [(253497, 253508, 'VAR'), (253512, 253525, 'TYPE')]
===== sent \end{proof}

\begin{lemma}
\label{lemma-geometrically-reduced-open}
Let $f : X \to Y$ be a flat proper morphism of finite presentation.
 ========

regex_match $f : X \to Y$
nn candidate X \to
nn candidate a flat proper morphism
nn a flat proper morphism
new_annotation [(254104, 254117, 'VAR'), (254121, 254143, 'TYPE')]
===== sent Algebra, Lemma \ref{algebra-lemma-integral-closure-Dedekind}.
After replacing $R$ by $R'$ we reduce to
Lemma \ref{lemma-proper-flat-over-dvr-reduced-fibre}.
\end{proof}







\section{Irreducible components of fibres}
\label{section-irreducible}

\begin{lemma}
\label{lemma-irreducible-components-in-neighbourhood}
Let $f : X \to Y$ be a morphism of schemes. ========

regex_match $f : X \to Y$
nn candidate X \to
nn candidate a morphism
nn a morphism
new_annotation [(256083, 256096, 'VAR'), (256100, 256110, 'TYPE')]
===== sent In particular this implies that $Y$ is integral, see
Properties, Lemma \ref{properties-lemma-characterize-integral}.
Let $X_\eta = X_{1, \eta} \cup \ldots \cup X_{n, \eta}$
be the decomposition of $X_\eta$ into irreducible components.
 ========

new_annotation []
===== sent Let $X_i \subset X$ be the reduced closed subscheme whose generic
fibre is $X_{i, \eta}$. Note that $Z_{i, j} = X_i \cap X_j$
is a closed subset of $X_i$ whose generic fibre $Z_{i, j, \eta}$
is nowhere dense in $X_{i, \eta}$. ========

regex_match $X_i \subset X$
nn candidate the reduced closed subscheme
nn the reduced closed subscheme
new_annotation [(256725, 256740, 'VAR'), (256744, 256772, 'TYPE')]
===== sent \end{proof}

\begin{lemma}
\label{lemma-base-change-fibres-geometrically-irreducible}
Let $f : X \to Y$ be a morphism of schemes.
Let $g : Y' \to Y$ be any morphism, and denote
$f' : X' \to Y'$ the base change of $f$.
Then
\begin{align*}
\{y' \in Y' \mid X'_{y'}\text{ is geometrically irreducible}\} \\
 ========

regex_match $f : X \to Y$
nn candidate X \to
nn candidate a morphism
nn a morphism
new_annotation [(257985, 257998, 'VAR'), (258002, 258012, 'TYPE')]
===== sent \end{proof}

\begin{lemma}
\label{lemma-base-change-fibres-geometrically-irreducible}
Let $f : X \to Y$ be a morphism of schemes.
Let $g : Y' \to Y$ be any morphism, and denote
$f' : X' \to Y'$ the base change of $f$.
Then
\begin{align*}
\{y' \in Y' \mid X'_{y'}\text{ is geometrically irreducible}\} \\
 ========

new_annotation []
===== sent This comes down to the statement that for $y' \in Y'$ with image
$y \in Y$ the fibre $X'_{y'} = X_y \times_y y'$ is geometrically
irreducible over $\kappa(y')$ if and only if $X_y$ is geometrically
irreducible over $\kappa(y)$. This follows from
Varieties,
Lemma \ref{varieties-lemma-geometrically-irreducible-check-after-extension}.
\end{proof}

\begin{lemma}
\label{lemma-base-change-fibres-nr-geometrically-irreducible-components}
Let $f : X \to Y$ be a morphism of schemes. ========

regex_match $f : X \to Y$
nn candidate X \to
nn candidate a morphism
nn a morphism
new_annotation [(258745, 258758, 'VAR'), (258762, 258772, 'TYPE')]
===== sent \label{lemma-irreducible-polynomial-over-domain}
Let $A$ be a domain with fraction field $K$.
Let $P \in A[x_1, \ldots, x_n]$.
Denote $\overline{K}$ the algebraic closure of $K$.
Assume $P$ is irreducible in $\overline{K}[x_1, \ldots, x_n]$.
Then there exists a $f \in A$ such that
$P^\varphi \in \kappa[x_1, \ldots, x_n]$ is irreducible for all
homomorphisms $\varphi : A_f \to \kappa$ into fields.
 ========

regex_match $A$
nn candidate a domain
nn a domain
new_annotation [(259806, 259809, 'VAR'), (259813, 259821, 'TYPE')]
===== sent \label{lemma-irreducible-polynomial-over-domain}
Let $A$ be a domain with fraction field $K$.
Let $P \in A[x_1, \ldots, x_n]$.
Denote $\overline{K}$ the algebraic closure of $K$.
Assume $P$ is irreducible in $\overline{K}[x_1, \ldots, x_n]$.
Then there exists a $f \in A$ such that
$P^\varphi \in \kappa[x_1, \ldots, x_n]$ is irreducible for all
homomorphisms $\varphi : A_f \to \kappa$ into fields.
 ========

new_annotation []
===== sent Thus we may assume that $P$ is monic in $x_n$ of degree $d > 0$.
For $i = 1, \ldots, n - 1$ let $d_i$ be the degree of $P$ in $x_i$.
Note that this implies that $P^\varphi$ is monic of degree $d$ in $x_n$
and has degree $\leq d_i$ in $x_i$ for every homomorphism
$\varphi : A \to \kappa$ where $\kappa$ is a field.
 ========

regex_match $d_i$
nn candidate the degree
nn the degree
new_annotation [(260546, 260551, 'VAR'), (260555, 260565, 'TYPE')]
===== sent \end{proof}

\begin{lemma}
\label{lemma-geom-irreducible-generic-fibre}
Let $f : X \to Y$ be a morphism of schemes.
 ========

regex_match $f : X \to Y$
nn candidate X \to
nn candidate a morphism
nn a morphism
new_annotation [(262711, 262724, 'VAR'), (262728, 262738, 'TYPE')]
===== sent \medskip\noindent
Let $x \in X_\eta$ be the generic point. ========

regex_match $x \in X_\eta$
nn candidate the generic point
nn the generic point
new_annotation [(264206, 264220, 'VAR'), (264224, 264241, 'TYPE')]
===== sent In particular the field extension $K \subset L$ is separable, see
Algebra, Lemma \ref{algebra-lemma-characterize-separable-field-extensions}.
Hence we can find $x_1, \ldots, x_{r + 1} \in L$ which generate $L$
over $K$ and such that $x_1, \ldots, x_r$ is a transcendence basis for
$L$ over $K$, see
Algebra, Lemma
\ref{algebra-lemma-generating-finitely-generated-separable-field-extensions}.
Let $P \in K(x_1, \ldots, x_r)[T]$ be the minimal polynomial for
$x_{r + 1}$. Clearing denominators we may assume that
$P$ has coefficients in $A[x_1, \ldots, x_r]$.
Note that as $L$ is geometrically reduced and geometrically irreducible
over $K$, the polynomial $P$ is irreducible in
$\overline{K}[x_1, \ldots, x_r, T]$ where $\overline{K}$ is the
algebraic closure of $K$. Denote
$$
B' = A[x_1, \ldots, x_{r + 1}]/(P(x_{r + 1}))
$$
and set $X' = \Spec(B')$. ========

regex_match $P \in K(x_1, \ldots, x_r)[T]$
nn candidate \in K(x_1
nn candidate the minimal polynomial
nn the minimal polynomial
new_annotation [(264991, 265021, 'VAR'), (265025, 265047, 'TYPE')]
===== sent Thus it suffices to prove the lemma for
$X' \to Y$ which is the content of
Lemma \ref{lemma-irreducible-polynomial-over-domain}.
\end{proof}

\begin{proof}[Second proof of Lemma \ref{lemma-geom-irreducible-generic-fibre}]
Let $Y' \subset Y$ be the reduction of $Y$. Let $X' \to X$ be the reduction
of $X$. Note that $X' \to X  \to Y$ factors through $Y'$, see
Schemes, Lemma \ref{schemes-lemma-map-into-reduction}.
As $Y' \to Y$ and $X' \to X$ are universal
homeomorphisms by
Morphisms, Lemma \ref{morphisms-lemma-reduction-universal-homeomorphism}
we see that it suffices to prove the lemma for $X' \to Y'$. ========

regex_match $Y' \subset Y$
nn candidate $Y
nn candidate the reduction
nn the reduction
new_annotation [(266333, 266347, 'VAR'), (266351, 266364, 'TYPE')]
===== sent Thus it suffices to prove the lemma for
$X' \to Y$ which is the content of
Lemma \ref{lemma-irreducible-polynomial-over-domain}.
\end{proof}

\begin{proof}[Second proof of Lemma \ref{lemma-geom-irreducible-generic-fibre}]
Let $Y' \subset Y$ be the reduction of $Y$. Let $X' \to X$ be the reduction
of $X$. Note that $X' \to X  \to Y$ factors through $Y'$, see
Schemes, Lemma \ref{schemes-lemma-map-into-reduction}.
As $Y' \to Y$ and $X' \to X$ are universal
homeomorphisms by
Morphisms, Lemma \ref{morphisms-lemma-reduction-universal-homeomorphism}
we see that it suffices to prove the lemma for $X' \to Y'$. ========

regex_match $X' \to X$
nn candidate $X
nn candidate the reduction
nn the reduction
new_annotation [(266377, 266387, 'VAR'), (266391, 266404, 'TYPE')]
===== sent Let $L = f.f.(B_K)$ be its fraction field. ========

new_annotation []
===== sent Let $K \subset K'$ be a finite purely inseparable
extension such that $(L \otimes_K K')_{red}$ is a separably generated field
extension, see
Algebra, Lemma \ref{algebra-lemma-make-separable}.
Choose $x_1, \ldots, x_n \in K'$ which generate the field extension
$K'$ over $K$, and such that $x_i^{q_i} \in A$ for some prime power
$q_i$ (proof existence $x_i$ omitted). ========

regex_match $K \subset K'$
nn candidate $
nn candidate a finite purely inseparable
extension
nn a finite purely inseparable
extension
new_annotation [(268269, 268283, 'VAR'), (268287, 268324, 'TYPE')]
===== sent Let $A'$ be the $A$-subalgebra
of $K'$ generated by $x_1, \ldots, x_n$. Then $A'$ is a finite
$A$-subalgebra $A' \subset K'$ whose fraction field is $K'$. Note that
$\Spec(A') \to \Spec(A)$ is a universal homeomorphism, see
Algebra, Lemma \ref{algebra-lemma-p-ring-map}.
Hence it suffices to prove the result after base changing to $\Spec(A')$.
We are going to replace $A$ by $A'$ and $B$ by $(B \otimes_A A')_{red}$
to arrive at the situation where $L$ is a separably generated field extension
of $K$. Of course it may happen that $(B \otimes_A A')_{red}$ is no longer
flat, or of finite presentation over $A'$, but this can be remedied by
replacing $A'$ by  $A'_f$ for a suitable $f \in A'$, see
Algebra, Lemma \ref{algebra-lemma-generic-flatness}.

\medskip\noindent
 ========

regex_match $A'$
nn candidate x_n$.
nn x_n$.
new_annotation [(268636, 268640, 'VAR'), (268698, 268703, 'TYPE')]
===== sent Thus it suffices to prove the lemma for
$X' \to Y$ which is the content of
Lemma \ref{lemma-irreducible-polynomial-over-domain}.
\end{proof}

\begin{lemma}
\label{lemma-nr-geom-irreducible-components-good}
Let $f : X \to Y$ be a morphism of schemes. ========

regex_match $f : X \to Y$
nn candidate X \to
nn candidate a morphism
nn a morphism
new_annotation [(271186, 271199, 'VAR'), (271203, 271213, 'TYPE')]
===== sent Let $y \in Y$ be a point. ========

regex_match $y \in Y$
nn candidate $y
nn candidate a point
nn a point
new_annotation [(271464, 271473, 'VAR'), (271477, 271484, 'TYPE')]
===== sent \end{lemma}

\begin{proof}
Let $Z$ be the reduced induced scheme structure on $\overline{\{y\}}$.
Let $f_Z : X_Z \to Z$ be the base change of $f$. Clearly it suffices to prove
the lemma for $f_Z$ and the generic point of $Z$. ========

regex_match $Z$
nn candidate the reduced induced scheme structure
nn the reduced induced scheme structure
new_annotation [(271616, 271619, 'VAR'), (271623, 271659, 'TYPE')]
===== sent \end{lemma}

\begin{proof}
Let $Z$ be the reduced induced scheme structure on $\overline{\{y\}}$.
Let $f_Z : X_Z \to Z$ be the base change of $f$. Clearly it suffices to prove
the lemma for $f_Z$ and the generic point of $Z$. ========

regex_match $f_Z : X_Z \to Z$
nn candidate X_Z \to
nn candidate the base change
nn the base change
new_annotation [(271687, 271704, 'VAR'), (271708, 271723, 'TYPE')]
===== sent In particular $n_{X/Y}(\eta) = n$.
Let $X_i$ be the closure of
$X_{i, \eta}$ in ========

regex_match $X_i$
nn candidate the closure
nn the closure
new_annotation [(272873, 272878, 'VAR'), (272882, 272893, 'TYPE')]
===== sent \end{proof}

\begin{lemma}
\label{lemma-nr-geom-irreducible-components-constructible}
Let $f : X \to Y$ be a morphism of schemes. ========

regex_match $f : X \to Y$
nn candidate X \to
nn candidate a morphism
nn a morphism
new_annotation [(273595, 273608, 'VAR'), (273612, 273622, 'TYPE')]
===== sent Then the level sets
$$
E_n = \{y \in Y \mid n_{X/Y}(y) = n\}
$$
of $n_{X/Y}$ are locally constructible in $Y$.
\end{lemma}

\begin{proof}
Fix $n$. Let $y \in Y$. We have to show that there exists an open neighbourhood
$V$ of $y$ in $Y$ such that $E_n \cap V$ is constructible in $V$. Thus we may
assume that $Y$ is affine. ========

new_annotation []
===== sent To see this let $Z \subset Y$ be an irreducible closed subscheme.
 ========

regex_match $Z \subset Y$
nn candidate $Z \subset
nn candidate an irreducible closed subscheme
nn an irreducible closed subscheme
new_annotation [(274896, 274909, 'VAR'), (274913, 274944, 'TYPE')]
===== sent We have to show that $E_n \cap Z$ either contains a nonempty open subset
or is not dense in $Z$. Let $\xi \in Z$ be the generic point. ========

regex_match $\xi \in Z$
nn candidate the generic point
nn the generic point
new_annotation [(275047, 275058, 'VAR'), (275062, 275079, 'TYPE')]
===== sent \end{proof}













\section{Connected components of fibres}
\label{section-connected}

\begin{lemma}
\label{lemma-connected-components-in-neighbourhood}
Let $f : X \to Y$ be a morphism of schemes. ========

regex_match $f : X \to Y$
nn candidate X \to
nn candidate a morphism
nn a morphism
new_annotation [(275407, 275420, 'VAR'), (275424, 275434, 'TYPE')]
===== sent In particular this implies that $Y$ is integral, see
Properties, Lemma \ref{properties-lemma-characterize-integral}.
Let $X_\eta = X_{1, \eta} \cup \ldots \cup X_{n, \eta}$
be the decomposition of $X_\eta$ into connected components.
 ========

new_annotation []
===== sent Let $X_i \subset X$ be the reduced closed subscheme whose generic
fibre is $X_{i, \eta}$. Note that $Z_{i, j} = X_i \cap X_j$
is a closed subset of $X$ whose generic fibre $Z_{i, j, \eta}$ is empty.
 ========

regex_match $X_i \subset X$
nn candidate the reduced closed subscheme
nn the reduced closed subscheme
new_annotation [(276043, 276058, 'VAR'), (276062, 276090, 'TYPE')]
===== sent \end{proof}

\begin{lemma}
\label{lemma-base-change-fibres-geometrically-connected}
Let $f : X \to Y$ be a morphism of schemes.
Let $g : Y' \to Y$ be any morphism, and denote
$f' : X' \to Y'$ the base change of $f$.
Then
\begin{align*}
\{y' \in Y' \mid X'_{y'}\text{ is geometrically connected}\} \\
 ========

regex_match $f : X \to Y$
nn candidate X \to
nn candidate a morphism
nn a morphism
new_annotation [(277204, 277217, 'VAR'), (277221, 277231, 'TYPE')]
===== sent \end{proof}

\begin{lemma}
\label{lemma-base-change-fibres-geometrically-connected}
Let $f : X \to Y$ be a morphism of schemes.
Let $g : Y' \to Y$ be any morphism, and denote
$f' : X' \to Y'$ the base change of $f$.
Then
\begin{align*}
\{y' \in Y' \mid X'_{y'}\text{ is geometrically connected}\} \\
 ========

new_annotation []
===== sent $\kappa(y)$. This follows from
Varieties,
Lemma \ref{varieties-lemma-geometrically-connected-check-after-extension}.
\end{proof}

\begin{lemma}
\label{lemma-base-change-fibres-nr-geometrically-connected-components}
Let $f : X \to Y$ be a morphism of schemes. ========

regex_match $f : X \to Y$
nn candidate X \to
nn candidate a morphism
nn a morphism
new_annotation [(277952, 277965, 'VAR'), (277969, 277979, 'TYPE')]
===== sent The result follows as the morphisms of schemes
$$
\xymatrix{
(X'_{y'})_{K'} &
(X'_{y'})_{K''} = (X_y)_{K''} \ar[l] \ar[r] &
(X_y)_K
}
$$
induce bijections between connected components, see
Varieties,
Lemma \ref{varieties-lemma-separably-closed-field-connected-components}.
\end{proof}

\begin{lemma}
\label{lemma-geometrically-connected-generic-fibre}
Let $f : X \to Y$ be a morphism of schemes.
 ========

regex_match $f : X \to Y$
nn candidate X \to
nn candidate a morphism
nn a morphism
new_annotation [(279010, 279023, 'VAR'), (279027, 279037, 'TYPE')]
===== sent Then, as $g$ is an open morphism by
Morphisms, Lemma \ref{morphisms-lemma-etale-open}
all the fibres of $f$ at points of the nonempty open $V = g(W)$ are
geometrically connected, see
Lemma \ref{lemma-base-change-fibres-nr-geometrically-connected-components}.
In this way we see that we may assume that the irreducible
components of the generic fibre $X_\eta$ are geometrically irreducible.

\medskip\noindent
Let $Y'$ be the reduction of $Y$, and set $X' = Y' \times_Y X$.
 ========

regex_match $Y'$
nn candidate the reduction
nn the reduction
new_annotation [(280324, 280328, 'VAR'), (280332, 280345, 'TYPE')]
===== sent Let $X_i$ be the closure of $X_{i, \eta}$ in $X$.
 ========

regex_match $X_i$
nn candidate the closure
nn the closure
new_annotation [(281077, 281082, 'VAR'), (281086, 281097, 'TYPE')]
===== sent After shrinking $Y$ we may assume that
$X = \bigcup X_i$, see
Lemma \ref{lemma-cover-generic-fibre-neighbourhood}.
Let $Z_{i, j} = X_i \cap X_j$.
Let
$$
\{1, \ldots, n\} \times \{1, \ldots, n\} = I \amalg J
$$
where $(i, j) \in I$ if $Z_{i, j, \eta} = \emptyset$ and
$(i, j) \in J$ if $Z_{i, j, \eta} \not = \emptyset$.
 ========

new_annotation []
===== sent \end{proof}

\begin{lemma}
\label{lemma-nr-geom-connected-components-good}
Let $f : X \to Y$ be a morphism of schemes. ========

regex_match $f : X \to Y$
nn candidate X \to
nn candidate a morphism
nn a morphism
new_annotation [(282768, 282781, 'VAR'), (282785, 282795, 'TYPE')]
===== sent Let $y \in Y$ be a point. ========

regex_match $y \in Y$
nn candidate $y
nn candidate a point
nn a point
new_annotation [(283042, 283051, 'VAR'), (283055, 283062, 'TYPE')]
===== sent \end{lemma}

\begin{proof}
Let $Z$ be the reduced induced scheme structure on $\overline{\{y\}}$.
Let $f_Z : X_Z \to Z$ be the base change of $f$. Clearly it suffices to prove
the lemma for $f_Z$ and the generic point of $Z$. ========

regex_match $Z$
nn candidate the reduced induced scheme structure
nn the reduced induced scheme structure
new_annotation [(283194, 283197, 'VAR'), (283201, 283237, 'TYPE')]
===== sent \end{lemma}

\begin{proof}
Let $Z$ be the reduced induced scheme structure on $\overline{\{y\}}$.
Let $f_Z : X_Z \to Z$ be the base change of $f$. Clearly it suffices to prove
the lemma for $f_Z$ and the generic point of $Z$. ========

regex_match $f_Z : X_Z \to Z$
nn candidate X_Z \to
nn candidate the base change
nn the base change
new_annotation [(283265, 283282, 'VAR'), (283286, 283301, 'TYPE')]
===== sent In particular $n_{X/Y}(\eta) = n$.
Let $X_i$ be the closure of
$X_{i, \eta}$ in ========

regex_match $X_i$
nn candidate the closure
nn the closure
new_annotation [(284629, 284634, 'VAR'), (284638, 284649, 'TYPE')]
===== sent \end{proof}

\begin{lemma}
\label{lemma-nr-geom-connected-components-constructible}
Let $f : X \to Y$ be a morphism of schemes. ========

regex_match $f : X \to Y$
nn candidate X \to
nn candidate a morphism
nn a morphism
new_annotation [(285350, 285363, 'VAR'), (285367, 285377, 'TYPE')]
===== sent Then the level sets
$$
E_n = \{y \in Y \mid n_{X/Y}(y) = n\}
$$
of $n_{X/Y}$ are locally constructible in $Y$.
\end{lemma}

\begin{proof}
Fix $n$. Let $y \in Y$. We have to show that there exists an open neighbourhood
$V$ of $y$ in $Y$ such that $E_n \cap V$ is constructible in $V$. Thus we may
assume that $Y$ is affine. ========

new_annotation []
===== sent To see this let $Z \subset Y$ be an irreducible closed subscheme.
 ========

regex_match $Z \subset Y$
nn candidate $Z \subset
nn candidate an irreducible closed subscheme
nn an irreducible closed subscheme
new_annotation [(286641, 286654, 'VAR'), (286658, 286689, 'TYPE')]
===== sent We have to show that $E_n \cap Z$ either contains a nonempty open subset
or is not dense in $Z$. Let $\xi \in Z$ be the generic point. ========

regex_match $\xi \in Z$
nn candidate the generic point
nn the generic point
new_annotation [(286792, 286803, 'VAR'), (286807, 286824, 'TYPE')]
===== sent \end{slogan}
Let $f : X \to S$ be a morphism of schemes.
 ========

regex_match $f : X \to S$
nn candidate X \to
nn candidate a morphism
nn a morphism
new_annotation [(287167, 287180, 'VAR'), (287184, 287194, 'TYPE')]
===== sent Write $Y = \Spec(R)$ and let $\pi \in R$ be a uniformizer.
 ========

regex_match $\pi \in R$
nn candidate a uniformizer
nn a uniformizer
new_annotation [(287495, 287506, 'VAR'), (287510, 287523, 'TYPE')]
===== sent This means there exists a nontrivial idempotent
$e \in \Gamma(X_\eta, \mathcal{O}_{X_\eta})$.
Let $U = \Spec(A)$ be any affine open in $X$.
Note that $\pi$ is a nonzerodivisor on $A$ as $A$ is flat over $R$, see
 ========

new_annotation []
===== sent Then $e|_{U_\eta}$ corresponds to an element $e \in A[1/\pi]$.
Let $z \in A$ be an element such that $e = z/\pi^n$ with $n \geq 0$ minimal.
 ========

regex_match $z \in A$
nn candidate $
nn candidate an element
nn an element
new_annotation [(287944, 287953, 'VAR'), (287957, 287967, 'TYPE')]
===== sent Finally, we set $X^0 = \bigcup_{y \in Y} X^0_y$.
\end{situation}

\begin{lemma}
\label{lemma-base-change-connected-along-section}
Let $f : X \to Y$, $s : Y \to X$ be as in
Situation \ref{situation-connected-along-section}.
If $g : Y' \to Y$ is any morphism, consider the base change diagram
$$
\xymatrix{
X' \ar[r]_{g'} \ar[d]^{f'} & X \ar[d]_f \\
Y' \ar@/^1pc/[u]^{s'} \ar[r]^g & Y \ar@/_1pc/[u]_s
}
$$
so that we obtain $(X')^0 \subset X'$.
Then $(X')^0 = (g')^{-1}(X^0)$.
\end{lemma}

\begin{proof}
Let $y' \in Y'$ with image $y \in Y$. We may think of
$X^0_y$ as a closed subscheme of $X_y$, see for example
Morphisms,
Definition \ref{morphisms-definition-scheme-structure-connected-component}.
As $s(y) \in X^0_y$ we conclude from
Varieties, Lemma
\ref{varieties-lemma-geometrically-connected-if-connected-and-point}
that $X_y^0$ is a geometrically connected scheme over $\kappa(y)$.
Hence $X_y^0 \times_y y' \to X'_{y'}$ is a connected closed subscheme
which contains $s'(y')$. Thus $X_y^0 \times_y y' \subset (X'_{y'})^0$.
The other inclusion $X_y^0 \times_y y' \supset (X'_{y'})^0$ is clear
as the image of $(X'_{y'})^0$ in $X_y$ is a connected subset of $X_y$ which
contains $s(y)$.
\end{proof}

\begin{lemma}
\label{lemma-connected-along-section-good}
 ========

new_annotation []
===== sent Finally, we set $X^0 = \bigcup_{y \in Y} X^0_y$.
\end{situation}

\begin{lemma}
\label{lemma-base-change-connected-along-section}
Let $f : X \to Y$, $s : Y \to X$ be as in
Situation \ref{situation-connected-along-section}.
If $g : Y' \to Y$ is any morphism, consider the base change diagram
$$
\xymatrix{
X' \ar[r]_{g'} \ar[d]^{f'} & X \ar[d]_f \\
Y' \ar@/^1pc/[u]^{s'} \ar[r]^g & Y \ar@/_1pc/[u]_s
}
$$
so that we obtain $(X')^0 \subset X'$.
Then $(X')^0 = (g')^{-1}(X^0)$.
\end{lemma}

\begin{proof}
Let $y' \in Y'$ with image $y \in Y$. We may think of
$X^0_y$ as a closed subscheme of $X_y$, see for example
Morphisms,
Definition \ref{morphisms-definition-scheme-structure-connected-component}.
As $s(y) \in X^0_y$ we conclude from
Varieties, Lemma
\ref{varieties-lemma-geometrically-connected-if-connected-and-point}
that $X_y^0$ is a geometrically connected scheme over $\kappa(y)$.
Hence $X_y^0 \times_y y' \to X'_{y'}$ is a connected closed subscheme
which contains $s'(y')$. Thus $X_y^0 \times_y y' \subset (X'_{y'})^0$.
The other inclusion $X_y^0 \times_y y' \supset (X'_{y'})^0$ is clear
as the image of $(X'_{y'})^0$ in $X_y$ is a connected subset of $X_y$ which
contains $s(y)$.
\end{proof}

\begin{lemma}
\label{lemma-connected-along-section-good}
 ========

new_annotation []
===== sent Let $f : X \to Y$, $s : Y \to X$ be as in
Situation \ref{situation-connected-along-section}.
Assume $f$ of finite type. ========

new_annotation []
===== sent Let $y \in Y$ be a point.
 ========

regex_match $y \in Y$
nn candidate $
nn candidate a point
nn a point
new_annotation [(290214, 290223, 'VAR'), (290227, 290234, 'TYPE')]
===== sent Then there exists a nonempty open $V \subset \overline{\{y\}}$ such that
the inverse image of $X^0$ in the base change $X_V$ is open and closed in
$X_V$.
\end{lemma}

\begin{proof}
Let $Z \subset Y$ be the induced reduced closed subscheme
structure on $\overline{\{y\}}$. Let $f_Z : X_Z \to Z$ and $s_Z : Z \to X_Z$
be the base changes of $f$ and $s$. By
Lemma \ref{lemma-base-change-connected-along-section}
we have $(X_Z)^0 = (X^0)_Z$. Hence it suffices to prove the lemma for
the morphism $X_Z \to Z$ and the point $x \in X_Z$ which maps to the generic
point of $Z$. ========

regex_match $Z \subset Y$
nn candidate $Z \subset
nn candidate closed subscheme
structure
nn closed subscheme
structure
new_annotation [(290421, 290434, 'VAR'), (290458, 290484, 'TYPE')]
===== sent Then there exists a nonempty open $V \subset \overline{\{y\}}$ such that
the inverse image of $X^0$ in the base change $X_V$ is open and closed in
$X_V$.
\end{lemma}

\begin{proof}
Let $Z \subset Y$ be the induced reduced closed subscheme
structure on $\overline{\{y\}}$. Let $f_Z : X_Z \to Z$ and $s_Z : Z \to X_Z$
be the base changes of $f$ and $s$. By
Lemma \ref{lemma-base-change-connected-along-section}
we have $(X_Z)^0 = (X^0)_Z$. Hence it suffices to prove the lemma for
the morphism $X_Z \to Z$ and the point $x \in X_Z$ which maps to the generic
point of $Z$. ========

new_annotation []
===== sent Next, let $T \subset X$ be the closure of $T_\eta$
and let $X^{00} \subset X$ be the closure of $X_\eta^0$. Note that
$T_\eta$, resp.\ $X^0_\eta$ is the generic fibre of $T$, resp.\ $X^{00}$,
see discussion preceding
Lemma \ref{lemma-cover-generic-fibre-neighbourhood}.
 ========

regex_match $T \subset X$
nn candidate $T \subset
nn candidate the closure
nn the closure
new_annotation [(291367, 291380, 'VAR'), (291384, 291395, 'TYPE')]
===== sent Next, let $T \subset X$ be the closure of $T_\eta$
and let $X^{00} \subset X$ be the closure of $X_\eta^0$. Note that
$T_\eta$, resp.\ $X^0_\eta$ is the generic fibre of $T$, resp.\ $X^{00}$,
see discussion preceding
Lemma \ref{lemma-cover-generic-fibre-neighbourhood}.
 ========

regex_match $X^{00} \subset X$
nn candidate \subset
nn candidate the closure
nn the closure
new_annotation [(291416, 291434, 'VAR'), (291438, 291449, 'TYPE')]
===== sent \end{proof}

\begin{lemma}
\label{lemma-connected-along-section-locally-constructible}
Let $f : X \to Y$, $s : Y \to X$ be as in
Situation \ref{situation-connected-along-section}.
If $f$ is of finite presentation then $X^0$ is locally constructible
in $X$.
\end{lemma}

\begin{proof}
Let $x \in X$. We have to show that there exists an open neighbourhood
$U$ of $x$ such that $X^0 \cap U$ is constructible in $U$.
This reduces us to the case where $Y$ is affine.
 ========

new_annotation []
===== sent \end{proof}

\begin{lemma}
\label{lemma-connected-along-section-locally-constructible}
Let $f : X \to Y$, $s : Y \to X$ be as in
Situation \ref{situation-connected-along-section}.
If $f$ is of finite presentation then $X^0$ is locally constructible
in $X$.
\end{lemma}

\begin{proof}
Let $x \in X$. We have to show that there exists an open neighbourhood
$U$ of $x$ such that $X^0 \cap U$ is constructible in $U$.
This reduces us to the case where $Y$ is affine.
 ========

new_annotation []
===== sent Since $f$ is of finite presentation,
i.e., of finite type, we see that $X$ is a Noetherian scheme too, see
Morphisms, Lemma \ref{morphisms-lemma-finite-type-noetherian}.
In order to prove the lemma in
this case it suffices to show that for every irreducible closed subset
$Z \subset X$ the intersection $Z \cap X^0$ either contains a nonempty
open of $Z$ or is not dense in $Z$, see
Topology, Lemma \ref{topology-lemma-characterize-constructible-Noetherian}.
Let $x \in Z$ be the generic point, and let $y = f(x)$. By
Lemma \ref{lemma-connected-along-section-good}
there exists a nonempty open subset $V \subset \overline{\{y\}}$ such
that $X^0 \cap X_V$ is open and closed in $X_V$. Since
$f(Z) \subset \overline{\{y\}}$ and $f(x) = y \in V$ we see that
$W = f^{-1}(V) \cap Z$ is a nonempty open subset of ========

regex_match $x \in Z$
nn candidate the generic point
nn the generic point
new_annotation [(294027, 294036, 'VAR'), (294040, 294057, 'TYPE')]
===== sent Since $f$ is of finite presentation,
i.e., of finite type, we see that $X$ is a Noetherian scheme too, see
Morphisms, Lemma \ref{morphisms-lemma-finite-type-noetherian}.
In order to prove the lemma in
this case it suffices to show that for every irreducible closed subset
$Z \subset X$ the intersection $Z \cap X^0$ either contains a nonempty
open of $Z$ or is not dense in $Z$, see
Topology, Lemma \ref{topology-lemma-characterize-constructible-Noetherian}.
Let $x \in Z$ be the generic point, and let $y = f(x)$. By
Lemma \ref{lemma-connected-along-section-good}
there exists a nonempty open subset $V \subset \overline{\{y\}}$ such
that $X^0 \cap X_V$ is open and closed in $X_V$. Since
$f(Z) \subset \overline{\{y\}}$ and $f(x) = y \in V$ we see that
$W = f^{-1}(V) \cap Z$ is a nonempty open subset of ========

new_annotation []
===== sent \end{proof}

\begin{lemma}
\label{lemma-connected-along-section-open-neighbourhood}
Let $f : X \to Y$, $s : Y \to X$ be as in
Situation \ref{situation-connected-along-section}.
Let $y \in Y$ be a point.
 ========

new_annotation []
===== sent \end{proof}

\begin{lemma}
\label{lemma-connected-along-section-open-neighbourhood}
Let $f : X \to Y$, $s : Y \to X$ be as in
Situation \ref{situation-connected-along-section}.
Let $y \in Y$ be a point.
 ========

regex_match $y \in Y$
nn candidate $y
nn candidate a point
nn a point
new_annotation [(294719, 294728, 'VAR'), (294732, 294739, 'TYPE')]
===== sent After possibly increasing $i$ we may also assume that $f_i$ is flat, see
Limits, Lemma \ref{limits-lemma-descend-flat-finite-presentation}.
Let $y_i$ be the image of $y$ in $Y_i$. Note that
$X_y = (X_{i, y_i}) \times_{y_i} y$. Hence $X_{i, y_i}$ is geometrically
reduced, see
Varieties, Lemma \ref{varieties-lemma-geometrically-reduced-upstairs}.
By
Lemma \ref{lemma-base-change-connected-along-section}
it suffices to prove the lemma for the system $f_i, s_i, y_i \in Y_i$.
 ========

regex_match $y_i$
nn candidate the image
nn the image
new_annotation [(295519, 295524, 'VAR'), (295528, 295537, 'TYPE')]
===== sent Since $f$ is of finite presentation,
i.e., of finite type, we see that $X$ is a Noetherian scheme too, see
Morphisms, Lemma \ref{morphisms-lemma-finite-type-noetherian}.
Let $x \in X^0$ be a point lying over $y$. By
Topology, Lemma \ref{topology-lemma-constructible-neighbourhood-Noetherian}
it suffices to prove that for any irreducible closed $Z \subset X$
passing through $x$ the intersection $X^0 \cap Z$ is dense in $Z$.
 ========

regex_match $x \in X^0$
nn candidate a point
nn a point
new_annotation [(296167, 296178, 'VAR'), (296182, 296189, 'TYPE')]
===== sent \end{proof}

\begin{lemma}
\label{lemma-connected-along-section-open}
Let $f : X \to Y$, $s : Y \to X$ be as in
Situation \ref{situation-connected-along-section}.
Assume
\begin{enumerate}
\item $f$ is of finite presentation and flat, and
\item all fibres of $f$ are geometrically reduced.
 ========

new_annotation []
===== sent This is an immediate consequence of
Lemma \ref{lemma-connected-along-section-open-neighbourhood}.
\end{proof}






\section{Dimension of fibres}
\label{section-dimension}

\begin{lemma}
\label{lemma-dimension-in-neighbourhood}
Let $f : X \to Y$ be a morphism of schemes. ========

regex_match $f : X \to Y$
nn candidate X \to
nn candidate a morphism
nn a morphism
new_annotation [(299005, 299018, 'VAR'), (299022, 299032, 'TYPE')]
===== sent If $X_\eta$ has dimension $n$,
then there exists a nonempty open $V \subset Y$
such that for all $y \in V$ the fibre $X_y$ has dimension $n$.
\end{lemma}

\begin{proof}
Let $Z = \{x \in X \mid \dim_x(X_{f(x)}) > n \}$. By
Morphisms, Lemma \ref{morphisms-lemma-openness-bounded-dimension-fibres}
this is a closed subset of $X$. By assumption $Z_\eta = \emptyset$.
 ========

new_annotation []
===== sent \label{lemma-base-change-dimension-fibres}
Let $f : X \to Y$ be a morphism of finite type. ========

regex_match $f : X \to Y$
nn candidate X \to
nn candidate a morphism
nn a morphism
new_annotation [(299971, 299984, 'VAR'), (299988, 299998, 'TYPE')]
===== sent Let
$$
n_{X/Y} : Y \to \{0, 1, 2, 3, \ldots, \infty\}
$$
be the function which associates to $y \in Y$ the dimension of $X_y$.
If $g : Y' \to Y$ is a morphism then
$$
n_{X'/Y'} = n_{X/Y} \circ g
$$
where $X' \to Y'$ is the base change of $f$.
\end{lemma}

\begin{proof}
This follows from
Morphisms, Lemma \ref{morphisms-lemma-dimension-fibre-after-base-change}.
\end{proof}

\begin{lemma}
\label{lemma-dimension-fibres-constructible}
Let $f : X \to Y$ be a morphism of schemes. ========

regex_match $f : X \to Y$
nn candidate X \to
nn candidate a morphism
nn a morphism
new_annotation [(300453, 300466, 'VAR'), (300470, 300480, 'TYPE')]
===== sent Then the level sets
$$
E_n = \{y \in Y \mid n_{X/Y}(y) = n\}
$$
of $n_{X/Y}$ are locally constructible in $Y$.
\end{lemma}

\begin{proof}
Fix $n$. Let $y \in Y$. We have to show that there exists an open neighbourhood
$V$ of $y$ in $Y$ such that $E_n \cap V$ is constructible in $V$. Thus we may
assume that $Y$ is affine. ========

new_annotation []
===== sent To see this let $Z \subset Y$ be an irreducible closed subscheme.
 ========

regex_match $Z \subset Y$
nn candidate $Z \subset
nn candidate an irreducible closed subscheme
nn an irreducible closed subscheme
new_annotation [(301654, 301667, 'VAR'), (301671, 301702, 'TYPE')]
===== sent We have to show that $E_n \cap Z$ either contains a nonempty open subset
or is not dense in $Z$. Let $\xi \in Z$ be the generic point. ========

regex_match $\xi \in Z$
nn candidate the generic point
nn the generic point
new_annotation [(301805, 301816, 'VAR'), (301820, 301837, 'TYPE')]
===== sent \end{proof}

\begin{lemma}
\label{lemma-dimension-fibres-flat}
Let $f : X \to Y$ be a flat morphism of schemes of finite presentation. ========

regex_match $f : X \to Y$
nn candidate X \to
nn candidate a flat morphism
nn a flat morphism
new_annotation [(302052, 302065, 'VAR'), (302069, 302084, 'TYPE')]
===== sent \end{lemma}

\begin{proof}
Let $W \subset X$, $W = \coprod_{d \geq 0} U_d$ be the open constructed in
Lemmas \ref{lemma-flat-finite-presentation-CM-open} and
\ref{lemma-flat-finite-presentation-CM-pieces}.
Let $y \in Y$ be a point. ========

new_annotation []
===== sent \end{lemma}

\begin{proof}
Let $W \subset X$, $W = \coprod_{d \geq 0} U_d$ be the open constructed in
Lemmas \ref{lemma-flat-finite-presentation-CM-open} and
\ref{lemma-flat-finite-presentation-CM-pieces}.
Let $y \in Y$ be a point. ========

regex_match $y \in Y$
nn candidate $y
nn candidate a point
nn a point
new_annotation [(302508, 302517, 'VAR'), (302521, 302528, 'TYPE')]
===== sent If $n_{X/Y}(y) = \dim(X_y) = n$, then
$y$ is in the image of $U_n \to Y$.
By Morphisms, Lemma \ref{morphisms-lemma-fppf-open}
we see that $f(U_n)$ is open in $Y$.
Hence there is an open neighbourhoof of $y$ where
$n_{X/Y}$ is $\geq n$.
\end{proof}

\begin{lemma}
\label{lemma-dimension-fibres-proper}
Let $f : X \to Y$ be a proper morphism of schemes. ========

regex_match $f : X \to Y$
nn candidate X \to
nn candidate a proper morphism
nn a proper morphism
new_annotation [(302835, 302848, 'VAR'), (302852, 302869, 'TYPE')]
===== sent \end{lemma}

\begin{proof}
Let $Z_d = \{x \in X \mid \dim_x(X_{f(x)}) > d\}$.
 ========

new_annotation []
===== sent \end{proof}

\begin{lemma}
\label{lemma-dimension-fibres-proper-flat}
Let $f : X \to Y$ be a proper, flat morphism of schemes of finite presentation.
 ========

regex_match $f : X \to Y$
nn candidate X \to
nn candidate a proper, flat morphism
nn a proper, flat morphism
new_annotation [(303449, 303462, 'VAR'), (303466, 303489, 'TYPE')]
===== sent Let $n_{X/Y}$ be the function on $Y$ giving the dimension of fibres of $f$
introduced in Lemma \ref{lemma-base-change-dimension-fibres}.
Then $n_{X/Y}$ is locally constant.
 ========

regex_match $n_{X/Y}$
nn candidate the function
nn the function
new_annotation [(303529, 303538, 'VAR'), (303542, 303554, 'TYPE')]
===== sent \begin{lemma}
\label{lemma-diagonal-picard-flat-proper}
Let $f : X \to S$ be a flat, proper morphism of finite presentation.
 ========

regex_match $f : X \to S$
nn candidate X \to
nn candidate a flat, proper morphism
nn a flat, proper morphism
new_annotation [(304135, 304148, 'VAR'), (304152, 304175, 'TYPE')]
===== sent Let $\mathcal{L}$ be an invertible $\mathcal{O}_X$-module.
 ========

regex_match $\mathcal{L}$
nn candidate an invertible $\mathcal{O}_X$-module
nn an invertible $\mathcal{O}_X$-module
new_annotation [(304204, 304217, 'VAR'), (304221, 304257, 'TYPE')]
===== sent Similarly for $E' = Rf_*\mathcal{L}^{\otimes -1}$.
Since there is never any cohomology in degrees $< 0$, we see that
$E$ and $E'$ have (locally) tor-amplitude in $[0, b]$ for some $b$.
Observe that for any $g : T \to S$ we have
$p_*(q^*\mathcal{L}) = H^0(Lg^*E)$ and
$p_*(q^*\mathcal{L}^{\otimes -1}) = H^0(Lg^*E')$.
Let $j : Z \to S$ and $j' : Z' \to S$ be the locally closed
immersions constructed in Derived Categories of Schemes, Lemma
\ref{perfect-lemma-locally-closed-where-H0-invertible}
for $E$ and $E'$ with $a = 0$; these are characterized
by the property that $H^0(Lj^*E)$ and $H^0((j')^*E')$
are invertible modules compatible with pullback.

 ========

new_annotation []
===== sent \medskip\noindent
Let $g : T \to S$ be a morphism. ========

regex_match $g : T \to S$
nn candidate T \to
nn candidate a morphism
nn a morphism
new_annotation [(305714, 305727, 'VAR'), (305731, 305741, 'TYPE')]
===== sent \end{proof}

\begin{lemma}
\label{lemma-get-a-closed}
Let $f : X \to S$ and $\mathcal{L}$ be as in
Lemma \ref{lemma-diagonal-picard-flat-proper}.
If moreover the geometric fibres of $f$ are
integral, then $Z$ is closed in $S$.
\end{lemma}

\begin{proof}
We first do a standard argument to reduce to the Noetherian case.
 ========

new_annotation []
===== sent Let $T \subset S_i$ be the locally closed subscheme of $S_i$
constructed in Derived Categories of Schemes, Lemma
\ref{perfect-lemma-locally-closed-where-H0-invertible}
for $Rf_{i, *}\mathcal{O}_{X_i}$ with $a = 0$.
By our assumption that $f_*\mathcal{O}_X = \mathcal{O}_S$ universally
we see that $S \to S_i$ factors through $T$.
Set $Y = X_i \times_{S_i} T \to T$ and $\mathcal{M} = \mathcal{L}_i|_Y$.
 ========

regex_match $T \subset S_i$
nn candidate $T \subset
nn candidate the locally closed subscheme
nn the locally closed subscheme
new_annotation [(309173, 309188, 'VAR'), (309192, 309220, 'TYPE')]
===== sent This means that the $Z(s)$ has codimension $\geq 2$ in $X$ which contradicts
Divisors, Lemma \ref{divisors-lemma-effective-Cartier-codimension-1}.
\end{proof}

\begin{lemma}
\label{lemma-H1-O-picard-flat-proper}
Consider a commutative diagram of schemes
$$
\xymatrix{
X' \ar[rr] \ar[dr]_{f'} & & X \ar[dl]^f \\
& S
}
$$
with $f' : X' \to S$ and $f : X \to S$ satisfying the hypotheses of
Lemma \ref{lemma-diagonal-picard-flat-proper}.
Let $\mathcal{L}$ be an invertible $\mathcal{O}_X$-module
and let $\mathcal{L}'$ be the pullback to $X'$. Let $Z \subset S$,
resp.\ $Z' \subset S$ be the locally closed subscheme constructed
in Lemma \ref{lemma-diagonal-picard-flat-proper}
for $(f, \mathcal{L})$, resp.\ $(f', \mathcal{L}')$
so that $Z \subset Z'$. If $s \in Z$ and
$$
H^1(X_s, \mathcal{O}) \longrightarrow H^1(X'_s, \mathcal{O})
$$
is injective, then $Z \cap U = Z' \cap U$ for some open neighbourhood
$U$ of $s$.
\end{lemma}

\begin{proof}
We may replace $S$ by ========

regex_match $\mathcal{L}$
nn candidate the pullback
nn the pullback
new_annotation [(311721, 311734, 'VAR'), (311801, 311813, 'TYPE')]
===== sent This means that the $Z(s)$ has codimension $\geq 2$ in $X$ which contradicts
Divisors, Lemma \ref{divisors-lemma-effective-Cartier-codimension-1}.
\end{proof}

\begin{lemma}
\label{lemma-H1-O-picard-flat-proper}
Consider a commutative diagram of schemes
$$
\xymatrix{
X' \ar[rr] \ar[dr]_{f'} & & X \ar[dl]^f \\
& S
}
$$
with $f' : X' \to S$ and $f : X \to S$ satisfying the hypotheses of
Lemma \ref{lemma-diagonal-picard-flat-proper}.
Let $\mathcal{L}$ be an invertible $\mathcal{O}_X$-module
and let $\mathcal{L}'$ be the pullback to $X'$. Let $Z \subset S$,
resp.\ $Z' \subset S$ be the locally closed subscheme constructed
in Lemma \ref{lemma-diagonal-picard-flat-proper}
for $(f, \mathcal{L})$, resp.\ $(f', \mathcal{L}')$
so that $Z \subset Z'$. If $s \in Z$ and
$$
H^1(X_s, \mathcal{O}) \longrightarrow H^1(X'_s, \mathcal{O})
$$
is injective, then $Z \cap U = Z' \cap U$ for some open neighbourhood
$U$ of $s$.
\end{lemma}

\begin{proof}
We may replace $S$ by ========

regex_match $\mathcal{L}'$
nn candidate the pullback
nn the pullback
new_annotation [(311783, 311797, 'VAR'), (311801, 311813, 'TYPE')]
===== sent This means that the $Z(s)$ has codimension $\geq 2$ in $X$ which contradicts
Divisors, Lemma \ref{divisors-lemma-effective-Cartier-codimension-1}.
\end{proof}

\begin{lemma}
\label{lemma-H1-O-picard-flat-proper}
Consider a commutative diagram of schemes
$$
\xymatrix{
X' \ar[rr] \ar[dr]_{f'} & & X \ar[dl]^f \\
& S
}
$$
with $f' : X' \to S$ and $f : X \to S$ satisfying the hypotheses of
Lemma \ref{lemma-diagonal-picard-flat-proper}.
Let $\mathcal{L}$ be an invertible $\mathcal{O}_X$-module
and let $\mathcal{L}'$ be the pullback to $X'$. Let $Z \subset S$,
resp.\ $Z' \subset S$ be the locally closed subscheme constructed
in Lemma \ref{lemma-diagonal-picard-flat-proper}
for $(f, \mathcal{L})$, resp.\ $(f', \mathcal{L}')$
so that $Z \subset Z'$. If $s \in Z$ and
$$
H^1(X_s, \mathcal{O}) \longrightarrow H^1(X'_s, \mathcal{O})
$$
is injective, then $Z \cap U = Z' \cap U$ for some open neighbourhood
$U$ of $s$.
\end{lemma}

\begin{proof}
We may replace $S$ by ========

new_annotation []
===== sent Let $E = Rf_*\mathcal{L}$ and $E' = Rf'_*\mathcal{L}' = Rf'_*\mathcal{O}_{X'}$.
These are perfect complexes whose formation commutes with arbitrary
change of base (Derived Categories of Schemes, Lemma
\ref{perfect-lemma-flat-proper-perfect-direct-image-general}).
 ========

new_annotation []
===== sent S$ satisfying the hypotheses of
Lemma \ref{lemma-diagonal-picard-flat-proper}.
Let $\mathcal{L}$ be an invertible $\mathcal{O}_X$-module
and let $\mathcal{L}_i$ be the pullback to $X_i$. Let $Z \subset S$,
resp.\ $Z_i \subset S$ be the locally closed subscheme constructed
in Lemma \ref{lemma-diagonal-picard-flat-proper}
for $(f, \mathcal{L})$, resp.\ $(f_i, \mathcal{L}_i)$
so that $Z \subset \bigcap_{i = 1, \ldots, n} Z_i$. If $s \in Z$ and
$$
H^1(X_s, \mathcal{O}) \longrightarrow
\bigoplus\nolimits_{i = 1, \ldots, n} H^1(X_{i, s}, \mathcal{O})
$$
is injective, then $Z \cap U = (\bigcap_{i = 1, \ldots, n} Z_i) \cap U$
(scheme theoretic intersection) for some open neighbourhood $U$ of $s$.
\end{lemma}

\begin{proof}
This lemma is a variant of Lemma \ref{lemma-H1-O-picard-flat-proper}
and we strongly urge the reader to read that proof first; this proof
is basically a copy of that proof with minor modifications. ========

regex_match $\mathcal{L}$
nn candidate the pullback
nn the pullback
new_annotation [(315261, 315274, 'VAR'), (315342, 315354, 'TYPE')]
===== sent S$ satisfying the hypotheses of
Lemma \ref{lemma-diagonal-picard-flat-proper}.
Let $\mathcal{L}$ be an invertible $\mathcal{O}_X$-module
and let $\mathcal{L}_i$ be the pullback to $X_i$. Let $Z \subset S$,
resp.\ $Z_i \subset S$ be the locally closed subscheme constructed
in Lemma \ref{lemma-diagonal-picard-flat-proper}
for $(f, \mathcal{L})$, resp.\ $(f_i, \mathcal{L}_i)$
so that $Z \subset \bigcap_{i = 1, \ldots, n} Z_i$. If $s \in Z$ and
$$
H^1(X_s, \mathcal{O}) \longrightarrow
\bigoplus\nolimits_{i = 1, \ldots, n} H^1(X_{i, s}, \mathcal{O})
$$
is injective, then $Z \cap U = (\bigcap_{i = 1, \ldots, n} Z_i) \cap U$
(scheme theoretic intersection) for some open neighbourhood $U$ of $s$.
\end{lemma}

\begin{proof}
This lemma is a variant of Lemma \ref{lemma-H1-O-picard-flat-proper}
and we strongly urge the reader to read that proof first; this proof
is basically a copy of that proof with minor modifications. ========

regex_match $\mathcal{L}_i$
nn candidate the pullback
nn the pullback
new_annotation [(315323, 315338, 'VAR'), (315342, 315354, 'TYPE')]
===== sent S$ satisfying the hypotheses of
Lemma \ref{lemma-diagonal-picard-flat-proper}.
Let $\mathcal{L}$ be an invertible $\mathcal{O}_X$-module
and let $\mathcal{L}_i$ be the pullback to $X_i$. Let $Z \subset S$,
resp.\ $Z_i \subset S$ be the locally closed subscheme constructed
in Lemma \ref{lemma-diagonal-picard-flat-proper}
for $(f, \mathcal{L})$, resp.\ $(f_i, \mathcal{L}_i)$
so that $Z \subset \bigcap_{i = 1, \ldots, n} Z_i$. If $s \in Z$ and
$$
H^1(X_s, \mathcal{O}) \longrightarrow
\bigoplus\nolimits_{i = 1, \ldots, n} H^1(X_{i, s}, \mathcal{O})
$$
is injective, then $Z \cap U = (\bigcap_{i = 1, \ldots, n} Z_i) \cap U$
(scheme theoretic intersection) for some open neighbourhood $U$ of $s$.
\end{lemma}

\begin{proof}
This lemma is a variant of Lemma \ref{lemma-H1-O-picard-flat-proper}
and we strongly urge the reader to read that proof first; this proof
is basically a copy of that proof with minor modifications. ========

new_annotation []
===== sent After shrinking
$S$ to an affine open neighbourhood of $s$ we may assume that
$\mathcal{L}_i = \mathcal{O}_{X_i}$ for $i = 1, \ldots, n$.
Let $E = Rf_*\mathcal{L}$ and
$E_i = Rf_{i, *}\mathcal{L}_i = Rf_{i, *}\mathcal{O}_{X_i}$.
These are perfect complexes whose formation commutes with arbitrary
change of base (Derived Categories of Schemes, Lemma
\ref{perfect-lemma-flat-proper-perfect-direct-image-general}).
 ========

new_annotation []
===== sent \end{proof}

\begin{lemma}
\label{lemma-pic-of-product}
Let $f : X \to S$ and $g : Y \to S$ be morphisms of schemes
satisfying the hypotheses of Lemma \ref{lemma-diagonal-picard-flat-proper}.
Let $\sigma : S \to X$ and $\tau : S \to Y$ be sections of
$f$ and $g$. Let $s \in ========

new_annotation []
===== sent \end{proof}

\begin{lemma}
\label{lemma-pic-of-product}
Let $f : X \to S$ and $g : Y \to S$ be morphisms of schemes
satisfying the hypotheses of Lemma \ref{lemma-diagonal-picard-flat-proper}.
Let $\sigma : S \to X$ and $\tau : S \to Y$ be sections of
$f$ and $g$. Let $s \in ========

new_annotation []
===== sent \end{proof}

\begin{lemma}
\label{lemma-pic-of-product}
Let $f : X \to S$ and $g : Y \to S$ be morphisms of schemes
satisfying the hypotheses of Lemma \ref{lemma-diagonal-picard-flat-proper}.
Let $\sigma : S \to X$ and $\tau : S \to Y$ be sections of
$f$ and $g$. Let $s \in ========

new_annotation []
===== sent Let $\mathcal{L}$ be an invertible sheaf on $X \times_S ========

regex_match $\mathcal{L}$
nn candidate an invertible sheaf
nn an invertible sheaf
new_annotation [(319687, 319700, 'VAR'), (319704, 319723, 'TYPE')]
===== sent \end{proof}

\begin{theorem}[Theorem of the cube]
\label{theorem-of-the-cube}
Let $k$ be a field. ========

regex_match $k$
nn candidate a field
nn a field
new_annotation [(320491, 320494, 'VAR'), (320498, 320505, 'TYPE')]
===== sent Let $X, Y, Z$ be varieties with
$k$-rational points $x, y, z$. Let $\mathcal{L}$ be an invertible
module on $X \times Y \times Z$. If
\begin{enumerate}
\item $\mathcal{L}$ is trivial over
$x \times Y \times Z$, $X \times y \times Z$, and $X \times Y \times z$, and
\item $X$ and $Y$ are geometrically integral and proper over $k$,
\end{enumerate}
then $\mathcal{L}$ is trivial.
 ========

new_annotation []
===== sent Let $X, Y, Z$ be varieties with
$k$-rational points $x, y, z$. Let $\mathcal{L}$ be an invertible
module on $X \times Y \times Z$. If
\begin{enumerate}
\item $\mathcal{L}$ is trivial over
$x \times Y \times Z$, $X \times y \times Z$, and $X \times Y \times z$, and
\item $X$ and $Y$ are geometrically integral and proper over $k$,
\end{enumerate}
then $\mathcal{L}$ is trivial.
 ========

regex_match $\mathcal{L}$
nn candidate an invertible
module
nn an invertible
module
new_annotation [(320574, 320587, 'VAR'), (320591, 320611, 'TYPE')]
===== sent \begin{lemma}
\label{lemma-Noetherian-approximation}
Let $f : X \to S$ be a morphism of affine schemes, which is of finite
presentation. ========

regex_match $f : X \to S$
nn candidate X \to
nn candidate a morphism
nn a morphism
new_annotation [(322352, 322365, 'VAR'), (322369, 322379, 'TYPE')]
===== sent Thus the lemma follows from
Algebra, Lemma \ref{algebra-lemma-limit-module-finite-presentation}.
\end{proof}

\begin{lemma}
\label{lemma-Noetherian-approximation-module}
Let $f : X \to S$ be a morphism of affine schemes, which is of finite
presentation. ========

regex_match $f : X \to S$
nn candidate X \to
nn candidate a morphism
nn a morphism
new_annotation [(323134, 323147, 'VAR'), (323151, 323161, 'TYPE')]
===== sent Let $\mathcal{F}$ be a quasi-coherent $\mathcal{O}_X$-module
of finite presentation. ========

regex_match $\mathcal{F}$
nn candidate finite presentation
nn finite presentation
new_annotation [(323218, 323231, 'VAR'), (323278, 323297, 'TYPE')]
===== sent Thus the lemma follows from
Algebra, Lemma \ref{algebra-lemma-limit-module-finite-presentation}.
\end{proof}

\begin{lemma}
\label{lemma-Noetherian-approximation-flat-module}
Let $f : X \to S$ be a morphism of affine schemes, which is of finite
presentation. ========

regex_match $f : X \to S$
nn candidate X \to
nn candidate a morphism
nn a morphism
new_annotation [(324146, 324159, 'VAR'), (324163, 324173, 'TYPE')]
===== sent Let $\mathcal{F}$ be a quasi-coherent $\mathcal{O}_X$-module
of finite presentation and flat over $S$. Then we may choose a diagram as in
Lemma \ref{lemma-Noetherian-approximation-module}
and sheaf $\mathcal{F}_0$ such that in addition $\mathcal{F}_0$
is flat over $S_0$.
\end{lemma}

\begin{proof}
Write $S = \Spec(A)$, $X = \Spec(B)$, and
$\mathcal{F} = \widetilde{M}$. As $f$ is of finite presentation we see that
$B$ is of finite presentation as an $A$-algebra, see
 ========

regex_match $\mathcal{F}$
nn candidate finite presentation
nn finite presentation
new_annotation [(324230, 324243, 'VAR'), (324290, 324309, 'TYPE')]
===== sent Thus the lemma follows from
Algebra, Lemma \ref{algebra-lemma-flat-finite-presentation-limit-flat}.
\end{proof}

\begin{lemma}
\label{lemma-Noetherian-approximation-flat}
Let $f : X \to S$ be a morphism of affine schemes, which is of finite
presentation and flat. ========

regex_match $f : X \to S$
nn candidate X \to
nn candidate a morphism
nn a morphism
new_annotation [(325285, 325298, 'VAR'), (325302, 325312, 'TYPE')]
===== sent \end{lemma}

\begin{proof}
This is a special case of
Lemma \ref{lemma-Noetherian-approximation-flat-module}.
\end{proof}

\begin{lemma}
\label{lemma-Noetherian-approximation-smooth}
Let $f : X \to S$ be a morphism of affine schemes, which is smooth.
 ========

regex_match $f : X \to S$
nn candidate X \to
nn candidate a morphism
nn a morphism
new_annotation [(325674, 325687, 'VAR'), (325691, 325701, 'TYPE')]
===== sent \end{lemma}

\begin{proof}
Write $S = \Spec(A)$, $X = \Spec(B)$, and
as $f$ is smooth we see that $B$ is smooth as an $A$-algebra, see
Morphisms,
Lemma \ref{morphisms-lemma-smooth-characterize}.
Hence the lemma follows from
Algebra, Lemma \ref{algebra-lemma-finite-presentation-fs-Noetherian}.
\end{proof}

\begin{lemma}
\label{lemma-Noetherian-approximation-geometrically-reduced}
Let $f : X \to S$ be a morphism of affine schemes, which is
of finite presentation with geometrically reduced fibres.
 ========

regex_match $f : X \to S$
nn candidate X \to
nn candidate a morphism
nn a morphism
new_annotation [(326240, 326253, 'VAR'), (326257, 326267, 'TYPE')]
===== sent \end{proof}

\begin{lemma}
\label{lemma-Noetherian-approximation-geometrically-irreducible}
Let $f : X \to S$ be a morphism of affine schemes, which is
of finite presentation with geometrically irreducible fibres.
 ========

regex_match $f : X \to S$
nn candidate X \to
nn candidate a morphism
nn a morphism
new_annotation [(327535, 327548, 'VAR'), (327552, 327562, 'TYPE')]
===== sent \end{proof}

\begin{lemma}
\label{lemma-Noetherian-approximation-geometrically-connected}
Let $f : X \to S$ be a morphism of affine schemes, which is
of finite presentation with geometrically connected fibres.
 ========

regex_match $f : X \to S$
nn candidate X \to
nn candidate a morphism
nn a morphism
new_annotation [(328857, 328870, 'VAR'), (328874, 328884, 'TYPE')]
===== sent \end{proof}

\begin{lemma}
\label{lemma-Noetherian-approximation-dimension-d}
Let $d \geq 0$ be an integer.
 ========

regex_match $d \geq 0$
nn candidate an integer
nn an integer
new_annotation [(330155, 330165, 'VAR'), (330169, 330179, 'TYPE')]
===== sent Let $f : X \to S$ be a morphism of affine schemes, which is
of finite presentation all of whose fibres have dimension $d$.
Then there exists a diagram as in
Lemma \ref{lemma-Noetherian-approximation}
such that in addition all fibres of $f_0$ have dimension $d$.
\end{lemma}

\begin{proof}
Apply
Lemma \ref{lemma-Noetherian-approximation}
to get a cartesian diagram
$$
\xymatrix{
X_0 \ar[d]_{f_0} & X \ar[l]^g \ar[d]^f \\
 ========

regex_match $f : X \to S$
nn candidate X \to
nn candidate a morphism
nn a morphism
new_annotation [(330185, 330198, 'VAR'), (330202, 330212, 'TYPE')]
===== sent After replacing $S_0$ by $\Spec(A_i)$ and
$X_0$ by $X_0 \times_{S_0} \Spec(A_i)$ we see that
all fibres of $f_0$ have dimension $d$.
\end{proof}

\begin{lemma}
\label{lemma-Noetherian-approximation-standard-syntomic}
Let $f : X \to S$ be a morphism of affine schemes, which is
standard syntomic (see
Morphisms, Definition \ref{morphisms-definition-syntomic}).
 ========

regex_match $f : X \to S$
nn candidate X \to
nn candidate a morphism
nn a morphism
new_annotation [(331446, 331459, 'VAR'), (331463, 331473, 'TYPE')]
===== sent Let $P$, $Q$ be properties of morphisms of schemes which are stable
under base change. ========

new_annotation []
===== sent Let $f : X \to S$ be a morphism of finite presentation
of affine schemes. ========

regex_match $f : X \to S$
nn candidate $
nn candidate X \to
nn candidate a morphism
nn a morphism
new_annotation [(332066, 332079, 'VAR'), (332083, 332093, 'TYPE')]
===== sent A_2 \ar[u] \ar[r] & A \ar[u]
}
}
$$
Let $A_0 \subset A$ be a finite type $\mathbf{Z}$-subalgebra of $A$
containing the image of both $A_1 \to A$ and $A_2 \to A$. ========

regex_match $A_0 \subset A$
nn candidate a finite type
nn a finite type
new_annotation [(333051, 333066, 'VAR'), (333070, 333083, 'TYPE')]
===== sent \begin{definition}
\label{definition-etale-neighbourhood}
Let $S$ be a scheme. ========

regex_match $S$
nn candidate a scheme
nn a scheme
new_annotation [(334302, 334305, 'VAR'), (334309, 334317, 'TYPE')]
===== sent Let $s \in S$ be a point.
 ========

regex_match $s \in S$
nn candidate $s
nn candidate a point
nn a point
new_annotation [(334323, 334332, 'VAR'), (334336, 334343, 'TYPE')]
===== sent \medskip\noindent
Note that $\kappa(s) \subset \kappa(u)$ is a finite separable extension
if $(U, u) \to (S, s)$ is an \'etale neighbourhood,
see Morphisms, Lemma \ref{morphisms-lemma-etale-at-point}.

\begin{lemma}
\label{lemma-realize-prescribed-residue-field-extension-etale}
Let $S$ be a scheme.
 ========

regex_match $S$
nn candidate a scheme
nn a scheme
new_annotation [(335947, 335950, 'VAR'), (335954, 335962, 'TYPE')]
===== sent Let $s \in ========

new_annotation []
===== sent Let $\kappa(s) \subset k$ be a finite separable field extension.
 ========

regex_match $\kappa(s) \subset k$
nn candidate \subset
nn candidate a finite separable field extension
nn a finite separable field extension
new_annotation [(335983, 336004, 'VAR'), (336008, 336042, 'TYPE')]
===== sent In this case the lemma follows from
Algebra, Lemma \ref{algebra-lemma-make-etale-map-prescribed-residue-field}.
\end{proof}

\begin{lemma}
\label{lemma-etale-neighbourhoods-not-quite-filtered}
Let $S$ be a scheme, and let $s$ be a point of $S$.
The category of \'etale neighborhoods has the following properties:
\begin{enumerate}
\item Let $(U_i, u_i)_{i=1, 2}$ be two \'etale neighborhoods of
$s$ in $S$. Then there exists a third \'etale neighborhood
$(U, u)$ and morphisms
$(U, u) \to (U_i, u_i)$, $i = 1, 2$.
\item ========

regex_match $S$
nn candidate a scheme
nn a scheme
new_annotation [(336460, 336463, 'VAR'), (336467, 336475, 'TYPE')]
===== sent In this case the lemma follows from
Algebra, Lemma \ref{algebra-lemma-make-etale-map-prescribed-residue-field}.
\end{proof}

\begin{lemma}
\label{lemma-etale-neighbourhoods-not-quite-filtered}
Let $S$ be a scheme, and let $s$ be a point of $S$.
The category of \'etale neighborhoods has the following properties:
\begin{enumerate}
\item Let $(U_i, u_i)_{i=1, 2}$ be two \'etale neighborhoods of
$s$ in $S$. Then there exists a third \'etale neighborhood
$(U, u)$ and morphisms
$(U, u) \to (U_i, u_i)$, $i = 1, 2$.
\item ========

regex_match $s$
nn candidate a point
nn a point
new_annotation [(336485, 336488, 'VAR'), (336492, 336499, 'TYPE')]
===== sent In this case the lemma follows from
Algebra, Lemma \ref{algebra-lemma-make-etale-map-prescribed-residue-field}.
\end{proof}

\begin{lemma}
\label{lemma-etale-neighbourhoods-not-quite-filtered}
Let $S$ be a scheme, and let $s$ be a point of $S$.
The category of \'etale neighborhoods has the following properties:
\begin{enumerate}
\item Let $(U_i, u_i)_{i=1, 2}$ be two \'etale neighborhoods of
$s$ in $S$. Then there exists a third \'etale neighborhood
$(U, u)$ and morphisms
$(U, u) \to (U_i, u_i)$, $i = 1, 2$.
\item ========

new_annotation []
===== sent Let $h_1, h_2: (U, u) \to (U', u')$ be two
morphisms between \'etale neighborhoods of $s$.
Assume $h_1$, $h_2$ induce the same map $\kappa(u') \to \kappa(u)$ of residue
fields. ========

new_annotation []
===== sent \end{proof}

\begin{lemma}
\label{lemma-elementary-etale-neighbourhoods}
Let $S$ be a scheme, and let $s$ be a point of $S$.
The category of elementary \'etale neighborhoods of $(S, s)$
is cofiltered (see
Categories, Definition \ref{categories-definition-codirected}).
 ========

regex_match $S$
nn candidate a scheme
nn a scheme
new_annotation [(338504, 338507, 'VAR'), (338511, 338519, 'TYPE')]
===== sent \end{proof}

\begin{lemma}
\label{lemma-elementary-etale-neighbourhoods}
Let $S$ be a scheme, and let $s$ be a point of $S$.
The category of elementary \'etale neighborhoods of $(S, s)$
is cofiltered (see
Categories, Definition \ref{categories-definition-codirected}).
 ========

regex_match $s$
nn candidate a point
nn a point
new_annotation [(338529, 338532, 'VAR'), (338536, 338543, 'TYPE')]
===== sent This is immediate from the definitions and
Lemma \ref{lemma-etale-neighbourhoods-not-quite-filtered}.
\end{proof}

\begin{lemma}
\label{lemma-describe-henselization}
Let $S$ be a scheme. ========

regex_match $S$
nn candidate a scheme
nn a scheme
new_annotation [(338893, 338896, 'VAR'), (338900, 338908, 'TYPE')]
===== sent Let $s \in ========

new_annotation []
===== sent Then we have
$$
\mathcal{O}_{S, s}^h =
\colim_{(U, u)} \mathcal{O}(U)
$$
where the colimit is over the filtered category which is opposite to the
category of elementary \'etale neighbourhoods $(U, u)$ of $(S, s)$.
\end{lemma}

\begin{proof}
Let $\Spec(A) \subset S$ be an affine neighbourhood of $s$.
Let $\mathfrak p \subset A$ be the prime ideal corresponding to $s$.
 ========

regex_match $\Spec(A) \subset S$
nn candidate \subset S$
nn candidate an affine neighbourhood
nn an affine neighbourhood
new_annotation [(339170, 339190, 'VAR'), (339194, 339217, 'TYPE')]
===== sent Then we have
$$
\mathcal{O}_{S, s}^h =
\colim_{(U, u)} \mathcal{O}(U)
$$
where the colimit is over the filtered category which is opposite to the
category of elementary \'etale neighbourhoods $(U, u)$ of $(S, s)$.
\end{lemma}

\begin{proof}
Let $\Spec(A) \subset S$ be an affine neighbourhood of $s$.
Let $\mathfrak p \subset A$ be the prime ideal corresponding to $s$.
 ========

regex_match $\mathfrak p \subset A$
nn candidate p \subset
nn candidate the prime ideal corresponding
nn the prime ideal corresponding
new_annotation [(339230, 339253, 'VAR'), (339257, 339286, 'TYPE')]
===== sent \end{slogan}
Let $X \to S$ be a morphism of schemes. ========

regex_match $X \to S$
nn candidate $X
nn candidate a morphism
nn a morphism
new_annotation [(340194, 340203, 'VAR'), (340207, 340217, 'TYPE')]
===== sent Let $x \in X$ with image $s \in ========

new_annotation []
===== sent S$.
Let $(V, v) \to (X_s, x)$ be an \'etale neighbourhood.
 ========

regex_match $(V, v) \to (X_s, x)$
nn candidate $(V
nn candidate an \'etale neighbourhood
nn an \'etale neighbourhood
new_annotation [(340270, 340291, 'VAR'), (340295, 340319, 'TYPE')]
===== sent We will rely
on Popescu's theorem, see
Smoothing Ring Maps, Theorem \ref{smoothing-theorem-popescu}.

\begin{lemma}
\label{lemma-map-approximation}
Let $S$ be a locally Noetherian scheme. ========

regex_match $S$
nn candidate a locally Noetherian scheme
nn a locally Noetherian scheme
new_annotation [(341784, 341787, 'VAR'), (341791, 341818, 'TYPE')]
===== sent Let $X$, $Y$ be
schemes locally of finite
type over $S$. Let $x \in X$ and $y \in Y$ be points lying over the
same point $s \in S$. Assume $\mathcal{O}_{S, s}$ is a G-ring.
 ========

new_annotation []
===== sent Let $X$, $Y$ be
schemes locally of finite
type over $S$. Let $x \in X$ and $y \in Y$ be points lying over the
same point $s \in S$. Assume $\mathcal{O}_{S, s}$ is a G-ring.
 ========

new_annotation []
===== sent Say $S = \Spec(R)$, $X = \Spec(A)$, $Y = \Spec(B)$.
Write $B = R[x_1, \ldots, x_n]/(f_1, \ldots, f_m)$.
Let $\mathfrak p \subset A$ be the prime ideal corresponding to $x$.
The local ring $\mathcal{O}_{X, x} = A_\mathfrak p$ is a G-ring by
More on Algebra, Proposition
\ref{more-algebra-proposition-finite-type-over-G-ring}.
The map $\varphi$ is a map
$$
B_\mathfrak q^\wedge \longrightarrow A_\mathfrak p^\wedge
$$
where $\mathfrak q \subset B$ is the prime corresponding to $y$.
Let $a_1, \ldots, a_n \in A_\mathfrak p^\wedge$ be the images
of $x_1, \ldots, x_n$ via
$R[x_1, \ldots, x_n] \to B \to B_\mathfrak q^\wedge \to A_\mathfrak p^\wedge$.
 ========

regex_match $\mathfrak p \subset A$
nn candidate p \subset
nn candidate the prime ideal corresponding
nn the prime ideal corresponding
new_annotation [(342748, 342771, 'VAR'), (342775, 342804, 'TYPE')]
===== sent Say $S = \Spec(R)$, $X = \Spec(A)$, $Y = \Spec(B)$.
Write $B = R[x_1, \ldots, x_n]/(f_1, \ldots, f_m)$.
Let $\mathfrak p \subset A$ be the prime ideal corresponding to $x$.
The local ring $\mathcal{O}_{X, x} = A_\mathfrak p$ is a G-ring by
More on Algebra, Proposition
\ref{more-algebra-proposition-finite-type-over-G-ring}.
The map $\varphi$ is a map
$$
B_\mathfrak q^\wedge \longrightarrow A_\mathfrak p^\wedge
$$
where $\mathfrak q \subset B$ is the prime corresponding to $y$.
Let $a_1, \ldots, a_n \in A_\mathfrak p^\wedge$ be the images
of $x_1, \ldots, x_n$ via
$R[x_1, \ldots, x_n] \to B \to B_\mathfrak q^\wedge \to A_\mathfrak p^\wedge$.
 ========

regex_match $a_1, \ldots, a_n \in A_\mathfrak p^\wedge$
nn candidate $a_1, \ldots
nn candidate the images
nn the images
new_annotation [(343125, 343168, 'VAR'), (343172, 343182, 'TYPE')]
===== sent This finishes the proof
by taking $U = \Spec(A') \to \Spec(B)$ as the morphism $f$
and $u = \mathfrak p'$.
\end{proof}

\begin{lemma}
\label{lemma-isomorphism-approximation}
Let $S$ be a locally Noetherian scheme. ========

regex_match $S$
nn candidate a locally Noetherian scheme
nn a locally Noetherian scheme
new_annotation [(343941, 343944, 'VAR'), (343948, 343975, 'TYPE')]
===== sent Let $X$, $Y$ be
schemes locally of finite
type over $S$. Let $x \in X$ and $y \in Y$ be points lying over the
same point $s \in S$. Assume $\mathcal{O}_{S, s}$ is a G-ring.
 ========

new_annotation []
===== sent Let $X$, $Y$ be
schemes locally of finite
type over $S$. Let $x \in X$ and $y \in Y$ be points lying over the
same point $s \in S$. Assume $\mathcal{O}_{S, s}$ is a G-ring.
 ========

new_annotation []
===== sent \end{proof}

\begin{lemma}
\label{lemma-control-agreement}
Let $T \to S$ be finite type morphisms of Noetherian schemes.
 ========

new_annotation []
===== sent Let $t \in T$ map to $s \in S$ and let
$\sigma : \mathcal{O}_{T, t} \to \mathcal{O}_{S, s}^\wedge$
be a local $\mathcal{O}_{S, s}$-algebra map. ========

new_annotation []
===== sent Say $S = \Spec(R)$ and
$T = \Spec(C)$. Let $c_1, \ldots, c_n \in C$ be generators
of $C$ as an $R$-algebra. ========

new_annotation []
===== sent Let $\mathfrak p \subset R$ be the
prime ideal corresponding to $s$. Say $\mathfrak p = (f_1, \ldots, f_m)$.
After replacing $R$ by a principal localization
(to clear denominators in $R_\mathfrak p$)
we may assume there exist
$r_1, \ldots, r_n \in R$ and $a_{i, I} \in \mathcal{O}_{S, s}^\wedge$
where $I = (i_1, \ldots, i_m)$ with $\sum i_j = N$ such that
$$
\sigma(c_i) = r_i + \sum\nolimits_I a_{i, I} f_1^{i_1} \ldots f_m^{i_m}
$$
in $\mathcal{O}_{S, s}^\wedge$. ========

new_annotation []
===== sent Then we consider
$$
C' = C[t_{i, I}]/
\left(c_i - r_i - \sum\nolimits_I t_{i, I} f_1^{i_1} \ldots f_m^{i_m}\right)
$$
with $\mathfrak p' = \mathfrak pC' + (t_{i, I})$ and factorization
of $\sigma : C \to \mathcal{O}_{S, s}^\wedge$ through $C'$
given by sending $t_{i, I}$ to $a_{i, I}$.
Taking $T' = \Spec(C')$ works because any $\sigma'$ as in the statement
of the lemma will send $c_i$ to $r_i$ modulo the maximal ideal
to the power $N$.
\end{proof}

\begin{lemma}
\label{lemma-control-graded}
Let $Y \to T \to S$ be finite type morphisms of Noetherian schemes.
 ========

new_annotation []
===== sent Let $t \in T$ map to $s \in S$ and let
$\sigma : \mathcal{O}_{T, t} \to \mathcal{O}_{S, s}^\wedge$
be a local $\mathcal{O}_{S, s}$-algebra map.
 ========

new_annotation []
===== sent Let $Y = Y_1 \cup \ldots \cup Y_n$
be an affine open covering. ========

new_annotation []
===== sent Write $S = \Spec(R)$, $T = \Spec(C)$, and $Y = \Spec(B)$.
Choose a presentation $B = C[x_1, \ldots, x_n]/(f_1, \ldots, f_m)$.
Denote $R^\wedge = \mathcal{O}_{S, s}^\wedge$.
Let $a_{kj} \in R^\wedge[x_1, \ldots, x_n]$ be polynomials
such that
$$
\sum\nolimits_{j = 1, \ldots, m} a_{kj}\sigma(f_j) = 0,\quad
\text{for }k = 1, \ldots, K
$$
is a set of generators for the module of relations among
the $\sigma(f_j) \in R^\wedge[x_1, \ldots, x_n]$.
 ========

new_annotation []
===== sent Thus we have an exact sequence
\begin{equation}
\label{equation-resolution}
R^\wedge[x_1, \ldots, x_n]^{\oplus K} \to
R^\wedge[x_1, \ldots, x_n]^{\oplus m} \to
R^\wedge[x_1, \ldots, x_n] \to B \otimes_{C, \sigma} R^\wedge \to 0
\end{equation}
Let $c$ be an integer which works in the Artin-Rees lemma for
both the first and the second map in this sequence and the ideal
$\mathfrak m_{R^\wedge}R^\wedge[x_1, \ldots, x_n]$ as defined in
More on Algebra, Section \ref{more-algebra-section-artin-rees}.
Write
$$
a_{kj} = \sum\nolimits_{I \in \Omega} a_{kj, I} x^I
\quad\text{and}\quad
f_j = \sum\nolimits_{I \in \Omega} f_{j, I} x^I
$$
in multiindex notation where $a_{kj, I} \in R^\wedge$, $f_{j, I} \in C$,
and $\Omega$ a finite set of multiindices. ========

regex_match $c$
nn candidate an integer
nn an integer
new_annotation [(359283, 359286, 'VAR'), (359290, 359300, 'TYPE')]
===== sent Thus $T' = \Spec(C')$ comes with a point $t' \in T'$ such that
$\sigma$ factors through $\mathcal{O}_{T, t} \to \mathcal{O}_{T', t'}$.
Let $t_{kj} = \sum t_{kj, I} x^I$ in $C'[x_1, \ldots, x_n]$.
Then we see that we have a complex
\begin{equation}
\label{equation-resolution-new}
C'[x_1, \ldots, x_n]^{\oplus K} \to
C'[x_1, \ldots, x_n]^{\oplus m} \to
C'[x_1, \ldots, x_n] \to B \otimes_C C' \to 0
\end{equation}
which is exact at $C'[x_1, \ldots, x_n]$ and whose base change
by $\tilde\sigma$ gives (\ref{equation-resolution}).

 ========

new_annotation []
===== sent \begin{lemma}
\label{lemma-nr-minimal-primes}
Let $R = \colim R_i$ be colimit of a directed system of rings
whose transition maps are faithfully flat.
 ========

new_annotation []
===== sent \end{proof}

\begin{lemma}
\label{lemma-nr-branches}
Let $X$ be a scheme and $x \in X$ a point. ========

regex_match $X$
nn candidate a scheme
nn a scheme
new_annotation [(367429, 367432, 'VAR'), (367436, 367444, 'TYPE')]
===== sent Let $\Spec(A)$ be an affine open neighbourhood
of $x$ and let $\mathfrak p \subset A$ be the prime ideal
corresponding to $x$. We may replace $X$ by $\Spec(A)$ and
it suffices to consider affine elementary \'etale neighbourhoods
$(U, u)$ in the supremum as they form a cofinal subsystem.
 ========

regex_match $\Spec(A)$
nn candidate an affine open neighbourhood
nn an affine open neighbourhood
new_annotation [(368452, 368462, 'VAR'), (368466, 368494, 'TYPE')]
===== sent Let $\Spec(A)$ be an affine open neighbourhood
of $x$ and let $\mathfrak p \subset A$ be the prime ideal
corresponding to $x$. We may replace $X$ by $\Spec(A)$ and
it suffices to consider affine elementary \'etale neighbourhoods
$(U, u)$ in the supremum as they form a cofinal subsystem.
 ========

regex_match $\mathfrak p \subset A$
nn candidate p \subset
nn candidate We
nn We
new_annotation [(368510, 368533, 'VAR'), (368575, 368577, 'TYPE')]
===== sent \begin{lemma}
\label{lemma-nr-branches-fibre}
Let $X \to S$ be a morphism of schemes and $x \in X$ a point with image $s$.
Then
\begin{enumerate}
\item the number of branches of the fibre $X_s$ at $x$ is equal to
the supremum of the number of irreducible components of the fibre $U_s$
passing through $u$ taken over elementary \'etale neighbourhoods
$(U, u) \to (X, x)$,
\item the number of geometric branches of the fibre $X_s$ at $x$ is equal to
the supremum of the number of irreducible components of the fibre $U_s$
passing through $u$ taken over \'etale neighbourhoods
$(U, u) \to (X, x)$,
\item the fibre $X_s$ is unibranch at $x$ if and only if for every
elementary \'etale neighbourhood $(U, u) \to (X, x)$ there is
exactly one irreducible component of the fibre $U_s$ passing through $u$, and
\item $X$ is geometrically unibranch at $x$ if and only if for every
\'etale neighbourhood $(U, u) \to (X, x)$ there
is exactly one irreducible component of $U_s$ passing through $u$.
\end{enumerate}
\end{lemma}

\begin{proof}
 ========

regex_match $X \to S$
nn candidate $X
nn candidate a morphism
nn a morphism
new_annotation [(370774, 370783, 'VAR'), (370787, 370797, 'TYPE')]
===== sent Combine Lemmas \ref{lemma-nr-branches} and
\ref{lemma-lift-etale-neighbourhood-fibre}.
\end{proof}

\begin{lemma}
\label{lemma-number-of-branches-and-smooth}
Let $X \to S$ be a smooth morphism of schemes.
 ========

regex_match $X \to S$
nn candidate $X
nn candidate a smooth morphism
nn a smooth morphism
new_annotation [(371915, 371924, 'VAR'), (371928, 371945, 'TYPE')]
===== sent Let $x \in X$ with image $s \in ========

new_annotation []
===== sent \begin{lemma}
\label{lemma-slice-smooth-given-element}
Let $f : X \to S$ be a morphism of schemes.
 ========

regex_match $f : X \to S$
nn candidate X \to
nn candidate a morphism
nn a morphism
new_annotation [(372959, 372972, 'VAR'), (372976, 372986, 'TYPE')]
===== sent Let $x \in X$ be a point with image $s \in ========

regex_match $x \in X$
nn candidate a point
nn a point
new_annotation [(373003, 373012, 'VAR'), (373016, 373023, 'TYPE')]
===== sent S$.
Let $h \in \mathfrak m_x \subset \mathcal{O}_{X, x}$.
Assume
\begin{enumerate}
\item $f$ is smooth at $x$, and
\item the image $\text{d}\overline{h}$ of $\text{d}h$ in
$$
\Omega_{X_s/s, x} \otimes_{\mathcal{O}_{X_s, x}} \kappa(x) =
\Omega_{X/S, x} \otimes_{\mathcal{O}_{X, x}} \kappa(x)
$$
is nonzero.
 ========

new_annotation []
===== sent \end{proof}

\begin{lemma}
\label{lemma-slice-smooth-once}
Let $f : X \to S$ be a morphism of schemes.
 ========

regex_match $f : X \to S$
nn candidate X \to
nn candidate a morphism
nn a morphism
new_annotation [(376060, 376073, 'VAR'), (376077, 376087, 'TYPE')]
===== sent Let $x \in X$ be a point with image $s \in ========

regex_match $x \in X$
nn candidate a point
nn a point
new_annotation [(376104, 376113, 'VAR'), (376117, 376124, 'TYPE')]
===== sent An example is the following: Let $k$ be a field
of characteristic $p > 0$ which is imperfect. ========

regex_match $k$
nn candidate a field
nn a field
new_annotation [(377565, 377568, 'VAR'), (377572, 377579, 'TYPE')]
===== sent Let $a \in k$ be an
element which is not a $p$th power. ========

new_annotation []
===== sent Let
$\mathfrak m = (x, y^p - a) \subset k[x, y]$. This corresponds to a closed
point $w$ of $X = \mathbf{A}^2_k$. Set $S = \mathbf{A}^1_k$ and
let $f : X \to S$ be the morphism corresponding to $k[x] \to k[x, y]$.
Then there does not exist any commutative diagram
$$
\xymatrix{
S' \ar[rr]_h \ar[rd]_g & & X \ar[ld]^f \\
& S
}
$$
with ========

regex_match $f : X \to S$
nn candidate X \to
nn candidate the morphism
nn the morphism
new_annotation [(377829, 377842, 'VAR'), (377846, 377858, 'TYPE')]
===== sent \begin{lemma}
\label{lemma-slice-smooth-once-separable-residue-field-extension}
Let $f : X \to S$ be a morphism of schemes.
 ========

regex_match $f : X \to S$
nn candidate X \to
nn candidate a morphism
nn a morphism
new_annotation [(378555, 378568, 'VAR'), (378572, 378582, 'TYPE')]
===== sent Let $x \in X$ be a point with image $s \in ========

regex_match $x \in X$
nn candidate a point
nn a point
new_annotation [(378599, 378608, 'VAR'), (378612, 378619, 'TYPE')]
===== sent \begin{lemma}
\label{lemma-slice-smooth}
Let $f : X \to S$ be a morphism of schemes.
 ========

regex_match $f : X \to S$
nn candidate X \to
nn candidate a morphism
nn a morphism
new_annotation [(380133, 380146, 'VAR'), (380150, 380160, 'TYPE')]
===== sent Let $x \in X$ be a point with image $s \in ========

regex_match $x \in X$
nn candidate a point
nn a point
new_annotation [(380177, 380186, 'VAR'), (380190, 380197, 'TYPE')]
===== sent \end{proof}

\begin{lemma}
\label{lemma-etale-nbhd-dominates-smooth}
Let $f : X \to S$ be a smooth morphism of schemes.
 ========

regex_match $f : X \to S$
nn candidate X \to
nn candidate a smooth morphism
nn a smooth morphism
new_annotation [(382886, 382899, 'VAR'), (382903, 382920, 'TYPE')]
===== sent Let $s \in S$ be a point in the image of $f$.
Then there exists an \'etale neighbourhood $(S', s') \to (S, s)$
and a $S$-morphism $S' \to X$.
\end{lemma}

\begin{proof}[First proof of Lemma \ref{lemma-etale-nbhd-dominates-smooth}]
 ========

regex_match $s \in S$
nn candidate $s
nn candidate a point
nn a point
new_annotation [(382937, 382946, 'VAR'), (382950, 382957, 'TYPE')]
===== sent Thus $W = \pi(V) \cap \mathbf{A}^d_s$ is a nonempty open subset of
$\mathbf{A}^d_s$. Let $w \in W$ be a point with $\kappa(s) \subset \kappa(w)$
finite separable, see
Varieties, Lemma \ref{varieties-lemma-affine-space-over-field}.
By
Algebra, Lemma \ref{algebra-lemma-dim-affine-space}
there exist $d$ elements
$\overline{f}_1, \ldots, \overline{f}_d \in \kappa(s)[x_1, \ldots, x_d]$
which generate the maximal ideal corresponding to $w$ in
$\kappa(s)[x_1, \ldots, x_d]$.
After replacing $R$ by a principal localization
we may assume there are $f_1, \ldots, f_d \in R[x_1, \ldots, x_d]$
which map to
$\overline{f}_1, \ldots, \overline{f}_d \in \kappa(s)[x_1, \ldots, x_d]$.
Consider the $R$-algebra
$$
R' = R[x_1, \ldots, x_d]/(f_1, \ldots, f_d)
$$
and set $S' = \Spec(R')$. ========

regex_match $w \in W$
nn candidate a point
nn a point
new_annotation [(384154, 384163, 'VAR'), (384167, 384174, 'TYPE')]
===== sent By construction the fibre of $S' \to V$ over $s$ is a single
point $s'$ whose residue field is finite separable over $\kappa(s)$.
Let $\mathfrak q' \subset R'$ be the corresponding prime. ========

regex_match $\mathfrak q' \subset R'$
nn candidate the corresponding prime
nn the corresponding prime
new_annotation [(385055, 385080, 'VAR'), (385084, 385107, 'TYPE')]
===== sent \begin{lemma}
\label{lemma-etale-dominates-smooth}
Let $S$ be a scheme. ========

regex_match $S$
nn candidate a scheme
nn a scheme
new_annotation [(386436, 386439, 'VAR'), (386443, 386451, 'TYPE')]
===== sent Let $\mathcal{U} = \{S_i \to S\}_{i \in I}$ be a smooth
covering of $S$, see
Topologies, Definition \ref{topologies-definition-smooth-covering}.
 ========

regex_match $\mathcal{U} = \{S_i \to S\}_{i \in I}$
nn candidate $\mathcal{U
nn candidate a smooth
covering
nn a smooth
covering
new_annotation [(386457, 386496, 'VAR'), (386500, 386517, 'TYPE')]
===== sent In this section we explain a result that roughly states that
\'etale coverings of a scheme $S$ can be refined by Zariski coverings
of finite locally free covers of $S$.

\begin{lemma}
\label{lemma-dominate-etale-neighbourhood-finite-flat}
Let $S$ be a scheme. ========

regex_match $S$
nn candidate a scheme
nn a scheme
new_annotation [(387574, 387577, 'VAR'), (387581, 387589, 'TYPE')]
===== sent Let $s \in ========

new_annotation []
===== sent Let $f : (U, u) \to (S, s)$ be an \'etale neighbourhood.
 ========

regex_match $f : (U, u) \to (S, s)$
nn candidate u) \to
nn candidate an \'etale neighbourhood
nn an \'etale neighbourhood
new_annotation [(387610, 387633, 'VAR'), (387637, 387661, 'TYPE')]
===== sent In this case the lemma (with $V = S$) follows from
Algebra, Lemma \ref{algebra-lemma-standard-etale-finite-flat-Zariski}.
\end{proof}

\begin{lemma}
\label{lemma-dominate-etale-affine-finite-flat}
Let $f : U \to S$ be a surjective \'etale morphism of affine schemes.
 ========

regex_match $f : U \to S$
nn candidate U \to
nn candidate a surjective \'etale morphism
nn a surjective \'etale morphism
new_annotation [(388559, 388572, 'VAR'), (388576, 388605, 'TYPE')]
===== sent Let $S$ be any scheme. ========

new_annotation []
===== sent Let $\{f_i : U_i \to S\}$ be an \'etale covering
of $S$. There exists a Zariski open covering $S = \bigcup V_j$,
for each $j$ a finite locally free, surjective morphism
$W_j \to V_j$, and for each $j$ a Zariski open covering
$\{W_{j, k} \to W_j\}$ such that the family
$\{W_{j, k} \to S\}$ refines the given \'etale covering
$\{f_i : U_i \to S\}$. What does this mean in practice?
 ========

regex_match $\{f_i : U_i \to S\}$
nn candidate an \'etale
nn an \'etale
new_annotation [(389310, 389331, 'VAR'), (389335, 389345, 'TYPE')]
===== sent \begin{lemma}
\label{lemma-etale-makes-quasi-finite-finite-at-point}
Let $f : X \to S$ be a morphism of schemes.
 ========

regex_match $f : X \to S$
nn candidate X \to
nn candidate a morphism
nn a morphism
new_annotation [(391074, 391087, 'VAR'), (391091, 391101, 'TYPE')]
===== sent Let $x \in X$. Set $s = f(x)$.
Assume that
\begin{enumerate}
\item $f$ is locally of finite type, and
\item $x \in X_s$ is isolated\footnote{In the presence of (1)
this means that $f$ is
quasi-finite at $x$, see
Morphisms, Lemma \ref{morphisms-lemma-quasi-finite-at-point-characterize}.}.
\end{enumerate}
Then there exist
\begin{enumerate}
\item[(a)] an elementary \'etale neighbourhood $(U, u) \to (S, s)$,
\item[(b)] an open subscheme $V \subset X_U$
(see \ref{equation-basic-diagram})
\end{enumerate}
such that
\begin{enumerate}
\item[(\romannumeral1)] $V \to U$ is a finite morphism,
\item[(\romannumeral2)] there is a unique point $v$ of $V$
mapping to $u$ in $U$, and
\item[(\romannumeral3)] the point $v$ maps to $x$
under the morphism $X_U \to X$, inducing $\kappa(x) = \kappa(v)$.
\end{enumerate}
 ========

new_annotation []
===== sent \end{lemma}

\begin{proof}
Let $Y \subset X$, $W \subset S$ be affine opens such that
$f(Y) \subset W$ and such that $x \in Y$. Note that $x$ is
also an isolated point of the fibre of the morphism $f|_Y : Y \to W$.
If we can prove the theorem for $f|_Y : Y \to W$, then the
theorem follows for $f$. ========

new_annotation []
===== sent \begin{lemma}
\label{lemma-etale-makes-quasi-finite-finite-multiple-points}
Let $f : X \to S$ be a morphism of schemes.
 ========

regex_match $f : X \to S$
nn candidate X \to
nn candidate a morphism
nn a morphism
new_annotation [(393069, 393082, 'VAR'), (393086, 393096, 'TYPE')]
===== sent Let $x_1, \ldots, x_n \in X$ be points having the same image $s$ in $S$.
Assume that
\begin{enumerate}
\item $f$ is locally of finite type, and
\item $x_i \in X_s$ is isolated for $i = 1, \ldots, n$.
\end{enumerate}
Then there exist
\begin{enumerate}
\item[(a)] an elementary \'etale neighbourhood $(U, u) \to (S, s)$,
\item[(b)] for each $i$ an open subscheme $V_i \subset X_U$,
\end{enumerate}
such that for each $i$ we have
\begin{enumerate}
\item[(\romannumeral1)] $V_i \to U$ is a finite morphism,
\item[(\romannumeral2)] there is a unique point $v_i$ of $V_i$
mapping to $u$ in $U$, and
\item[(\romannumeral3)] the point $v_i$ maps to $x_i$ in $X$ and
$\kappa(x_i) = \kappa(v_i)$.
\end{enumerate}
\end{lemma}

\begin{proof}
We will use induction on $n$.
 ========

new_annotation []
===== sent \begin{lemma}
\label{lemma-etale-makes-quasi-finite-finite-multiple-points-var}
Let $f : X \to S$ be a morphism of schemes.
 ========

regex_match $f : X \to S$
nn candidate X \to
nn candidate a morphism
nn a morphism
new_annotation [(394868, 394881, 'VAR'), (394885, 394895, 'TYPE')]
===== sent Let $x_1, \ldots, x_n \in X$ be points having the same image $s$ in $S$.
Assume that
\begin{enumerate}
\item $f$ is locally of finite type, and
\item $x_i \in X_s$ is isolated for $i = 1, \ldots, n$.
\end{enumerate}
Then there exist
\begin{enumerate}
\item[(a)] an \'etale neighbourhood $(U, u) \to (S, s)$,
\item[(b)] for each $i$ an integer $m_i$ and
open subschemes $V_{i, j} \subset X_U$, $j = 1, \ldots, m_i$
\end{enumerate}
such that we have
\begin{enumerate}
\item[(\romannumeral1)] each $V_{i, j} \to U$ is a finite morphism,
\item[(\romannumeral2)] there is a unique point $v_{i, j}$ of $V_{i, j}$
mapping to $u$ in $U$ with $\kappa(u) \subset \kappa(v_{i, j})$
finite purely inseparable,
\item[(\romannumeral4)] if $v_{i, j} = v_{i', j'}$, then $i = i'$ and
$j = j'$, and
\item[(\romannumeral3)] the points $v_{i, j}$ map to $x_i$ in $X$ and
no other points of $(X_U)_u$ map to $x_i$.
\end{enumerate}
\end{lemma}

\begin{proof}
This proof is a variant of the proof of
Algebra, Lemma \ref{algebra-lemma-etale-makes-quasi-finite-finite-variant}
in the language of schemes.
 ========

new_annotation []
===== sent For each $i$, let $\kappa(s) \subset L_i ========

new_annotation []
===== sent Choose a finite Galois
extension $\kappa(s) \subset L$ such that there exist
$\kappa(s)$-embeddings $L_i \to L$ for $i = 1, \ldots, n$.
Choose an \'etale neighbourhood $(U, u) \to (S, s)$ such that
$L \cong \kappa(u)$ as $\kappa(s)$-extensions
(Lemma \ref{lemma-realize-prescribed-residue-field-extension-etale}).

\medskip\noindent
Let $y_{i, j}$, $j = 1, \ldots, m_i$ be the points of $X_U$
lying over $x_i \in X$ and $u ========

new_annotation []
===== sent \end{proof}

\begin{lemma}
\label{lemma-etale-splits-off-quasi-finite-part-technical}
Let $f : X \to S$ be a morphism of schemes.
 ========

regex_match $f : X \to S$
nn candidate X \to
nn candidate a morphism
nn a morphism
new_annotation [(398116, 398129, 'VAR'), (398133, 398143, 'TYPE')]
===== sent Let $s \in S$. Let $x_1, \ldots, x_n \in X_s$. Assume that
\begin{enumerate}
\item $f$ is locally of finite type,
\item $f$ is separated, and
\item $x_1, \ldots, x_n$ are pairwise distinct isolated points of $X_s$.
\end{enumerate}
 ========

new_annotation []
===== sent Let $s \in S$. Let $x_1, \ldots, x_n \in X_s$. Assume that
\begin{enumerate}
\item $f$ is locally of finite type,
\item $f$ is separated, and
\item $x_1, \ldots, x_n$ are pairwise distinct isolated points of $X_s$.
\end{enumerate}
 ========

new_annotation []
===== sent \begin{lemma}
\label{lemma-etale-splits-off-quasi-finite-part-technical-variant}
Let $f : X \to S$ be a morphism of schemes.
 ========

regex_match $f : X \to S$
nn candidate X \to
nn candidate a morphism
nn a morphism
new_annotation [(399774, 399787, 'VAR'), (399791, 399801, 'TYPE')]
===== sent Let $s \in S$. Let $x_1, \ldots, x_n \in X_s$. Assume that
\begin{enumerate}
\item $f$ is locally of finite type,
\item $f$ is separated, and
\item $x_1, \ldots, x_n$ are pairwise distinct isolated points of $X_s$.
\end{enumerate}
 ========

new_annotation []
===== sent Let $s \in S$. Let $x_1, \ldots, x_n \in X_s$. Assume that
\begin{enumerate}
\item $f$ is locally of finite type,
\item $f$ is separated, and
\item $x_1, \ldots, x_n$ are pairwise distinct isolated points of $X_s$.
\end{enumerate}
 ========

new_annotation []
===== sent \begin{lemma}
\label{lemma-etale-splits-off-quasi-finite-part}
Let $f : X \to S$ be a morphism of schemes.
 ========

regex_match $f : X \to S$
nn candidate X \to
nn candidate a morphism
nn a morphism
new_annotation [(401035, 401048, 'VAR'), (401052, 401062, 'TYPE')]
===== sent Let $s \in S$. Assume that
\begin{enumerate}
\item $f$ is locally of finite type,
\item $f$ is separated, and
\item $X_s$ has at most finitely many isolated points.
 ========

new_annotation []
===== sent \begin{lemma}
\label{lemma-etale-makes-integral-split}
Let $R \to S$ be an integral ring map. ========

regex_match $R \to S$
nn candidate an integral ring map
nn an integral ring map
new_annotation [(402115, 402124, 'VAR'), (402128, 402148, 'TYPE')]
===== sent Let $\mathfrak p \subset R$ be a prime
ideal. ========

regex_match $\mathfrak p \subset R$
nn candidate p \subset
nn candidate a prime
ideal
nn a prime
ideal
new_annotation [(402154, 402177, 'VAR'), (402181, 402194, 'TYPE')]
===== sent Let $S' \subset S$ be the $R$-subalgebra generated by our $x_i$.
Since $\Spec(S) \to \Spec(S')$ is surjective
(Algebra, Lemma \ref{algebra-lemma-integral-overring-surjective})
we conclude that
$\mathfrak q_i' = S' \cap \mathfrak q_i$ are the primes of
$S'$ over $\mathfrak p$. By our choice of $x_i$ we conclude
these primes are distinct that and
$\kappa(\mathfrak q'_i)_{sep} = \kappa(\mathfrak q_i)_{sep}$.
 ========

regex_match $S' \subset S$
nn candidate S
nn candidate \to
nn \to
new_annotation [(403723, 403737, 'VAR'), (403800, 403803, 'TYPE')]
===== sent In the general case, let $R^{sh}$ be the strict henselization of
$R_\mathfrak p$. Then we can apply the result of the first paragraph
to $R^{sh} \to S \otimes_R R^{sh}$. Consider the $m$ mutually orthogonal
idempotents in $S \otimes_R R^{sh}$ corresponding to the product
decomposition. ========

regex_match $R^{sh}$
nn candidate the strict henselization
nn the strict henselization
new_annotation [(405600, 405608, 'VAR'), (405612, 405636, 'TYPE')]
===== sent \begin{lemma}
\label{lemma-finite-type-separated}
Let $f : X \to S$ be a morphism of schemes.
 ========

regex_match $f : X \to S$
nn candidate X \to
nn candidate a morphism
nn a morphism
new_annotation [(407191, 407204, 'VAR'), (407208, 407218, 'TYPE')]
===== sent Let $S'$ be the normalization of $S$ in $X$, see
Morphisms, Definition \ref{morphisms-definition-normalization-X-in-Y}.
Picture:
$$
\xymatrix{
X \ar[rd]_f \ar[rr]_{f'} & & S' \ar[ld]^\nu \\
& ========

regex_match $S'$
nn candidate the normalization
nn the normalization
new_annotation [(407279, 407283, 'VAR'), (407287, 407304, 'TYPE')]
===== sent \end{enumerate}
Let $x \in U$ be arbitrary. ========

new_annotation []
===== sent This implies $(f')^{-1}V \subset U$ and $V \subset U'$. Since $x$ was
arbitrary we see that (a), (b), and (c) are true.

\medskip\noindent
Let $s = f(x)$. Let $(T, t) \to (S, s)$ be an elementary \'etale
neighbourhood. ========

new_annotation []
===== sent This implies $(f')^{-1}V \subset U$ and $V \subset U'$. Since $x$ was
arbitrary we see that (a), (b), and (c) are true.

\medskip\noindent
Let $s = f(x)$. Let $(T, t) \to (S, s)$ be an elementary \'etale
neighbourhood. ========

regex_match $(T, t) \to (S, s)$
nn candidate $(T
nn candidate an elementary \'etale
neighbourhood
nn an elementary \'etale
neighbourhood
new_annotation [(408901, 408920, 'VAR'), (408924, 408959, 'TYPE')]
===== sent Denote by a subscript ${}_T$ the base change to $T$.
Let $y = (x, t) \in X_T$ be the unique point in
the fibre $X_t$ lying over $x$. Note that $U_T \subset X_T$
is the set of points where $f_T$ is quasi-finite, see
Morphisms, Lemma \ref{morphisms-lemma-base-change-quasi-finite}.
Note that
$$
X_T \xrightarrow{f'_T} S'_T \xrightarrow{\nu_T} T
$$
is the normalization of $T$ in $X_T$, see
Lemma \ref{lemma-normalization-smooth-localization}.
Suppose that the claim holds for $y \in U_T \subset X_T \to S'_T \to T$, i.e.,
suppose that we can find an open neighbourhood
$f'_T(y) \in V' \subset S'_T$ such that $(f'_T)^{-1}V' \to V'$ is an
isomorphism. ========

regex_match $y = (x, t) \in X_T$
nn candidate y
nn candidate the unique point
nn the unique point
new_annotation [(409018, 409038, 'VAR'), (409042, 409058, 'TYPE')]
===== sent \end{proof}

\begin{lemma}
\label{lemma-quasi-finite-separated-quasi-affine}
Let $f : X \to S$ be a morphism of schemes.
 ========

regex_match $f : X \to S$
nn candidate X \to
nn candidate a morphism
nn a morphism
new_annotation [(411089, 411102, 'VAR'), (411106, 411116, 'TYPE')]
===== sent Let $S'$ be the normalization of $S$ in $X$, see
Morphisms, Definition \ref{morphisms-definition-normalization-X-in-Y}.
Picture:
$$
\xymatrix{
X \ar[rd]_f \ar[rr]_{f'} & & S' \ar[ld]^\nu \\
& S &
}
$$
Then $f'$ is a quasi-compact open immersion and $\nu$ is integral.
 ========

regex_match $S'$
nn candidate the normalization
nn the normalization
new_annotation [(411175, 411179, 'VAR'), (411183, 411200, 'TYPE')]
===== sent It follows that $f$ is quasi-affine by
Morphisms, Lemma \ref{morphisms-lemma-characterize-quasi-affine}.
\end{proof}

\begin{lemma}[Zariski's Main Theorem]
\label{lemma-quasi-finite-separated-pass-through-finite}
\begin{reference}
\cite[IV Corollary 18.12.13]{EGA}
\end{reference}
Let $f : X \to S$ be a morphism of schemes.
 ========

regex_match $f : X \to S$
nn candidate X \to
nn candidate a morphism
nn a morphism
new_annotation [(412178, 412191, 'VAR'), (412195, 412205, 'TYPE')]
===== sent \end{lemma}

\begin{proof}
Let $X \to S' \to S$ be as in the conclusion of
Lemma \ref{lemma-quasi-finite-separated-quasi-affine}.
By
Properties, Lemma
\ref{properties-lemma-integral-algebra-directed-colimit-finite}
we can write
$\nu_*\mathcal{O}_{S'} = \colim_{i \in I} \mathcal{A}_i$ as a
directed colimit of finite quasi-coherent $\mathcal{O}_X$-algebras
$\mathcal{A}_i \subset \nu_*\mathcal{O}_{S'}$. Then
$\pi_i : T_i = \underline{\Spec}_S(\mathcal{A}_i) \to S$
is a finite morphism for each $i$.
Note that the transition morphisms $T_{i'} \to T_i$ are affine
and that $S' = \lim T_i$.

\medskip\noindent
By Limits, Lemma \ref{limits-lemma-descend-opens}
there exists an $i$ and a quasi-compact open
$U_i \subset T_i$ whose inverse image in $S'$ equals
$f'(X)$. For $i' \geq i$ let $U_{i'}$ be the inverse image
of $U_i$ in $T_{i'}$. Then $X \cong f'(X) = \lim_{i' \geq i} U_{i'}$, see
Limits, Lemma \ref{limits-lemma-directed-inverse-system-has-limit}.
By Limits, Lemma \ref{limits-lemma-finite-type-eventually-closed} we see that
$X \to U_{i'}$ is a closed immersion for some $i' \geq i$.
(In fact $X \cong U_{i'}$ for sufficiently
large $i'$ but we don't need this.) ========

new_annotation []
===== sent \end{lemma}

\begin{proof}
Let $X \to S' \to S$ be as in the conclusion of
Lemma \ref{lemma-quasi-finite-separated-quasi-affine}.
By
Properties, Lemma
\ref{properties-lemma-integral-algebra-directed-colimit-finite}
we can write
$\nu_*\mathcal{O}_{S'} = \colim_{i \in I} \mathcal{A}_i$ as a
directed colimit of finite quasi-coherent $\mathcal{O}_X$-algebras
$\mathcal{A}_i \subset \nu_*\mathcal{O}_{S'}$. Then
$\pi_i : T_i = \underline{\Spec}_S(\mathcal{A}_i) \to S$
is a finite morphism for each $i$.
Note that the transition morphisms $T_{i'} \to T_i$ are affine
and that $S' = \lim T_i$.

\medskip\noindent
By Limits, Lemma \ref{limits-lemma-descend-opens}
there exists an $i$ and a quasi-compact open
$U_i \subset T_i$ whose inverse image in $S'$ equals
$f'(X)$. For $i' \geq i$ let $U_{i'}$ be the inverse image
of $U_i$ in $T_{i'}$. Then $X \cong f'(X) = \lim_{i' \geq i} U_{i'}$, see
Limits, Lemma \ref{limits-lemma-directed-inverse-system-has-limit}.
By Limits, Lemma \ref{limits-lemma-finite-type-eventually-closed} we see that
$X \to U_{i'}$ is a closed immersion for some $i' \geq i$.
(In fact $X \cong U_{i'}$ for sufficiently
large $i'$ but we don't need this.) ========

regex_match $U_{i'}$
nn candidate the inverse image
nn the inverse image
new_annotation [(413270, 413278, 'VAR'), (413282, 413299, 'TYPE')]
===== sent \end{proof}

\begin{lemma}
\label{lemma-characterize-finite}
Let $f : X \to S$ be a morphism of schemes.
 ========

regex_match $f : X \to S$
nn candidate X \to
nn candidate a morphism
nn a morphism
new_annotation [(413968, 413981, 'VAR'), (413985, 413995, 'TYPE')]
===== sent \begin{lemma}
\label{lemma-proper-finite-fibre-finite-in-neighbourhood}
Let $f : X \to S$ be a morphism of schemes.
 ========

regex_match $f : X \to S$
nn candidate X \to
nn candidate a morphism
nn a morphism
new_annotation [(416014, 416027, 'VAR'), (416031, 416041, 'TYPE')]
===== sent Let $s \in ========

new_annotation []
===== sent X$.
Let $Z = X \setminus U$. Then $s \not \in f(Z)$. Since $f$ is proper
the set $f(Z) \subset S$ is closed. ========

new_annotation []
===== sent Hence it is quasi-finite
(Morphisms, Lemma \ref{morphisms-lemma-quasi-finite-locally-quasi-compact}),
hence has finite fibres
(Morphisms, Lemma \ref{morphisms-lemma-quasi-finite}), hence
is finite by Lemma \ref{lemma-characterize-finite}.
\end{proof}

\begin{lemma}
\label{lemma-flat-proper-family-cannot-collapse-fibre}
Consider a commutative diagram of schemes
$$
\xymatrix{
X \ar[rr]_h \ar[rd]_f & & Y \ar[ld]^g \\
& S
}
$$
Let $s \in S$. Assume
\begin{enumerate}
\item $X \to S$ is a proper morphism,
\item $Y \to S$ is separated and locally of finite type, and
\item the image of $X_s \to Y_s$ is finite.
 ========

new_annotation []
===== sent \end{enumerate}
Then there is an open
subspace $U \subset S$ containing $s$ such that $X_U \to Y_U$
factors through a closed subscheme $Z \subset Y_U$ finite over $U$.
\end{lemma}

\begin{proof}
Let $Z \subset Y$ be the scheme theoretic image of $h$, see
Morphisms, Section \ref{morphisms-section-scheme-theoretic-image}.
 ========

regex_match $Z \subset Y$
nn candidate $Z \subset
nn candidate the scheme theoretic image
nn the scheme theoretic image
new_annotation [(417595, 417608, 'VAR'), (417612, 417638, 'TYPE')]
===== sent Hence $Z \to S$ is finite in an open neighbourhood of $s$ by
Lemma \ref{lemma-proper-finite-fibre-finite-in-neighbourhood}.
\end{proof}

\begin{lemma}
\label{lemma-quasi-finite-finite-over-dense-open}
Let $f : Y \to X$ be a quasi-finite morphism.
 ========

regex_match $f : Y \to X$
nn candidate : Y \to X$
nn candidate a quasi-finite morphism
nn a quasi-finite morphism
new_annotation [(418142, 418155, 'VAR'), (418159, 418182, 'TYPE')]
===== sent Let $\eta \in X$ be a generic point of an irreducible
component of $X$. We see from
Morphisms, Lemmas
\ref{morphisms-lemma-quasi-finite} and \ref{morphisms-lemma-generically-finite}
that there exists an open neighbourhood $\eta \in U_\eta$ such that
$f^{-1}(U_\eta) \cap V_j \to U_\eta$ is finite. ========

regex_match $\eta \in X$
nn candidate $
nn candidate a generic point
nn a generic point
new_annotation [(418897, 418909, 'VAR'), (418913, 418928, 'TYPE')]
===== sent To see this let $U' \subset U$ be an affine open.
 ========

regex_match $U' \subset U$
nn candidate $U
nn candidate an affine
nn an affine
new_annotation [(420435, 420449, 'VAR'), (420453, 420462, 'TYPE')]
===== sent \end{proof}

\begin{lemma}
\label{lemma-stratify-flat-fp-lqf-universally-bounded}
Let $f : X \to S$ be flat, locally of finite presentation, separated,
locally quasi-finite with universally bounded fibres. ========

new_annotation []
===== sent \end{lemma}

\begin{proof}
Let $n$ be an integer bounding the degree of the fibres of $X \to S$.
 ========

regex_match $n$
nn candidate an integer
nn an integer
new_annotation [(422240, 422243, 'VAR'), (422247, 422257, 'TYPE')]
===== sent \medskip\noindent
Let $S' = S \setminus S_n$ endowed with the reduced induced scheme
structure and set $X' = X \times_S S'$. Note that the degrees of fibres
of $X' \to S'$ are universally bounded by $n - 1$. By induction we find a
stratification $S' = S_0 \amalg \ldots \amalg S_{n - 1}$ adapted
to the morphism $X' \to S'$. We claim that $S = \coprod_{r = 0, \ldots, n} S_r$
works for the morphism $X \to S$. Let $g : T \to S$ be a morphism of schemes
and assume that $X \times_S T \to T$ is finite locally free of degree $r$.
As remarked above this implies that $r \leq n$. If $r = n$, then it is
clear that $T \to S$ factors through $S_n$. If $r ========

new_annotation []
===== sent \medskip\noindent
Let $S' = S \setminus S_n$ endowed with the reduced induced scheme
structure and set $X' = X \times_S S'$. Note that the degrees of fibres
of $X' \to S'$ are universally bounded by $n - 1$. By induction we find a
stratification $S' = S_0 \amalg \ldots \amalg S_{n - 1}$ adapted
to the morphism $X' \to S'$. We claim that $S = \coprod_{r = 0, \ldots, n} S_r$
works for the morphism $X \to S$. Let $g : T \to S$ be a morphism of schemes
and assume that $X \times_S T \to T$ is finite locally free of degree $r$.
As remarked above this implies that $r \leq n$. If $r = n$, then it is
clear that $T \to S$ factors through $S_n$. If $r ========

regex_match $g : T \to S$
nn candidate T \to
nn candidate a morphism
nn a morphism
new_annotation [(424286, 424299, 'VAR'), (424303, 424313, 'TYPE')]
===== sent Finally, the degree is $r$ as all the fibres have degree $r$.
\end{proof}

\begin{lemma}
\label{lemma-stratify-flat-fp-qf}
Let $f : X \to S$ be a morphism of schemes which is flat, locally of
finite presentation, separated, and quasi-finite. ========

regex_match $f : X \to S$
nn candidate X \to
nn candidate a morphism
nn a morphism
new_annotation [(425989, 426002, 'VAR'), (426006, 426016, 'TYPE')]
===== sent \end{proof}

\begin{lemma}
\label{lemma-stratify-flat-fp-lqf}
Let $f : X \to S$ be a flat, locally of finite presentation, separated, and
locally quasi-finite morphism of schemes. ========

regex_match $f : X \to S$
nn candidate X \to
nn candidate finite presentation
nn finite presentation
new_annotation [(428173, 428186, 'VAR'), (428209, 428228, 'TYPE')]
===== sent \end{proof}

\begin{lemma}
\label{lemma-go-down-with-annihilators}
Let $f : X \to S$ be a morphism of schemes which is flat, locally of
finite presentation, and locally quasi-finite. ========

regex_match $f : X \to S$
nn candidate X \to
nn candidate a morphism
nn a morphism
new_annotation [(429001, 429014, 'VAR'), (429018, 429028, 'TYPE')]
===== sent Let $T \subset X$ be the scheme theoretic support of the finite
$\mathcal{O}_X$-module $\Im(g : \mathcal{O}_X \to \mathcal{O}_X)$.
Note that $T$ is the support of $g$ as a section of $\mathcal{O}_X$
(Modules, Definition \ref{modules-definition-support}) and
for any open $V \subset X$ we have $g|_V \not = 0$ if and only if
$V \cap T \not = \emptyset$.
Let $r$ be the smallest integer such that $f(T) \subset Z_r$
set theoretically. ========

regex_match $T \subset X$
nn candidate $T \subset
nn candidate the scheme theoretic support
nn the scheme theoretic support
new_annotation [(429834, 429847, 'VAR'), (429851, 429879, 'TYPE')]
===== sent Let $T \subset X$ be the scheme theoretic support of the finite
$\mathcal{O}_X$-module $\Im(g : \mathcal{O}_X \to \mathcal{O}_X)$.
Note that $T$ is the support of $g$ as a section of $\mathcal{O}_X$
(Modules, Definition \ref{modules-definition-support}) and
for any open $V \subset X$ we have $g|_V \not = 0$ if and only if
$V \cap T \not = \emptyset$.
Let $r$ be the smallest integer such that $f(T) \subset Z_r$
set theoretically. ========

regex_match $r$
nn candidate the smallest integer
nn the smallest integer
new_annotation [(430187, 430190, 'VAR'), (430194, 430214, 'TYPE')]
===== sent Let $\xi \in T$ be a generic point of an irreducible
component of $T$ such that $f(\xi) \not \in Z_{r - 1}$ (and hence
$f(\xi) \in Z_r$). ========

regex_match $\xi \in T$
nn candidate $
nn candidate a generic point
nn a generic point
new_annotation [(430267, 430278, 'VAR'), (430282, 430297, 'TYPE')]
===== sent We may replace $S$ by an affine neighbourhood of
$f(\xi)$ contained in $S \setminus Z_{r - 1}$. Write $S = \Spec(A)$
and let $I = (a_1, \ldots, a_m) \subset A$ be a finitely generated ideal
such that $V(I) = Z_r$ (set theoretically, see
Algebra, Lemma \ref{algebra-lemma-qc-open}).
 ========

regex_match $I = (a_1, \ldots, a_m) \subset A$
nn candidate $I
nn candidate Algebra
nn Algebra
new_annotation [(430526, 430560, 'VAR'), (430638, 430645, 'TYPE')]
===== sent The lemma follows by taking $\mathcal{F}$ the
quasi-coherent sheaf associated to the $A$-module $A[I]$ and
the map $\mathcal{F}^{\oplus r} \to \pi_*\mathcal{O}_V$ the
one corresponding to $A[I]^{\oplus r} \subset A^{\oplus r} \to B$.
\end{proof}

\begin{lemma}
\label{lemma-separated-locally-quasi-finite-over-affine}
Let $f : X \to Y$ be a separated, locally quasi-finite morphism
with $Y$ affine. ========

regex_match $f : X \to Y$
nn candidate X \to
nn candidate $Y$ affine
nn $Y$ affine
new_annotation [(431898, 431911, 'VAR'), (431963, 431973, 'TYPE')]
===== sent Then every finite set of points of $X$ is contained
in an open affine of $X$.
\end{lemma}

\begin{proof}
Let $x_1, \ldots, x_n \in X$. Choose a quasi-compact open
$U \subset X$ with $x_i \in U$. ========

new_annotation []
===== sent Then $U \to Y$ is quasi-affine by
Lemma \ref{lemma-quasi-finite-separated-quasi-affine}.
Hence there exists an affine open $V \subset U$ containing
$x_1, \ldots, x_n$ by
Properties, Lemma \ref{properties-lemma-ample-finite-set-in-affine}.
\end{proof}

\begin{lemma}
\label{lemma-there-is-a-scheme-integral-over}
Let $U \to X$ be a surjective \'etale morphism of schemes. ========

regex_match $U \to X$
nn candidate $U
nn candidate a surjective \'etale morphism
nn a surjective \'etale morphism
new_annotation [(432486, 432495, 'VAR'), (432499, 432528, 'TYPE')]
===== sent $Z'$.
After doing this we have $Z' \times_Y V = Z$. Finally,
let $T \subset Y$ be the induced reduced closed subscheme structure
on $Y \setminus V$. Consider the morphism
$$
Z' \amalg T \longrightarrow X
$$
This is a surjective integral morphism by construction.
 ========

regex_match $T \subset Y$
nn candidate $T \subset
nn candidate the induced reduced closed subscheme structure
nn the induced reduced closed subscheme structure
new_annotation [(435083, 435096, 'VAR'), (435100, 435146, 'TYPE')]
===== sent Since $T \subset U$ it is clear that the morphism $T \to X$
factors through $U$. On the other hand, let $z \in Z'$
be a point. ========

new_annotation []
===== sent You can find these facts as
Morphisms, Lemmas
\ref{morphisms-lemma-base-change-flat},
\ref{morphisms-lemma-base-change-finite-presentation},
\ref{morphisms-lemma-base-change-finite-locally-free},
\ref{morphisms-lemma-composition-flat},
\ref{morphisms-lemma-composition-finite-presentation},
\ref{morphisms-lemma-composition-finite-locally-free},
\ref{morphisms-lemma-fppf-open}, and
\ref{morphisms-lemma-finite-proper}.

\medskip\noindent
Note that $X_Z \to Z$ is flat morphism of finite presentation
which has a section $s$ coming from $\sigma$. Let $X_Z^0$ denote
the subset of $X_Z$ defined in
Situation \ref{situation-connected-along-section}.
By
Lemma \ref{lemma-connected-along-section-open}
it is an open subset of $X_Z$.

\medskip\noindent
The pullback $X_{Z \times_Y Z}$ of $X$ to $Z \times_Y Z$ comes equipped
with two sections $s_0, s_1$, namely the base changes of $s$ by
$\text{pr}_0, \text{pr}_1 : Z \times_Y Z \to Z$. The construction of
Situation \ref{situation-connected-along-section}
gives two subsets $(X_{Z \times_Y Z})_{s_0}^0$ and
$(X_{Z \times_Y Z})_{s_1}^0$. By
Lemma \ref{lemma-base-change-connected-along-section}
these are the inverse images of $X_Z^0$ under the morphisms
$1_X \times \text{pr}_0, 1_X \times \text{pr}_1 : X_{Z \times_Y Z} \to X_Z$.
 ========

new_annotation []
===== sent \medskip\noindent
Let $(Z \times_Y Z)_y = \{z_1, \ldots, z_n\}$.
As $X_y$ is geometrically connected, we see that the fibres of
$(X_{Z \times_Y Z})_{s_0}^0$ and $(X_{Z \times_Y Z})_{s_1}^0$
over each $z_i$ agree (being equal to the whole fibre). ========

new_annotation []
===== sent \end{proof}

\begin{lemma}
\label{lemma-fibre-geometrically-connected-reduced}
Let $h : Y \to S$ be a morphism of schemes.
 ========

regex_match $h : Y \to S$
nn candidate a morphism
nn a morphism
new_annotation [(440710, 440723, 'VAR'), (440727, 440737, 'TYPE')]
===== sent Let $s \in S$ be a point.
 ========

regex_match $s \in S$
nn candidate $s
nn candidate a point
nn a point
new_annotation [(440754, 440763, 'VAR'), (440767, 440774, 'TYPE')]
===== sent Let $T \subset Y_s$ be an open subscheme.
 ========

regex_match $T \subset Y_s$
nn candidate $T \subset Y_s$
nn candidate an open subscheme
nn an open subscheme
new_annotation [(440780, 440795, 'VAR'), (440799, 440816, 'TYPE')]
===== sent \end{proof}

\begin{lemma}
\label{lemma-normal-morphism-irreducible}
Let $h : Y \to S$ be a morphism of schemes.
 ========

regex_match $h : Y \to S$
nn candidate a morphism
nn a morphism
new_annotation [(443258, 443271, 'VAR'), (443275, 443285, 'TYPE')]
===== sent Let $s \in S$ be a point.
 ========

regex_match $s \in S$
nn candidate $s
nn candidate a point
nn a point
new_annotation [(443302, 443311, 'VAR'), (443315, 443322, 'TYPE')]
===== sent Let $T \subset Y_s$ be an open subscheme.
 ========

regex_match $T \subset Y_s$
nn candidate $T \subset Y_s$
nn candidate an open subscheme
nn an open subscheme
new_annotation [(443328, 443343, 'VAR'), (443347, 443364, 'TYPE')]
===== sent \begin{lemma}
\label{lemma-local-structure-finite-type}
Let $f : X \to S$ be a morphism. ========

regex_match $f : X \to S$
nn candidate X \to
nn candidate a morphism
nn a morphism
new_annotation [(445428, 445441, 'VAR'), (445445, 445455, 'TYPE')]
===== sent Let $x \in X$ and set $s = f(x)$.
Assume that $f$ is locally of finite type and that $n = \dim_x(X_s)$.
Then there exists a commutative diagram
$$
\xymatrix{
X \ar[dd] & X' \ar[l]^g \ar[d]^\pi & x \ar@{|->}[dd] &
x' \ar@{|->}[l]  \ar@{|->}[d] \\
& Y \ar[d]^h & & y \ar@{|->}[d] \\
 ========

new_annotation []
===== sent \end{slogan}
Let $f : X \to S$ be a morphism. ========

regex_match $f : X \to S$
nn candidate X \to
nn candidate a morphism
nn a morphism
new_annotation [(447319, 447332, 'VAR'), (447336, 447346, 'TYPE')]
===== sent Let $x \in X$ and set $s = f(x)$.
Assume that $f$ is locally of finite type and that $n = \dim_x(X_s)$.
Then there exists a commutative diagram
$$
\xymatrix{
X \ar[dd] & X' \ar[l]^g \ar[d]^\pi & x \ar@{|->}[dd] &
x' \ar@{|->}[l] \ar@{|->}[d] \\
& Y' \ar[d]^h & & y' \ar@{|->}[d] \\
 ========

new_annotation []
===== sent Let $T$ be the connected component of $Y_s$ containing $y$.
As $Y_s$ is Noetherian we see that $T$ is open.
 ========

regex_match $T$
nn candidate the connected component
nn the connected component
new_annotation [(449067, 449070, 'VAR'), (449074, 449097, 'TYPE')]
===== sent At this point we have verified all assumption of
Lemma \ref{lemma-normal-morphism-irreducible}
hold for the morphism $h : Y \to S$ and open $T \subset Y_s$.
As a result of applying
Lemma \ref{lemma-normal-morphism-irreducible}
we obtain $e : S' \to S$, $s' \in S'$, $Y'$ as
in the commutative diagram
$$
\xymatrix{
X \ar[dd] & X' \ar[l]^g \ar[d]^\pi & X' \times_Y Y' \ar[l] \ar[d] &
x \ar@{|->}[dd] & x' \ar@{|->}[l] \ar@{|->}[d] &
(x', s') \ar@{|->}[l] \ar@{|->}[d] \\
& Y \ar[d]^h & Y' \ar[d] \ar[l] & & y \ar@{|->}[d] &
(y, s') \ar@{|->}[l] \ar@{|->}[d] \\
S \ar@{=}[r] & S & S' \ar[l]_e & s & s \ar@{=}[l] & s' \ar@{|->}[l]
}
$$
where $e : (S', s') \to (S, s)$ is an elementary \'etale neighbourhood,
and where $Y' \subset Y_{S'}$ is an open neighbourhood all of whose fibres
over $S'$ are geometrically irreducible, such that $Y'_{s'} = T$ via
the identification $Y_s = Y_{S', s'}$. Let $(y, s') \in Y'$ be the point
corresponding to $y \in T$; this is also the unique point of $Y \times_S S'$
lying over $y$ with residue field equal to $\kappa(y)$ which maps to $s'$
in $S'$. ========

regex_match $(y, s') \in Y'$
nn candidate the point
nn the point
new_annotation [(450638, 450654, 'VAR'), (450658, 450667, 'TYPE')]
===== sent Similarly, let $(x', s') \in X' \times_Y Y' \subset X' \times_S S'$
be the unique point over $x'$ with residue field equal to $\kappa(x')$
lying over $s'$. Then the outer part of this diagram is a solution to the
problem posed in the lemma. ========

new_annotation []
===== sent Let $W' = h(V')$. ========

new_annotation []
===== sent \begin{lemma}
\label{lemma-finite-morphism-single-point-in-fibre}
Let $\pi : X \to Y$ be a finite morphism.
 ========

regex_match $\pi : X \to Y$
nn candidate X \to
nn candidate a finite morphism
nn a finite morphism
new_annotation [(452390, 452405, 'VAR'), (452409, 452426, 'TYPE')]
===== sent Let $x \in X$ with $y = \pi(x)$ such that $\pi^{-1}(\{y\}) = \{x\}$.
Then
\begin{enumerate}
\item For every neighbourhood $U \subset X$ of $x$ in $X$, there
exists a neighbourhood ========

new_annotation []
===== sent \begin{lemma}
\label{lemma-dominate-fppf-etale-locally}
Let $S$ be a scheme. ========

regex_match $S$
nn candidate a scheme
nn a scheme
new_annotation [(454573, 454576, 'VAR'), (454580, 454588, 'TYPE')]
===== sent Let $\{S_i \to S\}_{i \in I}$ be an fppf covering.
 ========

regex_match $\{S_i \to S\}_{i \in I}$
nn candidate $
nn candidate \to S\}_{i \in I}$
nn candidate an fppf
nn an fppf
new_annotation [(454594, 454619, 'VAR'), (454623, 454630, 'TYPE')]
===== sent \end{proof}

\begin{lemma}
\label{lemma-dominate-fppf}
Let $S$ be a scheme. ========

regex_match $S$
nn candidate a scheme
nn a scheme
new_annotation [(456210, 456213, 'VAR'), (456217, 456225, 'TYPE')]
===== sent Let $\{S_i \to S\}_{i \in I}$ be an fppf covering.
 ========

regex_match $\{S_i \to S\}_{i \in I}$
nn candidate $
nn candidate \to S\}_{i \in I}$
nn candidate an fppf
nn an fppf
new_annotation [(456231, 456256, 'VAR'), (456260, 456267, 'TYPE')]
===== sent Then there exist
\begin{enumerate}
\item a Zariski open covering $S = \bigcup U_j$,
\item surjective finite locally free morphisms $W_j \to U_j$,
\item Zariski open coverings $W_j = \bigcup_k W_{j, k}$,
\item surjective finite locally free morphisms $T_{j, k} \to W_{j, k}$
\end{enumerate}
such that the fppf covering $\{T_{j, k} \to S\}$ refines the given
covering $\{S_i \to S\}$.
\end{lemma}

\begin{proof}
Let $\{V_a \to S\}_{a \in A}$ be the fppf covering found in
Lemma \ref{lemma-dominate-fppf-etale-locally}.
 ========

regex_match $\{V_a \to S\}_{a \in A}$
nn candidate S\}_{a \in
nn candidate the fppf
nn the fppf
new_annotation [(456692, 456717, 'VAR'), (456721, 456729, 'TYPE')]
===== sent \end{proof}

\begin{lemma}
\label{lemma-extend-integral-surjective-morphisms}
Let $S$ be a scheme. ========

regex_match $S$
nn candidate a scheme
nn a scheme
new_annotation [(457596, 457599, 'VAR'), (457603, 457611, 'TYPE')]
===== sent If $U \subset S$ is open and $V \to U$ is a surjective
integral morphism, then there exists a surjective integral
morphism $\overline{V} \to S$ with $\overline{V} \times_S U$
isomorphic to $V$ as schemes over $U$.
\end{lemma}

\begin{proof}
Let $V' \to S$ be the normalization of $S$ in $U$, see
Morphisms, Section \ref{morphisms-section-normalization-X-in-Y}.
By construction $V' \to S$ is integral.
 ========

regex_match $V' \to S$
nn candidate $V
nn candidate the normalization
nn the normalization
new_annotation [(457858, 457868, 'VAR'), (457872, 457889, 'TYPE')]
===== sent By Morphisms, Lemmas \ref{morphisms-lemma-normalization-localization} and
\ref{morphisms-lemma-normalization-in-integral} we see that
the inverse image of $U$ in $V'$ is $V$. Let $Z$ be the reduced
induced scheme structure on $S \setminus U$. Then
$\overline{V} = V' \amalg Z$ works.
 ========

regex_match $Z$
nn candidate the reduced
induced scheme structure
nn the reduced
induced scheme structure
new_annotation [(458193, 458196, 'VAR'), (458200, 458236, 'TYPE')]
===== sent \end{proof}

\begin{lemma}
\label{lemma-extend-finite-surjective-morphisms}
Let $S$ be a quasi-compact and quasi-separated scheme.
 ========

regex_match $S$
nn candidate a quasi-compact and quasi-separated scheme
nn a quasi-compact and quasi-separated scheme
new_annotation [(458378, 458381, 'VAR'), (458385, 458427, 'TYPE')]
===== sent $S$. After replacing $V'$ by the scheme theoretic image
of $V$ we may assume that $V$ is dense in $V'$.
It follows that $V' \times_S U = V$ because $V \to V' \times_S U$
is closed as $V$ is finite over $U$. Let $Z$ be the reduced
induced scheme structure on $S \setminus U$. Then
$\overline{V} = V' \amalg Z$ works.
 ========

regex_match $Z$
nn candidate the reduced
induced scheme structure
nn the reduced
induced scheme structure
new_annotation [(459053, 459056, 'VAR'), (459060, 459096, 'TYPE')]
===== sent \end{proof}

\begin{lemma}
\label{lemma-dominate-fppf-integral}
Let $S$ be a scheme. ========

regex_match $S$
nn candidate a scheme
nn a scheme
new_annotation [(459226, 459229, 'VAR'), (459233, 459241, 'TYPE')]
===== sent Let $\{S_i \to S\}_{i \in I}$ be an fppf covering.
 ========

regex_match $\{S_i \to S\}_{i \in I}$
nn candidate $
nn candidate \to S\}_{i \in I}$
nn candidate an fppf
nn an fppf
new_annotation [(459247, 459272, 'VAR'), (459276, 459283, 'TYPE')]
===== sent Finally, the morphism $S' \to S$ is integral and surjective
by Limits, Lemmas \ref{limits-lemma-infinite-product-integral} and
\ref{limits-lemma-infinite-product-surjective}.
\end{proof}

\begin{lemma}
\label{lemma-dominate-fppf-finite}
Let $S$ be a quasi-compact and quasi-separated scheme.
 ========

regex_match $S$
nn candidate a quasi-compact and quasi-separated scheme
nn a quasi-compact and quasi-separated scheme
new_annotation [(460783, 460786, 'VAR'), (460790, 460832, 'TYPE')]
===== sent Let $\{S_i \to S\}_{i \in I}$ be an fppf covering.
 ========

regex_match $\{S_i \to S\}_{i \in I}$
nn candidate \to S\}_{i \in I}$
nn candidate an fppf
nn an fppf
new_annotation [(460838, 460863, 'VAR'), (460867, 460874, 'TYPE')]
===== sent Then there exists a surjective finite morphism $S' \to S$
of finite presentation and an
open covering $S' = \bigcup U'_\alpha$ such that for each $\alpha$ the
morphism $U'_\alpha \to S$ factors through $S_i \to S$ for some $i$.
\end{lemma}

\begin{proof}
Let $Y \to X$ be the integral surjective morphism found in
Lemma \ref{lemma-dominate-fppf-integral}.
Choose a finite affine open covering $Y = \bigcup V_j$
such that $V_j \to X$ factors through $S_{i(j)}$.
We can write $Y = \lim Y_\lambda$ with
$Y_\lambda \to X$ finite and of finite presentation, see
Limits, Lemma \ref{limits-lemma-integral-limit-finite-and-finite-presentation}.
 ========

regex_match $Y \to X$
nn candidate $Y
nn candidate the integral surjective morphism
nn the integral surjective morphism
new_annotation [(461144, 461153, 'VAR'), (461157, 461189, 'TYPE')]
===== sent \end{lemma}

\begin{proof}
Let $\{T_i \to T\}$ be an fppf covering of schemes, see
Topologies, Definition \ref{topologies-definition-fppf-covering}.
Observe that $T_i \to T$ is locally of finite type.
 ========

regex_match $\{T_i \to T\}$
nn candidate an fppf
nn an fppf
new_annotation [(462113, 462128, 'VAR'), (462132, 462139, 'TYPE')]
===== sent Let $U \subset T$ be an affine open.
 ========

regex_match $U \subset T$
nn candidate $U
nn candidate an affine
nn an affine
new_annotation [(462287, 462300, 'VAR'), (462304, 462313, 'TYPE')]
===== sent \begin{lemma}
\label{lemma-quasi-projective}
Let $S$ be a scheme which has an ample invertible sheaf.
 ========

regex_match $S$
nn candidate a scheme
nn a scheme
new_annotation [(464063, 464066, 'VAR'), (464070, 464078, 'TYPE')]
===== sent Let $f : X \to S$ be a morphism of schemes. ========

regex_match $f : X \to S$
nn candidate X \to
nn candidate a morphism
nn a morphism
new_annotation [(464120, 464133, 'VAR'), (464137, 464147, 'TYPE')]
===== sent Choose an immersion $i : X \to \mathbf{P}^n_S$ over $S$.
Let $\mathcal{L}$ be an ample invertible sheaf on $S$. ========

regex_match $\mathcal{L}$
nn candidate an ample invertible sheaf
nn an ample invertible sheaf
new_annotation [(465783, 465796, 'VAR'), (465800, 465825, 'TYPE')]
===== sent Let $s_i \in \Gamma(\mathbf{P}^n_S, \mathcal{O}(1))$
be the global section corresponding to the section $T_i$
of $\mathcal{O}_S T_0 \oplus \ldots \oplus \mathcal{O}_S ========

new_annotation []
===== sent \medskip\noindent
The equivalence of (1) and (5) follows from
Morphisms, Lemmas \ref{morphisms-lemma-ample-very-ample} and
\ref{morphisms-lemma-finite-type-ample-very-ample}.
\end{proof}

\begin{lemma}
\label{lemma-category-quasi-projective}
Let $S$ be a scheme which has an ample invertible sheaf.
 ========

regex_match $S$
nn candidate a scheme
nn a scheme
new_annotation [(467039, 467042, 'VAR'), (467046, 467054, 'TYPE')]
===== sent Let $\text{QP}_S$ be the full subcategory of the
category of schemes over $S$ satisfying the equivalent
conditions of Lemma \ref{lemma-quasi-projective}.
\begin{enumerate}
\item if $S' \to S$ is a morphism of schemes and $S'$ has
an ample invertible sheaf, then base change determines
a functor $\text{QP}_S \to \text{QP}_{S'}$,
\item if $X \in \text{QP}_S$ and $Y \in \text{QP}_X$, then $Y \in \text{QP}_S$,
\item the category $\text{QP}_S$ is closed under fibre products,
\item the category $\text{QP}_S$ is closed under
finite disjoint unions,
\item if $X \to S$ is projective, then $X \in \text{QP}_S$,
\item if $X \to S$ is quasi-affine of finite type, then
$X$ is in $\text{QP}_S$,
\item if $X \to S$ is quasi-finite and separated, then
$X \in \text{QP}_S$,
\item if $X \to S$ is a quasi-compact immersion, then
$X \in \text{QP}_S$,
\item add more here.
 ========

regex_match $\text{QP}_S$
nn candidate the full subcategory
nn the full subcategory
new_annotation [(467096, 467109, 'VAR'), (467113, 467133, 'TYPE')]
===== sent \begin{lemma}
\label{lemma-projective}
Let $S$ be a scheme which has an ample invertible sheaf.
 ========

regex_match $S$
nn candidate a scheme
nn a scheme
new_annotation [(469431, 469434, 'VAR'), (469438, 469446, 'TYPE')]
===== sent Let $f : X \to S$ be a morphism of schemes. ========

regex_match $f : X \to S$
nn candidate X \to
nn candidate a morphism
nn a morphism
new_annotation [(469488, 469501, 'VAR'), (469505, 469515, 'TYPE')]
===== sent \end{proof}

\begin{lemma}
\label{lemma-category-projective}
Let $S$ be a scheme which has an ample invertible sheaf.
 ========

regex_match $S$
nn candidate a scheme
nn a scheme
new_annotation [(472219, 472222, 'VAR'), (472226, 472234, 'TYPE')]
===== sent Let $\text{P}_S$ be the full subcategory of the
category of schemes over $S$ satisfying the equivalent
conditions of Lemma \ref{lemma-projective}.
\begin{enumerate}
\item if $S' \to S$ is a morphism of schemes and $S'$ has
an ample invertible sheaf, then base change determines
a functor $\text{P}_S \to \text{P}_{S'}$,
\item if $X \in \text{P}_S$ and $Y \in \text{P}_X$, then $Y \in \text{P}_S$,
\item the category $\text{P}_S$ is closed under fibre products,
\item the category $\text{P}_S$ is closed under
finite disjoint unions,
\item if $X \to S$ is finite, then $X$ is in $\text{P}_S$,
\item add more here.
 ========

regex_match $\text{P}_S$
nn candidate the full subcategory
nn the full subcategory
new_annotation [(472276, 472288, 'VAR'), (472292, 472312, 'TYPE')]
===== sent \begin{lemma}
\label{lemma-ample-in-neighbourhood}
\begin{reference}
\cite[IV Corollary 9.6.4]{EGA}
\end{reference}
Let $f : X \to Y$ be a proper morphism of schemes.
 ========

regex_match $f : X \to Y$
nn candidate X \to
nn candidate a proper morphism
nn a proper morphism
new_annotation [(474041, 474054, 'VAR'), (474058, 474075, 'TYPE')]
===== sent Let $\mathcal{L}$ be an invertible $\mathcal{O}_X$-module.
 ========

regex_match $\mathcal{L}$
nn candidate an invertible $\mathcal{O}_X$-module
nn an invertible $\mathcal{O}_X$-module
new_annotation [(474092, 474105, 'VAR'), (474109, 474145, 'TYPE')]
===== sent Let $y \in Y$ be a point such that $\mathcal{L}_y$ is ample
on $X_y$. Then there is an open neighbourhood $V \subset Y$
of $y$ such that $\mathcal{L}|_{f^{-1}(V)}$ is ample on $f^{-1}(V)/V$.
\end{lemma}

\begin{proof}
We may assume $Y$ is affine. ========

regex_match $y \in Y$
nn candidate $y
nn candidate a point
nn a point
new_annotation [(474151, 474160, 'VAR'), (474164, 474171, 'TYPE')]
===== sent Then we find a directed set $I$
and an inverse system of morphisms $X_i \to Y_i$ of schemes
with $Y_i$ of finite type over $\mathbf{Z}$, with affine
transition morphisms $X_i \to X_{i'}$ and $Y_i \to Y_{i'}$,
with $X_i \to Y_i$ proper, such that $X \to Y = \lim (X_i \to Y_i)$.
See Limits, Lemma
\ref{limits-lemma-proper-limit-of-proper-finite-presentation-noetherian}.
After shrinking $I$ we can assume we have a compatible system of
invertible $\mathcal{O}_{X_i}$-modules $\mathcal{L}_i$
pulling back to $\mathcal{L}$, see
Limits, Lemma \ref{limits-lemma-descend-invertible-modules}.
Let $y_i \in Y_i$ be the image of $y$.
Then $\kappa(y) = \colim \kappa(y_i)$.
Hence for some $i$ we have $\mathcal{L}_{i, y_i}$
is ample on $X_{i, y_i}$ by
Limits, Lemma \ref{limits-lemma-limit-ample}.
 ========

regex_match $y_i \in Y_i$
nn candidate the image
nn the image
new_annotation [(474984, 474997, 'VAR'), (475001, 475010, 'TYPE')]
===== sent \end{proof}











\section{Closed points in fibres}
\label{section-closed-points-fibres}

\noindent
Some of the material in this section is taken from the preprint
\cite{Osserman-Payne}.

\begin{lemma}
\label{lemma-locally-principal-vertical}
Let $f : X \to S$ be a morphism of schemes.
 ========

regex_match $f : X \to S$
nn candidate X \to
nn candidate a morphism
nn a morphism
new_annotation [(476019, 476032, 'VAR'), (476036, 476046, 'TYPE')]
===== sent Let $Z \subset X$ be a closed subscheme.
 ========

regex_match $Z \subset X$
nn candidate $Z \subset
nn candidate a closed subscheme
nn a closed subscheme
new_annotation [(476063, 476076, 'VAR'), (476080, 476098, 'TYPE')]
===== sent Let $s \in ========

new_annotation []
===== sent S$.
Assume
\begin{enumerate}
\item $S$ is irreducible with generic point $\eta$,
\item $X$ is irreducible,
\item $f$ is dominant,
\item $f$ is locally of finite type,
\item $\dim(X_s) \leq \dim(X_\eta)$,
\item $Z$ is locally principal in $X$, and
\item $Z_\eta = \emptyset$.
\end{enumerate}
Then the fibre $Z_s$ is (set theoretically) a union of
irreducible components of $X_s$.
\end{lemma}

\begin{proof}
Let $X_{red}$ denote the reduction of $X$. Then $Z \cap X_{red}$ is
a locally principal closed subscheme of $X_{red}$, see
Divisors, Lemma \ref{divisors-lemma-pullback-locally-principal}.
 ========

new_annotation []
===== sent Hence after replacing $X$ by an
open neighbourhood of $x$ we may assume that
$\dim_z(Z_{f(z)}) \leq d - 1$ for all $z \in Z$, see
Morphisms, Lemma \ref{morphisms-lemma-openness-bounded-dimension-fibres}.
Let $\xi' \in Z$ be a generic point of an irreducible component of $Z$
and set $s' = f(\xi)$. As $Z \not = X$ is locally principal we see that
$\dim(\mathcal{O}_{X, \xi}) = 1$, see
Algebra, Lemma \ref{algebra-lemma-minimal-over-1}
(this is where we use $X$ is Noetherian).
 ========

regex_match $\xi' \in Z$
nn candidate $\xi' \in
nn candidate a generic point
nn a generic point
new_annotation [(478416, 478428, 'VAR'), (478432, 478447, 'TYPE')]
===== sent Let $\xi \in X$ be the generic point of $X$ and
let $\xi_1$ be a generic point of any irreducible component
of $X_{s'}$ which contains $\xi'$. Then we see that we have
the specializations
$$
\xi \leadsto \xi_1 \leadsto \xi'.
 ========

regex_match $\xi \in X$
nn candidate the generic point
nn the generic point
new_annotation [(478689, 478700, 'VAR'), (478704, 478721, 'TYPE')]
===== sent Let $\xi \in X$ be the generic point of $X$ and
let $\xi_1$ be a generic point of any irreducible component
of $X_{s'}$ which contains $\xi'$. Then we see that we have
the specializations
$$
\xi \leadsto \xi_1 \leadsto \xi'.
 ========

regex_match $\xi_1$
nn candidate a generic point
nn a generic point
new_annotation [(478737, 478744, 'VAR'), (478748, 478763, 'TYPE')]
===== sent If the lemma is false then there exists a point
$x \in X$ lying over $s$ such that $x$ is a generic point of an
irreducible component of $Z_s$, but
not a generic point of any of the irreducible components of $X_s$.
Let $U \subset S$ be an affine neighbourhood of $s$ and let
$V \subset X$ be an affine neighbourhood of $x$ with $f(V) \subset U$.
Write ========

regex_match $U \subset S$
nn candidate $U \subset S$
nn candidate an affine neighbourhood
nn an affine neighbourhood
new_annotation [(479684, 479697, 'VAR'), (479701, 479724, 'TYPE')]
===== sent Let $\mathfrak q \subset B$,
resp.\ $\mathfrak p \subset A$ be the prime corresponding to $x$, resp.\ $s$.
After possibly shrinking $V$ we may assume $Z \cap V$ is cut out by
some element $g \in B$. Denote $K = f.f.(A)$. What we know at this point
is the following:
\begin{enumerate}
\item $A \subset B$ is a finitely generated extension of domains,
\item the element $g \otimes 1$ is invertible in $B \otimes_A K$,
\item $d = \dim(B \otimes_A K) = \dim(B \otimes_A \kappa(\mathfrak p))$,
\item $g \otimes 1$ is not a unit of $B \otimes_A \kappa(\mathfrak p)$, and
\item $g \otimes 1$ is not in any of the minimal primes of
$B \otimes_A \kappa(\mathfrak p)$.
\end{enumerate}
We are seeking a contradiction.

 ========

new_annotation []
===== sent For a finitely generated $\mathbf{Z}$-algebra $A_0 \subset A$
let $B_0 \subset B$ be the $A_0$-subalgebra generated by
$x_1, \ldots, x_n$, denote $K_0 = f.f.(A_0)$, and set
$\mathfrak p_0 = A_0 \cap \mathfrak p$.
We claim that when $A_0$ is large enough then (1) -- (5) also hold for
the system $(A_0 \subset B_0, g, \mathfrak p_0)$.

\medskip\noindent
We prove each of the conditions in turn. ========

regex_match $B_0 \subset B$
nn candidate the $A_0$-subalgebra
nn the $A_0$-subalgebra
new_annotation [(480760, 480775, 'VAR'), (480779, 480799, 'TYPE')]
===== sent The fourth assumption of the lemma holds if $A \to B$ is flat, see
Lemma \ref{lemma-equality-dimensions}.

\begin{lemma}
\label{lemma-horizontal}
Let $A \to B$ be a local homomorphism of local rings, and
$g \in \mathfrak m_B$. Assume
\begin{enumerate}
\item $A$ and $B$ are domains and $A \subset B$,
\item $B$ is essentially of finite type over $A$,
\item $g$ is not contained in any minimal prime over $\mathfrak m_AB$, and
\item $\dim(B/\mathfrak m_AB) +
 ========

regex_match $A \to B$
nn candidate a local homomorphism
nn a local homomorphism
new_annotation [(483269, 483278, 'VAR'), (483282, 483302, 'TYPE')]
===== sent \Spec(A)$.
\end{lemma}

\begin{proof}
Note that the two assertions are equivalent by
Algebra, Lemma \ref{algebra-lemma-image-dense-generic-points}.
To start the proof let $C$ be an $A$-algebra of finite type
and $\mathfrak q$ a prime of $C$ such that $B = C_{\mathfrak q}$.
Of course we may assume that $C$ is a domain and that $g \in C$.
 ========

regex_match $C$
nn candidate $C$
nn candidate finite type
nn finite type
new_annotation [(483956, 483959, 'VAR'), (483981, 483992, 'TYPE')]
===== sent \end{proof}

\begin{lemma}
\label{lemma-equality-dimensions}
Let $A \to B$ be a local homomorphism of local rings. ========

regex_match $A \to B$
nn candidate a local homomorphism
nn a local homomorphism
new_annotation [(485692, 485701, 'VAR'), (485705, 485725, 'TYPE')]
===== sent \end{enumerate}
Then we have
$$
\dim(B/\mathfrak m_AB) +
\text{trdeg}_{\kappa(\mathfrak m_A)}(\kappa(\mathfrak m_B)) =
\text{trdeg}_A(B).
$$
\end{lemma}

\begin{proof}
Let $C$ be an $A$-algebra of finite type and $\mathfrak q$ a prime of $C$
such that $B = C_{\mathfrak q}$. We may assume $C$ is a domain.
We have
$\dim_{\mathfrak q}(C/\mathfrak m_AC) = \dim(B/\mathfrak m_AB) +
 ========

regex_match $C$
nn candidate finite type
nn finite type
new_annotation [(486070, 486073, 'VAR'), (486095, 486106, 'TYPE')]
===== sent \medskip\noindent
Let $A \subset B$ be as in the lemma with $A$ a valuation ring.
 ========

new_annotation []
===== sent \end{proof}

\begin{lemma}
\label{lemma-closed-point-nearby-fibre}
Let $f : X \to S$ be a morphism of schemes.
 ========

regex_match $f : X \to S$
nn candidate X \to
nn candidate a morphism
nn a morphism
new_annotation [(488135, 488148, 'VAR'), (488152, 488162, 'TYPE')]
===== sent Let $x \leadsto x'$ be a specialization of points in $X$.
Set $s = f(x)$ and $s' = f(x')$.
Assume
\begin{enumerate}
\item $x'$ is a closed point of $X_{s'}$, and
\item $f$ is locally of finite type.
 ========

regex_match $x \leadsto x'$
nn candidate a specialization
nn a specialization
new_annotation [(488179, 488194, 'VAR'), (488198, 488214, 'TYPE')]
===== sent Let $A = B \cap \kappa(s)$. Note that this is a valuation ring (see
Algebra, Lemma \ref{algebra-lemma-valuation-ring-cap-field})
which dominates the image of $\mathcal{O}_{S, s'} \to \kappa(s)$.
Consider the commutative diagram
$$
\xymatrix{
\Spec(B) \ar[rd] \ar[r] &
X_A \ar[d] \ar[r] & X \ar[d] \\
& \Spec(A) \ar[r] & S
}
$$
The generic (resp.\ closed) point of $B$ maps to a point $x_A$
(resp.\ $x'_A$) of $X_A$ lying over the generic (resp.\ closed)
point of $\Spec(A)$. Note that $x'_A$ is a closed point
of the special fibre of $X_A$ by
Morphisms,
Lemma \ref{morphisms-lemma-base-change-closed-point-fibre-locally-finite-type}.
Note that the generic fibre of $X_A \to \Spec(A)$ is isomorphic
to $X_s$. ========

new_annotation []
===== sent Assume $\dim_x(X_\eta) > 0$.

\medskip\noindent
Let $X' \subset X$ be the reduced induced scheme structure on
the irreducible closed subscheme $\overline{\{x\}}$ of $X$, see
 ========

regex_match $X' \subset X$
nn candidate $X
nn candidate the reduced induced scheme structure
nn the reduced induced scheme structure
new_annotation [(490136, 490150, 'VAR'), (490154, 490190, 'TYPE')]
===== sent Note that in
this case $\dim_x(X_\eta) = \dim(X_\eta) = \dim(X_{s'})$, and that in fact
$X_{s'}$ is equidimensional, see
Algebra,
Lemma \ref{algebra-lemma-finite-type-domain-over-valuation-ring-dim-fibres}.

\medskip\noindent
Let $W \subset X_\eta$ be a proper closed subset (this is the
subset we want to ``avoid''). ========

regex_match $W \subset X_\eta$
nn candidate $W
nn candidate a proper closed subset
nn a proper closed subset
new_annotation [(490890, 490908, 'VAR'), (490912, 490934, 'TYPE')]
===== sent Let $\mathfrak q \subset ========

new_annotation []
===== sent B$
be the maximal ideal corresponding to the point $x'$.
Let $\mathfrak p_1, \ldots, \mathfrak p_s \subset B$ be the
minimal primes lying over $\mathfrak m_AB$. There are finitely
many as these correspond to the irreducible components of the
Noetherian scheme $X_{s'}$. ========

new_annotation []
===== sent \mathfrak p_i$
for all $i$, see
Algebra, Lemma \ref{algebra-lemma-silly}.
Denote $Z \subset X$ the locally principal closed subscheme defined by $g$.
Let $Z_\eta = Z_{1, \eta} \cup \ldots \cup Z_{n, \eta}$, $n \geq 0$
be the decomposition of the generic fibre of $Z$ into irreducible
components (finitely many as the generic fibre is Noetherian).
 ========

new_annotation []
===== sent \begin{lemma}
\label{lemma-quasi-finite-quasi-section-meeting-nearby-open}
Let $\varphi : A \to B$ be a local ring map of local rings.
 ========

regex_match $\varphi : A \to B$
nn candidate a local ring map
nn a local ring map
new_annotation [(494079, 494098, 'VAR'), (494102, 494118, 'TYPE')]
===== sent Let $V \subset \Spec(B)$ be an open subscheme
which contains at least one prime not lying over $\mathfrak m_A$.
Assume $A$ is Noetherian, $\varphi$ essentially of finite type, and
$A/\mathfrak m_A \subset B/\mathfrak m_B$ is finite.
 ========

regex_match $V \subset \Spec(B)$
nn candidate $V
nn candidate an open subscheme
nn an open subscheme
new_annotation [(494139, 494159, 'VAR'), (494163, 494180, 'TYPE')]
===== sent \end{proof}

\begin{lemma}
\label{lemma-quasi-finite-quasi-section-meeting-nearby-open-X}
Let $f : X \to S$ be a morphism of schemes.
 ========

regex_match $f : X \to S$
nn candidate X \to
nn candidate a morphism
nn a morphism
new_annotation [(495695, 495708, 'VAR'), (495712, 495722, 'TYPE')]
===== sent Let $x \in X$ with image $s \in ========

new_annotation []
===== sent S$.
Let $U \subset X$ be an open subscheme.
 ========

regex_match $U \subset X$
nn candidate $U
nn candidate an open subscheme
nn an open subscheme
new_annotation [(495775, 495788, 'VAR'), (495792, 495809, 'TYPE')]
===== sent Let $z \in X$ be the
image of the point $\mathfrak q$ under the canonical
morphism $\Spec(B) \to X$. Set $Z = \overline{\{z\}}$
with the induced reduced scheme structure. ========

new_annotation []
===== sent \begin{lemma}
\label{lemma-stein-universally-closed}
Let $S$ be a scheme. ========

regex_match $S$
nn candidate a scheme
nn a scheme
new_annotation [(499691, 499694, 'VAR'), (499698, 499706, 'TYPE')]
===== sent Let $f : X \to S$ be a universally closed and
quasi-separated morphism. ========

regex_match $f : X \to S$
nn candidate $
nn candidate X \to
nn candidate a universally closed and
quasi-separated morphism
nn a universally closed and
quasi-separated morphism
new_annotation [(499712, 499725, 'VAR'), (499729, 499778, 'TYPE')]
===== sent \end{proof}

\begin{lemma}
\label{lemma-characterize-geometrically-connected-fibres}
Let $f : X \to S$ be a morphism of schemes.
 ========

regex_match $f : X \to S$
nn candidate X \to
nn candidate a morphism
nn a morphism
new_annotation [(503575, 503588, 'VAR'), (503592, 503602, 'TYPE')]
===== sent Let $s \in ========

new_annotation []
===== sent \end{proof}

\begin{theorem}[Stein factorization; Noetherian case]
\label{theorem-stein-factorization-Noetherian}
Let $S$ be a locally Noetherian scheme.
 ========

regex_match $S$
nn candidate a locally Noetherian scheme
nn a locally Noetherian scheme
new_annotation [(504557, 504560, 'VAR'), (504564, 504591, 'TYPE')]
===== sent Let $f : X \to S$ be a proper morphism.
 ========

regex_match $f : X \to S$
nn candidate X \to
nn candidate a proper morphism
nn a proper morphism
new_annotation [(504597, 504610, 'VAR'), (504614, 504631, 'TYPE')]
===== sent There exists a factorization
$$
\xymatrix{
X \ar[rr]_{f'} \ar[rd]_f & & S' \ar[dl]^\pi \\
& S &
}
$$
with the following properties:
\begin{enumerate}
\item the morphism $f'$ is proper with geometrically connected fibres,
\item the morphism $\pi : S' \to S$ is finite,
\item we have $f'_*\mathcal{O}_X = \mathcal{O}_{S'}$,
\item we have $S' = \underline{\Spec}_S(f_*\mathcal{O}_X)$, and
\item $S'$ is the normalization of $S$ in $X$, see
Morphisms, Definition \ref{morphisms-definition-normalization-X-in-Y}.
\end{enumerate}
\end{theorem}

\begin{proof}
Let $f = \pi \circ f'$ be the factorization of
Lemma \ref{lemma-stein-universally-closed}. Note that besides the
conclusions of Lemma \ref{lemma-stein-universally-closed} we
also have that $f'$ is separated
(Schemes, Lemma \ref{schemes-lemma-compose-after-separated})
and finite type
(Morphisms, Lemma \ref{morphisms-lemma-permanence-finite-type}).
 ========

regex_match $f = \pi \circ f'$
nn candidate the factorization
nn the factorization
new_annotation [(505190, 505208, 'VAR'), (505212, 505229, 'TYPE')]
===== sent Let $s \in S$ be a point of $S$. We have to show that $X_s$ is
geometrically connected. ========

regex_match $s \in S$
nn candidate $s
nn candidate a point
nn a point
new_annotation [(506106, 506115, 'VAR'), (506119, 506126, 'TYPE')]
===== sent \end{proof}

\begin{theorem}[Stein factorization; general case]
\label{theorem-stein-factorization-general}
Let $S$ be a scheme.
 ========

regex_match $S$
nn candidate a scheme
nn a scheme
new_annotation [(507988, 507991, 'VAR'), (507995, 508003, 'TYPE')]
===== sent Let $f : X \to S$ be a proper morphism.
 ========

regex_match $f : X \to S$
nn candidate X \to
nn candidate a proper morphism
nn a proper morphism
new_annotation [(508009, 508022, 'VAR'), (508026, 508043, 'TYPE')]
===== sent Then $S' = \Spec(R')$.
Thus we may replace $S$ by $S'$ and assume that
$S = \Spec(R)$ is affine $R = \Gamma(X, \mathcal{O}_X)$.
Next, let $s \in S$ be a point. ========

regex_match $s \in S$
nn candidate a point
nn a point
new_annotation [(509334, 509343, 'VAR'), (509347, 509354, 'TYPE')]
===== sent Let $U \to S$ be an \'etale morphism
of affine schemes and let $u \in U$ be a point mapping to $s$.
Let $X_U \to U$ be the base change of $X$. By
Lemma \ref{lemma-characterize-geometrically-connected-fibres}
it suffices to show that the fibre of $X_U \to U$ over $u$ is
connected. ========

regex_match $U \to S$
nn candidate $
nn candidate an \'etale morphism
nn an \'etale morphism
new_annotation [(509360, 509369, 'VAR'), (509373, 509392, 'TYPE')]
===== sent Let $U \to S$ be an \'etale morphism
of affine schemes and let $u \in U$ be a point mapping to $s$.
Let $X_U \to U$ be the base change of $X$. By
Lemma \ref{lemma-characterize-geometrically-connected-fibres}
it suffices to show that the fibre of $X_U \to U$ over $u$ is
connected. ========

regex_match $u \in U$
nn candidate a point mapping
nn a point mapping
new_annotation [(509419, 509428, 'VAR'), (509432, 509447, 'TYPE')]
===== sent Let $U \to S$ be an \'etale morphism
of affine schemes and let $u \in U$ be a point mapping to $s$.
Let $X_U \to U$ be the base change of $X$. By
Lemma \ref{lemma-characterize-geometrically-connected-fibres}
it suffices to show that the fibre of $X_U \to U$ over $u$ is
connected. ========

regex_match $X_U \to U$
nn candidate the base change
nn the base change
new_annotation [(509460, 509471, 'VAR'), (509475, 509490, 'TYPE')]
===== sent Say $S_i = \Spec(R_i)$. Let $R'_i = \Gamma(X_i, \mathcal{O}_{X_i})$.
Observe that we have ring maps $R_i \to R_i' \to R$. ========

new_annotation []
===== sent $S'_i = \Spec(R'_i)$.
Let $s'_i \in S'_i$ be the image of $s$.
We have $X_s = \lim X_{i, s'_i}$ because $X = \lim X_i$,
$S = \lim S'_i$, and $\kappa(s) = \colim \kappa(s'_i)$.
Now let $X_s = U \amalg V$ with $U$ and $V$ open and closed.
 ========

regex_match $s'_i \in S'_i$
nn candidate the image
nn the image
new_annotation [(510746, 510761, 'VAR'), (510765, 510774, 'TYPE')]
===== sent $S'_i = \Spec(R'_i)$.
Let $s'_i \in S'_i$ be the image of $s$.
We have $X_s = \lim X_{i, s'_i}$ because $X = \lim X_i$,
$S = \lim S'_i$, and $\kappa(s) = \colim \kappa(s'_i)$.
Now let $X_s = U \amalg V$ with $U$ and $V$ open and closed.
 ========

new_annotation []
===== sent \begin{lemma}
\label{lemma-geometrically-connected-fibres-towards-normal}
Let $f : X \to Y$ be a morphism of schemes. ========

regex_match $f : X \to Y$
nn candidate X \to
nn candidate a morphism
nn a morphism
new_annotation [(511364, 511377, 'VAR'), (511381, 511391, 'TYPE')]
===== sent \begin{lemma}
\label{lemma-proper-flat-nr-geom-conn-comps-lower-semicontinuous}
Let $X \to S$ be a flat proper morphism of finite presentation. ========

regex_match $X \to S$
nn candidate $X
nn candidate a flat proper morphism
nn a flat proper morphism
new_annotation [(512798, 512807, 'VAR'), (512811, 512833, 'TYPE')]
===== sent \end{lemma}

\begin{proof}
Let $s \in S$. Set $n = n_{X/S}(s)$. Note that $n < \infty$ as the geometric
fibre of $X \to S$ at $s$ is a proper scheme over a field, hence Noetherian,
hence has a finite number of connected components. ========

new_annotation []
===== sent We have to find an open
neighbourhood $V$ of $s$ such that $n_{X/S}|_V \geq n$.
Let $X \to S' \to S$ be the Stein factorization as in
Theorem \ref{theorem-stein-factorization-general}.
 ========

regex_match $X \to S' \to S$
nn candidate $X
nn candidate the Stein factorization
nn the Stein factorization
new_annotation [(513414, 513430, 'VAR'), (513434, 513457, 'TYPE')]
===== sent Thus in a neighbourhood of $s$ we see that $n_{X/S}$ is
at least $n$.
\end{proof}

\begin{lemma}
\label{lemma-proper-flat-geom-red}
Let $f : X \to S$ be a morphism of schemes. ========

regex_match $f : X \to S$
nn candidate X \to
nn candidate a morphism
nn a morphism
new_annotation [(514373, 514386, 'VAR'), (514390, 514400, 'TYPE')]
===== sent \begin{lemma}
\label{lemma-split-off-proper-part-henselian}
Let $(A, I)$ be a henselian pair. ========

regex_match $(A, I)$
nn candidate $(A
nn candidate a henselian pair
nn a henselian pair
new_annotation [(515711, 515719, 'VAR'), (515723, 515739, 'TYPE')]
===== sent Let $X \to \Spec(A)$
be separated and of finite type. ========

new_annotation []
===== sent Set $X_0 = X \times_{\Spec(A)} \Spec(A/I)$.
Let $Y \subset X_0$ be an open and closed subscheme such that
$Y ========

regex_match $Y \subset X_0$
nn candidate $Y
nn candidate an open and closed subscheme
nn an open and closed subscheme
new_annotation [(515843, 515858, 'VAR'), (515862, 515890, 'TYPE')]
===== sent This is an open and closed subscheme
of $X'_0$. Suppose the lemma holds for $(X', Y')$. Let $W' \subset X'$
be the open and closed subscheme proper over $A$ such that $Y' = W'_0$.
 ========

new_annotation []
===== sent Let $\overline{X}$ be the scheme theoretic image of $j$.
Since $j$ is a quasi-compact morphism
(Schemes, Lemma \ref{schemes-lemma-quasi-compact-permanence})
we see that $j : X \to \overline{X}$ is an open immersion
(Morphisms, Lemma \ref{morphisms-lemma-quasi-compact-immersion}).
 ========

regex_match $\overline{X}$
nn candidate the scheme theoretic image
nn the scheme theoretic image
new_annotation [(517299, 517313, 'VAR'), (517317, 517343, 'TYPE')]
===== sent Suppose that the lemma holds for $(\overline{X}, j_0(Y))$.
Let $\overline{W} \subset \overline{X}$ be the
corresponding open and closed subscheme proper over $A$
such that $j_0(Y) = \overline{W}_0$.
 ========

new_annotation []
===== sent Let $Z = X_0 \setminus Y$. This is an open and closed
subscheme of $X_0$ and $X_0 = Y \amalg Z$.
 ========

new_annotation []
===== sent Let $X \to X' \to \Spec(A)$ be the Stein factorization as in
Theorem \ref{theorem-stein-factorization-general}.
Let $Y' \subset X'_0$ and $Z' \subset X'_0$ be the images of
$Y$ and $Z$.
 ========

regex_match $X \to X' \to \Spec(A)$
nn candidate $X
nn candidate the Stein factorization
nn the Stein factorization
new_annotation [(518784, 518807, 'VAR'), (518811, 518834, 'TYPE')]
===== sent Let $X \to X' \to \Spec(A)$ be the Stein factorization as in
Theorem \ref{theorem-stein-factorization-general}.
Let $Y' \subset X'_0$ and $Z' \subset X'_0$ be the images of
$Y$ and $Z$.
 ========

new_annotation []
===== sent Hence by
More on Algebra, Lemma \ref{more-algebra-lemma-characterize-henselian-pair}
we see that we may write $X' = W' \amalg V'$
with $W'$ and $V'$ open and closed and
with $Y' = W'_0$ and $Z' = V'_0$.
Let $W$ be the inverse image in $X$
to finish the proof.
 ========

regex_match $W$
nn candidate the inverse image
nn the inverse image
new_annotation [(519610, 519613, 'VAR'), (519617, 519634, 'TYPE')]
===== sent \begin{lemma}
\label{lemma-separated-locally-quasi-finite-morphisms-fppf-descend}
Let $S$ be a scheme.
 ========

regex_match $S$
nn candidate a scheme
nn a scheme
new_annotation [(520429, 520432, 'VAR'), (520436, 520444, 'TYPE')]
===== sent Let $\{X_i \to S\}_{i\in I}$ be an fppf covering, see
Topologies, Definition \ref{topologies-definition-fppf-covering}.
Let $(V_i/X_i, \varphi_{ij})$ be a descent datum
relative to $\{X_i \to S\}$. If each morphism
$V_i \to X_i$ is separated and locally quasi-finite,
then the descent datum is effective.
 ========

regex_match $\{X_i \to S\}_{i\in I}$
nn candidate \to S\}_{i\in I}$
nn candidate an fppf
nn an fppf
new_annotation [(520450, 520474, 'VAR'), (520478, 520485, 'TYPE')]
===== sent Let $\{X_i \to S\}_{i\in I}$ be an fppf covering, see
Topologies, Definition \ref{topologies-definition-fppf-covering}.
Let $(V_i/X_i, \varphi_{ij})$ be a descent datum
relative to $\{X_i \to S\}$. If each morphism
$V_i \to X_i$ is separated and locally quasi-finite,
then the descent datum is effective.
 ========

regex_match $(V_i/X_i, \varphi_{ij})$
nn candidate $(V_i/X_i
nn candidate a descent datum
nn a descent datum
new_annotation [(520570, 520595, 'VAR'), (520599, 520614, 'TYPE')]
===== sent Let $(V, \varphi)$ be a descent datum relative to $X$ over $S$
and assume that $\pi : V \to X$ is separated and
locally quasi-finite.

 ========

regex_match $(V, \varphi)$
nn candidate $(V
nn candidate a descent datum
nn a descent datum
new_annotation [(521376, 521390, 'VAR'), (521394, 521409, 'TYPE')]
===== sent \medskip\noindent
Let $W^1 \subset V$ be any affine open.
 ========

new_annotation []
===== sent This means the original descent datum is effective by glueing the
schemes obtained from descending the opens $W_i$, see
Descent, Lemma \ref{descent-lemma-effective-for-fpqc-is-local-upstairs}.
\end{proof}






\section{Relative finite presentation}
\label{section-finite-type-finite-presentation}

\noindent
Let $R \to A$ be a finite type ring map. ========

regex_match $R \to A$
nn candidate a finite type ring map
nn a finite type ring map
new_annotation [(523411, 523420, 'VAR'), (523424, 523446, 'TYPE')]
===== sent Let $M$ be an $A$-module. ========

regex_match $M$
new_annotation []
===== sent \begin{definition}
\label{definition-relatively-finitely-presented-sheaf}
Let $f : X \to S$ be a morphism of schemes which is locally of finite type.
 ========

regex_match $f : X \to S$
nn candidate X \to
nn candidate a morphism
nn a morphism
new_annotation [(523847, 523860, 'VAR'), (523864, 523874, 'TYPE')]
===== sent Let $\mathcal{F}$ be a quasi-coherent $\mathcal{O}_X$-module. ========

regex_match $\mathcal{F}$
nn candidate a quasi-coherent $\mathcal{O}_X$-module
nn a quasi-coherent $\mathcal{O}_X$-module
new_annotation [(523923, 523936, 'VAR'), (523940, 523979, 'TYPE')]
===== sent We will see that $X \to S$ is locally of finite presentation if
and only if $\mathcal{O}_X$ is of finite presentation relative to $S$.

\begin{lemma}
\label{lemma-relative-finite-presentation-characterize}
Let $f : X \to S$ be a morphism of schemes which is locally of finite type.
 ========

regex_match $f : X \to S$
nn candidate X \to
nn candidate a morphism
nn a morphism
new_annotation [(524840, 524853, 'VAR'), (524857, 524867, 'TYPE')]
===== sent Let $\mathcal{F}$ be a quasi-coherent $\mathcal{O}_X$-module. ========

regex_match $\mathcal{F}$
nn candidate a quasi-coherent $\mathcal{O}_X$-module
nn a quasi-coherent $\mathcal{O}_X$-module
new_annotation [(524916, 524929, 'VAR'), (524933, 524972, 'TYPE')]
===== sent Let $S = \bigcup V_i$ and $f^{-1}(V_i) = \bigcup U_{ij}$ be
affine open coverings as in
Definition \ref{definition-relatively-finitely-presented-sheaf}.
 ========

new_annotation []
===== sent Let $U \subset X$ and $V \subset S$ be as in (2).
 ========

new_annotation []
===== sent \end{proof}

\begin{lemma}
\label{lemma-relative-finite-presentation}
Let $f : X \to S$ be a morphism of schemes which is locally of finite
type. ========

regex_match $f : X \to S$
nn candidate X \to
nn candidate a morphism
nn a morphism
new_annotation [(527646, 527659, 'VAR'), (527663, 527673, 'TYPE')]
===== sent Let $\mathcal{F}$ be a quasi-coherent $\mathcal{O}_X$-module.
 ========

regex_match $\mathcal{F}$
nn candidate a quasi-coherent $\mathcal{O}_X$-module
nn a quasi-coherent $\mathcal{O}_X$-module
new_annotation [(527722, 527735, 'VAR'), (527739, 527778, 'TYPE')]
===== sent \item The morphism $f$ is locally of finite presentation if and only
if $\mathcal{O}_X$ is of finite presentation relative to $S$.
\end{enumerate}
\end{lemma}

\begin{proof}
Follows immediately from the definitions, see
discussion following
More on Algebra, Definition
\ref{more-algebra-definition-relatively-finitely-presented}.
\end{proof}

\begin{lemma}
\label{lemma-finite-morphism-relative-finite-presentation}
Let $\pi : X \to Y$ be a finite morphism of schemes locally of finite
type over a base scheme $S$. Let $\mathcal{F}$ be a quasi-coherent
$\mathcal{O}_X$-module. ========

regex_match $\pi : X \to Y$
nn candidate Let $\pi
nn candidate X \to
nn candidate a finite morphism
nn a finite morphism
new_annotation [(528383, 528398, 'VAR'), (528402, 528419, 'TYPE')]
===== sent \item The morphism $f$ is locally of finite presentation if and only
if $\mathcal{O}_X$ is of finite presentation relative to $S$.
\end{enumerate}
\end{lemma}

\begin{proof}
Follows immediately from the definitions, see
discussion following
More on Algebra, Definition
\ref{more-algebra-definition-relatively-finitely-presented}.
\end{proof}

\begin{lemma}
\label{lemma-finite-morphism-relative-finite-presentation}
Let $\pi : X \to Y$ be a finite morphism of schemes locally of finite
type over a base scheme $S$. Let $\mathcal{F}$ be a quasi-coherent
$\mathcal{O}_X$-module. ========

regex_match $\mathcal{F}$
nn candidate a quasi-coherent
$\mathcal{O}_X$-module
nn a quasi-coherent
$\mathcal{O}_X$-module
new_annotation [(528482, 528495, 'VAR'), (528499, 528538, 'TYPE')]
===== sent \end{proof}

\begin{lemma}
\label{lemma-base-change-relative-finite-presentation}
Let $f : X \to S$ be a morphism of schemes which is locally of finite
type. ========

regex_match $f : X \to S$
nn candidate X \to
nn candidate a morphism
nn a morphism
new_annotation [(528915, 528928, 'VAR'), (528932, 528942, 'TYPE')]
===== sent Let $\mathcal{F}$ be a quasi-coherent $\mathcal{O}_X$-module.
 ========

regex_match $\mathcal{F}$
nn candidate a quasi-coherent $\mathcal{O}_X$-module
nn a quasi-coherent $\mathcal{O}_X$-module
new_annotation [(528991, 529004, 'VAR'), (529008, 529047, 'TYPE')]
===== sent Let $S' \to S$ be a morphism of schemes, set $X' = X \times_S S'$
and denote $\mathcal{F}'$ the pullback of $\mathcal{F}$ to $X'$.
If $\mathcal{F}$ is of finite presentation relative to $S$, then
$\mathcal{F}'$ is of finite presentation relative to $S'$.
\end{lemma}

\begin{proof}
Translation of the result of
More on Algebra, Lemma
\ref{more-algebra-lemma-base-change-relative-finite-presentation}
into the language of schemes.
 ========

regex_match $S' \to S$
nn candidate S
nn candidate a morphism
nn a morphism
new_annotation [(529053, 529063, 'VAR'), (529067, 529077, 'TYPE')]
===== sent \end{proof}

\begin{lemma}
\label{lemma-pull-relative-finite-presentation}
Let $X \to Y \to S$ be morphisms of schemes which are locally of finite
type. ========

new_annotation []
===== sent Let $\mathcal{G}$ be a quasi-coherent $\mathcal{O}_Y$-module.
 ========

regex_match $\mathcal{G}$
nn candidate a quasi-coherent $\mathcal{O}_Y$-module
nn a quasi-coherent $\mathcal{O}_Y$-module
new_annotation [(529636, 529649, 'VAR'), (529653, 529692, 'TYPE')]
===== sent \end{proof}

\begin{lemma}
\label{lemma-composition-relative-finite-presentation}
Let $X \to Y \to S$ be morphisms of schemes which are locally of finite
type. ========

new_annotation []
===== sent Let $\mathcal{F}$ be a quasi-coherent $\mathcal{O}_X$-module.
 ========

regex_match $\mathcal{F}$
nn candidate a quasi-coherent $\mathcal{O}_X$-module
nn a quasi-coherent $\mathcal{O}_X$-module
new_annotation [(530200, 530213, 'VAR'), (530217, 530256, 'TYPE')]
===== sent \end{proof}

\begin{lemma}
\label{lemma-ses-relatively-finite-presentation}
Let $X \to S$ be a morphism of schemes which is locally of finite type.
 ========

regex_match $X \to S$
nn candidate $X
nn candidate a morphism
nn a morphism
new_annotation [(530683, 530692, 'VAR'), (530696, 530706, 'TYPE')]
===== sent Let $0 \to \mathcal{F}' \to \mathcal{F} \to \mathcal{F}'' \to 0$
be a short exact sequence of quasi-coherent $\mathcal{O}_X$-modules.
 ========

new_annotation []
===== sent \end{slogan}
Let $X \to S$ be a morphism of schemes which is locally of finite type.
 ========

regex_match $X \to S$
nn candidate $X
nn candidate a morphism
nn a morphism
new_annotation [(531581, 531590, 'VAR'), (531594, 531604, 'TYPE')]
===== sent Let $\mathcal{F}, \mathcal{F}'$ be quasi-coherent $\mathcal{O}_X$-modules.
 ========

new_annotation []
===== sent Hence, on a first reading we suggest the reader work exclusively
with objects in $D_\QCoh(\mathcal{O}_X)$.

\begin{lemma}
\label{lemma-relatively-pseudo-coherent}
Let $X \to S$ be a finite type morphism of affine schemes.
 ========

regex_match $X \to S$
nn candidate $X
nn candidate a finite type morphism
nn a finite type morphism
new_annotation [(533215, 533224, 'VAR'), (533228, 533250, 'TYPE')]
===== sent Let $E$ be an object of $D(\mathcal{O}_X)$.
Let $m \in \mathbf{Z}$.
The following are equivalent
\begin{enumerate}
\item for some closed immersion $i : X \to \mathbf{A}^n_S$
the object $Ri_*E$ of $D(\mathcal{O}_{\mathbf{A}^n_S})$
is $m$-pseudo-coherent, and
\item for all closed immersions $i : X \to \mathbf{A}^n_S$
the object $Ri_*E$ of $D(\mathcal{O}_{\mathbf{A}^n_S})$
is $m$-pseudo-coherent.
 ========

regex_match $E$
nn candidate an object
nn an object
new_annotation [(533274, 533277, 'VAR'), (533281, 533290, 'TYPE')]
===== sent Let $E$ be an object of $D(\mathcal{O}_X)$.
Let $m \in \mathbf{Z}$.
The following are equivalent
\begin{enumerate}
\item for some closed immersion $i : X \to \mathbf{A}^n_S$
the object $Ri_*E$ of $D(\mathcal{O}_{\mathbf{A}^n_S})$
is $m$-pseudo-coherent, and
\item for all closed immersions $i : X \to \mathbf{A}^n_S$
the object $Ri_*E$ of $D(\mathcal{O}_{\mathbf{A}^n_S})$
is $m$-pseudo-coherent.
 ========

new_annotation []
===== sent \end{enumerate}
\end{lemma}

\begin{proof}
Say $S = \Spec(R)$ and $X = \Spec(A)$. Let $i$ correspond to the surjection
$\alpha : R[x_1, \ldots, x_n] \to A$ and let $X \to \mathbf{A}^m_S$
correspond to $\beta : R[y_1, \ldots, y_m] \to A$.
 ========

new_annotation []
===== sent \end{enumerate}
\end{lemma}

\begin{proof}
Say $S = \Spec(R)$ and $X = \Spec(A)$. Let $i$ correspond to the surjection
$\alpha : R[x_1, \ldots, x_n] \to A$ and let $X \to \mathbf{A}^m_S$
correspond to $\beta : R[y_1, \ldots, y_m] \to A$.
 ========

new_annotation []
===== sent \begin{definition}
\label{definition-relative-pseudo-coherence}
Let $f : X \to S$ be a morphism of schemes which is locally of finite type.
 ========

regex_match $f : X \to S$
nn candidate X \to
nn candidate a morphism
nn a morphism
new_annotation [(535677, 535690, 'VAR'), (535694, 535704, 'TYPE')]
===== sent Let $E$ be an object of $D(\mathcal{O}_X)$. Let $\mathcal{F}$ be an
$\mathcal{O}_X$-module. ========

regex_match $E$
nn candidate an object
nn an object
new_annotation [(535753, 535756, 'VAR'), (535760, 535769, 'TYPE')]
===== sent Let $E$ be an object of $D(\mathcal{O}_X)$. Let $\mathcal{F}$ be an
$\mathcal{O}_X$-module. ========

new_annotation []
===== sent \begin{lemma}
\label{lemma-relative-pseudo-coherence}
Let $f : X \to S$ be a morphism of schemes which is locally of finite type ========

regex_match $f : X \to S$
nn candidate X \to
nn candidate a morphism
nn a morphism
new_annotation [(537000, 537013, 'VAR'), (537017, 537027, 'TYPE')]
===== sent \begin{lemma}
\label{lemma-localize-relative-pseudo-coherent}
Let $S$ be an affine scheme. ========

regex_match $S$
nn candidate an affine scheme
nn an affine scheme
new_annotation [(538575, 538578, 'VAR'), (538582, 538598, 'TYPE')]
===== sent Let $V \subset S$ be a standard open.
 ========

regex_match $V \subset S$
nn candidate $
nn candidate a standard
nn a standard
new_annotation [(538604, 538617, 'VAR'), (538621, 538631, 'TYPE')]
===== sent Let $X \to V$ be a finite type morphism of affine schemes.
 ========

regex_match $X \to V$
nn candidate a finite type morphism
nn a finite type morphism
new_annotation [(538642, 538651, 'VAR'), (538655, 538677, 'TYPE')]
===== sent Let $U \subset X$ be an affine open. ========

regex_match $U \subset X$
nn candidate $U
nn candidate an affine
nn an affine
new_annotation [(538701, 538714, 'VAR'), (538718, 538727, 'TYPE')]
===== sent Let $E$ be an object of
$D(\mathcal{O}_X)$. If the equivalent conditions of
Lemma \ref{lemma-relatively-pseudo-coherent}
are satisfied for the pair $(X \to V, E)$, then
the equivalent conditions of
Lemma \ref{lemma-relatively-pseudo-coherent}
are satisfied for the pair $(U \to S, E|_U)$.
\end{lemma}

\begin{proof}
Write $S = \Spec(R)$, $V = D(f)$, $X = \Spec(A)$, and $U = D(g)$.
Assume the equivalent conditions of
Lemma \ref{lemma-relatively-pseudo-coherent}
are satisfied for the pair $(X \to V, E)$.

\medskip\noindent
Choose $R_f[x_1, \ldots, x_n] \to A$ surjective. ========

regex_match $E$
nn candidate an object
nn an object
new_annotation [(538738, 538741, 'VAR'), (538745, 538754, 'TYPE')]
===== sent \end{proof}

\begin{lemma}
\label{lemma-glue-relative-pseudo-coherent}
Let $X \to S$ be a finite type morphism of affine schemes. ========

regex_match $X \to S$
nn candidate $X
nn candidate a finite type morphism
nn a finite type morphism
new_annotation [(540434, 540443, 'VAR'), (540447, 540469, 'TYPE')]
===== sent Let $E$ be an
object of $D(\mathcal{O}_X)$. Let $m \in \mathbf{Z}$.
Let $X = \bigcup U_i$ be a standard affine open covering.
 ========

new_annotation []
===== sent Let $E$ be an
object of $D(\mathcal{O}_X)$. Let $m \in \mathbf{Z}$.
Let $X = \bigcup U_i$ be a standard affine open covering.
 ========

new_annotation []
===== sent Let $E$ be an
object of $D(\mathcal{O}_X)$. Let $m \in \mathbf{Z}$.
Let $X = \bigcup U_i$ be a standard affine open covering.
 ========

regex_match $X = \bigcup U_i$
nn candidate $X
nn candidate affine
nn affine
new_annotation [(540561, 540578, 'VAR'), (540593, 540599, 'TYPE')]
===== sent \end{proof}

\begin{lemma}
\label{lemma-relative-pseudo-coherence-characterize}
Let $f : X \to S$ be a morphism of schemes which is locally of finite type.
 ========

regex_match $f : X \to S$
nn candidate X \to
nn candidate a morphism
nn a morphism
new_annotation [(542209, 542222, 'VAR'), (542226, 542236, 'TYPE')]
===== sent Let $E$ be an object of $D(\mathcal{O}_X)$.
Fix $m \in \mathbf{Z}$. The following are equivalent
\begin{enumerate}
\item $E$ is $m$-pseudo-coherent relative to $S$,
\item for every affine opens $U \subset X$ and $V \subset S$
with $f(U) \subset V$ the equivalent conditions of
Lemma \ref{lemma-relatively-pseudo-coherent}
are satisfied for the pair $(U \to V, E|_U)$.
\end{enumerate}
 ========

regex_match $E$
nn candidate an object
nn an object
new_annotation [(542285, 542288, 'VAR'), (542292, 542301, 'TYPE')]
===== sent Let $S = \bigcup V_i$ and $f^{-1}(V_i) = \bigcup U_{ij}$ be
affine open coverings as in
Definition ========

new_annotation []
===== sent \ref{definition-relative-pseudo-coherence}.
Let $U \subset X$ and $V \subset S$ be as in (2).
 ========

new_annotation []
===== sent By Lemma \ref{lemma-localize-relative-pseudo-coherent}
the assumption that the equivalent conditions of
Lemma \ref{lemma-relatively-pseudo-coherent}
are satisfied for the pair $(U_{ij} \to V_i, E|_{U_{ij}})$
implies that the equivalent conditions of
Lemma \ref{lemma-relatively-pseudo-coherent}
are satisfied for the pair $(U' \to V, E|_{U'})$.
\end{proof}

\noindent
For objects of the derived category whose cohomology sheaves are
quasi-coherent, we can relate relative $m$-pseudo-coherence
to the notion defined in More on Algebra, Definition
\ref{more-algebra-definition-relatively-pseudo-coherent}.
We will use the fact that for an affine scheme
$U = \Spec(A)$ the functor $R\Gamma(U, -)$ induces an equivalence
between $D_\QCoh(\mathcal{O}_U)$ and $D(A)$, see
Derived Categories of Schemes, Lemma
\ref{perfect-lemma-affine-compare-bounded}.
This functor is compatible with pullbacks:
if $E$ is an object of $D_\QCoh(\mathcal{O}_U)$
and $A \to B$ is a ring map corresponding to a morphism of affine
schemes $g : V = \Spec(B) \to \Spec(A) = U$, then
$R\Gamma(V, Lg^*E) = R\Gamma(U, E) \otimes_A^\mathbf{L} B$.
See Derived Categories of Schemes, Lemma
\ref{perfect-lemma-quasi-coherence-pullback}.

\begin{lemma}
\label{lemma-qcoh-relative-pseudo-coherence-characterize}
Let $f : X \to S$ be a morphism of schemes which is locally of finite type.
 ========

regex_match $f : X \to S$
nn candidate X \to
nn candidate a morphism
nn a morphism
new_annotation [(545523, 545536, 'VAR'), (545540, 545550, 'TYPE')]
===== sent Let $E$ be an object of $D_\QCoh(\mathcal{O}_X)$.
Fix $m \in \mathbf{Z}$. The following are equivalent
\begin{enumerate}
\item $E$ is $m$-pseudo-coherent relative to $S$,
\item there exists an affine open covering $S = \bigcup V_i$ and
for each $i$ an affine open covering $f^{-1}(V_i) = \bigcup U_{ij}$
such that the complex of $\mathcal{O}_X(U_{ij})$-modules
$R\Gamma(U_{ij}, E)$ is $m$-pseudo-coherent relative to
$\mathcal{O}_S(V_i)$, and
\item for every affine opens $U \subset X$ and $V \subset S$
with $f(U) \subset V$ the complex of $\mathcal{O}_X(U)$-modules
$R\Gamma(U, E)$ is $m$-pseudo-coherent relative to $\mathcal{O}_S(V)$.
\end{enumerate}
\end{lemma}

\begin{proof}
Let $U$ and $V$ be as in (2) and choose a closed immersion
$i : U \to \mathbf{A}^n_V$. A formal argument, using
Lemma \ref{lemma-relative-pseudo-coherence-characterize}, shows it
suffices to prove that
$Ri_*(E|_U)$ is $m$-pseudo-coherent if and only if $R\Gamma(U, E)$
is $m$-pseudo-coherent relative to $\mathcal{O}_S(V)$.
Say $U = \Spec(A)$, $V = \Spec(R)$, and
$\mathbf{A}^n_V = \Spec(R[x_1, \ldots, x_n]$. By the remarks
preceding the lemma, $E|_U$ is quasi-isomorphic to the
complex of quasi-coherent sheaves on $U$ associated to the object
$R\Gamma(U, E)$ of $D(A)$. Note that
$R\Gamma(U, E) = R\Gamma(\mathbf{A}^n_V, Ri_*(E|_U))$ as $i$ is a
closed immersion (and hence $i_*$ is exact). ========

regex_match $E$
nn candidate an object
nn an object
new_annotation [(545599, 545602, 'VAR'), (545606, 545615, 'TYPE')]
===== sent Let $E$ be an object of $D_\QCoh(\mathcal{O}_X)$.
Fix $m \in \mathbf{Z}$. The following are equivalent
\begin{enumerate}
\item $E$ is $m$-pseudo-coherent relative to $S$,
\item there exists an affine open covering $S = \bigcup V_i$ and
for each $i$ an affine open covering $f^{-1}(V_i) = \bigcup U_{ij}$
such that the complex of $\mathcal{O}_X(U_{ij})$-modules
$R\Gamma(U_{ij}, E)$ is $m$-pseudo-coherent relative to
$\mathcal{O}_S(V_i)$, and
\item for every affine opens $U \subset X$ and $V \subset S$
with $f(U) \subset V$ the complex of $\mathcal{O}_X(U)$-modules
$R\Gamma(U, E)$ is $m$-pseudo-coherent relative to $\mathcal{O}_S(V)$.
\end{enumerate}
\end{lemma}

\begin{proof}
Let $U$ and $V$ be as in (2) and choose a closed immersion
$i : U \to \mathbf{A}^n_V$. A formal argument, using
Lemma \ref{lemma-relative-pseudo-coherence-characterize}, shows it
suffices to prove that
$Ri_*(E|_U)$ is $m$-pseudo-coherent if and only if $R\Gamma(U, E)$
is $m$-pseudo-coherent relative to $\mathcal{O}_S(V)$.
Say $U = \Spec(A)$, $V = \Spec(R)$, and
$\mathbf{A}^n_V = \Spec(R[x_1, \ldots, x_n]$. By the remarks
preceding the lemma, $E|_U$ is quasi-isomorphic to the
complex of quasi-coherent sheaves on $U$ associated to the object
$R\Gamma(U, E)$ of $D(A)$. Note that
$R\Gamma(U, E) = R\Gamma(\mathbf{A}^n_V, Ri_*(E|_U))$ as $i$ is a
closed immersion (and hence $i_*$ is exact). ========

new_annotation []
===== sent \end{proof}

\begin{lemma}
\label{lemma-closed-morphism-relative-pseudo-coherence}
Let $i : X \to Y$ morphism of schemes locally of finite type over a
base scheme $S$. Assume that $i$ induces a homeomorphism of $X$ with a closed
subset of $Y$. Let $E$ be an object of $D(\mathcal{O}_X)$.
Then $E$ is $m$-pseudo-coherent relative to $S$ if and only if
$Ri_*E$ is $m$-pseudo-coherent relative to $S$.
\end{lemma}

\begin{proof}
 ========

new_annotation []
===== sent \end{proof}

\begin{lemma}
\label{lemma-closed-morphism-relative-pseudo-coherence}
Let $i : X \to Y$ morphism of schemes locally of finite type over a
base scheme $S$. Assume that $i$ induces a homeomorphism of $X$ with a closed
subset of $Y$. Let $E$ be an object of $D(\mathcal{O}_X)$.
Then $E$ is $m$-pseudo-coherent relative to $S$ if and only if
$Ri_*E$ is $m$-pseudo-coherent relative to $S$.
\end{lemma}

\begin{proof}
 ========

regex_match $E$
nn candidate an object
nn an object
new_annotation [(547646, 547649, 'VAR'), (547653, 547662, 'TYPE')]
===== sent Again by
Lemma \ref{lemma-relative-pseudo-coherence-characterize}
this holds if and only if $Ri_*E$ is $m$-pseudo-coherent relative to $S$.
\end{proof}

\begin{lemma}
\label{lemma-finite-morphism-relative-pseudo-coherence}
Let $\pi : X \to Y$ be a finite morphism of schemes locally of finite
type over a base scheme $S$. Let $E$ be an object of
$D_\QCoh(\mathcal{O}_X)$. Then $E$ is $m$-pseudo-coherent
relative to $S$ if and only if $R\pi_*E$ is $m$-pseudo-coherent
relative to $S$.
\end{lemma}

\begin{proof}
Translation of the result of
More on Algebra, Lemma
\ref{more-algebra-lemma-finite-extension-pseudo-coherent}
into the language of schemes. ========

regex_match $\pi : X \to Y$
nn candidate X \to
nn candidate a finite morphism
nn a finite morphism
new_annotation [(549268, 549283, 'VAR'), (549287, 549304, 'TYPE')]
===== sent Again by
Lemma \ref{lemma-relative-pseudo-coherence-characterize}
this holds if and only if $Ri_*E$ is $m$-pseudo-coherent relative to $S$.
\end{proof}

\begin{lemma}
\label{lemma-finite-morphism-relative-pseudo-coherence}
Let $\pi : X \to Y$ be a finite morphism of schemes locally of finite
type over a base scheme $S$. Let $E$ be an object of
$D_\QCoh(\mathcal{O}_X)$. Then $E$ is $m$-pseudo-coherent
relative to $S$ if and only if $R\pi_*E$ is $m$-pseudo-coherent
relative to $S$.
\end{lemma}

\begin{proof}
Translation of the result of
More on Algebra, Lemma
\ref{more-algebra-lemma-finite-extension-pseudo-coherent}
into the language of schemes. ========

regex_match $E$
nn candidate an object
nn an object
new_annotation [(549367, 549370, 'VAR'), (549374, 549383, 'TYPE')]
===== sent \ref{perfect-lemma-quasi-coherence-direct-image}.
To do the translation use
Lemma \ref{lemma-relative-pseudo-coherence-characterize}.
\end{proof}

\begin{lemma}
\label{lemma-cone-relatively-pseudo-coherent}
Let $f : X \to S$ be a morphism of schemes which is locally of finite type.
 ========

regex_match $f : X \to S$
nn candidate X \to
nn candidate a morphism
nn a morphism
new_annotation [(550033, 550046, 'VAR'), (550050, 550060, 'TYPE')]
===== sent Let $(E, E', E'')$ be a distinguished triangle of
$D(\mathcal{O}_X)$. Let $m \in \mathbf{Z}$.
\begin{enumerate}
\item If $E$ is $(m + 1)$-pseudo-coherent relative to $S$ and
$E'$ is $m$-pseudo-coherent relative to $S$ then $E''$ is
$m$-pseudo-coherent relative to $S$.
\item If $E, E''$ are $m$-pseudo-coherent relative to $S$,
then $E'$ is $m$-pseudo-coherent relative to $S$.
\item If $E'$ is $(m + 1)$-pseudo-coherent relative to $S$
and $E''$ is $m$-pseudo-coherent relative to $S$, then
$E$ is $(m + 1)$-pseudo-coherent relative to $S$.
\end{enumerate}
 ========

regex_match $(E, E', E'')$
nn candidate a distinguished triangle
nn a distinguished triangle
new_annotation [(550109, 550123, 'VAR'), (550127, 550151, 'TYPE')]
===== sent Let $(E, E', E'')$ be a distinguished triangle of
$D(\mathcal{O}_X)$. Let $m \in \mathbf{Z}$.
\begin{enumerate}
\item If $E$ is $(m + 1)$-pseudo-coherent relative to $S$ and
$E'$ is $m$-pseudo-coherent relative to $S$ then $E''$ is
$m$-pseudo-coherent relative to $S$.
\item If $E, E''$ are $m$-pseudo-coherent relative to $S$,
then $E'$ is $m$-pseudo-coherent relative to $S$.
\item If $E'$ is $(m + 1)$-pseudo-coherent relative to $S$
and $E''$ is $m$-pseudo-coherent relative to $S$, then
$E$ is $(m + 1)$-pseudo-coherent relative to $S$.
\end{enumerate}
 ========

new_annotation []
===== sent \end{lemma}

\begin{proof}
Immediate from Lemma \ref{lemma-relative-pseudo-coherence-characterize} and
Cohomology, Lemma \ref{cohomology-lemma-cone-pseudo-coherent}.
\end{proof}

\begin{lemma}
\label{lemma-rel-n-pseudo-module}
Let $X \to S$ be a morphism of schemes which is locally of finite type.
 ========

regex_match $X \to S$
nn candidate $X
nn candidate a morphism
nn a morphism
new_annotation [(550998, 551007, 'VAR'), (551011, 551021, 'TYPE')]
===== sent Let $\mathcal{F}$ be an $\mathcal{O}_X$-module. ========

regex_match $\mathcal{F}$
nn candidate an $\mathcal{O}_X$-module
nn an $\mathcal{O}_X$-module
new_annotation [(551070, 551083, 'VAR'), (551087, 551112, 'TYPE')]
===== sent \end{proof}

\begin{lemma}
\label{lemma-summands-relative-pseudo-coherent}
Let $X \to S$ be a morphism of schemes which is locally of finite type.
 ========

regex_match $X \to S$
nn candidate $X
nn candidate a morphism
nn a morphism
new_annotation [(552499, 552508, 'VAR'), (552512, 552522, 'TYPE')]
===== sent Let $m \in \mathbf{Z}$. Let $E, K$ be objects of $D(\mathcal{O}_X)$.
If $E \oplus K$ is $m$-pseudo-coherent relative to $S$ so are $E$ and $K$.
\end{lemma}

\begin{proof}
Follows from
Cohomology, Lemma \ref{cohomology-lemma-summands-pseudo-coherent}
and the definitions.
 ========

new_annotation []
===== sent Let $m \in \mathbf{Z}$. Let $E, K$ be objects of $D(\mathcal{O}_X)$.
If $E \oplus K$ is $m$-pseudo-coherent relative to $S$ so are $E$ and $K$.
\end{lemma}

\begin{proof}
Follows from
Cohomology, Lemma \ref{cohomology-lemma-summands-pseudo-coherent}
and the definitions.
 ========

new_annotation []
===== sent \end{proof}

\begin{lemma}
\label{lemma-complex-relative-pseudo-coherent-modules}
Let $X \to S$ be a morphism of schemes which is locally of finite type.
 ========

regex_match $X \to S$
nn candidate $X
nn candidate a morphism
nn a morphism
new_annotation [(552924, 552933, 'VAR'), (552937, 552947, 'TYPE')]
===== sent Let $m \in \mathbf{Z}$. Let $\mathcal{F}^\bullet$ be a (locally) bounded
above complex of $\mathcal{O}_X$-modules such that $\mathcal{F}^i$ is
$(m - i)$-pseudo-coherent relative to $S$ for all $i$. Then
$\mathcal{F}^\bullet$ is $m$-pseudo-coherent relative to $S$.
\end{lemma}

\begin{proof}
Follows from
Cohomology, Lemma \ref{cohomology-lemma-complex-pseudo-coherent-modules}
and the definitions.
 ========

new_annotation []
===== sent Let $m \in \mathbf{Z}$. Let $\mathcal{F}^\bullet$ be a (locally) bounded
above complex of $\mathcal{O}_X$-modules such that $\mathcal{F}^i$ is
$(m - i)$-pseudo-coherent relative to $S$ for all $i$. Then
$\mathcal{F}^\bullet$ is $m$-pseudo-coherent relative to $S$.
\end{lemma}

\begin{proof}
Follows from
Cohomology, Lemma \ref{cohomology-lemma-complex-pseudo-coherent-modules}
and the definitions.
 ========

regex_match $\mathcal{F}^\bullet$
nn candidate complex
nn complex
new_annotation [(553020, 553041, 'VAR'), (553071, 553078, 'TYPE')]
===== sent \end{proof}

\begin{lemma}
\label{lemma-cohomology-relative-pseudo-coherent}
Let $X \to S$ be a morphism of schemes which is locally of finite type.
 ========

regex_match $X \to S$
nn candidate $X
nn candidate a morphism
nn a morphism
new_annotation [(553472, 553481, 'VAR'), (553485, 553495, 'TYPE')]
===== sent Let $m \in \mathbf{Z}$. Let $E$ be an object of $D(\mathcal{O}_X)$.
If $E$ is (locally) bounded above and $H^i(E)$ is $(m - i)$-pseudo-coherent
relative to $S$ for all $i$, then $E$ is $m$-pseudo-coherent relative to $S$.
\end{lemma}

\begin{proof}
Follows from
Cohomology, Lemma \ref{cohomology-lemma-cohomology-pseudo-coherent}
and the definitions.
 ========

new_annotation []
===== sent Let $m \in \mathbf{Z}$. Let $E$ be an object of $D(\mathcal{O}_X)$.
If $E$ is (locally) bounded above and $H^i(E)$ is $(m - i)$-pseudo-coherent
relative to $S$ for all $i$, then $E$ is $m$-pseudo-coherent relative to $S$.
\end{lemma}

\begin{proof}
Follows from
Cohomology, Lemma \ref{cohomology-lemma-cohomology-pseudo-coherent}
and the definitions.
 ========

regex_match $E$
nn candidate an object
nn an object
new_annotation [(553568, 553571, 'VAR'), (553575, 553584, 'TYPE')]
===== sent \end{proof}

\begin{lemma}
\label{lemma-base-change-relative-pseudo-coherent}
Let $X \to S$ be a morphism of schemes which is locally of finite type.
 ========

regex_match $X \to S$
nn candidate $X
nn candidate a morphism
nn a morphism
new_annotation [(553973, 553982, 'VAR'), (553986, 553996, 'TYPE')]
===== sent Let $m \in \mathbf{Z}$. Let $E$ be an object of $D(\mathcal{O}_X)$
which is $m$-pseudo-coherent relative to $S$. Let $S' \to S$ be a
morphism of schemes. ========

new_annotation []
===== sent Let $m \in \mathbf{Z}$. Let $E$ be an object of $D(\mathcal{O}_X)$
which is $m$-pseudo-coherent relative to $S$. Let $S' \to S$ be a
morphism of schemes. ========

regex_match $E$
nn candidate an object
nn an object
new_annotation [(554069, 554072, 'VAR'), (554076, 554085, 'TYPE')]
===== sent Let $m \in \mathbf{Z}$. Let $E$ be an object of $D(\mathcal{O}_X)$
which is $m$-pseudo-coherent relative to $S$. Let $S' \to S$ be a
morphism of schemes. ========

new_annotation []
===== sent We conclude the lemma holds
by an application of
Cohomology, Lemma \ref{cohomology-lemma-pseudo-coherent-pullback}.
\end{proof}

\begin{lemma}
\label{lemma-pull-relative-pseudo-coherent}
Let $f : X \to Y$ be a morphism of schemes locally of finite type
over a base $S$. Let $m \in \mathbf{Z}$. Let $E$ be an object of
$D(\mathcal{O}_Y)$. Assume
\begin{enumerate}
\item $\mathcal{O}_X$ is pseudo-coherent relative to $Y$\footnote{This
means $f$ is pseudo-coherent, see
Definition \ref{definition-pseudo-coherent}.}, and
\item $E$ is $m$-pseudo-coherent relative to $S$.
\end{enumerate}
Then $Lf^*E$ is $m$-pseudo-coherent relative to $S$.
\end{lemma}

\begin{proof}
The problem is local on $X$. Thus we may assume $X$, $Y$, and $S$ are affine.
 ========

regex_match $f : X \to Y$
nn candidate X \to
nn candidate a morphism
nn a morphism
new_annotation [(555753, 555766, 'VAR'), (555770, 555780, 'TYPE')]
===== sent We conclude the lemma holds
by an application of
Cohomology, Lemma \ref{cohomology-lemma-pseudo-coherent-pullback}.
\end{proof}

\begin{lemma}
\label{lemma-pull-relative-pseudo-coherent}
Let $f : X \to Y$ be a morphism of schemes locally of finite type
over a base $S$. Let $m \in \mathbf{Z}$. Let $E$ be an object of
$D(\mathcal{O}_Y)$. Assume
\begin{enumerate}
\item $\mathcal{O}_X$ is pseudo-coherent relative to $Y$\footnote{This
means $f$ is pseudo-coherent, see
Definition \ref{definition-pseudo-coherent}.}, and
\item $E$ is $m$-pseudo-coherent relative to $S$.
\end{enumerate}
Then $Lf^*E$ is $m$-pseudo-coherent relative to $S$.
\end{lemma}

\begin{proof}
The problem is local on $X$. Thus we may assume $X$, $Y$, and $S$ are affine.
 ========

new_annotation []
===== sent We conclude the lemma holds
by an application of
Cohomology, Lemma \ref{cohomology-lemma-pseudo-coherent-pullback}.
\end{proof}

\begin{lemma}
\label{lemma-pull-relative-pseudo-coherent}
Let $f : X \to Y$ be a morphism of schemes locally of finite type
over a base $S$. Let $m \in \mathbf{Z}$. Let $E$ be an object of
$D(\mathcal{O}_Y)$. Assume
\begin{enumerate}
\item $\mathcal{O}_X$ is pseudo-coherent relative to $Y$\footnote{This
means $f$ is pseudo-coherent, see
Definition \ref{definition-pseudo-coherent}.}, and
\item $E$ is $m$-pseudo-coherent relative to $S$.
\end{enumerate}
Then $Lf^*E$ is $m$-pseudo-coherent relative to $S$.
\end{lemma}

\begin{proof}
The problem is local on $X$. Thus we may assume $X$, $Y$, and $S$ are affine.
 ========

regex_match $E$
nn candidate an object
nn an object
new_annotation [(555860, 555863, 'VAR'), (555867, 555876, 'TYPE')]
===== sent \end{proof}

\begin{lemma}
\label{lemma-composition-relative-pseudo-coherent}
Let $f : X \to Y$ be a morphism of schemes locally of finite type
over a base $S$. Let $m \in \mathbf{Z}$. Let $E$ be an object of
$D(\mathcal{O}_X)$. Assume $\mathcal{O}_Y$ is pseudo-coherent relative
to $S$\footnote{This means $Y \to S$ is pseudo-coherent, see
Definition \ref{definition-pseudo-coherent}.}.
Then the following are equivalent
\begin{enumerate}
\item $E$ is $m$-pseudo-coherent relative to $Y$, and
\item $E$ is $m$-pseudo-coherent relative to $S$.
\end{enumerate}
\end{lemma}

\begin{proof}
The question is local on $X$, hence we may assume $X$, $Y$, and $S$ are affine.
 ========

regex_match $f : X \to Y$
nn candidate X \to
nn candidate a morphism
nn a morphism
new_annotation [(558939, 558952, 'VAR'), (558956, 558966, 'TYPE')]
===== sent \end{proof}

\begin{lemma}
\label{lemma-composition-relative-pseudo-coherent}
Let $f : X \to Y$ be a morphism of schemes locally of finite type
over a base $S$. Let $m \in \mathbf{Z}$. Let $E$ be an object of
$D(\mathcal{O}_X)$. Assume $\mathcal{O}_Y$ is pseudo-coherent relative
to $S$\footnote{This means $Y \to S$ is pseudo-coherent, see
Definition \ref{definition-pseudo-coherent}.}.
Then the following are equivalent
\begin{enumerate}
\item $E$ is $m$-pseudo-coherent relative to $Y$, and
\item $E$ is $m$-pseudo-coherent relative to $S$.
\end{enumerate}
\end{lemma}

\begin{proof}
The question is local on $X$, hence we may assume $X$, $Y$, and $S$ are affine.
 ========

new_annotation []
===== sent \end{proof}

\begin{lemma}
\label{lemma-composition-relative-pseudo-coherent}
Let $f : X \to Y$ be a morphism of schemes locally of finite type
over a base $S$. Let $m \in \mathbf{Z}$. Let $E$ be an object of
$D(\mathcal{O}_X)$. Assume $\mathcal{O}_Y$ is pseudo-coherent relative
to $S$\footnote{This means $Y \to S$ is pseudo-coherent, see
Definition \ref{definition-pseudo-coherent}.}.
Then the following are equivalent
\begin{enumerate}
\item $E$ is $m$-pseudo-coherent relative to $Y$, and
\item $E$ is $m$-pseudo-coherent relative to $S$.
\end{enumerate}
\end{lemma}

\begin{proof}
The question is local on $X$, hence we may assume $X$, $Y$, and $S$ are affine.
 ========

regex_match $E$
nn candidate an object
nn an object
new_annotation [(559046, 559049, 'VAR'), (559053, 559062, 'TYPE')]
===== sent Let $E$
be an object of $D(\mathcal{O}_X)$. Then the following
are equivalent
\begin{enumerate}
\item $E$ is $m$-pseudo-coherent relative to $S$,
\item ========

new_annotation []
===== sent \begin{lemma}
\label{lemma-pseudo-coherent}
Let $f : X \to S$ be a morphism of schemes. ========

regex_match $f : X \to S$
nn candidate X \to
nn candidate a morphism
nn a morphism
new_annotation [(562040, 562053, 'VAR'), (562057, 562067, 'TYPE')]
===== sent This translates into the following algebra result:
Let $A \to B$ be a pseudo-coherent ring map.
 ========

regex_match $A \to B$
nn candidate a pseudo-coherent ring map
nn a pseudo-coherent ring map
new_annotation [(564464, 564473, 'VAR'), (564477, 564503, 'TYPE')]
===== sent Let $A \to A'$ be flat. ========

new_annotation []
===== sent This follows from the fact that a flat ring map of finite presentation is
pseudo-coherent (and even perfect), see
More on Algebra,
Lemma \ref{more-algebra-lemma-flat-finite-presentation-perfect}.
\end{proof}

\begin{lemma}
\label{lemma-permanence-pseudo-coherent}
Let $f : X \to Y$ be a morphism of schemes pseudo-coherent
over a base scheme $S$. ========

regex_match $f : X \to Y$
nn candidate X \to
nn candidate a morphism
nn a morphism
new_annotation [(565858, 565871, 'VAR'), (565875, 565885, 'TYPE')]
===== sent This follows from
More on Algebra,
Lemma \ref{more-algebra-lemma-composition-relative-pseudo-coherent}.
\end{proof}

\begin{lemma}
\label{lemma-finite-pseudo-coherent}
Let $f : X \to S$ be a finite morphism of schemes.
 ========

regex_match $f : X \to S$
nn candidate X \to
nn candidate a finite morphism
nn a finite morphism
new_annotation [(566337, 566350, 'VAR'), (566354, 566371, 'TYPE')]
===== sent This follows from
the more general
More on Algebra, Lemma
\ref{more-algebra-lemma-finite-extension-pseudo-coherent}.
\end{proof}

\begin{lemma}
\label{lemma-Noetherian-pseudo-coherent}
Let $f : X \to S$ be a morphism of schemes.
 ========

regex_match $f : X \to S$
nn candidate X \to
nn candidate a morphism
nn a morphism
new_annotation [(566994, 567007, 'VAR'), (567011, 567021, 'TYPE')]
===== sent The final hypothesis (3) of
Descent, Lemma \ref{descent-lemma-descending-properties-morphisms}
translates into the following algebra statement:
Let $A \to B$ be a faithfully flat ring map.
 ========

regex_match $A \to B$
nn candidate a faithfully flat ring map
nn a faithfully flat ring map
new_annotation [(568393, 568402, 'VAR'), (568406, 568432, 'TYPE')]
===== sent Let $C = A[x_1, \ldots, x_n]/I$ be an $A$-algebra.
 ========

regex_match $C = A[x_1, \ldots, x_n]/I$
nn candidate $C
new_annotation []
===== sent This is
More on Algebra, Lemma \ref{more-algebra-lemma-flat-descent-pseudo-coherent}.
\end{proof}

\begin{lemma}
\label{lemma-quotient-of-flat-finitely-presented}
Let $A \to B$ be a flat ring map of finite presentation.
 ========

regex_match $A \to B$
nn candidate a flat ring map
nn a flat ring map
new_annotation [(568789, 568798, 'VAR'), (568802, 568817, 'TYPE')]
===== sent Let $I \subset B$ be an ideal. ========

regex_match $I \subset B$
nn candidate $I
nn candidate an ideal
nn an ideal
new_annotation [(568846, 568859, 'VAR'), (568863, 568871, 'TYPE')]
===== sent \end{lemma}

\begin{proof}
Let $f : X \to S$ be a morphism of schemes.
 ========

regex_match $f : X \to S$
nn candidate X \to
nn candidate a morphism
nn a morphism
new_annotation [(572986, 572999, 'VAR'), (573003, 573013, 'TYPE')]
===== sent Let $\{g_i : X_i \to X\}$ be an fppf covering such that each composition
$f \circ g_i$ is pseudo-coherent. ========

regex_match $\{g_i : X_i \to X\}$
nn candidate an fppf
nn an fppf
new_annotation [(573030, 573051, 'VAR'), (573055, 573062, 'TYPE')]
===== sent \begin{lemma}
\label{lemma-perfect}
Let $f : X \to S$ be a morphism of schemes which is locally of finite type.
 ========

regex_match $f : X \to S$
nn candidate X \to
nn candidate a morphism
nn a morphism
new_annotation [(576157, 576170, 'VAR'), (576174, 576184, 'TYPE')]
===== sent \end{enumerate}
\end{lemma}

\begin{proof}
Assume (1) and let $U, V$ be as in (2).
 ========

new_annotation []
===== sent This translates into the following algebra result:
Let $A \to B$ be a perfect ring map.
 ========

regex_match $A \to B$
nn candidate a perfect ring map
nn a perfect ring map
new_annotation [(577931, 577940, 'VAR'), (577944, 577962, 'TYPE')]
===== sent Let $A \to A'$ be flat. ========

new_annotation []
===== sent \end{proof}

\begin{lemma}
\label{lemma-flat-finite-presentation-perfect}
Let $f : X \to S$ be a morphism of schemes.
 ========

regex_match $f : X \to S$
nn candidate X \to
nn candidate a morphism
nn a morphism
new_annotation [(579170, 579183, 'VAR'), (579187, 579197, 'TYPE')]
===== sent \end{enumerate}
\end{lemma}

\begin{proof}
The implication (2) $\Rightarrow$ (1) is
More on Algebra,
Lemma \ref{more-algebra-lemma-flat-finite-presentation-perfect}.
The converse follows from the fact that a pseudo-coherent morphism
is locally of finite presentation, see
Lemma \ref{lemma-pseudo-coherent-finite-presentation}.
\end{proof}

\begin{lemma}
\label{lemma-regular-target-perfect}
Let $f : X \to S$ be a morphism of schemes.
 ========

regex_match $f : X \to S$
nn candidate X \to
nn candidate a morphism
nn a morphism
new_annotation [(579741, 579754, 'VAR'), (579758, 579768, 'TYPE')]
===== sent The final hypothesis (3) of
Descent, Lemma \ref{descent-lemma-descending-properties-morphisms}
translates into the following algebra statement:
Let $A \to B$ be a faithfully flat ring map.
 ========

regex_match $A \to B$
nn candidate a faithfully flat ring map
nn a faithfully flat ring map
new_annotation [(582908, 582917, 'VAR'), (582921, 582947, 'TYPE')]
===== sent Let $C = A[x_1, \ldots, x_n]/I$ be an $A$-algebra.
 ========

regex_match $C = A[x_1, \ldots, x_n]/I$
nn candidate $C
new_annotation []
===== sent This is
More on Algebra, Lemma \ref{more-algebra-lemma-flat-descent-perfect}.
\end{proof}

\begin{lemma}
\label{lemma-check-perfect-stalks}
Let $f : X \to S$ be a pseudo-coherent morphism of schemes.
 ========

regex_match $f : X \to S$
nn candidate X \to
nn candidate a pseudo-coherent morphism
nn a pseudo-coherent morphism
new_annotation [(583265, 583278, 'VAR'), (583282, 583308, 'TYPE')]
===== sent \end{proof}

\begin{lemma}
\label{lemma-perfect-proper-perfect-direct-image}
Let $S$ be a Noetherian scheme. ========

regex_match $S$
nn candidate a Noetherian scheme
nn a Noetherian scheme
new_annotation [(585545, 585548, 'VAR'), (585552, 585571, 'TYPE')]
===== sent Let $f : X \to S$ be a perfect proper
morphism of schemes. ========

regex_match $f : X \to S$
nn candidate $
nn candidate X \to
nn candidate a perfect proper
morphism
nn a perfect proper
morphism
new_annotation [(585577, 585590, 'VAR'), (585594, 585619, 'TYPE')]
===== sent Let $E \in D(\mathcal{O}_X)$ be perfect. ========

new_annotation []
===== sent Let $\{g_i : X_i \to X\}_{i \in I}$ be an fppf covering of schemes and let
$f : X \to S$ be a morphism such that each $f \circ g_i$ is
perfect. ========

regex_match $\{g_i : X_i \to X\}_{i \in I}$
nn candidate $\{g_i
nn candidate an fppf
nn an fppf
new_annotation [(586414, 586445, 'VAR'), (586449, 586456, 'TYPE')]
===== sent The desired conclusion follows from
More on Algebra, Lemma \ref{more-algebra-lemma-flat-descent-tor-amplitude}.
\end{proof}

\begin{lemma}
\label{lemma-factor-regular-immersion}
Let $i : Z \to Y$ and $j : Y \to X$ be immersions of schemes.
 ========

new_annotation []
===== sent \begin{lemma}
\label{lemma-koszul-independence-factorization}
Let $S$ be a scheme. ========

regex_match $S$
nn candidate a scheme
nn a scheme
new_annotation [(589056, 589059, 'VAR'), (589063, 589071, 'TYPE')]
===== sent Let $U$, $P$, $P'$ be schemes over $S$.
Let $u \in U$. Let $i : U \to P$, $i' : ========

new_annotation []
===== sent Let $U$, $P$, $P'$ be schemes over $S$.
Let $u \in U$. Let $i : U \to P$, $i' : ========

new_annotation []
===== sent Let $U$, $P$, $P'$ be schemes over $S$.
Let $u \in U$. Let $i : U \to P$, $i' : ========

new_annotation []
===== sent Let $f : X \to S$ be a morphism of schemes which is locally
of finite type. ========

regex_match $f : X \to S$
nn candidate $
nn candidate X \to
nn candidate a morphism
nn a morphism
new_annotation [(590001, 590014, 'VAR'), (590018, 590028, 'TYPE')]
===== sent Let $x \in ========

new_annotation []
===== sent Picture
$$
\xymatrix{
X \ar[rd] & U \ar[l] \ar[d] \ar[r]_-i & \mathbf{A}^n_S = P \ar[ld]^\pi \\
& S
}
$$
In fact you can do this with any affine open neighbourhood
$U$ of $x$ in $X$, see
Morphisms, Lemma \ref{morphisms-lemma-quasi-affine-finite-type-over-S}.

\begin{definition}
\label{definition-lci}
Let $f : X \to S$ be a morphism of schemes.
 ========

regex_match $f : X \to S$
nn candidate X \to
nn candidate a morphism
nn a morphism
new_annotation [(590606, 590619, 'VAR'), (590623, 590633, 'TYPE')]
===== sent \begin{enumerate}
\item Let $x \in X$. We say that $f$ is {\it Koszul at $x$} if $f$
is of finite type at $x$ and there exists an open neighbourhood
and a factorization of $f|_U$ as $\pi \circ i$ where ========

new_annotation []
===== sent \end{enumerate}
\end{definition}

\noindent
We have seen above that the choice of the factorization
$f|_U = \pi \circ i$ is irrelevant, i.e., given a factorization
of $f|_U$ as an immersion $i$ followed by a smooth morphism $\pi$, whether or
not $i$ is Koszul regular in a neighbourhood of $x$ is an intrinsic
property of $f$ at $x$. Let us record this here explicitly as a lemma
so that we can refer to it

\begin{lemma}
\label{lemma-lci}
Let $f : X \to S$ be a local complete intersection morphism.
 ========

regex_match $f : X \to S$
nn candidate X \to
nn candidate a local complete intersection morphism
nn a local complete intersection morphism
new_annotation [(591505, 591518, 'VAR'), (591522, 591560, 'TYPE')]
===== sent Let $P$ be a scheme smooth over $S$. Let $U \subset X$ be an open subscheme
and $i : U \to P$ an immersion of schemes over $S$.
Then $i$ is a Koszul-regular immersion.
 ========

regex_match $P$
nn candidate $U \subset X$
nn $U \subset X$
new_annotation [(591566, 591569, 'VAR'), (591603, 591616, 'TYPE')]
===== sent Let $P$ be a scheme smooth over $S$. Let $U \subset X$ be an open subscheme
and $i : U \to P$ an immersion of schemes over $S$.
Then $i$ is a Koszul-regular immersion.
 ========

regex_match $U \subset X$
nn candidate $U \subset X$
nn candidate an open subscheme
nn an open subscheme
new_annotation [(591603, 591616, 'VAR'), (591620, 591637, 'TYPE')]
===== sent \begin{lemma}
\label{lemma-lci-properties}
Let $f : X \to S$ be a local complete intersection morphism.
 ========

regex_match $f : X \to S$
nn candidate X \to
nn candidate a local complete intersection morphism
nn a local complete intersection morphism
new_annotation [(592018, 592031, 'VAR'), (592035, 592073, 'TYPE')]
===== sent Finally a composition of perfect morphisms is perfect, see
Lemma \ref{lemma-composition-perfect}.
\end{proof}

\begin{lemma}
\label{lemma-affine-lci}
Let $f : X = \Spec(B) \to S = \Spec(A)$ be a morphism of affine schemes.
 ========

regex_match $f : X = \Spec(B) \to S = \Spec(A)$
nn candidate X
nn candidate a morphism
nn a morphism
new_annotation [(593030, 593065, 'VAR'), (593069, 593079, 'TYPE')]
===== sent \end{lemma}

\begin{proof}
Let $g : Y \to S$ and $f : X \to Y$ be local complete intersection
morphisms. ========

new_annotation []
===== sent Let $x \in X$ and set $y = f(x)$. Choose an open neighbourhood
$V \subset Y$ of $y$ and a factorization $g|_V = \pi \circ i$ for some
Koszul-regular immersion $i : V \to P$ and smooth morphism $\pi : P \to S$.
Next choose an open neighbourhood $U$ of $x \in X$ and a factorization
$f|_U = \pi' \circ i'$ for some Koszul-regular immersion $i' : U \to P'$
and smooth morphism $\pi' : P' \to Y$. ========

new_annotation []
===== sent \end{slogan}
Let $f : X \to S$ be a morphism of schemes.
 ========

regex_match $f : X \to S$
nn candidate X \to
nn candidate a morphism
nn a morphism
new_annotation [(595498, 595511, 'VAR'), (595515, 595525, 'TYPE')]
===== sent Let $x \in X$ be a
point and consider the corresponding commutative diagram of local
rings
$$
\xymatrix{
J \ar[r] &
\mathcal{O}_{\mathbf{A}^{n + m}_S, x} \ar[r] &
\mathcal{O}_{X, x} \\
 ========

new_annotation []
===== sent \end{lemma}

\begin{proof}
Let $f : X \to S$ be a morphism of schemes.
 ========

regex_match $f : X \to S$
nn candidate X \to
nn candidate a morphism
nn a morphism
new_annotation [(601375, 601388, 'VAR'), (601392, 601402, 'TYPE')]
===== sent Let $\{S_i \to S\}$ be an fpqc covering of $S$.
Assume that each base change $f_i : X_i \to S_i$ of $f$ is
a local complete intersection morphism.
 ========

regex_match $\{S_i \to S\}$
nn candidate an fpqc
nn an fpqc
new_annotation [(601419, 601434, 'VAR'), (601438, 601445, 'TYPE')]
===== sent Note that this implies in particular that $f$ is locally of finite
type, see
Lemma \ref{lemma-lci-properties}
and
Descent, Lemma \ref{descent-lemma-descending-property-locally-finite-type}.
Let $x \in X$. Choose an open neighbourhood $U$ of $x$ and
an immersion $j : U \to \mathbf{A}^n_S$ over $S$ (see
discussion preceding
Definition \ref{definition-lci}).
 ========

new_annotation []
===== sent \end{proof}

\begin{lemma}
\label{lemma-base-change-lci-fibres}
Let $S$ be a scheme. ========

regex_match $S$
nn candidate a scheme
nn a scheme
new_annotation [(605354, 605357, 'VAR'), (605361, 605369, 'TYPE')]
===== sent Let $f : X \to Y$ be a morphism of schemes over $S$.
Assume both $X$ and $Y$ are flat and locally of finite presentation over $S$.
 ========

regex_match $f : X \to Y$
nn candidate $
nn candidate X \to
nn candidate a morphism
nn a morphism
new_annotation [(605375, 605388, 'VAR'), (605392, 605402, 'TYPE')]
===== sent Let $S' \to S$ be a morphism of schemes. ========

regex_match $S' \to S$
nn candidate S
nn candidate a morphism
nn a morphism
new_annotation [(605741, 605751, 'VAR'), (605755, 605765, 'TYPE')]
===== sent Set $X' = S' \times_S X$,
$Y' = S' \times_S Y$, and denote $f' : X' \to Y'$ the base change of $f$.
Let $x' \in X'$ be a point such that $f'$ is Koszul at $x'$. Denote
$s' \in S'$, $x \in X$, $y' \in Y'$ , $y \in Y$, $s \in ========

regex_match $x' \in X'$
nn candidate x
nn candidate a point
nn a point
new_annotation [(605882, 605893, 'VAR'), (605897, 605904, 'TYPE')]
===== sent \end{proof}

\begin{lemma}
\label{lemma-unramified-lci}
Let $f : X \to Y$ be a local complete intersection morphism of schemes.
 ========

regex_match $f : X \to Y$
nn candidate X \to
nn candidate a local complete intersection morphism
nn a local complete intersection morphism
new_annotation [(607868, 607881, 'VAR'), (607885, 607923, 'TYPE')]
===== sent By
Lemma \ref{lemma-two-unramified-morphisms-formally-smooth}
we see that $\mathcal{C}_{X/Y}$ is a locally direct summand of
$\mathcal{C}_{X/V}$ which is finite locally free as $i$ is a Koszul-regular
(hence quasi-regular) immersion, see
Divisors, Lemma \ref{divisors-lemma-quasi-regular-immersion}.
\end{proof}

\begin{lemma}
\label{lemma-transitivity-conormal-lci}
Let $Z \to Y \to X$ be formally unramified morphisms of schemes.
 ========

new_annotation []
===== sent \medskip\noindent
In the sequences below each of the maps
are as constructed in either
Morphisms, Lemma \ref{morphisms-lemma-functoriality-differentials}
or
Lemma \ref{lemma-universal-thickening-functorial}.
Let $g : Z \to Y$ and $f : Y \to X$ be morphisms of schemes.
 ========

new_annotation []
===== sent \begin{lemma}
\label{lemma-check-weakly-etale-stalks}
Let $f : X \to Y$ be a morphism of schemes. ========

regex_match $f : X \to Y$
nn candidate X \to
nn candidate a morphism
nn a morphism
new_annotation [(613499, 613512, 'VAR'), (613516, 613526, 'TYPE')]
===== sent Let $x \in X$ with image $y = f(x)$ in $Y$.
There are canonical maps of rings
$$
\mathcal{O}_{X, x} \otimes_{\mathcal{O}_{Y, y}} \mathcal{O}_{X, x}
\longrightarrow
\mathcal{O}_{X \times_Y X, \Delta_{X/Y}(x)}
\longrightarrow
\mathcal{O}_{X, x}
$$
where the first map is a localization (hence flat) and the second map is a
surjection (hence an epimorphism of rings).
 ========

new_annotation []
===== sent These conditions are equivalent by
Algebra, Lemma \ref{algebra-lemma-composition-flat} and
More on Algebra, Lemma \ref{more-algebra-lemma-key}.
\end{proof}

\begin{lemma}
\label{lemma-key}
Let $X \to Y$ be a morphism of schemes such that
$X \to X \times_Y X$ is flat. ========

regex_match $X \to Y$
nn candidate $X
nn candidate a morphism
nn a morphism
new_annotation [(614559, 614568, 'VAR'), (614572, 614582, 'TYPE')]
===== sent Let $\mathcal{F}$ be an $\mathcal{O}_X$-module.
 ========

regex_match $\mathcal{F}$
nn candidate an $\mathcal{O}_X$-module
nn an $\mathcal{O}_X$-module
new_annotation [(614638, 614651, 'VAR'), (614655, 614680, 'TYPE')]
===== sent If $\mathcal{F}$ is flat over $Y$, then $\mathcal{F}$ is flat over $X$.
\end{lemma}

\begin{proof}
Let $x \in X$ with image $y = f(x)$ in $Y$.
Since $X \to X \times_Y X$ is flat, we see that
$\mathcal{O}_{X, x} \otimes_{\mathcal{O}_{Y, y}} \mathcal{O}_{X, x} \to
\mathcal{O}_{X, x}$ is flat. ========

new_annotation []
===== sent \end{proof}

\begin{lemma}
\label{lemma-weakly-etale-characterize}
Let $f : X \to S$ be a morphism of schemes. ========

regex_match $f : X \to S$
nn candidate X \to
nn candidate a morphism
nn a morphism
new_annotation [(615148, 615161, 'VAR'), (615165, 615175, 'TYPE')]
===== sent \end{proof}

\begin{lemma}
\label{lemma-composition-weakly-etale}
Let $X \to Y \to Z$ be morphisms of schemes.
 ========

new_annotation []
===== sent \end{proof}

\begin{lemma}
\label{lemma-base-change-weakly-etale}
Let $X \to Y$ and $Y' \to Y$ be morphisms of schemes and let
$X' = Y' \times_Y X$ be the base change of $X$.
\begin{enumerate}
\item If $X \to X \times_Y X$ is flat, then $X' \to X' \times_{Y'} X'$
is flat.
 ========

new_annotation []
===== sent \end{proof}

\begin{lemma}
\label{lemma-go-down}
Let $X \to Y \to Z$ be morphisms of schemes. ========

new_annotation []
===== sent By
Morphisms, Lemma \ref{morphisms-lemma-flat-permanence}
we see that $Y$ is flat over $Y \times_Z Y$.
\end{proof}

\begin{lemma}
\label{lemma-weakly-etale-formally-unramified}
Let $f : X \to Y$ be a weakly \'etale morphism of schemes.
 ========

regex_match $f : X \to Y$
nn candidate X \to
nn candidate a weakly \'etale morphism
nn a weakly \'etale morphism
new_annotation [(619098, 619111, 'VAR'), (619115, 619140, 'TYPE')]
===== sent Then $f$ is formally unramified, i.e., $\Omega_{X/Y} = 0$.
\end{lemma}

\begin{proof}
Recall that $f$ is formally unramified if and only if $\Omega_{X/Y} = 0$ by
Lemma \ref{lemma-formally-unramified-differentials}.
Via Lemma \ref{lemma-weakly-etale-characterize} and
Morphisms, Lemma \ref{morphisms-lemma-differentials-affine}
this follows from the case of rings which is
More on Algebra, Lemma \ref{more-algebra-lemma-formally-unramified}.
\end{proof}

\begin{lemma}
\label{lemma-when-weakly-etale}
Let $f : X \to Y$ be a morphism of schemes. ========

regex_match $f : X \to Y$
nn candidate X \to
nn candidate a morphism
nn a morphism
new_annotation [(619657, 619670, 'VAR'), (619674, 619684, 'TYPE')]
===== sent \end{proof}

\begin{lemma}
\label{lemma-reduced-goes-up}
Let $f : X \to Y$ be a morphism of schemes.
 ========

regex_match $f : X \to Y$
nn candidate X \to
nn candidate a morphism
nn a morphism
new_annotation [(620487, 620500, 'VAR'), (620504, 620514, 'TYPE')]
===== sent \begin{lemma}
\label{lemma-weakly-etale-strictly-henselian-local-rings}
Let $f : X \to Y$ be a morphism of schemes.
 ========

regex_match $f : X \to Y$
nn candidate X \to
nn candidate a morphism
nn a morphism
new_annotation [(620973, 620986, 'VAR'), (620990, 621000, 'TYPE')]
===== sent \end{enumerate}
\end{lemma}

\begin{proof}
Let $x \in X$ be a point with image $y = f(x)$ in $Y$.
Choose a separable algebraic closure $\kappa^{sep}$ of $\kappa(x)$.
Let $\mathcal{O}_{X, x}^{sh}$ be the strict henselization
corresponding to $\kappa^{sep}$ and $\mathcal{O}_{Y, y}^{sh}$
the strict henselization relative to the separable algebraic
closure of $\kappa(y)$ in $\kappa^{sep}$.
 ========

regex_match $x \in X$
nn candidate a point
nn a point
new_annotation [(621275, 621284, 'VAR'), (621288, 621295, 'TYPE')]
===== sent \end{enumerate}
\end{lemma}

\begin{proof}
Let $x \in X$ be a point with image $y = f(x)$ in $Y$.
Choose a separable algebraic closure $\kappa^{sep}$ of $\kappa(x)$.
Let $\mathcal{O}_{X, x}^{sh}$ be the strict henselization
corresponding to $\kappa^{sep}$ and $\mathcal{O}_{Y, y}^{sh}$
the strict henselization relative to the separable algebraic
closure of $\kappa(y)$ in $\kappa^{sep}$.
 ========

regex_match $\mathcal{O}_{X, x}^{sh}$
nn candidate $\mathcal{O}_{X
nn candidate the strict henselization
nn the strict henselization
new_annotation [(621398, 621423, 'VAR'), (621427, 621451, 'TYPE')]
===== sent Thus (2) implies (1) by Lemma \ref{lemma-check-weakly-etale-stalks}.
\end{proof}

\begin{lemma}
\label{lemma-normal-goes-up}
Let $f : X \to Y$ be a morphism of schemes. If $Y$ is a normal scheme
and $f$ weakly \'etale, then $X$ is a normal scheme.
 ========

regex_match $f : X \to Y$
nn candidate X \to
nn candidate a morphism
nn a morphism
new_annotation [(623156, 623169, 'VAR'), (623173, 623183, 'TYPE')]
===== sent Hence the lemma follows from
Lemma \ref{lemma-weakly-etale-strictly-henselian-local-rings}.
\end{proof}

\begin{lemma}
\label{lemma-weakly-etale-permanence}
Let $S$ be a scheme. ========

regex_match $S$
nn candidate a scheme
nn a scheme
new_annotation [(623660, 623663, 'VAR'), (623667, 623675, 'TYPE')]
===== sent Let $f : X \to Y$ be a morphism of schemes over $S$.
If $X$, $Y$ are weakly \'etale over $S$, then $f$ is weakly \'etale.
 ========

regex_match $f : X \to Y$
nn candidate $
nn candidate X \to
nn candidate a morphism
nn a morphism
new_annotation [(623681, 623694, 'VAR'), (623698, 623708, 'TYPE')]
===== sent Although the proof of the result is kind of
laborious, in essence it follows in a straightforward manner from
Epp's result on eliminating ramification, see
More on Algebra, Theorem \ref{more-algebra-theorem-epp}.

\medskip\noindent
Let $A$ be a Dedekind domain with fraction field $K$.
Let $X$ be a scheme flat and of finite type over $A$.
 ========

regex_match $A$
nn candidate a Dedekind domain
nn a Dedekind domain
new_annotation [(624878, 624881, 'VAR'), (624885, 624902, 'TYPE')]
===== sent Although the proof of the result is kind of
laborious, in essence it follows in a straightforward manner from
Epp's result on eliminating ramification, see
More on Algebra, Theorem \ref{more-algebra-theorem-epp}.

\medskip\noindent
Let $A$ be a Dedekind domain with fraction field $K$.
Let $X$ be a scheme flat and of finite type over $A$.
 ========

regex_match $X$
nn candidate a scheme
nn a scheme
new_annotation [(624932, 624935, 'VAR'), (624939, 624947, 'TYPE')]
===== sent Let $L$ be a finite extension of $K$. Let $B$ be the integral
closure of $A$ in $L$. Then $B$ is a Dedekind domain
(Algebra, Lemma \ref{algebra-lemma-integral-closure-Dedekind}).
 ========

regex_match $L$
nn candidate a finite extension
nn a finite extension
new_annotation [(624986, 624989, 'VAR'), (624993, 625011, 'TYPE')]
===== sent Let $L$ be a finite extension of $K$. Let $B$ be the integral
closure of $A$ in $L$. Then $B$ is a Dedekind domain
(Algebra, Lemma \ref{algebra-lemma-integral-closure-Dedekind}).
 ========

regex_match $B$
nn candidate the integral
closure
nn the integral
closure
new_annotation [(625024, 625027, 'VAR'), (625031, 625051, 'TYPE')]
===== sent Let $X_B = X \times_{\Spec(A)} \Spec(B)$ be the base change.
 ========

regex_match $X_B = X \times_{\Spec(A)} \Spec(B)$
nn candidate the base change
nn the base change
new_annotation [(625165, 625201, 'VAR'), (625205, 625220, 'TYPE')]
===== sent More precisely, if $K \subset L \subset M$ are finite extensions
of fields with integral closures $A \subset B \subset C$
then the normalized base change $Z$ of $Y \to \Spec(B)$
relative to $L \subset M$ is equal to the normalized base change
of $X \to \Spec(A)$ relative to $K \subset M$.

\begin{theorem}
\label{theorem-normalized-base-change-with-reduced-fibre}
Let $A$ be a Dedekind ring with fraction field $K$.
Let $X$ be a scheme flat and of finite type over $A$.
 ========

regex_match $A$
nn candidate a Dedekind ring
nn a Dedekind ring
new_annotation [(626567, 626570, 'VAR'), (626574, 626589, 'TYPE')]
===== sent More precisely, if $K \subset L \subset M$ are finite extensions
of fields with integral closures $A \subset B \subset C$
then the normalized base change $Z$ of $Y \to \Spec(B)$
relative to $L \subset M$ is equal to the normalized base change
of $X \to \Spec(A)$ relative to $K \subset M$.

\begin{theorem}
\label{theorem-normalized-base-change-with-reduced-fibre}
Let $A$ be a Dedekind ring with fraction field $K$.
Let $X$ be a scheme flat and of finite type over $A$.
 ========

regex_match $X$
nn candidate a scheme
nn a scheme
new_annotation [(626619, 626622, 'VAR'), (626626, 626634, 'TYPE')]
===== sent This implies that $U$ is dense in all but
finitely many fibres, see Lemma \ref{lemma-nowhere-dense-generic-fibre}.
Let $x_1, \ldots, x_r \in X \setminus U$ be the finitely many generic
points of irreducible components of $X \setminus U$ which are moreover
generic points of irreducible components of fibres of $X \to \Spec(A)$.
Set $\mathcal{O}_i = \mathcal{O}_{X, x_i}$. Let $A_i$ be the localization of
$A$ at the maximal ideal corresponding to the image of $x_i$ in $\Spec(A)$.
 ========

regex_match $x_1, \ldots, x_r \in X \setminus U$
nn candidate x_r \in X \setminus
nn candidate the finitely many generic
points
nn the finitely many generic
points
new_annotation [(628184, 628220, 'VAR'), (628224, 628256, 'TYPE')]
===== sent This implies that $U$ is dense in all but
finitely many fibres, see Lemma \ref{lemma-nowhere-dense-generic-fibre}.
Let $x_1, \ldots, x_r \in X \setminus U$ be the finitely many generic
points of irreducible components of $X \setminus U$ which are moreover
generic points of irreducible components of fibres of $X \to \Spec(A)$.
Set $\mathcal{O}_i = \mathcal{O}_{X, x_i}$. Let $A_i$ be the localization of
$A$ at the maximal ideal corresponding to the image of $x_i$ in $\Spec(A)$.
 ========

regex_match $A_i$
nn candidate the localization
nn the localization
new_annotation [(628441, 628446, 'VAR'), (628450, 628466, 'TYPE')]
===== sent By
More on Algebra, Proposition
\ref{more-algebra-proposition-epp-essentially-finite-type}
there exist finite extensions
$K \subset K_i$ which are solutions for the extension of discrete valuation
rings $A_i \to \mathcal{O}_i$. Let $K \subset L$ be a finite extension
dominating all of the extensions $K \subset K_i$. ========

regex_match $K \subset L$
nn candidate a finite extension
nn a finite extension
new_annotation [(628778, 628791, 'VAR'), (628795, 628813, 'TYPE')]
===== sent Thus $Y \to X_B$ is an isomorphism over $U_B$.
Let $y \in Y$ be a generic point of an irreducible
component of a fibre of $Y \to \Spec(B)$ lying over the maximal
ideal $\mathfrak m \subset B$. Assume that $y \not \in U_B$.
 ========

regex_match $y \in Y$
nn candidate $y
nn candidate a generic point
nn a generic point
new_annotation [(629311, 629320, 'VAR'), (629324, 629339, 'TYPE')]
===== sent \end{proof}

\begin{lemma}[Variant over curves]
\label{lemma-normalized-base-change-with-reduced-fibre-over-curve}
Let $f : X \to S$ be a flat, finite type morphism of schemes.
 ========

regex_match $f : X \to S$
nn candidate X \to
nn candidate a flat, finite type morphism
nn a flat, finite type morphism
new_annotation [(630296, 630309, 'VAR'), (630313, 630341, 'TYPE')]
===== sent \end{lemma}

\begin{proof}
Choose a finite affine open covering $S = \bigcup \Spec(A_i)$.
Then $K$ is equal to the fraction field of $A_i$ for all $i$.
Let $X_i = X \times_S \Spec(A_i)$.
Choose $L_i/K$ as in
Theorem \ref{theorem-normalized-base-change-with-reduced-fibre}
for the morphism $X_i \to \Spec(A_i)$.
 ========

new_annotation []
===== sent Let $B_i \subset L_i$ be the integral closure of $A_i$ and
let $Y_i$ be the normalized base change of $X$ to $B_i$.
Let $L/K$ be a finite extension dominating each $L_i$.
Let $T_i \subset T$ be the inverse image of $\Spec(A_i)$.
For each $i$ we get a commutative diagram
$$
\xymatrix{
g^{-1}(T_i) \ar[r] \ar[d] & Y_i \ar[r] \ar[d] & X \times_S \Spec(A_i) \ar[d] \\
 ========

regex_match $B_i \subset L_i$
nn candidate the integral closure
nn the integral closure
new_annotation [(631090, 631107, 'VAR'), (631111, 631131, 'TYPE')]
===== sent Let $B_i \subset L_i$ be the integral closure of $A_i$ and
let $Y_i$ be the normalized base change of $X$ to $B_i$.
Let $L/K$ be a finite extension dominating each $L_i$.
Let $T_i \subset T$ be the inverse image of $\Spec(A_i)$.
For each $i$ we get a commutative diagram
$$
\xymatrix{
g^{-1}(T_i) \ar[r] \ar[d] & Y_i \ar[r] \ar[d] & X \times_S \Spec(A_i) \ar[d] \\
 ========

regex_match $Y_i$
nn candidate the normalized base change
nn the normalized base change
new_annotation [(631149, 631154, 'VAR'), (631158, 631184, 'TYPE')]
===== sent Let $B_i \subset L_i$ be the integral closure of $A_i$ and
let $Y_i$ be the normalized base change of $X$ to $B_i$.
Let $L/K$ be a finite extension dominating each $L_i$.
Let $T_i \subset T$ be the inverse image of $\Spec(A_i)$.
For each $i$ we get a commutative diagram
$$
\xymatrix{
g^{-1}(T_i) \ar[r] \ar[d] & Y_i \ar[r] \ar[d] & X \times_S \Spec(A_i) \ar[d] \\
 ========

regex_match $L/K$
nn candidate a finite extension
nn a finite extension
new_annotation [(631206, 631211, 'VAR'), (631215, 631233, 'TYPE')]
===== sent Let $B_i \subset L_i$ be the integral closure of $A_i$ and
let $Y_i$ be the normalized base change of $X$ to $B_i$.
Let $L/K$ be a finite extension dominating each $L_i$.
Let $T_i \subset T$ be the inverse image of $\Spec(A_i)$.
For each $i$ we get a commutative diagram
$$
\xymatrix{
g^{-1}(T_i) \ar[r] \ar[d] & Y_i \ar[r] \ar[d] & X \times_S \Spec(A_i) \ar[d] \\
 ========

regex_match $T_i \subset T$
nn candidate the inverse image
nn the inverse image
new_annotation [(631261, 631276, 'VAR'), (631280, 631297, 'TYPE')]
===== sent \end{proof}

\begin{lemma}[Variant with separable extension]
\label{lemma-normalized-base-change-with-reduced-fibre-separable}
Let $A$ be a Dedekind ring with fraction field $K$.
Let $X$ be a scheme flat and of finite type over $A$.
 ========

regex_match $A$
nn candidate a Dedekind ring
nn a Dedekind ring
new_annotation [(631988, 631991, 'VAR'), (631995, 632010, 'TYPE')]
===== sent \end{proof}

\begin{lemma}[Variant with separable extension]
\label{lemma-normalized-base-change-with-reduced-fibre-separable}
Let $A$ be a Dedekind ring with fraction field $K$.
Let $X$ be a scheme flat and of finite type over $A$.
 ========

regex_match $X$
nn candidate a scheme
nn a scheme
new_annotation [(632040, 632043, 'VAR'), (632047, 632055, 'TYPE')]
===== sent This implies that $U$ is dense in all but
finitely many fibres, see Lemma \ref{lemma-nowhere-dense-generic-fibre}.
Let $x_1, \ldots, x_r \in X \setminus U$ be the finitely many generic
points of irreducible components of $X \setminus U$ which are moreover
generic points of irreducible components of fibres of $X \to \Spec(A)$.
Set $\mathcal{O}_i = \mathcal{O}_{X, x_i}$. Observe that the fraction
field of $\mathcal{O}_i$ is the residue field of a generic point of $X$.
Let $A_i$ be the localization of $A$ at the maximal ideal corresponding
to the image of $x_i$ in $\Spec(A)$. We may apply More on Algebra, Lemma
\ref{more-algebra-lemma-epp-essentially-finite-type-separable}
and we find finite separable extensions $K \subset K_i$ which are
solutions for $A_i \to \mathcal{O}_i$. Let $K \subset L$ be a finite
separable extension dominating all of the extensions $K \subset K_i$.
Then $K \subset L$ is still a solution for $A_i \to \mathcal{O}_i$ by
More on Algebra, Lemma \ref{more-algebra-lemma-formally-smooth-goes-up}.

\medskip\noindent
Consider the diagram (\ref{equation-normalized-base-change})
with the extension $L/K$ we just produced. ========

regex_match $x_1, \ldots, x_r \in X \setminus U$
nn candidate x_r \in X \setminus
nn candidate the finitely many generic
points
nn the finitely many generic
points
new_annotation [(633995, 634031, 'VAR'), (634035, 634067, 'TYPE')]
===== sent This implies that $U$ is dense in all but
finitely many fibres, see Lemma \ref{lemma-nowhere-dense-generic-fibre}.
Let $x_1, \ldots, x_r \in X \setminus U$ be the finitely many generic
points of irreducible components of $X \setminus U$ which are moreover
generic points of irreducible components of fibres of $X \to \Spec(A)$.
Set $\mathcal{O}_i = \mathcal{O}_{X, x_i}$. Observe that the fraction
field of $\mathcal{O}_i$ is the residue field of a generic point of $X$.
Let $A_i$ be the localization of $A$ at the maximal ideal corresponding
to the image of $x_i$ in $\Spec(A)$. We may apply More on Algebra, Lemma
\ref{more-algebra-lemma-epp-essentially-finite-type-separable}
and we find finite separable extensions $K \subset K_i$ which are
solutions for $A_i \to \mathcal{O}_i$. Let $K \subset L$ be a finite
separable extension dominating all of the extensions $K \subset K_i$.
Then $K \subset L$ is still a solution for $A_i \to \mathcal{O}_i$ by
More on Algebra, Lemma \ref{more-algebra-lemma-formally-smooth-goes-up}.

\medskip\noindent
Consider the diagram (\ref{equation-normalized-base-change})
with the extension $L/K$ we just produced. ========

regex_match $A_i$
nn candidate the localization
nn the localization
new_annotation [(634351, 634356, 'VAR'), (634360, 634376, 'TYPE')]
===== sent This implies that $U$ is dense in all but
finitely many fibres, see Lemma \ref{lemma-nowhere-dense-generic-fibre}.
Let $x_1, \ldots, x_r \in X \setminus U$ be the finitely many generic
points of irreducible components of $X \setminus U$ which are moreover
generic points of irreducible components of fibres of $X \to \Spec(A)$.
Set $\mathcal{O}_i = \mathcal{O}_{X, x_i}$. Observe that the fraction
field of $\mathcal{O}_i$ is the residue field of a generic point of $X$.
Let $A_i$ be the localization of $A$ at the maximal ideal corresponding
to the image of $x_i$ in $\Spec(A)$. We may apply More on Algebra, Lemma
\ref{more-algebra-lemma-epp-essentially-finite-type-separable}
and we find finite separable extensions $K \subset K_i$ which are
solutions for $A_i \to \mathcal{O}_i$. Let $K \subset L$ be a finite
separable extension dominating all of the extensions $K \subset K_i$.
Then $K \subset L$ is still a solution for $A_i \to \mathcal{O}_i$ by
More on Algebra, Lemma \ref{more-algebra-lemma-formally-smooth-goes-up}.

\medskip\noindent
Consider the diagram (\ref{equation-normalized-base-change})
with the extension $L/K$ we just produced. ========

regex_match $K \subset L$
nn candidate a finite
separable extension
nn a finite
separable extension
new_annotation [(634664, 634677, 'VAR'), (634681, 634709, 'TYPE')]
===== sent Thus $Y \to X_B$ is an isomorphism over $U_B$.
Let $y \in Y$ be a generic point of an irreducible
component of a fibre of $Y \to \Spec(B)$ lying over the maximal
ideal $\mathfrak m \subset B$. Assume that $y \not \in U_B$.
 ========

regex_match $y \in Y$
nn candidate $y
nn candidate a generic point
nn a generic point
new_annotation [(635207, 635216, 'VAR'), (635220, 635235, 'TYPE')]
===== sent \end{proof}

\begin{lemma}[Variant with separable extensions over curves]
\label{lemma-normalized-base-change-with-reduced-fibre-over-curve-separable}
Let $f : X \to S$ be a flat, finite type morphism of schemes.
 ========

regex_match $f : X \to S$
nn candidate X \to
nn candidate a flat, finite type morphism
nn a flat, finite type morphism
new_annotation [(636228, 636241, 'VAR'), (636245, 636273, 'TYPE')]
===== sent \end{lemma}

\begin{proof}
Let $f : X \to Y$ be an ind-quasi-affine morphism. ========

regex_match $f : X \to Y$
nn candidate X \to
nn candidate an ind-quasi-affine morphism
nn an ind-quasi-affine morphism
new_annotation [(638172, 638185, 'VAR'), (638189, 638217, 'TYPE')]
===== sent Let $Z$ be an affine
scheme and let $Z \to Y$ be a morphism. ========

regex_match $Z$
nn candidate an affine
scheme
nn an affine
scheme
new_annotation [(638223, 638226, 'VAR'), (638230, 638246, 'TYPE')]
===== sent Let $Z$ be an affine
scheme and let $Z \to Y$ be a morphism. ========

regex_match $Z \to Y$
nn candidate a morphism
nn a morphism
new_annotation [(638255, 638264, 'VAR'), (638268, 638278, 'TYPE')]
===== sent Let $W \subset Z \times_Y X$ be a quasi-compact open.
 ========

regex_match $W \subset Z \times_Y X$
nn candidate $
nn candidate \subset Z \times_Y
nn candidate a quasi-compact open
nn a quasi-compact open
new_annotation [(638329, 638353, 'VAR'), (638357, 638377, 'TYPE')]
===== sent \end{lemma}

\begin{proof}
Let $f : X \to Y$ be a morphism of schemes. ========

regex_match $f : X \to Y$
nn candidate X \to
nn candidate a morphism
nn a morphism
new_annotation [(639220, 639233, 'VAR'), (639237, 639247, 'TYPE')]
===== sent Let $\{g_i : Y_i \to Y\}$
be an fpqc covering such that the base change $f_i : X_i \to Y_i$
is ind-quasi-affine for all $i$. We will show $f$ is ind-quasi-affine.
 ========

new_annotation []
===== sent Namely, let $U \subset X$ be a quasi-compact open mapping into
an affine open $V \subset Y$. We have to show that $U$ is quasi-affine.
 ========

regex_match $U \subset X$
nn candidate $U
nn candidate a quasi-compact open mapping
nn a quasi-compact open mapping
new_annotation [(639435, 639448, 'VAR'), (639452, 639480, 'TYPE')]
===== sent Let $V_j \subset Y_{i_j}$, $j = 1, \ldots, m$ be affine opens
such that $V = \bigcup g_{i_j}(V_j)$ (exist by definition of
fpqc coverings). ========

new_annotation []
===== sent \end{lemma}

\begin{proof}
Let $f : X \to Y$ be a separated locally quasi-finite morphism of schemes.
Let $V \subset Y$ be affine and $U \subset f^{-1}(V)$ quasi-compact
open. ========

regex_match $f : X \to Y$
nn candidate X \to
nn candidate a separated locally quasi-finite morphism
nn a separated locally quasi-finite morphism
new_annotation [(640475, 640488, 'VAR'), (640492, 640533, 'TYPE')]
===== sent \end{lemma}

\begin{proof}
Let $f : X \to Y$ be a separated locally quasi-finite morphism of schemes.
Let $V \subset Y$ be affine and $U \subset f^{-1}(V)$ quasi-compact
open. ========

new_annotation []
===== sent The material in this section is discussed in the correct
generality in Criteria for Representability, Section
\ref{criteria-section-relative-morphisms}.

\medskip\noindent
Let $S$ be a scheme. ========

regex_match $S$
nn candidate a scheme
nn a scheme
new_annotation [(641301, 641304, 'VAR'), (641308, 641316, 'TYPE')]
===== sent Let $Z$ and $X$ be schemes over $S$.
Given a scheme $T$ over $S$ we can consider morphisms
$b : T \times_S Z \to T \times_S X$ over $S$. Picture
\begin{equation}
\label{equation-hom}
\vcenter{
\xymatrix{
T \times_S Z \ar[rd] \ar[rr]_b & &
T \times_S X \ar[ld] & Z \ar[rd] & & X \ar[ld] \\
& T \ar[rrr] & & & S
}
}
\end{equation}
Of course, we can also think of $b$ as a morphism
$b : T \times_S Z \to X$ such that
$$
\xymatrix{
T \times_S Z \ar[r] \ar[d] \ar@/^1pc/[rrr]_-b &
Z \ar[rd] & & X \ar[ld] \\
T \ar[rr] & & S
}
$$
commutes. ========

new_annotation []
===== sent \begin{lemma}
\label{lemma-hom-from-finite-free-into-affine}
Let $Z \to S$ and $X \to S$ be morphisms of affine schemes.
 ========

new_annotation []
===== sent \end{proof}

\begin{lemma}
\label{lemma-hom-from-finite-locally-free-into-affine}
Let $Z \to S$ and $X \to S$ be morphisms of schemes.
If $Z \to S$ is finite locally free and $X \to S$ is affine,
then $\mathit{Mor}_S(Z, X)$ is representable by a scheme
affine over $S$.
\end{lemma}

 ========

new_annotation []
===== sent \begin{proof}
Choose an affine open covering $S = \bigcup U_i$ such that
$\Gamma(Z \times_S U_i, \mathcal{O}_{Z \times_S U_i})$ is
finite free over $\mathcal{O}_S(U_i)$. Let $F_i \subset \mathit{Mor}_S(Z, X)$
be the subfunctor which assigns to $T/S$ the empty set if
$T \to S$ does not factor through $U_i$ and $\mathit{Mor}_S(Z, X)(T)$
otherwise. ========

new_annotation []
===== sent It holds if one of the following
is true: $X$ is quasi-affine, $f$ is quasi-affine, $f$ is quasi-projective,
$f$ is locally projective, there exists an ample invertible sheaf on $X$,
there exists an $f$-ample invertible sheaf on $X$, or
there exists an $f$-very ample invertible sheaf on $X$.

\begin{lemma}
\label{lemma-hom-from-finite-locally-free-representable}
Let $Z \to S$ and $X \to S$ be morphisms of schemes.
 ========

new_annotation []
===== sent \end{proof}

\begin{lemma}
\label{lemma-hom-from-finite-locally-free-separated-lqf}
Let $Z \to S$ and $X \to S$ be morphisms of schemes.
 ========

new_annotation []
===== sent X' \ar[r] & S'
}
$$
Let $S \to S'$ be a morphism. Denote by $X$ and $Y$ the base
changes of $X'$ and $Y'$ to $S$.
Assume $Y' \to S'$ and $Z' \to X'$ are flat.
 ========

regex_match $S \to S'$
nn candidate a morphism
nn a morphism
new_annotation [(649324, 649334, 'VAR'), (649338, 649348, 'TYPE')]
===== sent \end{proof}

\begin{lemma}[Derived Chow's lemma]
\label{lemma-derived-chow}
Let $A$ be a ring. ========

regex_match $A$
nn candidate a ring
nn a ring
new_annotation [(650527, 650530, 'VAR'), (650534, 650540, 'TYPE')]
===== sent Let $X$ be a separated scheme of finite presentation
over $A$. ========

regex_match $X$
nn candidate a separated scheme
nn a separated scheme
new_annotation [(650546, 650549, 'VAR'), (650553, 650571, 'TYPE')]
===== sent Let $x \in ========

new_annotation []
===== sent Let $x' \in X'$ be the image of $x$.
If we can prove the lemma for $x' \in X'/A'$, then
the lemma follows for $x \in X/A$.
 ========

regex_match $x' \in X'$
nn candidate x
nn candidate the image
nn the image
new_annotation [(651617, 651628, 'VAR'), (651632, 651641, 'TYPE')]
===== sent Namely, if $U', n', V', Z', z', E'$ provide the solution
for $x' \in X'/A'$, then we can let
$U \subset X$ be the inverse image of $U'$,
let $n = n'$,
let $V \subset \mathbf{P}^n_A$ be the inverse image of $V'$,
let $Z \subset X \times \mathbf{P}^n$ be
the scheme theoretic inverse image of $Z'$,
let $z \in Z$ be the unique point mapping to $x$, and
let $E$ be the derived pullback of $E'$.
Observe that $E$ is pseudo-coherent by
Cohomology, Lemma \ref{cohomology-lemma-pseudo-coherent-pullback}.
It only remains to check (5). ========

new_annotation []
===== sent Namely, if $U', n', V', Z', z', E'$ provide the solution
for $x' \in X'/A'$, then we can let
$U \subset X$ be the inverse image of $U'$,
let $n = n'$,
let $V \subset \mathbf{P}^n_A$ be the inverse image of $V'$,
let $Z \subset X \times \mathbf{P}^n$ be
the scheme theoretic inverse image of $Z'$,
let $z \in Z$ be the unique point mapping to $x$, and
let $E$ be the derived pullback of $E'$.
Observe that $E$ is pseudo-coherent by
Cohomology, Lemma \ref{cohomology-lemma-pseudo-coherent-pullback}.
It only remains to check (5). ========

regex_match $V \subset \mathbf{P}^n_A$
nn candidate $V
nn candidate the inverse image
nn the inverse image
new_annotation [(651891, 651917, 'VAR'), (651921, 651938, 'TYPE')]
===== sent Namely, if $U', n', V', Z', z', E'$ provide the solution
for $x' \in X'/A'$, then we can let
$U \subset X$ be the inverse image of $U'$,
let $n = n'$,
let $V \subset \mathbf{P}^n_A$ be the inverse image of $V'$,
let $Z \subset X \times \mathbf{P}^n$ be
the scheme theoretic inverse image of $Z'$,
let $z \in Z$ be the unique point mapping to $x$, and
let $E$ be the derived pullback of $E'$.
Observe that $E$ is pseudo-coherent by
Cohomology, Lemma \ref{cohomology-lemma-pseudo-coherent-pullback}.
It only remains to check (5). ========

new_annotation []
===== sent Namely, if $U', n', V', Z', z', E'$ provide the solution
for $x' \in X'/A'$, then we can let
$U \subset X$ be the inverse image of $U'$,
let $n = n'$,
let $V \subset \mathbf{P}^n_A$ be the inverse image of $V'$,
let $Z \subset X \times \mathbf{P}^n$ be
the scheme theoretic inverse image of $Z'$,
let $z \in Z$ be the unique point mapping to $x$, and
let $E$ be the derived pullback of $E'$.
Observe that $E$ is pseudo-coherent by
Cohomology, Lemma \ref{cohomology-lemma-pseudo-coherent-pullback}.
It only remains to check (5). ========

regex_match $z \in Z$
nn candidate $
nn candidate the unique point mapping
nn the unique point mapping
new_annotation [(652037, 652046, 'VAR'), (652050, 652074, 'TYPE')]
===== sent Namely, if $U', n', V', Z', z', E'$ provide the solution
for $x' \in X'/A'$, then we can let
$U \subset X$ be the inverse image of $U'$,
let $n = n'$,
let $V \subset \mathbf{P}^n_A$ be the inverse image of $V'$,
let $Z \subset X \times \mathbf{P}^n$ be
the scheme theoretic inverse image of $Z'$,
let $z \in Z$ be the unique point mapping to $x$, and
let $E$ be the derived pullback of $E'$.
Observe that $E$ is pseudo-coherent by
Cohomology, Lemma \ref{cohomology-lemma-pseudo-coherent-pullback}.
It only remains to check (5). ========

regex_match $E$
nn candidate the derived pullback
nn the derived pullback
new_annotation [(652091, 652094, 'VAR'), (652098, 652118, 'TYPE')]
===== sent Let $Z$ be the scheme theoretic closure of
$$
(\text{id}_U, j) : U \longrightarrow X \times_A ========

regex_match $Z$
nn candidate the scheme theoretic closure
nn the scheme theoretic closure
new_annotation [(653512, 653515, 'VAR'), (653519, 653547, 'TYPE')]
===== sent \mathbf{P}^n_A
$$
Since the projection $X \times \mathbf{P}^n \to X$ is separated,
we conclude from Morphisms, Lemma
\ref{morphisms-lemma-scheme-theoretic-image-of-partial-section}
that $b : Z \to X$ is an isomorphism over $U$.
Let $z \in Z$ be the unique point lying over $x$.

\medskip\noindent
 ========

regex_match $z \in Z$
nn candidate $
nn candidate the unique point
nn the unique point
new_annotation [(653834, 653843, 'VAR'), (653847, 653863, 'TYPE')]
===== sent Let $Y \subset \mathbf{P}^n_A$ be the scheme theoretic
closure of $j$. Then it is clear that $Z \subset X \times_A ========

regex_match $Y \subset \mathbf{P}^n_A$
nn candidate $Y
nn candidate the scheme theoretic
closure
nn the scheme theoretic
closure
new_annotation [(653903, 653929, 'VAR'), (653933, 653961, 'TYPE')]
===== sent Let $A$, $x \in X$, and
$U, n, V, Z, z, E$ be as in Lemma \ref{lemma-derived-chow}.
For any $K \in D_\QCoh(\mathcal{O}_X)$ we have
$$
Rq_*(Lp^*K \otimes^\mathbf{L} E)|_V = R(U \to V)_*K|_U
$$
where $p : X \times_A \mathbf{P}^n_A \to X$ and
$q : X \times_A \mathbf{P}^n_A \to \mathbf{P}^n_A$ are
the projections and where the morphism $U \to V$ is
the finitely presented closed immersion $c \circ (b|_U)^{-1}$.
\end{lemma}

\begin{proof}
Since $b^{-1}(U) = c^{-1}(V)$ and since $c$ is a closed immersion
over $V$, we see that $c \circ (b|_U)^{-1}$ is a closed immersion.
 ========

new_annotation []
===== sent \end{proof}

\begin{lemma}
\label{lemma-characterize-pseudo-coherent}
Let $A$ be a ring. ========

regex_match $A$
nn candidate a ring
nn a ring
new_annotation [(656379, 656382, 'VAR'), (656386, 656392, 'TYPE')]
===== sent Let $X$ be a scheme separated and
of finite presentation over $A$. ========

regex_match $X$
nn candidate a scheme
nn a scheme
new_annotation [(656398, 656401, 'VAR'), (656405, 656413, 'TYPE')]
===== sent Let $K \in D_\QCoh(\mathcal{O}_X)$.
If $R\Gamma(X, E \otimes^\mathbf{L} K)$ is pseudo-coherent
in $D(A)$ for every pseudo-coherent $E$ in $D(\mathcal{O}_X)$,
then $K$ is pseudo-coherent relative to $A$.
 ========

new_annotation []
===== sent \end{lemma}

\begin{proof}
Assume $K \in D_\QCoh(\mathcal{O}_X)$ and
$R\Gamma(X, E \otimes^\mathbf{L} K)$ is pseudo-coherent
in $D(A)$ for every pseudo-coherent $E$ in $D(\mathcal{O}_X)$.
Let $x \in X$. We will show that $K$ is pseudo-coherent relative to $A$
in a neighbourhood of $x$ and this will prove the lemma.

 ========

new_annotation []
===== sent By Lemma \ref{lemma-compute-Fourier-Mukai-for-derived-chow}
we have
$$
Rq_*(Lp^*K \otimes^\mathbf{L} E)|_{X \times_A V} =
R(U \to V)_*K|_U
$$
Since $U \to V$ is a closed immersion into an open subscheme of
$\mathbf{P}^n_A$ this means $K|_U$ is pseudo-coherent relative to $A$
by Lemma \ref{lemma-check-relative-pseudo-coherence-on-charts}.
\end{proof}







\section{Descent finiteness properties of complexes}
\label{section-descent-finiteness}

\noindent
This section is the continuation of Derived Categories of Schemes,
Section \ref{perfect-section-descent-finiteness}.

\begin{lemma}
\label{lemma-relative-pseudo-coherent-descends-fppf}
Let $X \to S$ be locally of finite type.
 ========

new_annotation []
===== sent Let $\{f_i : X_i \to X\}$ be an fppf covering of schemes.
 ========

regex_match $\{f_i : X_i \to X\}$
nn candidate an fppf
nn an fppf
new_annotation [(658859, 658880, 'VAR'), (658884, 658891, 'TYPE')]
===== sent Let $E \in D_\QCoh(\mathcal{O}_X)$. Let $m \in \mathbf{Z}$.
Then $E$ is $m$-pseudo-coherent relative to $S$
if and only if each $Lf_i^*E$ is $m$-pseudo-coherent relative to $S$.
\end{lemma}

\begin{proof}
 ========

new_annotation []
===== sent Let $E \in D_\QCoh(\mathcal{O}_X)$. Let $m \in \mathbf{Z}$.
Then $E$ is $m$-pseudo-coherent relative to $S$
if and only if each $Lf_i^*E$ is $m$-pseudo-coherent relative to $S$.
\end{lemma}

\begin{proof}
 ========

new_annotation []
===== sent Hence we conclude by
Lemma \ref{lemma-summands-relative-pseudo-coherent}.
\end{proof}

\begin{lemma}
\label{lemma-relative-pseudo-coherent-post-compose}
Let $X \to T \to S$ be morphisms of schemes. ========

new_annotation []
===== sent Let $E \in D(\mathcal{O}_X)$. Let $m \in \mathbf{Z}$.
Then $E$ is $m$-pseudo-coherent relative to $S$
if and only if $E$ is $m$-pseudo-coherent relative to $T$.
\end{lemma}

\begin{proof}
 ========

new_annotation []
===== sent Let $E \in D(\mathcal{O}_X)$. Let $m \in \mathbf{Z}$.
Then $E$ is $m$-pseudo-coherent relative to $S$
if and only if $E$ is $m$-pseudo-coherent relative to $T$.
\end{lemma}

\begin{proof}
 ========

new_annotation []
===== sent \end{proof}

\begin{lemma}
\label{lemma-relative-pseudo-coherent-descends-fppf-base}
Let $f : X \to S$ be locally of finite type.
 ========

new_annotation []
===== sent Let $\{S_i \to S\}$ be an fppf covering of schemes.
 ========

regex_match $\{S_i \to S\}$
nn candidate an fppf
nn an fppf
new_annotation [(661781, 661796, 'VAR'), (661800, 661807, 'TYPE')]
===== sent Let $E \in D_\QCoh(\mathcal{O}_X)$. Let $m \in \mathbf{Z}$.
Then $E$ is $m$-pseudo-coherent relative to $S$
if and only if each $Lg_i^*E$ is $m$-pseudo-coherent relative to $S_i$.
\end{lemma}

\begin{proof}
 ========

new_annotation []
===== sent Let $E \in D_\QCoh(\mathcal{O}_X)$. Let $m \in \mathbf{Z}$.
Then $E$ is $m$-pseudo-coherent relative to $S$
if and only if each $Lg_i^*E$ is $m$-pseudo-coherent relative to $S_i$.
\end{lemma}

\begin{proof}
 ========

new_annotation []
===== sent \end{proof}






\section{Relatively perfect objects}
\label{section-relatively-perfect}

\noindent
This section is a continuation of the discussion in
Derived Categories of Schemes, Section
\ref{perfect-section-relatively-perfect}.

\begin{lemma}
\label{lemma-thickening-pseudo-coherent}
Let $i : X \to X'$ be a finite order thickening of schemes. ========

regex_match $i : X \to X'$
nn candidate X \to
nn candidate a finite order thickening
nn a finite order thickening
new_annotation [(662978, 662992, 'VAR'), (662996, 663021, 'TYPE')]
===== sent To do this, we may reduce to the case of a first order thickening, see
Section \ref{section-thickenings}. Let $\mathcal{I} \subset \mathcal{O}_{X'}$
be the quasi-coherent sheaf of ideals cutting out $X$.
Tensoring the short exact sequence
$$
0 \to \mathcal{I} \to \mathcal{O}_{X'} \to i_*\mathcal{O}_X \to 0
$$
with $K'$ we obtain a distinguished triangle
$$
K' \otimes_{\mathcal{O}_{X'}}^\mathbf{L} \mathcal{I}
\to K' \to
K' \otimes_{\mathcal{O}_{X'}}^\mathbf{L} i_*\mathcal{O}_X
\to
(K' \otimes_{\mathcal{O}_{X'}}^\mathbf{L} \mathcal{I})[1]
$$
Since $i_* = Ri_*$ and since we may view $\mathcal{I}$
as a quasi-coherent $\mathcal{O}_X$-module (as we have a first
order thickening) we may rewrite this as
$$
i_*(K \otimes_{\mathcal{O}_X}^\mathbf{L} \mathcal{I})
\to K' \to
i_*K \to
i_*(K \otimes_{\mathcal{O}_X}^\mathbf{L} \mathcal{I})[1]
$$
Please use Cohomology, Lemma
\ref{cohomology-lemma-projection-formula-closed-immersion}
to identify the terms. ========

new_annotation []
===== sent Let $E' \in D(\mathcal{O}_{X'})$. If $E = Li^*(E')$ is $Y$-perfect,
then $E'$ is $Y'$-perfect.
 ========

new_annotation []
===== sent In particular, $E'$ is in $D_\QCoh(\mathcal{O}_{X'})$, see
Derived Categories of Schemes, Lemma \ref{perfect-lemma-pseudo-coherent}.
To prove that $E'$ locally has finite tor dimension
we may work locally on $X'$. Hence we may assume
$X'$, $S'$, $X$, $S$ are affine, say given by
rings $A'$, $R'$, $A$, $R$.
Then we reduce to the commutative algebra version by
Derived Categories of Schemes,
Lemma \ref{perfect-lemma-affine-locally-rel-perfect}.
The commutative algebra version in More on Algebra, Lemma
\ref{more-algebra-lemma-thickening-relatively-perfect}.
\end{proof}

\begin{lemma}
\label{lemma-henselian-relatively-perfect}
Let $(R, I)$ be a pair consisting of a ring and an ideal $I$
contained in the Jacobson radical. ========

regex_match $(R, I)$
nn candidate $(R
nn candidate a pair
nn a pair
new_annotation [(666675, 666683, 'VAR'), (666687, 666693, 'TYPE')]
===== sent Set $S = \Spec(R)$ and $S_0 = \Spec(R/I)$.
Let $f : X \to S$ be proper, flat, and of finite presentation.
 ========

new_annotation []
===== sent Denote $X_0 = S_0 \times_S X$. Let $E \in D(\mathcal{O}_X)$
be pseudo-coherent. ========

new_annotation []
===== sent \end{lemma}

\begin{proof}
Choose a finite affine open covering $X = U_1 \cup \ldots \cup U_n$.
For each $i$ we can choose a closed immersion $U_i \to \mathbf{A}^{d_i}_S$.
Set $U_{i, 0} = S_0 \times_S U_i$.
For each $i$ the complex $E_0|_{U_{i, 0}}$ has tor amplitude
in $[a_i, b_i]$ for some $a_i, b_i \in \mathbf{Z}$.
Let $x \in X$ be a point.
 ========

regex_match $x \in X$
nn candidate a point
nn a point
new_annotation [(667369, 667378, 'VAR'), (667382, 667389, 'TYPE')]
===== sent The restriction $E|_{U_i}$ corresponds
(by Derived Categories of Schemes, Lemma
\ref{perfect-lemma-affine-compare-bounded} and
\ref{perfect-lemma-pseudo-coherent-affine})
to a pseudo-coherent object $K$ of $D(A)$.
Observe that $E_0$ corresponds to $K \otimes_A^\mathbf{L} A/IA$.
Let $\mathfrak q \subset ========

new_annotation []
===== sent Let $I \subset \mathfrak p_0 \subset R$ be the prime
ideal corresponding to $f(x_0)$.
Then we have
\begin{align*}
K \otimes_R^\mathbf{L} \kappa(\mathfrak p_0)
& =
(K \otimes_R^\mathbf{L} R/I) \otimes_{R/I}^\mathbf{L}
\kappa(\mathfrak p_0) \\
& =
(K \otimes_A^\mathbf{L} A/IA) \otimes_{R/I}^\mathbf{L} \kappa(\mathfrak p_0)
\end{align*}
the second equality because $R \to A$ is flat.
 ========

regex_match $I \subset \mathfrak p_0 \subset R$
nn candidate I
nn candidate we
nn we
new_annotation [(668753, 668788, 'VAR'), (668840, 668842, 'TYPE')]
===== sent To understand the title of this section, please take a look at
Algebraic Curves, Sections
\ref{curves-section-contracting-rational-tails},
\ref{curves-section-contracting-rational-bridges}, and
\ref{curves-section-contracting-to-stable}.

\begin{lemma}
\label{lemma-check-h1-fibre-zero}
Let $f : X \to Y$ be a proper morphism of schemes. ========

regex_match $f : X \to Y$
nn candidate X \to
nn candidate a proper morphism
nn a proper morphism
new_annotation [(669928, 669941, 'VAR'), (669945, 669962, 'TYPE')]
===== sent Let $y \in ========

new_annotation []
===== sent Let $y' \in Y'$ map to $y$. If we can show that
$H^1(X'_{y'}, \mathcal{O}_{X'_{y'}}) = 0$, then it follows
that $H^1(X_y, \mathcal{O}_{X_y}) = 0$ by flat base change
(Cohomology of Schemes, Lemma
\ref{coherent-lemma-flat-base-change-cohomology})
as $X'_{y'} = X_y \otimes_{\kappa(y)} \kappa(y')$.
 ========

new_annotation []
===== sent \end{proof}

\begin{lemma}
\label{lemma-h1-fibre-zero}
Let $f : X \to Y$ be a proper morphism of schemes. ========

regex_match $f : X \to Y$
nn candidate X \to
nn candidate a proper morphism
nn a proper morphism
new_annotation [(671813, 671826, 'VAR'), (671830, 671847, 'TYPE')]
===== sent Let $y \in ========

new_annotation []
===== sent Let $y_0 \in Y_0$ be the image of $y$.
Then $H^1(X_{0, y_0}, \mathcal{O}_{X_{0, y_0}}) = 0$
by flat base change (see above).
 ========

regex_match $y_0 \in Y_0$
nn candidate $y_0
nn candidate the image
nn the image
new_annotation [(674793, 674806, 'VAR'), (674810, 674819, 'TYPE')]
===== sent \end{proof}

\begin{lemma}
\label{lemma-globally-generated-vanishing}
Let $f : X \to Y$ be a proper morphism of schemes such
that $\dim(X_y) \leq 1$ and $H^1(X_y, \mathcal{O}_{X_y}) = 0$
for $y \in Y$. Let $\mathcal{F}$ be quasi-coherent on $X$. Then
\begin{enumerate}
\item $R^pf_*\mathcal{F} = 0$ for $p > 1$, and
\item $R^1f_*\mathcal{F} = 0$ if there is a surjection
$f^*\mathcal{G} \to \mathcal{F}$ with $\mathcal{G}$ quasi-coherent
on $Y$.
\end{enumerate}
If $Y$ is affine, then we also have
\begin{enumerate}
\item[(3)] $H^p(X, \mathcal{F}) = 0$ for $p \not \in \{0, 1\}$, and
\item[(4)] $H^1(X, \mathcal{F}) = 0$ if $\mathcal{F}$ is globally generated.
 ========

regex_match $f : X \to Y$
nn candidate X \to
nn candidate a proper morphism
nn a proper morphism
new_annotation [(678948, 678961, 'VAR'), (678965, 678982, 'TYPE')]
===== sent \end{proof}

\begin{lemma}
\label{lemma-globally-generated-vanishing}
Let $f : X \to Y$ be a proper morphism of schemes such
that $\dim(X_y) \leq 1$ and $H^1(X_y, \mathcal{O}_{X_y}) = 0$
for $y \in Y$. Let $\mathcal{F}$ be quasi-coherent on $X$. Then
\begin{enumerate}
\item $R^pf_*\mathcal{F} = 0$ for $p > 1$, and
\item $R^1f_*\mathcal{F} = 0$ if there is a surjection
$f^*\mathcal{G} \to \mathcal{F}$ with $\mathcal{G}$ quasi-coherent
on $Y$.
\end{enumerate}
If $Y$ is affine, then we also have
\begin{enumerate}
\item[(3)] $H^p(X, \mathcal{F}) = 0$ for $p \not \in \{0, 1\}$, and
\item[(4)] $H^1(X, \mathcal{F}) = 0$ if $\mathcal{F}$ is globally generated.
 ========

new_annotation []
===== sent \end{proof}

\begin{lemma}
\label{lemma-h1-fibre-zero-check-h0-kappa}
Let $f : X \to Y$ be a proper morphism of schemes. ========

regex_match $f : X \to Y$
nn candidate X \to
nn candidate a proper morphism
nn a proper morphism
new_annotation [(680965, 680978, 'VAR'), (680982, 680999, 'TYPE')]
===== sent By flat base change
(Cohomology of Schemes, Lemma \ref{coherent-lemma-flat-base-change-cohomology})
we see that the result holds for $X'' \to Y''$.
Thus we may assume $Y'$ is a closed subscheme of $Y$.
Let $\mathcal{I} \subset \mathcal{O}_Y$ be the ideal cutting out $Y'$.
Then there is a short exact sequence
$$
0 \to \mathcal{I}\mathcal{O}_X \to
\mathcal{O}_X \to \mathcal{O}_{X'} \to 0
$$
where we view $\mathcal{O}_{X'}$ as a quasi-coherent module on $X$.
 ========

regex_match $\mathcal{I} \subset \mathcal{O}_Y$
nn candidate $\mathcal{I
nn candidate the ideal
nn the ideal
new_annotation [(681673, 681708, 'VAR'), (681712, 681721, 'TYPE')]
===== sent Let $s \in S$ be a point. ========

regex_match $s \in S$
nn candidate $s
nn candidate a point
nn a point
new_annotation [(682612, 682621, 'VAR'), (682625, 682632, 'TYPE')]
===== sent \in V$,
(c) $R^1f_*\mathcal{O}_X|_V = 0$,
(d) $\mathcal{O}_V \to f_*\mathcal{O}_X|_V$
is surjective,
and (b), (c), and (d) remain true after base change by any $Y' \to V$.
\end{lemma}

\begin{proof}
Let $y \in Y$. ========

new_annotation []
===== sent After increasing $0$ we may assume $Y_0$
is affine and $X_0 \to Y_0$ proper, see
Limits, Lemmas \ref{limits-lemma-eventually-proper} and
\ref{limits-lemma-limit-affine}.
Let $s_0 \in S_0$ be the image of $s$.
As $Y_s$ is affine, we see that
$R^1f_{s,*}\mathcal{O}_{X_s} = 0$ is equivalent
to $H^1(X_s, \mathcal{O}_{X_s}) = 0$.
Since $X_s$ is the base change of $X_{0, s_0}$ by
the faithfully flat map $\kappa(s_0) \to \kappa(s)$
we see that $H^1(X_{0, s_0}, \mathcal{O}_{X_{0, s_0}}) = 0$
and hence $R^1f_{0, *}\mathcal{O}_{X_{0, s_0}} = 0$.
Similarly, the surjectivity of
$\mathcal{O}_{Y_s} \to f_{s, *}\mathcal{O}_{X_s}$
implies the surjectivity of
$\mathcal{O}_{Y_{0, s_0}} \to f_{0, *}\mathcal{O}_{X_{0, s_0}}$
 ========

regex_match $s_0 \in S_0$
nn candidate the image
nn the image
new_annotation [(686804, 686817, 'VAR'), (686821, 686830, 'TYPE')]
===== sent Let $y \in Y_s$ be a point. ========

regex_match $y \in Y_s$
nn candidate $
nn candidate a point
nn a point
new_annotation [(688316, 688327, 'VAR'), (688331, 688338, 'TYPE')]
===== sent Let $s \in S$ be a point. ========

regex_match $s \in S$
nn candidate $s
nn candidate a point
nn a point
new_annotation [(692451, 692460, 'VAR'), (692464, 692471, 'TYPE')]
===== sent \in V$,
(d) $R^1f_*\mathcal{O}_X|_V = 0$,
(e) $\mathcal{O}_V \to f_*\mathcal{O}_X|_V$
is an isomorphism, and (a) -- (e)
remain true after base change of $f^{-1}(V) \to V$ by any $S' \to S$.
\end{lemma}

\begin{proof}
Let $y \in Y_s$. We may always replace $Y$ by an open neighbourhood of $y$.
Thus we may assume $Y$ and $S$ affine. ========

new_annotation []
===== sent After increasing $0$ we may assume $Y_0$
is affine and $X_0 \to S_0$ proper, see
Limits, Lemmas \ref{limits-lemma-eventually-proper} and
\ref{limits-lemma-limit-affine}.
Let $s_0 \in S_0$ be the image of $s$.
As $Y_s$ is affine, we see that
$R^1f_{s, *}\mathcal{O}_{X_s} = 0$ is equivalent
to $H^1(X_s, \mathcal{O}_{X_s}) = 0$.
Since $X_s$ is the base change of $X_{0, s_0}$ by
the faithfully flat map $\kappa(s_0) \to \kappa(s)$
we see that $H^1(X_{0, s_0}, \mathcal{O}_{X_{0, s_0}}) = 0$
and hence $R^1f_{0, *}\mathcal{O}_{X_{0, s_0}} = 0$.
Similarly, as $\mathcal{O}_{Y_s} \to f_{s, *}\mathcal{O}_{X_s}$
is an isomorphism, so is
$\mathcal{O}_{Y_{0, s_0}} \to f_{0, *}\mathcal{O}_{X_{0, s_0}}$.
 ========

regex_match $s_0 \in S_0$
nn candidate the image
nn the image
new_annotation [(694143, 694156, 'VAR'), (694160, 694169, 'TYPE')]
===== sent Let $y \in Y_s$ be a point. ========

regex_match $y \in Y_s$
nn candidate $y
nn candidate a point
nn a point
new_annotation [(695260, 695271, 'VAR'), (695275, 695282, 'TYPE')]
===== sent \end{proof}

\begin{lemma}
\label{lemma-bijection-on-Pic}
Let $f : X \to Y$ be a proper morphism of Noetherian schemes
such that $f_*\mathcal{O}_X = \mathcal{O}_Y$, such that
the fibres of $f$ have dimension $\leq 1$, and such that
$H^1(X_y, \mathcal{O}_{X_y}) = 0$ for $y \in Y$.
 ========

regex_match $f : X \to Y$
nn candidate X \to
nn candidate a proper morphism
nn a proper morphism
new_annotation [(696696, 696709, 'VAR'), (696713, 696730, 'TYPE')]

In [245]:

    

annotated_data_with_filenames = list(zip(annotated_data, list_of_texs))
random.shuffle(annotated_data_with_filenames)
train_data = [ann_data[0] for ann_data in annotated_data_with_filenames[:-1]]
test_data = [ann_data[0] for ann_data in annotated_data_with_filenames[-1:]] #we hold out one tex file for testing


    

In [246]:

    

def train_ner(nlp, train_data, entity_types):
    # Add new words to vocab.
    for raw_text, _ in train_data:
        doc = nlp.make_doc(raw_text)
        for word in doc:
            _ = nlp.vocab[word.orth]

    # Train NER.
    ner = EntityRecognizer(nlp.vocab, entity_types=entity_types)
    for itn in range(5):
        random.shuffle(train_data)
        for raw_text, entity_offsets in train_data:
            doc = nlp.make_doc(raw_text)
            gold = GoldParse(doc, entities=entity_offsets)
            ner.update(doc, gold)
    return ner


    

In [247]:

    

ner = train_ner(nlp, train_data, ['VAR', 'TYPE'])


    

In [249]:

    

#first test on a simple sentence
doc = nlp.make_doc('Let $S$ be a scheme and let $something here$ be a great thing you know.')  
nlp.tagger(doc)
ner(doc)

for ent in doc.ents:
    ent.merge()
    
for word in doc:
    if word.ent_type:
        print('(' + word.text + ':' + word.ent_type_ +')', end=word.whitespace_)
    else:
        print(word.text_with_ws, end='')


    

    




Let ($S$:VAR) be (a scheme:TYPE) and let ($something here$:VAR) be (a great thing:TYPE) you know.

In [250]:

    

#then test on the hold out tex file and save in a new tex file
doc = nlp.make_doc(annotated_data_with_filenames[-1][0][0])  
nlp.tagger(doc)
ner(doc)

for ent in doc.ents:
    ent.merge()

new_filename = 'a-' + annotated_data_with_filenames[-1][1] 
with open( 'annotated_tex_files/'+ new_filename, 'w') as f:
    for word in doc:
        if word.ent_type:
            f.write('(' + word.text + ':' + word.ent_type_ +')' + word.whitespace_)
        else:
            f.write(word.text_with_ws)


    

In [ ]: