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 [193]:

    

annotated_data=[]

directory = os.fsencode('tex_files/')
list_of_texs = [os.fsdecode(file) for file in os.listdir(directory)[0:3]]
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')]

In [201]:

    

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 [207]:

    

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 [208]:

    

ner = train_ner(nlp, train_data, ['VAR', 'TYPE'])


    

In [209]:

    

#first test on a simple sentence
doc = nlp.make_doc('Let $S$ be a scheme and let $bubu here$ be a great thing you know.')  
nlp.tagger(doc)
ner(doc)

for ent in doc.ents:
    print(ent)
    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='')


    

    




$S$
a scheme
a great thing
Let ($S$:VAR) be (a scheme:TYPE) and let $bubu here$ be (a great thing:TYPE) you know.

In [214]:

    

#then test on the hold out tex file and save in a new tex file
doc = nlp.make_doc(test_data[0][0])  
print(doc)
nlp.tagger(doc)
ner(doc)

for ent in doc.ents:
    ent.merge()

new_filename = 'a-' + annotated_data_with_filenames[-1][1] + '.tex'
with open( 'annotated_tex_files/'+ '.tex', 'w') as f:
    for word in doc:
        print(word)
        if word.ent_type:
            f.write('(' + word.text + ':' + word.ent_type_ +')' + word.whitespace_)
        else:
            f.write(word.text_with_ws)


    

    




\
\

In [221]:

    

test_data[0]


    

    
Out[221]:





"\\input{preamble}\n\n% OK, start here.\n%\n\\begin{document}\n\n\\title{Intersection Theory}\n\n\n\\maketitle\n\n\\phantomsection\n\\label{section-phantom}\n\n\\tableofcontents\n\n\n\\section{Introduction}\n\\label{section-introduction}\n\n\\noindent\nIn this chapter we construct the intersection product on the Chow groups\nmodulo rational equivalence on a nonsingular projective variety over an\nalgebraically closed field. Our tools are Serre's Tor formula\n(see \\cite[Chapter V]{Serre_algebre_locale}), reduction to the diagonal,\nand the moving lemma.\n\n\\medskip\\noindent\nWe first recall cycles and how to construct proper pushforward and\nflat pullback of cycles. Next, we introduce rational equivalence of cycles\nwhich gives us the Chow groups $A_*(X)$. Proper pushforward and flat pullback\nfactor through rational equivalence to give operations on Chow groups.\nThis takes up Sections \n\\ref{section-cycles},\n\\ref{section-cycle-of-closed},\n\\ref{section-cycle-of-coherent-sheaf},\n\\ref{section-pushforward},\n\\ref{section-flat-pullback},\n\\ref{section-rational-equivalence},\n\\ref{section-alternative},\n\\ref{section-pushforward-and-rational-equivalence}, and\n\\ref{section-flat-pullback-and-rational-equivalence}.\nFor proofs we mostly refer to the chapter on Chow homology\nwhere these results have been proven in the setting of\nschemes locally of finite type over a universally catenary Noetherian base, see\nChow Homology, Section \\ref{chow-section-setup} ff.\n\n\\medskip\\noindent\nSince we work on a nonsingular projective $X$ any irreducible component\nof the intersection $V \\cap W$ of two irreducible closed subvarieties\nhas dimension at least $\\dim(V) + \\dim(W) - \\dim(X)$. We say $V$ and $W$\nintersect properly if equality holds for every irreducible component $Z$.\nIn this case we define the intersection multiplicity\n$e_Z = e(X, V \\cdot W, Z)$ by the formula\n$$\ne_Z = \\sum\\nolimits_i\n(-1)^i\n\\text{length}_{\\mathcal{O}_{X, Z}}\n\\text{Tor}_i^{\\mathcal{O}_{X, Z}}(\\mathcal{O}_{W, Z}, \\mathcal{O}_{V, Z})\n$$\nWe need to do a little bit of commutative algebra to show that these\nintersection multiplicities agree with intuition in simple cases,\nnamely, that sometimes\n$$\ne_Z = \\text{length}_{\\mathcal{O}_{X, Z}} \\mathcal{O}_{V \\cap W, Z},\n$$\nin other words, only $\\text{Tor}_0$ contributes. This happens when\n$V$ and $W$ are Cohen-Macaulay in the generic point of $Z$ or when\n$W$ is cut out by a regular sequence in $\\mathcal{O}_{X, Z}$ which\nalso defines a regular sequence on $\\mathcal{O}_{V, Z}$. However,\nExample \\ref{example-naive-multiplicity-wrong} shows that higher\ntors are necessary in general. Moreover, there is a relationship\nwith the Samuel multiplicity. These matters are discussed in\nSections\n\\ref{section-intersect-properly},\n\\ref{section-tor-formula},\n\\ref{section-multiplicities},\n\\ref{section-computing-intersection-multiplicities}, and\n\\ref{section-intersection-product}. \n\n\\medskip\\noindent\nReduction to the diagonal is the statement that we can intersect\n$V$ and $W$ by intersecting $V \\times W$ with the diagonal in $X \\times X$.\nThis innocuous statement, which is clear on the level of scheme\ntheoretic intersections, reduces an intersection of a general pair\nof closed subschemes, to the case where one of the two is locally cut\nout by a regular sequence. We use this, following Serre, to obtain positivity\nof intersection multiplicities. Moreover, reduction to the diagonal\nleads to additivity of intersection multiplicities, associativity, and a\nprojection formula. This can be found in Sections\n\\ref{section-exterior-product},\n\\ref{section-reduction-diagonal},\n\\ref{section-associative},\n\\ref{section-flat-pullback-and-intersection-products}, and\n\\ref{section-projection-formula-flat}.\n\n\\medskip\\noindent\nFinally, we come to the moving lemmas and applications. There are two\nparts to the moving lemma. The first is that given closed subvarieties\n$$\nZ \\subset X \\subset \\mathbf{P}^N\n$$\nwith $X$ nonsingular, we can find a subvariety $C \\subset \\mathbf{P}^N$\nintersecting $X$ properly such that\n$$\nC \\cdot X = [Z] + \\sum m_j [Z_j]\n$$\nand such that the other components $Z_j$ are ``more general'' than $Z$.\nThe second part is that one can move $C \\subset \\mathbf{P}^N$ over\na rational curve to a subvariety in general position with respect to\nany given list of subvarieties. Combined these results imply that it suffices\nto define the intersection product of cycles on $X$ which intersect\nproperly which was done above. Of course this only leads to an intersection\nproduct on $A_*(X)$ if one can show, as we do in the text, that these products\npass through rational equivalence. This and some applications are discussed \nin Sections\n\\ref{section-projection},\n\\ref{section-moving-lemma},\n\\ref{section-intersections-and-rational-equivalence},\n\\ref{section-chow-rings},\n\\ref{section-general-pullback}, and\n\\ref{section-pullback-cycles}.\n\n\n\n\\section{Conventions}\n\\label{section-conventions}\n\n\\noindent\nWe fix an algebraically closed ground field $\\mathbf{C}$ of any\ncharacteristic. All schemes and varieties are over $\\mathbf{C}$ and all\nmorphisms are over $\\mathbf{C}$. A variety $X$ is\n{\\it nonsingular} if $X$ is a regular scheme (see\nProperties, Definition \\ref{properties-definition-regular}).\nIn our case this means that the morphism $X \\to \\Spec(\\mathbf{C})$\nis smooth (see\nVarieties, Lemma \\ref{varieties-lemma-geometrically-regular-smooth}).\n\n\n\\section{Cycles}\n\\label{section-cycles}\n\n\\noindent\nLet $X$ be a variety. A {\\it closed subvariety} of $X$ is an integral\nclosed subscheme $Z \\subset X$. A {\\it $k$-cycle} on $X$ is a finite\nformal sum $\\sum n_i [Z_i]$ where each $Z_i$ is a closed subvariety\nof dimension $k$. Whenever we use the notation $\\alpha = \\sum n_i[Z_i]$\nfor a $k$-cycle we always assume the subvarieties $Z_i$ are pairwise\ndistinct and $n_i \\not = 0$ for all $i$. In this case the\n{\\it support} of $\\alpha$ is the closed subset\n$$\n\\text{Supp}(\\alpha) = \\bigcup Z_i \\subset X\n$$\nof dimension $k$. The group of $k$-cycles is denoted $Z_k(X)$.\nSee Chow Homology, Section \\ref{chow-section-cycles}.\n\n\n\\section{Cycle associated to closed subscheme}\n\\label{section-cycle-of-closed}\n\n\\noindent\nSuppose that $X$ is a variety and that $Z \\subset X$ be a closed subscheme\nwith $\\dim(Z) \\leq k$. Let $Z_i$ be the irreducible components of $Z$ of\ndimension $k$ and let $n_i$ be the {\\it multiplicity of $Z_i$ in $Z$}\ndefined as\n$$\nn_i = \\text{length}_{\\mathcal{O}_{X, Z_i}} \\mathcal{O}_{Z, Z_i}\n$$\nwhere $\\mathcal{O}_{X, Z_i}$, resp.\\ $\\mathcal{O}_{Z, Z_i}$ is the\nlocal ring of $X$, resp.\\ $Z$ at the generic point of $Z_i$.\nWe define the $k$-cycle associated to $Z$ to be the $k$-cycle\n$$\n[Z]_k = \\sum n_i [Z_i].\n$$\nSee Chow Homology, Section \\ref{chow-section-cycle-of-closed-subscheme}.\n\n\n\\section{Cycle associated to a coherent sheaf}\n\\label{section-cycle-of-coherent-sheaf}\n\n\\noindent\nSuppose that $X$ is a variety and that\n$\\mathcal{F}$ is a coherent $\\mathcal{O}_X$-module with\n$\\dim(\\text{Supp}(\\mathcal{F})) \\leq k$.\nLet $Z_i$ be the irreducible components of $\\text{Supp}(\\mathcal{F})$\nof dimension $k$ and let $n_i$ be the\n{\\it multiplicity of $Z_i$ in $\\mathcal{F}$} defined as\n$$\nn_i = \\text{length}_{\\mathcal{O}_{X, Z_i}} \\mathcal{F}_{\\xi_i}\n$$\nwhere $\\mathcal{O}_{X, Z_i}$ is the\nlocal ring of $X$ at the generic point $\\xi_i$ of $Z_i$\nand $\\mathcal{F}_{\\xi_i}$ is the stalk of $\\mathcal{F}$ at this point.\nWe define the $k$-cycle associated to $\\mathcal{F}$ to be the $k$-cycle\n$$\n[\\mathcal{F}]_k = \\sum n_i [Z_i].\n$$\nSee Chow Homology, Section \\ref{chow-section-cycle-of-coherent-sheaf}.\nNote that, if $Z \\subset X$ is a closed subscheme with $\\dim(Z) \\leq k$, then\n$[Z]_k = [\\mathcal{O}_Z]_k$ by definition.\n\n\n\\section{Proper pushforward}\n\\label{section-pushforward}\n\n\\noindent\nSuppose that $f : X \\to Y$ is a proper morphism of varieties.\nLet $Z \\subset X$ be a $k$-dimensional\nclosed subvariety. We define $f_*[Z]$ to be $0$ if $\\dim(f(Z))<k$\nand $d \\cdot [f(Z)]$ if $\\dim(f(Z)) = k$ where\n$$\nd = [\\mathbf{C}(Z) : \\mathbf{C}(f(Z))] = \\deg(Z/f(Z))\n$$\nis the degree of the dominant morphism $Z \\to f(Z)$, see\nMorphisms, Definition \\ref{morphisms-definition-degree}.\nLet $\\alpha = \\sum n_i [Z_i]$ be a $k$-cycle on $Y$. The\n{\\it pushforward} of $\\alpha$ is the sum $f_* \\alpha = \\sum n_i f_*[Z_i]$\nwhere each $f_*[Z_i]$ is defined as above. This defines a homomorphism\n$$\nf_* : Z_k(X) \\longrightarrow Z_k(Y)\n$$\nSee Chow Homology, Section \\ref{chow-section-proper-pushforward}.\n\n\\begin{lemma}\n\\label{lemma-push-coherent}\n\\begin{reference}\nSee \\cite[Chapter V]{Serre_algebre_locale}.\n\\end{reference}\nSuppose that $f : X \\to Y$ is a proper morphism of varieties.\nLet $\\mathcal{F}$ be a coherent sheaf with\n$\\dim(\\text{Supp}(\\mathcal{F})) \\leq k$, then\n$f_*[\\mathcal{F}]_k = [f_*\\mathcal{F}]_k$. In particular, if\n$Z \\subset X$ is a closed subscheme of dimension $\\leq k$, then\n$f_*[Z] = [f_*\\mathcal{O}_Z]_k$.\n\\end{lemma}\n\n\\begin{proof}\nSee Chow Homology, Lemma \\ref{chow-lemma-cycle-push-sheaf}.\n\\end{proof}\n\n\\begin{lemma}\n\\label{lemma-compose-pushforward}\nLet $f : X \\to Y$ and $g : Y \\to Z$ be proper morphisms of\nvarieties. Then $g_* \\circ f_* = (g \\circ f)_*$ as maps $Z_k(X) \\to Z_k(Z)$.\n\\end{lemma}\n\n\\begin{proof}\nSpecial case of Chow Homology, Lemma \\ref{chow-lemma-compose-pushforward}.\n\\end{proof}\n\n\n\n\\section{Flat pullback}\n\\label{section-flat-pullback}\n\n\\noindent\nSuppose that $f : X \\to Y$ is a flat morphism of varieties.\nBy Morphisms, Lemma\n\\ref{morphisms-lemma-dimension-fibre-at-a-point-additive} \nevery fibre of $f$ has dimension $r = \\dim(X) - \\dim(Y)$\\footnote{Conversely,\nif $f : X \\to Y$ is a dominant morphism of varieties,\n$X$ is Cohen-Macaulay, $Y$ is nonsingular, and all fibres have\nthe same dimension $r$, then $f$ is flat. This follows from\nAlgebra, Lemma \\ref{algebra-lemma-CM-over-regular-flat} and\nVarieties, Lemma \\ref{varieties-lemma-dimension-fibres-locally-algebraic}\nshowing $\\dim(X) = \\dim(Y) + r$.}.\nLet $Z \\subset X$ be a $k$-dimensional closed subvariety. We define\n$f^*[Z]$ to be the $(k + r)$-cycle associated to the scheme theoretic\ninverse image: $f^*[Z] = [f^{-1}(Z)]_{k + r}$. Let\n$\\alpha = \\sum n_i [Z_i]$ be a $k$-cycle on $Y$. The {\\it pullback} of\n$\\alpha$ is the sum $f_* \\alpha = \\sum n_i f^*[Z_i]$ where each $f^*[Z_i]$\nis defined as above. This defines a homomorphism\n$$\nf^* : Z_k(Y) \\longrightarrow Z_{k + r}(X)\n$$\nSee Chow Homology, Section \\ref{chow-section-flat-pullback}.\n\n\\begin{lemma}\n\\label{lemma-pullback}\nLet $f : X \\to Y$ be a flat morphism of varieties. Set $r = \\dim(X) - \\dim(Y)$.\nThen $f^*[\\mathcal{F}]_k = [f^*\\mathcal{F}]_{k + r}$\nif $\\mathcal{F}$ is a coherent sheaf on $Y$ and the dimension of the\nsupport of $\\mathcal{F}$ is at most $k$.\n\\end{lemma}\n\n\\begin{proof}\nSee Chow Homology, Lemma \\ref{chow-lemma-pullback-coherent}.\n\\end{proof}\n\n\\begin{lemma}\n\\label{lemma-compose-flat-pullback}\nLet $f : X \\to Y$ and $g : Y \\to Z$ be flat morphisms of\nvarieties. Then $g \\circ f$ is flat and $f^* \\circ g^* = (g \\circ f)^*$\nas maps $Z_k(Z) \\to Z_{k + \\dim(X) - \\dim(Z)}(X)$.\n\\end{lemma}\n\n\\begin{proof}\nSpecial case of Chow Homology, Lemma \\ref{chow-lemma-compose-flat-pullback}.\n\\end{proof}\n\n\n\\section{Rational Equivalence}\n\\label{section-rational-equivalence}\n\n\\noindent\nWe are going to define rational equivalence in a way which at first\nglance may seem different from what you are used to, or from what\nis in \\cite[Chapter I]{F} or\nChow Homology, Section \\ref{chow-section-rational-equivalence}.\nHowever, in Section \\ref{section-alternative} we will show that\nthe two notions agree.\n\n\\medskip\\noindent\nLet $X$ be a variety. Let $W \\subset X \\times \\mathbf{P}^1$\nbe a closed subvariety of dimension $k + 1$. Let $a, b$ be distinct closed\npoints of $\\mathbf{P}^1$. Assume that $X \\times a$, $X \\times b$ and $W$\nintersect properly:\n$$\n\\dim (W \\cap X \\times a) \\leq k,\\quad\n\\dim (W \\cap X \\times b) \\leq k.\n$$\nThis is true as soon as $W \\to \\mathbf{P}^1$ is dominant or if $W$ is\ncontained in a fibre of the projection over a closed point different from\n$a$ or $b$ (this is an uninteresting case which we will discard). In this\nsituation the scheme theoretic fibre $W_a$ of the morphism\n$W \\to \\mathbf{P}^1$ is equal to the scheme theoretic intersection\n$W \\cap X \\times a$ in $X \\times \\mathbf{P}^1$. Identifying $X \\times a$\nand $X \\times b$ with $X$ we may think of the fibres $W_a$ and $W_b$\nas closed subschemes of $X$ of dimension $\\leq k$\\footnote{We will sometimes\nthink of $W_a$ as a closed subscheme of $X \\times \\mathbf{P}^1$ and sometimes\nas a closed subscheme of $X$. It should always be clear from context which\npoint of view is taken.}. A basic example of a\nrational equivalence is\n$$\n[W_a]_k \\sim_{rat} [W_b]_k\n$$\nThe cycles $[W_a]_k$ and $[W_b]_k$ are easy to compute in practice\n(given $W$) because they are obtained by proper intersection with\na Cartier divisor (we will see this in\nSection \\ref{section-intersection-product}).\nSince the automorphism group of $\\mathbf{P}^1$ is $2$-transitive we may\nmove the pair of closed points $a, b$ to any pair we like. A traditional\nchoice is to choose $a = 0$ and $b = \\infty$.\n\n\\medskip\\noindent\nMore generally, let $\\alpha = \\sum n_i [W_i]$ be a $(k + 1)$-cycle on\n$X \\times \\mathbf{P}^1$.  Let $a_i, b_i$ be pairs of distinct closed points of\n$\\mathbf{P}^1$. Assume that $X \\times a_i$, $X \\times b_i$ and $W_i$ intersect\nproperly, in other words, each $W_i, a_i, b_i$ satisfies the condition\ndiscussed above. A {\\it cycle rationally equivalent to zero} is any cycle\nof the form\n$$\n\\sum n_i([W_{i, a_i}]_k - [W_{i, b_i}]_k).\n$$\nThis is indeed a $k$-cycle. The collection of $k$-cycles rationally\nequivalent to zero is an additive subgroup of the group of $k$-cycles.\nWe say two $k$-cycles are {\\it rationally equivalent}, notation\n$\\alpha \\sim_{rat} \\alpha'$, if $\\alpha - \\alpha'$ is a cycle rationally\nequivalent to zero.\n\n\\medskip\\noindent\nWe define\n$$\nA_k(X) = Z_k(X)/ \\sim_{rat}\n$$\nto be the {\\it Chow group of $k$-cycles on $X$}. We will see in\nLemma \\ref{lemma-rational-equivalence}\nthat this agrees with the Chow group as defined in\nChow Homology, Definition \\ref{chow-definition-rational-equivalence}.\n\n\n\\section{Rational equivalence and rational functions}\n\\label{section-alternative}\n\n\\noindent\nLet $X$ be a variety. Let $W \\subset X$ be a subvariety\nof dimension $k + 1$. Let $f \\in \\mathbf{C}(W)^*$ be a nonzero rational\nfunction on $W$. For every subvariety $Z \\subset W$ of dimension $k$\none can define the order of vanishing $\\text{ord}_{W, Z}(f)$ of $f$ at\nat $Z$. If $f$ is an element of the local ring $\\mathcal{O}_{W, Z}$,\nthen one has\n$$\n\\text{ord}_{W, Z}(f) =\n\\text{length}_{\\mathcal{O}_{X, z}} \\mathcal{O}_{W, Z}/f\\mathcal{O}_{W, Z}\n$$\nwhere $\\mathcal{O}_{X, Z}$, resp.\\ $\\mathcal{O}_{W, Z}$ is the\nlocal ring of $X$, resp.\\ $W$ at the generic point of $Z$. In general one\nextends the definition by multiplicativity. The {\\it principal divisor\nassociated to $f$} is\n$$\n\\text{div}_W(f) = \\sum \\text{ord}_{W, Z}(f)[Z]\n$$\nin $Z_k(W)$. Since $W \\subset X$ is a closed subvariety we may think\nof $\\text{div}_W(f)$ as a cycle on $X$.\nSee Chow Homology, Section \\ref{chow-section-principal-divisors}.\n\n\\begin{lemma}\n\\label{lemma-rational-equivalence}\nLet $X$ be a variety. Let $W \\subset X$ be a subvariety\nof dimension $k + 1$. Let $f \\in \\mathbf{C}(W)^*$ be a nonzero rational\nfunction on $W$. Then $\\text{div}_W(f)$ is rationally equivalent to zero on\n$X$. Conversely, these principal divisors generate the abelian group of\ncycles rationally equivalent to zero on $X$.\n\\end{lemma}\n\n\\begin{proof}\nThe first assertion follows from\nChow Homology, Lemma \\ref{chow-lemma-rational-function}.\nMore precisely, let $W' \\subset X \\times \\mathbf{P}^1$ be the closure\nof the graph of $f$. Then $\\text{div}_W(f) = [W'_0]_k - [W'_\\infty]$\nin $Z_k(W) \\subset Z_k(X)$, see part (6) of\nChow Homology, Lemma \\ref{chow-lemma-rational-function}.\n\n\\medskip\\noindent\nFor the second, let $W' \\subset X \\times \\mathbf{P}^1$ be a closed\nsubvariety of dimension $k + 1$ which dominates $\\mathbf{P}^1$.\nWe will show that $[W'_0]_k - [W'_\\infty]_k$ is a principal divisor\nwhich will finish the proof. Let $W \\subset X$ be the image of $W'$\nunder the projection to $X$. Then $W' \\to W$ is proper and generically\nfinite\\footnote{If $W' \\to W$ is birational, then the result follows\nfrom Chow Homology, Lemma \\ref{chow-lemma-rational-function}.\nOur task is to show that even if $W' \\to W$\nhas degree $>1$ the basic rational equivalence\n$[W'_0]_k \\sim_{rat} [W'_\\infty]_k$ comes from a principal divisor\non a subvariety of $X$.}. Let $f$ denote the projection $W' \\to \\mathbf{P}^1$\nviewed as an element of $\\mathbf{C}(W')^*$. Let\n$g = \\text{Nm}(f) \\in \\mathbf{C}(W)^*$ be the norm. By\nChow Homology, Lemma \\ref{chow-lemma-proper-pushforward-alteration}\nwe have\n$$\n\\text{div}_W(g) = \\text{pr}_{X, *}\\text{div}_{W'}(f)\n$$\nSince it is clear that \n$\\text{div}_{W'}(f) = [W'_0]_k - [W'_\\infty]_k$ the proof is complete.\n\\end{proof}\n\n\n\\section{Proper pushforward and rational equivalence}\n\\label{section-pushforward-and-rational-equivalence}\n\n\\noindent\nSuppose that $f : X \\to Y$ is a proper morphism of varieties.\nLet $\\alpha \\sim_{rat} 0$ be a $k$-cycle on\n$X$ rationally equivalent to $0$. Then the {pushforward}\nof $\\alpha$ is rationally equivalent to zero:\n$f_* \\alpha \\sim_{rat} 0$. See Chapter I of \\cite{F} or\nChow Homology, Lemma \\ref{chow-lemma-proper-pushforward-rational-equivalence}.\n\n\\medskip\\noindent\nTherefore we obtain a commutative diagram\n$$\n\\xymatrix{\nZ_k(X) \\ar[r] \\ar[d]_{f_*} & A_k(X) \\ar[d]^{f_*} \\\\\nZ_k(Y) \\ar[r] & A_k(Y)\n}\n$$\nof groups of $k$-cycles.\n\n\n\\section{Flat pullback and rational equivalence}\n\\label{section-flat-pullback-and-rational-equivalence}\n\n\\noindent\nSuppose that $f : X \\to Y$ is a flat morphism of varieties.\nSet $r = \\dim(X) - \\dim(Y)$.\nLet $\\alpha \\sim_{rat} 0$ be a $k$-cycle on\n$Y$ rationally equivalent to $0$. Then the pullback\nof $\\alpha$ is rationally equivalent to zero:\n$f^* \\alpha \\sim_{rat} 0$. See Chapter I of \\cite{F} or\nChow Homology, Lemma \\ref{chow-lemma-flat-pullback-rational-equivalence}.\n\n\\medskip\\noindent\nTherefore we obtain a commutative diagram\n$$\n\\xymatrix{\nZ_{k + r}(X) \\ar[r] & A_{k + r}(X) \\\\\nZ_k(Y) \\ar[r] \\ar[u]^{f^*} & A_k(Y) \\ar[u]_{f^*}\n}\n$$\nof groups of $k$-cycles.\n\n\n\\section{The short exact sequence for an open}\n\\label{section-ses}\n\n\\noindent\nLet $X$ be a variety and let $U \\subset X$ be an open subvariety.\nLet $X \\setminus U = \\bigcup Z_i$ be the decomposition into irreducible\ncomponents\\footnote{Since in this chapter we only consider Chow groups\nof varieties, we are prohibited from taking $Z_k(X \\setminus U)$\nand $A_k(X \\setminus U)$, hence the approach using the varieties $Z_i$.}.\nThen for each $k \\geq 0$ there exists a commutative diagram\n$$\n\\xymatrix{\n\\bigoplus Z_k(Z_i) \\ar[r] \\ar[d] &\nZ_k(X) \\ar[r] \\ar[d] &\nZ_k(U) \\ar[d] \\ar[r] &\n0 \\\\\n\\bigoplus A_k(Z_i) \\ar[r] &\nA_k(X) \\ar[r] &\nA_k(U) \\ar[r] &\n0\n}\n$$\nwith exact rows. Here the vertical arrows are the canonical quotient maps.\nThe left horizontal arrows are given by proper pushforward along the closed\nimmersions $Z_i \\to X$. The right horizontal arrows are given by flat\npullback along the open immersion $j : U \\to X$. Since we have seen that\nthese maps factor through rational equivalence we obtain the commutativity\nof the squares. The top row is exact simply because every subvariety\nof $X$ is either contained in some $Z_i$ or has irreducible intersection\nwith $U$. The bottom row is exact because every principal divisor\n$\\text{div}_W(f)$ on $U$ is the restriction of a principal divisor on $X$.\nMore precisely, if $W \\subset U$ is a $(k + 1)$-dimensional closed subvariety\nand $f \\in \\mathbf{C}(W)^*$, then denote $\\overline{W}$ the closure of $W$\nin $X$. Then $W \\subset \\overline{W}$ is an open immersion, so\n$\\mathbf{C}(W) = \\mathbf{C}(\\overline{W})$ and we may think of $f$\nas a nonconstant rational function on $\\overline{W}$. Then clearly\n$$\nj^*\\text{div}_{\\overline{W}}(f) = \\text{div}_W(f)\n$$\nin $Z_k(X)$. The exactness of the lower row follows easily from this.\nFor details see Chow Homology, Lemma \\ref{chow-lemma-restrict-to-open}.\n\n\n\\section{Proper intersections}\n\\label{section-intersect-properly}\n\n\\noindent\nFirst a few lemmas to get dimension estimates.\n\n\\begin{lemma}\n\\label{lemma-dimension-product-varieties}\nLet $X$ and $Y$ be varieties. Then $X \\times Y$ is a variety and\n$\\dim(X \\times Y) = \\dim(X) + \\dim(Y)$.\n\\end{lemma}\n\n\\begin{proof}\nThe scheme $X \\times Y = X \\times_{\\Spec(\\mathbf{C})} Y$ is a variety by\nVarieties, Lemma \\ref{varieties-lemma-product-varieties}.\nThe statement on dimension is\nVarieties, Lemma \\ref{varieties-lemma-dimension-product-locally-algebraic}.\n\\end{proof}\n\n\\noindent\nRecall that a regular immersion $i : X \\to Y$ of schemes\nis a closed immersion whose\ncorresponding sheaf of ideals is locally generated by a regular sequence, see\nDivisors, Section \\ref{divisors-section-regular-immersions}.\nMoreover, the conormal sheaf $\\mathcal{C}_{X/Y}$ is finite locally free of\nrank equal to the length of the regular sequence. Let us say $i$ is a\n{\\it regular immersion of codimension $c$}\nif $\\mathcal{C}_{X/Y}$ is locally free of rank $c$.\n\n\\medskip\\noindent\nMore generally, recall\n(More on Morphisms, Section \\ref{more-morphisms-section-lci})\nthat $f : X \\to Y$ is a local complete intersection\nmorphism if we can cover $X$ by opens $U$ such that we can factor\n$f|_U$ as\n$$\n\\xymatrix{\nU \\ar[rr]_i \\ar[rd] & & \\mathbf{A}^n_Y \\ar[ld] \\\\\n& Y\n}\n$$\nwhere $i$ is a Koszul regular immersion (if $Y$ is locally Noetherian\nthis is the same as asking $i$ to be a regular immersion, see\nDivisors, Lemma \\ref{divisors-lemma-regular-immersion-noetherian}).\nLet us say that $f$ is a {\\it local complete intersection morphism\nof relative dimension $r$} if for any factorization as above, the\nclosed immersion $i$ has conormal sheaf of rank $n - r$ (in other\nwords if $i$ is a Koszul-regular immersion of codimension $n - r$\nwhich in the Noetherian case just means it is regular immersion of\ncodimension $n - r$).\n\n\\begin{lemma}\n\\label{lemma-pullback-by-regular-immersion}\nLet $f : X \\to Y$ be a morphism of varieties.\n\\begin{enumerate}\n\\item If $Z \\subset Y$ is a subvariety dimension $d$ and $f$ is a regular\nimmersion of codimension $c$, then every irreducible component\nof $f^{-1}(Z)$ has dimension $\\geq d - c$.\n\\item If $Z \\subset Y$ is a subvariety of dimension $d$ and\n$f$ is a local complete intersection morphism of relative dimension $r$,\nthen every irreducible component of $f^{-1}(Z)$ has dimension $\\geq d + r$.\n\\end{enumerate}\n\\end{lemma}\n\n\\begin{proof}\nProof of (1). We may work locally, hence we may assume that\n$Y = \\Spec(A)$ and $X = V(f_1, \\ldots, f_c)$ where $f_1, \\ldots, f_c$\nis a regular sequence in $A$. If $Z = \\Spec(A/\\mathfrak p)$, then\nwe see that $f^{-1}(Z) = \\Spec(A/\\mathfrak p + (f_1, \\ldots, f_c))$.\nIf $V$ is an irreducible component of $f^{-1}(Z)$, then we can\nchoose a closed point $v \\in V$ not contained in any other irreducible\ncomponent of $f^{-1}(Z)$. Then\n$$\n\\dim(Z) = \\dim \\mathcal{O}_{Z, v}\n\\quad\\text{and}\\quad\n\\dim(V) = \\dim \\mathcal{O}_{V, v} = \\dim \\mathcal{O}_{Z, v}/(f_1, \\ldots, f_c)\n$$\nThe first equality for example by\nAlgebra, Lemma \\ref{algebra-lemma-dimension-prime-polynomial-ring}\nand the second equality by our choice of closed point.\nThe result now follows from the fact that dividing by one element\nin the maximal ideal decreases the dimension by at most $1$, see\nAlgebra, Lemma \\ref{algebra-lemma-one-equation}.\n\n\\medskip\\noindent\nProof of (2). Choose a factorization as in the definition of a\nlocal complete intersection and apply (1). Some details omitted.\n\\end{proof}\n\n\\begin{lemma}\n\\label{lemma-diagonal-regular-immersion}\nLet $X$ be a nonsingular variety. Then the diagonal\n$\\Delta : X \\to X \\times X$ is a regular immersion of codimension $\\dim(X)$.\n\\end{lemma}\n\n\\begin{proof}\nIn fact, any closed immersion between nonsingular projective\nvarieties is a regular immersion, see Divisors,\nLemma \\ref{divisors-lemma-immersion-smooth-into-smooth-regular-immersion}.\n\\end{proof}\n\n\\noindent\nThe following lemma demonstrates how reduction to the diagonal works.\n\n\\begin{lemma}\n\\label{lemma-intersect-in-smooth}\nLet $X$ be a nonsingular variety and let $W,V \\subset X$\nbe closed subvarieties with $\\dim(W) = s$ and $\\dim(V) = r$. Then every\nirreducible component $Z$ of $V \\cap W$ has dimension $\\geq r + s - \\dim(X)$.\n\\end{lemma}\n\n\\begin{proof}\nSince $V \\cap W = \\Delta^{-1}(V \\times W)$ (scheme theoretically)\nwe conclude by Lemmas \\ref{lemma-diagonal-regular-immersion} and\n\\ref{lemma-pullback-by-regular-immersion}.\n\\end{proof}\n\n\\noindent\nThis lemma suggests the following definition.\n\n\\begin{definition}\n\\label{definition-proper-intersection}\nLet $X$ be a nonsingular variety.\n\\begin{enumerate}\n\\item Let $W,V \\subset X$ be closed subvarieties with\n$\\dim(W) = s$ and $\\dim(V) = r$. We say that $W$ and $V$\n{\\it intersect properly} if $\\dim(V \\cap W) \\leq r + s - \\dim(X)$.\n\\item Let $\\alpha = \\sum n_i [W_i]$ be an $s$-cycle,\nand $\\beta = \\sum_j m_j [V_j]$ be an $r$-cycle on $X$. We say\nthat $\\alpha$ and $\\beta$ {\\it intersect properly} if\n$W_i$ and $V_j$ intersect properly for all $i$ and $j$.\n\\end{enumerate}\n\\end{definition}\n\n\n\\section{Intersection multiplicities using Tor formula}\n\\label{section-tor-formula}\n\n\\noindent\nA basic fact we will use frequently is that given sheaves of\nmodules $\\mathcal{F}$, $\\mathcal{G}$ on a ringed space $(X, \\mathcal{O}_X)$\nand a point $x \\in X$ we have\n$$\n\\text{Tor}_p^{\\mathcal{O}_X}(\\mathcal{F}, \\mathcal{G})_x =\n\\text{Tor}_p^{\\mathcal{O}_{X, x}}(\\mathcal{F}_x, \\mathcal{G}_x)\n$$\nas $\\mathcal{O}_{X, x}$-modules. This can be seen in several ways\nfrom our construction of derived tensor products in\nCohomology, Section \\ref{cohomology-section-flat}, for example it follows from\nCohomology, Lemma \\ref{cohomology-lemma-check-K-flat-stalks}.\nMoreover, if $X$ is a scheme and $\\mathcal{F}$ and $\\mathcal{G}$\nare quasi-coherent, then the modules\n$\\text{Tor}_p^{\\mathcal{O}_X}(\\mathcal{F}, \\mathcal{G})$ are\nquasi-coherent too, see\nDerived Categories of Schemes, Lemma\n\\ref{perfect-lemma-quasi-coherence-tensor-product}.\nMore important for our purposes is the following result.\n\n\\begin{lemma}\n\\label{lemma-tensor-coherent}\nLet $X$ be a locally Noetherian scheme.\n\\begin{enumerate}\n\\item If $\\mathcal{F}$ and $\\mathcal{G}$ are coherent $\\mathcal{O}_X$-modules,\nthen $\\text{Tor}_p^{\\mathcal{O}_X}(\\mathcal{F}, \\mathcal{G})$ is too.\n\\item If $L$ and $K$ are in $D^-_{\\textit{Coh}}(\\mathcal{O}_X)$, then\nso is $L \\otimes_{\\mathcal{O}_X}^\\mathbf{L} K$.\n\\end{enumerate}\n\\end{lemma}\n\n\\begin{proof}\nLet us explain how to prove (1) in a more elementary way and part (2)\nusing previously developed general theory.\n\n\\medskip\\noindent\nProof of (1). Since formation of $\\text{Tor}$ commutes with localization\nwe may assume $X$ is affine. Hence $X = \\Spec(A)$ for some Noetherian\nring $A$ and $\\mathcal{F}$, $\\mathcal{G}$ correspond to finite $A$-modules\n$M$ and $N$ (Cohomology of Schemes, Lemma\n\\ref{coherent-lemma-coherent-Noetherian}).\nBy Derived Categories of Schemes, Lemma\n\\ref{perfect-lemma-quasi-coherence-tensor-product} we may\ncompute the $\\text{Tor}$'s by first computing the $\\text{Tor}$'s\nof $M$ and $N$ over $A$, and then taking the associated $\\mathcal{O}_X$-module.\nSince the modules $\\text{Tor}_p^A(M, N)$ are finite by\nAlgebra, Lemma \\ref{algebra-lemma-tor-noetherian}\nwe conclude.\n\n\\medskip\\noindent\nBy Derived Categories of Schemes, Lemma\n\\ref{perfect-lemma-identify-pseudo-coherent-noetherian}\nthe assumption is equivalent to asking $L$ and $K$ to be\n(locally) pseudo-coherent. Then $L \\otimes_{\\mathcal{O}_X}^\\mathbf{L} K$\nis pseudo-coherent by\nCohomology, Lemma \\ref{cohomology-lemma-tensor-pseudo-coherent}.\n\\end{proof}\n\n\\begin{lemma}\n\\label{lemma-compute-tor-nonsingular}\nLet $X$ be a nonsingular variety.\nLet $\\mathcal{F}$, $\\mathcal{G}$ be coherent $\\mathcal{O}_X$-modules.\nThe $\\mathcal{O}_X$-module\n$\\text{Tor}_p^{\\mathcal{O}_X}(\\mathcal{F}, \\mathcal{G})$\nis coherent, has stalk at $x$ equal to\n$\\text{Tor}_p^{\\mathcal{O}_{X, x}}(\\mathcal{F}_x, \\mathcal{G}_x)$,\nis supported on\n$\\text{Supp}(\\mathcal{F}) \\cap \\text{Supp}(\\mathcal{G})$, and\nis nonzero only for $p \\in \\{0, \\ldots, \\dim(X)\\}$.\n\\end{lemma}\n\n\\begin{proof}\nThe result on stalks was discussed above and it implies the support\ncondition. The $\\text{Tor}$'s are coherent by\nLemma \\ref{lemma-tensor-coherent}. The vanishing of negative\n$\\text{Tor}$'s is immediate from the construction. The\nvanishing of $\\text{Tor}_p$ for $p > \\dim(X)$ can be seen as follows:\nhe local rings $\\mathcal{O}_{X, x}$ are regular\n(as $X$ is nonsingular) of dimension $\\leq \\dim(X)$\n(Algebra, Lemma \\ref{algebra-lemma-dimension-prime-polynomial-ring}),\nhence $\\mathcal{O}_{X, x}$ has finite global dimension $\\leq \\dim(X)$\n(Algebra, Lemma \\ref{algebra-lemma-finite-gl-dim-finite-dim-regular})\nwhich implies that $\\text{Tor}$-groups of modules vanish beyond the dimension\n(More on Algebra, Lemma \\ref{more-algebra-lemma-finite-gl-dim-tor-dimension}).\n\\end{proof}\n\n\\noindent\nLet $X$ be a nonsingular variety and $W, V \\subset X$\nbe closed subvarieties with $\\dim(W) = s$ and $\\dim(V) = r$.\nAssume $V$ and $W$ intersect properly.\nIn this case Lemma \\ref{lemma-intersect-in-smooth} tells us all irreducible\ncomponents of $V \\cap W$ have dimension equal to $r + s - \\dim(X)$.\nThe sheaves $\\text{Tor}_j^{\\mathcal{O}_X}(\\mathcal{O}_W, \\mathcal{O}_V)$ are\ncoherent, supported on $V \\cap W$, and zero if $j < 0$ or $j > \\dim(X)$\n(Lemma \\ref{lemma-compute-tor-nonsingular}).\nWe define the {\\it intersection product} as\n$$\nW \\cdot V = \\sum\\nolimits_i (-1)^i\n[\\text{Tor}_i^{\\mathcal{O}_X}(\\mathcal{O}_W, \\mathcal{O}_V)]_{r + s - \\dim(X)}.\n$$\nWe stress that this makes sense only because of our assumption that\n$V$ and $W$ intersect properly. This fact will necessitate a moving\nlemma in order to define the intersection product in general.\n\n\\medskip\\noindent\nWith this notation, the cycle $V \\cdot W$ is a formal linear\ncombination $\\sum e_Z Z$ of the irreducible components $Z$\nof the intersection $V \\cap W$. The integers $e_Z$ are called\nthe {\\it intersection multiplicities}\n$$\ne_Z = e(X, V \\cdot W, Z) =\n\\sum\\nolimits_i\n(-1)^i\n\\text{length}_{\\mathcal{O}_{X, Z}}\n\\text{Tor}_i^{\\mathcal{O}_{X, Z}}(\\mathcal{O}_{W, Z}, \\mathcal{O}_{V, Z})\n$$\nwhere $\\mathcal{O}_{X, Z}$, resp.\\ $\\mathcal{O}_{W, Z}$,\nresp.\\ $\\mathcal{O}_{V, Z}$ denotes the local ring of $X$, resp.\\ $W$,\nresp.\\ $V$ at the generic point of $Z$.\nThese alternating sums of lengths of $\\text{Tor}$'s satisfy many good\nproperties, as we will see later on.\n\n\\medskip\\noindent\nIn the case of transversal intersections, the intersection number is $1$.\n\n\\begin{lemma}\n\\label{lemma-transversal}\nLet $X$ be a nonsingular variety. Let $V, W \\subset X$ be\nclosed subvarieties which intersect properly. Let $Z$ be an irreducible\ncomponent of $V \\cap W$ and assume that the multiplicity\n(in the sense of Section \\ref{section-cycle-of-closed}) of $Z$\nin the closed subscheme $V \\cap W$ is $1$.\nThen $e(X, V \\cdot W, Z) = 1$ and $V$ and $W$ are smooth\nin a general point of $Z$.\n\\end{lemma}\n\n\\begin{proof}\nLet $(A, \\mathfrak m, \\kappa) =\n(\\mathcal{O}_{X, \\xi}, \\mathfrak m_\\xi, \\kappa(\\xi))$ where $\\xi \\in Z$\nis the generic point. Then $\\dim(A) = \\dim(X) - \\dim(Z)$, see\nVarieties, Lemma \\ref{varieties-lemma-dimension-locally-algebraic}.\nLet $I, J \\subset A$ cut out the trace of $V$ and $W$\nin $\\Spec(A)$. Set $\\overline{I} = I + \\mathfrak m^2/\\mathfrak m^2$.\nThen $\\dim_\\kappa \\overline{I} \\leq \\dim(X) - \\dim(V)$ with equality\nif and only if $A/I$ is regular (this follows from the lemma cited\nabove and the definition of regular rings, see\nAlgebra, Definition \\ref{algebra-definition-regular-local}\nand the discussion preceding it). Similarly for $\\overline{J}$.\nIf the multiplicity is $1$, then\n$\\text{length}_A(A/I + J) = 1$, hence $I + J = \\mathfrak m$, hence\n$\\overline{I} + \\overline{J} = \\mathfrak m/\\mathfrak m^2$.\nThen we get equality everywhere (because the intersection is\nproper). Hence we find $f_1, \\ldots, f_a \\in I$ and $g_1, \\ldots g_b \\in J$\nsuch that $\\overline{f}_1, \\ldots, \\overline{g}_b$ is a basis\nfor $\\mathfrak m/\\mathfrak m^2$. Then $f_1, \\ldots, g_b$ is a\nregular system of parameters and a regular sequence\n(Algebra, Lemma \\ref{algebra-lemma-regular-ring-CM}).\nThe same lemma shows $A/(f_1, \\ldots, f_a)$ is a regular local ring\nof dimension $\\dim(X) - \\dim(V)$, hence $A/(f_1, \\ldots, f_a) \\to A/I$\nis an isomorphism (if the kernel is nonzero, then the dimension\nof $A/I$ is strictly less, see\nAlgebra, Lemmas \\ref{algebra-lemma-regular-domain} and\n\\ref{algebra-lemma-one-equation}).\nWe conclude $I = (f_1, \\ldots, f_a)$ and $J = (g_1, \\ldots, g_b)$\nby symmetry. Thus the Koszul complex $K_\\bullet(A, f_1, \\ldots, f_a)$\non $f_1, \\ldots, f_a$ is a resolution of $A/I$, see\nMore on Algebra, Lemma \\ref{more-algebra-lemma-regular-koszul-regular}.\nHence\n\\begin{align*}\n\\text{Tor}_p^A(A/I, A/J)\n& =\nH_p(K_\\bullet(A, f_1, \\ldots, f_a) \\otimes_A A/J) \\\\\n& =\nH_p(K_\\bullet(A/J, f_1 \\bmod J, \\ldots, f_a \\bmod J))\n\\end{align*}\nSince we've seen above that $f_1 \\bmod J, \\ldots, f_a \\bmod J$ is\na regular system of parameters in the regular local ring $A/J$\nwe conclude that there is only one cohomology group, namely\n$H_0 = A/(I + J) = \\kappa$. This finishes the proof.\n\\end{proof}\n\n\\begin{example}\n\\label{example-naive-multiplicity-wrong}\nIn this example we show that it is necessary to use the higher\ntors in the formula for the intersection multiplicities above.\nLet $X$ be a nonsingular variety of dimension $4$.\nLet $p \\in X$ be a closed point. Let $V, W \\subset X$\nbe closed subvarieties in $X$. Assume that there is an\nisomorphism\n$$\n\\mathcal{O}_{X, p}^\\wedge \\cong \\mathbf{C}[[x, y, z, w]]\n$$\nsuch that the ideal of $V$ is $(xz, xw, yz, yw)$ and the ideal\nof $W$ is $(x - z, y - w)$. Then a computation shows that\n$$\n\\text{length}\\ \\mathbf{C}[[x, y, z, w]]/\n(xz, xw, yz, yw, x - z, y - w) = 3\n$$\nOn the other hand, the multiplicity $e(X, V \\cdot W, p) = 2$\nas can be seen from the fact that formal locally $V$ is the\nunion of two smooth planes $x = y = 0$ and $z = w = 0$ at $p$,\neach of which has intersection multiplicity $1$ with the plane\n$x - z = y - w = 0$ (Lemma \\ref{lemma-transversal}). To make an\nactual example, take\na general morphism $f : \\mathbf{P}^2 \\to \\mathbf{P}^4$ given by\n$5$ homogeneous polynomials of degree $> 1$. The image\n$V \\subset \\mathbf{P}^4 = X$ will have singularities of the type\ndescribed above, because there will be $p_1, p_2 \\in \\mathbf{P}^2$\nwith $f(p_1) = f(p_2)$. To find $W$ take a general plane passing\nthrough such a point.\n\\end{example}\n\n\n\n\n\n\n\\section{Algebraic multiplicities}\n\\label{section-multiplicities}\n\n\\noindent\nLet $(A, \\mathfrak m, \\kappa)$ be a Noetherian local ring.\nLet $M$ be a finite $A$-module and let $I \\subset A$ be an ideal\nof definition (Algebra, Definition \\ref{algebra-definition-ideal-definition}).\nRecall that the function\n$$\n\\chi_{I, M}(n) = \\text{length}_A(M/I^nM) =\n\\sum\\nolimits_{p = 0, \\ldots, n - 1} \\text{length}_A(I^pM/I^{p + 1}M)\n$$\nis a numerical polynomial\n(Algebra, Proposition \\ref{algebra-proposition-hilbert-function-polynomial}).\nThe degree of this polynomial is equal to $\\dim(\\text{Supp}(M))$ by\nAlgebra, Lemma \\ref{algebra-lemma-support-dimension-d}.\n\n\\begin{definition}\n\\label{definition-multiplicity}\nIn the situation above, if $d \\geq \\dim(\\text{Supp}(M))$, then we set\n$e_I(M, d)$ equal to $0$ if $d > \\dim(\\text{Supp}(M))$\nand equal to $d!$ times the\nleading coefficient of the numerical polynomial $\\chi_{I, M}$ so that\n$$\n\\chi_{I, M}(n) \\sim e_I(M, d) \\frac{n^d}{d!} + \\text{lower order terms}\n$$\nThe {\\it multiplicity of $M$ for the ideal of definition $I$}\nis $e_I(M) = e_I(M, \\dim(\\text{Supp}(M)))$.\n\\end{definition}\n\n\\noindent\nWe have the following properties of these multiplicities.\n\n\\begin{lemma}\n\\label{lemma-multiplicity-ses}\nLet $A$ be a Noetherian local ring. Let $I \\subset A$ be an ideal of\ndefinition. Let $0 \\to M' \\to M \\to M'' \\to 0$ be a short exact sequence\nof finite $A$-modules. Let $d \\geq \\dim(\\text{Supp}(M))$. Then\n$$\ne_I(M, d) = e_I(M', d) + e_I(M'', d)\n$$\n\\end{lemma}\n\n\\begin{proof}\nImmediate from the definitions and\nAlgebra, Lemma \\ref{algebra-lemma-hilbert-ses-chi}.\n\\end{proof}\n\n\\begin{lemma}\n\\label{lemma-multiplicity-as-a-sum}\nLet $A$ be a Noetherian local ring. Let $I \\subset A$ be an ideal of\ndefinition. Let $M$ be a finite $A$-module. Let $d \\geq \\dim(\\text{Supp}(M))$.\nThen\n$$\ne_I(M, d) =\n\\sum \\text{length}_{A_\\mathfrak p}(M_\\mathfrak p) e_I(A/\\mathfrak p, d)\n$$\nwhere the sum is over primes $\\mathfrak p \\subset A$ with\n$\\dim(A/\\mathfrak p) = d$.\n\\end{lemma}\n\n\\begin{proof}\nBoth the left and side and the right hand side are additive in short\nexact sequences of modules of dimension $\\leq d$, see\nLemma \\ref{lemma-multiplicity-ses} and\nAlgebra, Lemma \\ref{algebra-lemma-length-additive}.\nHence by Algebra, Lemma \\ref{algebra-lemma-filter-Noetherian-module}\nit suffices to prove this when $M = A/\\mathfrak q$ for some\nprime $\\mathfrak q$ of $A$ with $\\dim(A/\\mathfrak q) \\leq d$.\nThis case is obvious.\n\\end{proof}\n\n\\begin{lemma}\n\\label{lemma-leading-coefficient}\nLet $P$ be a polynomial of degree $r$ with leading coefficient $a$.\nThen\n$$\nr! a = \\sum\\nolimits_{i = 0, \\ldots, r} (-1)^i{r \\choose i} P(t - i)\n$$\nfor any $t$.\n\\end{lemma}\n\n\\begin{proof}\nLet us write $\\Delta$ the operator which to a polynomial $P$ associates\nthe polynomial $\\Delta(P) = P(t) - P(t - 1)$. We claim that\n$$\n\\Delta^r(P) = \\sum\\nolimits_{i = 0, \\ldots, r} (-1)^i {r \\choose i} P(t - i)\n$$\nThis is true for $r = 0, 1$ by inspection. Assume it is true for $r$.\nThen we compute\n\\begin{align*}\n\\Delta^{r + 1}(P)\n& =\n\\sum\\nolimits_{i = 0, \\ldots, r} (-1)^i {r \\choose i} \\Delta(P)(t - i) \\\\\n& =\n\\sum\\nolimits_{n = -r, \\ldots, 0} (-1)^i {r \\choose i}\n(P(t - i) - P(t - i - 1))\n\\end{align*}\nThus the claim follows from the equality\n$$\n{r + 1 \\choose i} = {r \\choose i} + {r \\choose i - 1}\n$$\nThe lemma follows from the fact that $\\Delta(P)$ is of degree $r - 1$\nwith leading coefficient $ra$ if the degree of $P$ is $r$.\n\\end{proof}\n\n\\noindent\nAn important fact is that one can compute the multiplicity in terms\nof the Koszul complex. Recall that if $R$ is a ring and\n$f_1, \\ldots, f_r \\in R$, then $K_\\bullet(f_1, \\ldots, f_r)$\ndenotes the Koszul complex, see\nMore on Algebra, Section \\ref{more-algebra-section-koszul}.\n\n\\begin{theorem}\n\\label{theorem-multiplicity-with-koszul}\n\\begin{reference}\n\\cite[Theorem 1 in part B of Chapter IV]{Serre_algebre_locale}\n\\end{reference}\nLet $A$ be a Noetherian local ring. Let $I = (f_1, \\ldots, f_r) \\subset A$\nbe an ideal of definition. Let $M$ be a finite $A$-module. Then\n$$\ne_I(M, r) = \\sum\n(-1)^i\\text{length}_A H_i(K_\\bullet(f_1, \\ldots, f_r) \\otimes_A M)\n$$\n\\end{theorem}\n\n\\begin{proof}\nLet us change the Koszul complex $K_\\bullet(f_1, \\ldots, f_r)$ into a cochain\ncomplex $K^\\bullet$ by setting $K^n = K_{-n}(f_1, \\ldots, f_r)$.\nThen $K^\\bullet$ is sitting in degrees $-r, \\ldots, 0$ and\n$H^i(K^\\bullet \\otimes_A M) = H_{-i}(K_\\bullet(f_1, \\ldots, f_r) \\otimes_A M)$.\nThe statement of the theorem makes sense as the modules\n$H^i(K^\\bullet \\otimes M)$ are annihilated by $f_1, \\ldots, f_r$\n(More on Algebra, Lemma \\ref{more-algebra-lemma-homotopy-koszul})\nhence have finite length.\nDefine a filtration on the complex $K^\\bullet$ by setting\n$$\nF^p(K^n \\otimes_A M) =\nI^{\\max(0, p + n)}(K^n \\otimes_A M),\\quad p \\in \\mathbf{Z}\n$$\nSince $f_i I^p \\subset I^{p + 1}$ this is a filtration by subcomplexes.\nThus we have a filtered complex and we obtain a spectral sequence, see\nHomology, Section \\ref{homology-section-filtered-complex}.\nWe have\n$$\nE_0 = \\bigoplus\\nolimits_{p, q} E_0^{p, q} =\n\\bigoplus\\nolimits_{p, q} \\text{gr}^p(K^{p + q} \\otimes_A M) =\n\\text{Gr}_I(K^\\bullet \\otimes_A M)\n$$\nSince $K^n$ is finite free we have\n$$\n\\text{Gr}_I(K^\\bullet \\otimes_A M) =\n\\text{Gr}_I(K^\\bullet) \\otimes_{\\text{Gr}_I(A)} \\text{Gr}_I(M)\n$$\nNote that $\\text{Gr}_I(K^\\bullet)$ is the Koszul\ncomplex over $\\text{Gr}_I(A)$ on the elements\n$\\overline{f}_1, \\ldots, \\overline{f}_r \\in I/I^2$.\nA simple calculation (omitted)\nshows that the differential $d_0$ on $E_0$\nagrees with the differential coming from the Koszul complex.\nSince $\\text{Gr}_I(M)$ is a finite $\\text{Gr}_I(A)$-module\nand since $\\text{Gr}_I(A)$ is Noetherian (as a quotient\nof $A/I[x_1, \\ldots, x_r]$ with $x_i \\mapsto \\overline{f}_i$), the\ncohomology module $E_1 = \\bigoplus E_1^{p, q}$\nis a finite $\\text{Gr}_I(A)$-module. However, as above\n$E_1$ is annihilated by $\\overline{f}_1, \\ldots, \\overline{f}_r$.\nWe conclude $E_1$ has finite length.\nIn particular we find that $\\text{Gr}^p_F(K^\\bullet \\otimes M)$ is\nacyclic for $p \\gg 0$.\n\n\\medskip\\noindent\nNext, we check that the spectral sequence above converges\nusing Homology, Lemma \\ref{homology-lemma-filtered-complex-ss-converges}.\nThe required equalities follow easily from the Artin-Rees lemma\nin the form stated in Algebra, Lemma \\ref{algebra-lemma-map-AR}.\nThus we see that\n\\begin{align*}\n\\sum (-1)^i\\text{length}_A(H^i(K^\\bullet \\otimes_A M))\n& =\n\\sum (-1)^{p + q} \\text{length}_A(E_\\infty^{p, q}) \\\\\n& =\n\\sum (-1)^{p + q} \\text{length}_A(E_1^{p, q})\n\\end{align*}\nbecause as we've seen above the length of $E_1$ is finite\n(of course this uses additivity of lengths). Pick $t$ so\nlarge that $\\text{Gr}^p_F(K^\\bullet \\otimes M)$\nis acyclic for $p \\geq t$ (see above). Using\nadditivity again we see that\n$$\n\\sum (-1)^{p + q} \\text{length}_A(E_1^{p, q}) =\n\\sum\\nolimits_n \\sum\\nolimits_{p \\leq t}\n(-1)^n \\text{length}_A(\\text{gr}^p(K^n \\otimes_A M))\n$$\nThis is equal to\n$$\n\\sum\\nolimits_{n = -r, \\ldots, 0} (-1)^n{r \\choose |n|} \\chi_{I, M}(t + n)\n$$\nby our choice of filtration above and the definition of $\\chi_{I, M}$ in\nAlgebra, Section \\ref{algebra-section-Noetherian-local}.\nThe lemma follows from Lemma \\ref{lemma-leading-coefficient}\nand the definition of $e_I(M, r)$.\n\\end{proof}\n\n\\begin{remark}[Trivial generalization]\n\\label{remark-trivial-generalization}\nLet $(A, \\mathfrak m, \\kappa)$ be a Noetherian local ring.\nLet $M$ be a finite $A$-module. Let $I \\subset A$ be an ideal.\nThe following are equivalent\n\\begin{enumerate}\n\\item $I' = I + \\text{Ann}(M)$ is an ideal of definition\n(Algebra, Definition \\ref{algebra-definition-ideal-definition}),\n\\item the image $\\overline{I}$ of $I$ in $\\overline{A} = A/\\text{Ann}(M)$\nis an ideal of definition,\n\\item $\\text{Supp}(M/IM) \\subset \\{\\mathfrak m\\}$,\n\\item $\\dim(\\text{Supp}(M/IM)) \\leq 0$, and\n\\item $\\text{length}_A(M/IM) < \\infty$.\n\\end{enumerate}\nThis follows from Algebra, Lemma \\ref{algebra-lemma-support-point}\n(details omitted). If this is the case we have $M/I^nM = M/(I')^nM$\nfor all $n$ and $M/I^nM = M/\\overline{I}^nM$ for all $n$\nif $M$ is viewed as an $\\overline{A}$-module.\nThus we can define\n$$\n\\chi_{I, M}(n) = \\text{length}_A(M/I^nM) =\n\\sum\\nolimits_{p = 0, \\ldots, n - 1} \\text{length}_A(I^pM/I^{p + 1}M)\n$$\nand we get\n$$\n\\chi_{I, M}(n) = \\chi_{I', M}(n) = \\chi_{\\overline{I}, M}(n)\n$$\nfor all $n$ by the equalities above.\nAll the results of Algebra, Section \\ref{algebra-section-Noetherian-local}\nand all the results in this section, have analogues in this setting.\nIn particular we can define multiplicities $e_I(M, d)$ for\n$d \\geq \\dim(\\text{Supp}(M))$ and we have\n$$\n\\chi_{I, M}(n) \\sim e_I(M, d) \\frac{n^d}{d!} + \\text{lower order terms}\n$$\nas in the case where $I$ is an ideal of definition.\n\\end{remark}\n\n\n\n\\section{Computing intersection multiplicities}\n\\label{section-computing-intersection-multiplicities}\n\n\\noindent\nIn this section we discuss some cases where the intersection multiplicities\ncan be computed by different means. Here is a first example.\n\n\\begin{lemma}\n\\label{lemma-intersection-multiplicity-CM}\nLet $X$ be a nonsingular variety and $W, V \\subset X$ closed\nsubvarieties which intersect properly. Let $Z$ be an irreducible component\nof $V \\cap W$ with generic point $\\xi$. Assume that $\\mathcal{O}_{W, \\xi}$\nand $\\mathcal{O}_{V, \\xi}$ are Cohen-Macaulay. Then\n$$\ne(X, V \\cdot W, Z) =\n\\text{length}_{\\mathcal{O}_{X, \\xi}}(\\mathcal{O}_{V \\cap W, \\xi})\n$$\nwhere $V \\cap W$ is the scheme theoretic intersection.\nIn particular, if both $V$ and $W$ are Cohen-Macaulay, then\n$V \\cdot W = [V \\cap W]_{\\dim(V) + \\dim(W) - \\dim(X)}$.\n\\end{lemma}\n\n\\begin{proof}\nSet $A = \\mathcal{O}_{X, \\xi}$, $B = \\mathcal{O}_{V, \\xi}$, and\n$C = \\mathcal{O}_{W, \\xi}$. By Auslander-Buchsbaum\n(Algebra, Proposition \\ref{algebra-proposition-Auslander-Buchsbaum})\nwe can find a finite free resolution $F_\\bullet \\to B$ of\nlength\n$$\n\\text{depth}(A) - \\text{depth}(B) =\n\\dim(A) - \\dim(B) = \\dim(C)\n$$\nFirst equality as $A$ and $B$ are Cohen-Macaulay and the second\nas $V$ and $W$ intersect properly. Then $F_\\bullet \\otimes_A C$ is a\ncomplex of finite free modules representing $B \\otimes_A^\\mathbf{L} C$\nhence has cohomology modules with support in $\\{\\mathfrak m_A\\}$.\nBy the Acyclicity lemma (Algebra, Lemma \\ref{algebra-lemma-acyclic})\nwhich applies as $C$ is Cohen-Macaulay\nwe conclude that $F_\\bullet \\otimes_A C$ has nonzero\ncohomology only in degree $0$. This finishes the proof.\n\\end{proof}\n\n\\begin{lemma}\n\\label{lemma-one-ideal-ci}\nLet $A$ be a Noetherian local ring. Let $I = (f_1, \\ldots, f_r)$ be an ideal\ngenerated by a regular sequence. Let $M$ be a finite $A$-module. Assume that\n$\\dim(\\text{Supp}(M/IM)) = 0$. Then\n$$\ne_I(M, r) = \\sum (-1)^i\\text{length}_A(\\text{Tor}_i^A(A/I, M))\n$$\nHere $e_I(M, r)$ is as in Remark \\ref{remark-trivial-generalization}.\n\\end{lemma}\n\n\\begin{proof}\nSince $f_1, \\ldots, f_r$ is a regular sequence the Koszul complex\n$K_\\bullet(f_1, \\ldots, f_r)$ is a resolution of $A/I$ over $A$, see\nMore on Algebra, Lemma\n\\ref{more-algebra-lemma-noetherian-finite-all-equivalent}.\nThus the right hand side is equal to\n$$\n\\sum (-1)^i\\text{length}_A H_i(K_\\bullet(f_1, \\ldots, f_r) \\otimes_A M)\n$$\nNow the result follows immediately from\nTheorem \\ref{theorem-multiplicity-with-koszul} if $I$ is an ideal\nof definition. In general, we replace $A$ by $\\overline{A} = A/\\text{Ann}(M)$\nand $f_1, \\ldots, f_r$ by $\\overline{f}_1, \\ldots, \\overline{f}_r$\nwhich is allowed because\n$$\nK_\\bullet(f_1, \\ldots, f_r) \\otimes_A M =\nK_\\bullet(\\overline{f}_1, \\ldots, \\overline{f}_r) \\otimes_{\\overline{A}} M\n$$\nSince $e_I(M, r) = e_{\\overline{I}}(M, r)$ where\n$\\overline{I} = (\\overline{f}_1, \\ldots, \\overline{f}_r) \\subset \\overline{A}$\nis an ideal of definition the result follows from\nTheorem \\ref{theorem-multiplicity-with-koszul} in this case as well.\n\\end{proof}\n\n\\begin{lemma}\n\\label{lemma-multiplicity-with-lci}\nLet $X$ be a nonsingular variety. Let $W,V \\subset X$ be\nclosed subvarieties which intersect properly. Let $Z$ be an irreducible\ncomponent of $V \\cap W$ with generic point $\\xi$.\nSuppose the ideal of $V$ in $\\mathcal{O}_{X, \\xi}$ is cut out by\na regular sequence $f_1, \\ldots, f_c \\in \\mathcal{O}_{X, \\xi}$.\nThen $e(X, V\\cdot W, Z)$ is equal to $c!$ times the leading coefficient in\nthe Hilbert polynomial\n$$\nt \\mapsto \\text{length}_{\\mathcal{O}_{X, \\xi}}\n\\mathcal{O}_{W, \\xi}/(f_1, \\ldots, f_c)^t,\\quad t \\gg 0.\n$$\nIn particular, this coefficient is $> 0$.\n\\end{lemma}\n\n\\begin{proof}\nThe equality\n$$\ne(X, V\\cdot W, Z) = e_{(f_1, \\ldots, f_c)}(\\mathcal{O}_{W, \\xi}, c)\n$$\nfollows from the more general Lemma \\ref{lemma-one-ideal-ci}.\nTo see that $e_{(f_1, \\ldots, f_c)}(\\mathcal{O}_{W, \\xi}, c)$ is\n$> 0$ or equivalently that $e_{(f_1, \\ldots, f_c)}(\\mathcal{O}_{W, \\xi}, c)$\nis the leading coefficient of the Hilbert polynomial\nit suffices to show that the\ndimension of $\\mathcal{O}_{W, \\xi}$ is $c$, because the degree of the\nHilbert polynomial is equal to the dimension by\nAlgebra, Proposition \\ref{algebra-proposition-dimension}.\nSay $\\dim(V) = r$, $\\dim(W) = s$, and $\\dim(X) = n$. Then\n$\\dim(Z) = r + s - n$ as the intersection is proper. Thus\nthe transcendence degree of $\\kappa(\\xi)$ over $\\mathbf{C}$ is\n$r + s - n$, see Algebra, Lemma\n\\ref{algebra-lemma-dimension-prime-polynomial-ring}.\nWe have $r + c = n$ because $V$ is cut out by a regular sequence\nin a neighbourhood of $\\xi$, see\nDivisors, Lemma \\ref{divisors-lemma-Noetherian-scheme-regular-ideal}\nand then Lemma \\ref{lemma-pullback-by-regular-immersion}\napplies (for example). Thus\n$$\n\\dim(\\mathcal{O}_{W, \\xi}) = s - (r + s - n) = s - ((n - c) + s - n) = c\n$$\nthe first equality by Algebra, Lemma\n\\ref{algebra-lemma-dimension-at-a-point-finite-type-field}.\n\\end{proof}\n\n\\begin{lemma}\n\\label{lemma-multiplicity-with-effective-Cartier-divisor}\nIn Lemma \\ref{lemma-multiplicity-with-lci} assume that  $c = 1$, i.e., $V$\nis an effective Cartier divisor. Then\n$$\ne(X, V \\cdot W, Z) =\n\\text{length}_{\\mathcal{O}_{X, \\xi}}\n(\\mathcal{O}_{W, \\xi}/f_1\\mathcal{O}_{W, \\xi}).\n$$\n\\end{lemma}\n\n\\begin{proof}\nIn this case the image of $f_1$ in $\\mathcal{O}_{W, \\xi}$ is nonzero by\nproperness of intersection, hence a nonzerodivisor divisor. Moreover,\n$\\mathcal{O}_{W, \\xi}$ is a Noetherian local domain of dimension $1$.\nThus\n$$\n\\text{length}_{\\mathcal{O}_{X, \\xi}}\n(\\mathcal{O}_{W, \\xi}/f_1^t\\mathcal{O}_{W, \\xi}) =\nt \\text{length}_{\\mathcal{O}_{X, \\xi}}\n(\\mathcal{O}_{W, \\xi}/f_1\\mathcal{O}_{W, \\xi})\n$$\nfor all $t \\geq 1$, see Algebra, Lemma \\ref{algebra-lemma-ord-additive}.\nThis proves the lemma.\n\\end{proof}\n\n\\begin{lemma}\n\\label{lemma-multiplicity-lci-CM}\nIn Lemma \\ref{lemma-multiplicity-with-lci} assume that\nthe local ring $\\mathcal{O}_{W, \\xi}$ is Cohen-Macaulay. Then we\nhave\n$$\ne(X, V \\cdot W, Z) =\n\\text{length}_{\\mathcal{O}_{X, \\xi}} (\\mathcal{O}_{W, \\xi}/\nf_1\\mathcal{O}_{W, \\xi} + \\ldots + f_c\\mathcal{O}_{W, \\xi}).\n$$\n\\end{lemma}\n\n\\begin{proof}\nThis follows immediately from Lemma \\ref{lemma-intersection-multiplicity-CM}.\nAlternatively, we can deduce it from Lemma \\ref{lemma-multiplicity-with-lci}.\nNamely, by Algebra, Lemma \\ref{algebra-lemma-reformulate-CM}\nwe see that $f_1, \\ldots, f_c$ is a regular sequence in\n$\\mathcal{O}_{W, \\xi}$. Then\nAlgebra, Lemma \\ref{algebra-lemma-regular-quasi-regular} shows that\n$f_1, \\ldots, f_c$ is a quasi-regular sequence.\nThis easily implies the length of\n$\\mathcal{O}_{W, \\xi}/(f_1, \\ldots, f_c)^t$ is\n$$\n{c + t \\choose c}\n\\text{length}_{\\mathcal{O}_{X, \\xi}} (\\mathcal{O}_{W, \\xi}/\nf_1\\mathcal{O}_{W, \\xi} + \\ldots + f_c\\mathcal{O}_{W, \\xi}).\n$$\nLooking at the leading coefficient we conclude.\n\\end{proof}\n\n\n\\section{Intersection product using Tor formula}\n\\label{section-intersection-product}\n\n\\noindent\nLet $X$ be a nonsingular variety. Let\n$\\alpha = \\sum n_i [W_i]$ be an $r$-cycle and\n$\\beta = \\sum_j m_j [V_j]$ be an $s$-cycle on $X$.\nAssume that $\\alpha$ and $\\beta$ intersect properly, see\nDefinition \\ref{definition-proper-intersection}.\nIn this case we define\n$$\n\\alpha \\cdot \\beta = \\sum\\nolimits_{i,j} n_i m_j W_i \\cdot V_j.\n$$\nwhere $W_i \\cdot V_j$ is as defined in Section \\ref{section-tor-formula}.\nIf $\\beta = [V]$ where $V$ is a closed subvariety of dimension $s$,\nthen we sometimes write $\\alpha \\cdot \\beta = \\alpha \\cdot V$.\n\n\\begin{lemma}\n\\label{lemma-rational-equivalence-and-intersection}\nLet $X$ be a nonsingular variety. Let $a, b \\in \\mathbf{P}^1$\nbe distinct closed points. Let $k \\geq 0$.\n\\begin{enumerate}\n\\item If $W \\subset X \\times \\mathbf{P}^1$ is a closed subvariety\nof dimension $k + 1$ which intersects $X \\times a$ properly, then\n\\begin{enumerate}\n\\item $[W_a]_k = W \\cdot X \\times a$ as cycles on $X \\times \\mathbf{P}^1$, and\n\\item $[W_a]_k = \\text{pr}_{X, *}(W \\cdot X \\times a)$ as cycles on $X$.\n\\end{enumerate}\n\\item Let $\\alpha$ be a $(k + 1)$-cycle on $X \\times \\mathbf{P}^1$\nwhich intersects $X \\times a$ and $X \\times b$ properly. Then\n$pr_{X,*}( \\alpha \\cdot X \\times a - \\alpha \\cdot X \\times b)$\nis rationally equivalent to zero.\n\\item Conversely, any $k$-cycle which is\nrationally equivalent to $0$ is of this form.\n\\end{enumerate}\n\\end{lemma}\n\n\\begin{proof}\nFirst we observe that $X \\times a$ is an effective Cartier divisor in\n$X \\times \\mathbf{P}^1$ and that $W_a$ is the scheme theoretic intersection\nof $W$ with $X \\times a$. Hence the equality in (1)(a) is\nimmediate from the definitions and the calculation of intersection\nmultiplicity in case of a Cartier divisor given in\nLemma \\ref{lemma-multiplicity-with-effective-Cartier-divisor}.\nPart (1)(b) holds because $W_a \\to X \\times \\mathbf{P}^1 \\to X$ maps\nisomorphically onto its image which is how we viewed $W_a$\nas a closed subscheme of $X$ in Section \\ref{section-rational-equivalence}.\nParts (2) and (3) are formal consequences of part (1) and the definitions.\n\\end{proof}\n\n\\noindent\nFor transversal intersections of closed subschemes the intersection\nmultiplicity is $1$.\n\n\\begin{lemma}\n\\label{lemma-transversal-subschemes}\nLet $X$ be a nonsingular variety. Let $r, s \\geq 0$ and let\n$Y, Z \\subset X$ be closed subschemes with $\\dim(Y) \\leq r$ and\n$\\dim(Z) \\leq s$. Assume $[Y]_r = \\sum n_i[Y_i]$ and\n$[Z]_s = \\sum m_j[Z_j]$ intersect properly.\nLet $T$ be an irreducible component of $Y_{i_0} \\cap Z_{j_0}$\nfor some $i_0$ and $j_0$ and assume that the multiplicity\n(in the sense of Section \\ref{section-cycle-of-closed}) of $T$\nin the closed subscheme $Y \\cap Z$ is $1$.\nThen\n\\begin{enumerate}\n\\item the coefficient of $T$ in $[Y]_r \\cdot [Z]_s$ is $1$,\n\\item $Y$ and $Z$ are nonsingular at the generic point of $Z$,\n\\item $n_{i_0} = 1$, $m_{j_0} = 1$, and\n\\item $T$ is not contained in $Y_i$ or $Z_j$ for $i \\not = i_0$ and\n$j \\not = j_0$.\n\\end{enumerate}\n\\end{lemma}\n\n\\begin{proof}\nSet $n = \\dim(X)$, $a = n - r$, $b = n - s$. Observe that\n$\\dim(T) = r + s - n = n - a - b$ by the assumption that the\nintersections are transversal. Let $(A, \\mathfrak m, \\kappa) =\n(\\mathcal{O}_{X, \\xi}, \\mathfrak m_\\xi, \\kappa(\\xi))$ where $\\xi \\in T$\nis the generic point. Then $\\dim(A) = a + b$, see\nVarieties, Lemma \\ref{varieties-lemma-dimension-locally-algebraic}.\nLet $I_0, I, J_0, J \\subset A$ cut out the trace of\n$Y_{i_0}$, $Y$, $Z_{j_0}$, $Z$ in $\\Spec(A)$.\nThen $\\dim(A/I) = \\dim(A/I_0) = b$ and $\\dim(A/J) = \\dim(A/J_0) = a$\nby the same reference. Set $\\overline{I} = I + \\mathfrak m^2/\\mathfrak m^2$.\nThen $I \\subset I_0 \\subset \\mathfrak m$ and\n$J \\subset J_0 \\subset \\mathfrak m$ and $I + J = \\mathfrak m$.\nBy Lemma \\ref{lemma-transversal} and its proof we see that\n$I_0 = (f_1, \\ldots, f_a)$ and $J_0 = (g_1, \\ldots, g_b)$\nwhere $f_1, \\ldots, g_b$ is a regular system of parameters\nfor the regular local ring $A$. Since $I + J = \\mathfrak m$, the map\n$$\nI \\oplus J \\to\n\\mathfrak m/\\mathfrak m^2 =\n\\kappa f_1\n\\oplus \\ldots \\oplus\n\\kappa f_a\n\\oplus\n\\kappa g_1\n\\oplus \\ldots \\oplus\n\\kappa g_b\n$$\nis surjective. We conclude that we can find\n$f_1', \\ldots, f_a' \\in I$ and $g'_1, \\ldots, g_b' \\in J$\nwhose residue classes in $\\mathfrak m/\\mathfrak m^2$ are equal to the\nresidue classes of $f_1, \\ldots, f_a$ and $g_1, \\ldots, g_b$.\nThen $f'_1, \\ldots, g'_b$ is a regular system of parameters of $A$.\nBy Algebra, Lemma \\ref{algebra-lemma-regular-ring-CM} we find that\n$A/(f'_1, \\ldots, f'_a)$ is a regular local ring of dimension $b$.\nThus any nontrivial quotient of $A/(f'_1, \\ldots, f'_a)$\nhas strictly smaller dimension\n(Algebra, Lemmas \\ref{algebra-lemma-regular-domain} and\n\\ref{algebra-lemma-one-equation}). Hence $I = (f'_1, \\ldots, f'_a) = I_0$.\nBy symmetry $J = J_0$. This proves (2), (3), and (4).\nFinally, the coefficient of $T$ in $[Y]_r \\cdot [Z]_s$\nis the coefficient of $T$ in $Y_{i_0} \\cdot Z_{j_0}$ which is\nis $1$ by Lemma \\ref{lemma-transversal}.\n\\end{proof}\n\n\n\n\\section{Exterior product}\n\\label{section-exterior-product}\n\n\\noindent\nLet $X$ and $Y$ be varieties.\nLet $V$, resp.\\ $W$ be a closed subvariety of $X$, resp.\\ $Y$.\nThe product $V\\times W$ is a closed subvariety of $X\\times Y$\n(Lemma \\ref{lemma-dimension-product-varieties}).\nFor a $k$-cycle $\\alpha = \\sum n_i [V_i]$ and a $l$-cycle\n$\\beta = \\sum m_j [V_j]$ on $Y$ we define the\n{\\it exterior product} of $\\alpha$ and $\\beta$ to be the cycle\n$\\alpha \\times \\beta = \\sum n_i m_j [V_i \\times W_j]$.\nExterior product defines a $\\mathbf{Z}$-linear map\n$$\nZ_r(X) \\otimes_\\mathbf{Z} Z_s(Y) \\longrightarrow Z_{r + s}(X \\times Y)\n$$\nLet us prove that exterior product factors through rational equivalence.\n\n\\begin{lemma}\n\\label{lemma-exterior-product-rational-equivalence}\nLet $X$ and $Y$ be varieties.\nLet $\\alpha \\in Z_r(X)$ and $\\beta \\in Z_s(Y)$.\nIf $\\alpha \\sim_{rat} 0$ or $\\beta \\sim_{rat} 0$, then\n$\\alpha \\times \\beta \\sim_{rat} 0$.\n\\end{lemma}\n\n\\begin{proof}\nBy linearity and symmetry in $X$ and $Y$, it suffices to prove this when\n$\\alpha = [V]$ for some subvariety $V \\subset X$ of dimension $s$ and\n$\\beta = [W_a]_s - [W_b]_s$ for some closed subvariety\n$W \\subset Y \\times \\mathbf{P}^1$ of dimension $s + 1$ which\nintersects $Y \\times a$ and $Y \\times b$ properly. In this case\nthe lemma follows if we can prove\n$$\n[(V \\times W)_a]_{r + s} = [V] \\times [W_a]_s\n$$\nand similarly with $a$ replaced by $b$. Namely, then we see that\n$\\alpha \\times \\beta = [(V \\times W)_a]_{r + s} - [(V \\times W)_b]_{r + s}$\nas desired. To see the displayed equality we note the equality\n$$\nV \\times W_a = (V \\times W)_a\n$$\nof schemes. The projection $V \\times W_a \\to W_a$ induces a bijection\nof irreducible components (see for example\nVarieties, Lemma \\ref{varieties-lemma-bijection-irreducible-components}).\nLet $W' \\subset W_a$ be an irreducible component with generic point $\\zeta$.\nThen $V \\times W'$ is the corresponding irreducible component of\n$V \\times W_a$ (see Lemma \\ref{lemma-dimension-product-varieties}).\nLet $\\xi$ be the generic point of $V \\times W'$. We have to show that\n$$\n\\text{length}_{\\mathcal{O}_{Y, \\zeta}}(\\mathcal{O}_{W_a, \\zeta}) =\n\\text{length}_{\\mathcal{O}_{X \\times Y, \\xi}}(\n\\mathcal{O}_{V \\times W_a, \\xi})\n$$\nIn this formula we may replace\n$\\mathcal{O}_{Y, \\zeta}$ by $\\mathcal{O}_{W_a, \\zeta}$ and\nwe may replace\n$\\mathcal{O}_{X \\times Y, \\zeta}$ by $\\mathcal{O}_{V \\times W_a, \\zeta}$\n(see Algebra, Lemma \\ref{algebra-lemma-length-independent}).\nAs $\\mathcal{O}_{W_a, \\zeta} \\to \\mathcal{O}_{V \\times W_a, \\xi}$ is flat,\nby Algebra, Lemma \\ref{algebra-lemma-pullback-module} it suffices\nto show that\n$$\n\\text{length}_{\\mathcal{O}_{V \\times W_a, \\xi}}(\n\\mathcal{O}_{V \\times W_a, \\xi}/\n\\mathfrak m_\\zeta\\mathcal{O}_{V \\times W_a, \\xi}) = 1\n$$\nThis is true because the quotient on the right is the local ring\n$\\mathcal{O}_{V \\times W', \\xi}$ of a variety at a generic point\nhence equal to $\\kappa(\\xi)$.\n\\end{proof}\n\n\\noindent\nWe conclude that exterior product defines a commutative diagram\n$$\n\\xymatrix{\nZ_r(X) \\otimes_\\mathbf{Z} Z_s(Y) \\ar[r] \\ar[d] &\nZ_{r + s}(X \\times Y) \\ar[d] \\\\\nA_r(X) \\otimes_\\mathbf{Z} A_s(Y) \\ar[r] &\nA_{r + s}(X \\times Y)\n}\n$$\nfor any pair of varieties $X$ and $Y$. For nonsingular varieties\nwe can think of the exterior product as an intersection product\nof pullbacks.\n\n\\begin{lemma}\n\\label{lemma-exterior-product}\nLet $X$ and $Y$ be nonsingular varieties.\nLet $\\alpha \\in Z_r(X)$ and $\\beta \\in Z_s(Y)$.\nThen\n\\begin{enumerate}\n\\item $\\text{pr}_Y^*(\\beta) = [X] \\times \\beta$ and\n$\\text{pr}_X^*(\\alpha) = \\alpha \\times [Y]$,\n\\item $\\alpha \\times [Y]$ and $[X]\\times \\beta$\nintersect properly on $X\\times Y$, and\n\\item we have\n$\\alpha \\times \\beta =\n(\\alpha \\times [Y])\\cdot ([X]\\times\\beta) =\npr_Y^*(\\alpha) \\cdot pr_X^*(\\beta)$\nin $Z_{r + s}(X \\times Y)$.\n\\end{enumerate}\n\\end{lemma}\n\n\\begin{proof}\nBy linearity we may assume $\\alpha = [V]$ and $\\beta = [W]$.\nThen (1) says that $\\text{pr}_Y^{-1}(W) = X \\times W$ and\n$\\text{pr}_X^{-1}(V) = V \\times Y$. This is clear.\nPart (2) holds because $X \\times W \\cap V \\times Y = V \\times W$ and\n$\\dim(V \\times W) = r + s$ by Lemma \\ref{lemma-dimension-product-varieties}.\n\n\\medskip\\noindent\nProof of (3).\nLet $\\xi$ be the generic point of $V \\times W$.\nSince the projections $X \\times W \\to W$ is smooth as a base change of\n$X \\to \\Spec(\\mathbf{C})$, we see that $X \\times W$ is nonsingular\nat every point lying over the generic point of $W$, in particular at $\\xi$.\nSimilarly for $V \\times Y$. Hence $\\mathcal{O}_{X \\times W, \\xi}$\nand $\\mathcal{O}_{V \\times Y, \\xi}$ are Cohen-Macaulay local rings\nand Lemma \\ref{lemma-intersection-multiplicity-CM} applies.\nSince $V \\times Y \\cap X \\times W = V \\times W$ scheme theoretically\nthe proof is complete.\n\\end{proof}\n\n\n\n\\section{Reduction to the diagonal}\n\\label{section-reduction-diagonal}\n\n\\noindent\nLet $X$ be a nonsingular variety. We will use $\\Delta$\nto denote either the diagonal morphism $\\Delta : X \\to X \\times X$\nor the image $\\Delta \\subset X \\times X$.\nReduction to the diagonal is the statement that\nintersection products on $X$ can be reduced to intersection products\nof exterior products with the diagonal on $X \\times X$.\n\n\\begin{lemma}\n\\label{lemma-tor-and-diagonal}\nLet $X$ be a nonsingular variety.\n\\begin{enumerate}\n\\item If $\\mathcal{F}$ and $\\mathcal{G}$ are coherent $\\mathcal{O}_X$-modules,\nthen there are canonical isomorphisms\n$$\n\\text{Tor}_i^{\\mathcal{O}_{X \\times X}}(\\mathcal{O}_\\Delta,\n\\text{pr}_1^*\\mathcal{F} \\otimes_{\\mathcal{O}_{X \\times X}}\n\\text{pr}_2^*\\mathcal{G})\n=\n\\Delta_*\\text{Tor}_i^{\\mathcal{O}_X}(\\mathcal{F}, \\mathcal{G})\n$$\n\\item If $K$ and $M$ are in $D_\\QCoh(\\mathcal{O}_X)$, then\nthere is a canonical isomorphism\n$$\nL\\Delta^* \\left(\nL\\text{pr}_1^*K \\otimes_{\\mathcal{O}_{X \\times X}}^\\mathbf{L} L\\text{pr}_2^*M\n\\right)\n= K \\otimes_{\\mathcal{O}_X}^\\mathbf{L} M\n$$\nin $D_\\QCoh(\\mathcal{O}_X)$ and a canonical isomorphism\n$$\n\\mathcal{O}_\\Delta \\otimes_{\\mathcal{O}_{X \\times X}}^\\mathbf{L}\nL\\text{pr}_1^*K \\otimes_{\\mathcal{O}_{X \\times X}}^\\mathbf{L} L\\text{pr}_2^*M\n= \\Delta_*(K \\otimes_{\\mathcal{O}_X}^\\mathbf{L} M)\n$$\nin $D_\\QCoh(X \\times X)$.\n\\end{enumerate}\n\\end{lemma}\n\n\\begin{proof}\nLet us explain how to prove (1) in a more elementary way and part (2) using\nmore general theory. As (2) implies (1) the reader can skip the proof\nof (1).\n\n\\medskip\\noindent\nProof of (1). Choose an affine open $\\Spec(A) \\subset X$.\nThen $A$ is a Noetherian $\\mathbf{C}$-algebra and\n$\\mathcal{F}$, $\\mathcal{G}$ correspond to finite $A$-modules\n$M$ and $N$ (Cohomology of Schemes, Lemma\n\\ref{coherent-lemma-coherent-Noetherian}).\nBy Derived Categories of Schemes, Lemma\n\\ref{perfect-lemma-quasi-coherence-tensor-product} we may\ncompute $\\text{Tor}_i$ over $\\mathcal{O}_X$\nby first computing the $\\text{Tor}$'s\nof $M$ and $N$ over $A$, and then taking the associated $\\mathcal{O}_X$-module.\nFor the $\\text{Tor}_i$ over $\\mathcal{O}_{X \\times X}$ we compute\nthe tor of $A$ and $M \\otimes_\\mathbf{C} N$ over $A \\otimes_\\mathbf{C} A$\nand then take the associated $\\mathcal{O}_{X \\times X}$-module.\nHence on this affine patch we have to prove that\n$$\n\\text{Tor}_i^{A \\otimes_\\mathbf{C} A}(A, M \\otimes_\\mathbf{C} N) =\n\\text{Tor}_i^A(M, N)\n$$\nTo see this choose resolutions $F_\\bullet \\to M$ and $G_\\bullet \\to M$\nby finite free $A$-modules\n(Algebra, Lemma \\ref{algebra-lemma-resolution-by-finite-free}).\nNote that $\\text{Tot}(F_\\bullet \\otimes_\\mathbf{C} G_\\bullet)$\nis a resolution of $M \\otimes_\\mathbf{C} N$ as it computes\nTor groups over $\\mathbf{C}$! Of course the terms of\n$F_\\bullet \\otimes_\\mathbf{C} G_\\bullet$ are finite free\n$A \\otimes_\\mathbf{C} A$-modules. Hence the left hand side\nof the displayed equation is the module\n$$\nH_i(A \\otimes_{A \\otimes_\\mathbf{C} A}\n\\text{Tot}(F_\\bullet \\otimes_\\mathbf{C} G_\\bullet))\n$$\nand the right hand side is the module\n$$\nH_i(\\text{Tot}(F_\\bullet \\otimes_A G_\\bullet))\n$$\nSince $A \\otimes_{A \\otimes_\\mathbf{C} A} (F_p \\otimes_\\mathbf{C} G_q)\n= F_p \\otimes_A G_q$ we see that these modules are equal.\nThis defines an isomorphism over the affine open $\\Spec(A) \\times \\Spec(A)$\n(which is good enough for the application to equality of intersection numbers).\nWe omit the proof that these isomorphisms glue.\n\n\\medskip\\noindent\nProof of (2). The second statement follows from the first by the\nprojection formula as stated in\nDerived Categories of Schemes, Lemma \\ref{perfect-lemma-cohomology-base-change}.\nTo see the first, represent $K$ and $M$ by K-flat complexes\n$\\mathcal{K}^\\bullet$ and $\\mathcal{M}^\\bullet$.\nSince pullback and tensor product preserve K-flat complexes\n(Cohomology, Lemmas \\ref{cohomology-lemma-tensor-product-K-flat} and\n\\ref{cohomology-lemma-pullback-K-flat})\nwe see that it suffices to show\n$$\n\\Delta^*\\text{Tot}(\n\\text{pr}_1^*\\mathcal{K}^\\bullet\n\\otimes_{\\mathcal{O}_{X \\times X}} \\text{pr}_2^*\\mathcal{M}^\\bullet)\n=\n\\text{Tot}(\n\\mathcal{K}^\\bullet \\otimes_{\\mathcal{O}_X} \\mathcal{M}^\\bullet)\n$$\nThus it suffices to see that there are canonical isomorphisms\n$$\n\\Delta^*(\\text{pr}_1^*\\mathcal{K}\n\\otimes_{\\mathcal{O}_{X \\times X}} \\text{pr}_2^*\\mathcal{M})\n\\longrightarrow\n\\mathcal{K} \\otimes_{\\mathcal{O}_X} \\mathcal{M}\n$$\nwhenever $\\mathcal{K}$ and $\\mathcal{M}$ are $\\mathcal{O}_X$-modules\n(not necessarily quasi-coherent or flat).\nWe omit the details.\n\\end{proof}\n\n\\begin{lemma}\n\\label{lemma-reduction-diagonal}\nLet $X$ be a nonsingular variety. Let $\\alpha$, resp.\\ $\\beta$\nbe an $r$-cycle, resp.\\ $s$-cycle on $X$. Assume $\\alpha$ and $\\beta$\nintersect properly. Then\n\\begin{enumerate}\n\\item $\\alpha \\times \\beta$ and $[\\Delta]$ intersect properly\n\\item we have $\\Delta_*(\\alpha \\cdot \\beta) = [\\Delta] \\cdot \\alpha\\times\\beta$\nas cycles on $X \\times X$,\n\\item if $X$ is proper, then\n$\\text{pr}_{1, *}([\\Delta] \\cdot \\alpha\\times\\beta) = \\alpha\\cdot\\beta$,\nwhere $pr_1 : X\\times X \\to X$ is the projection.\n\\end{enumerate}\n\\end{lemma}\n\n\\begin{proof}\nBy linearity it suffices to prove this when $\\alpha = [V]$ and $\\beta = [W]$\nfor some closed subvarieties $V \\subset X$ and $W \\subset Y$ which intersect\nproperly. Recall that $V \\times W$ is a closed subvariety of dimension $r + s$.\nObserve that scheme theoretically we have\n$V \\cap W = \\Delta^{-1}(V \\times W)$ as well as\n$\\Delta(V \\cap W) = \\Delta \\cap V \\times W$.\nThis proves (1).\n\n\\medskip\\noindent\nProof of (2). Let $Z \\subset V \\cap W$ be an irreducible component\nwith generic point $\\xi$. We have to show that the coefficient of\n$Z$ in $\\alpha \\cdot \\beta$ is the same as the coefficient of\n$\\Delta(Z)$ in $[\\Delta] \\cdot \\alpha \\times \\beta$. The first is given\nby the integer\n$$\n\\sum (-1)^i\n\\text{length}_{\\mathcal{O}_{X, \\xi}}\n\\text{Tor}_i^{\\mathcal{O}_X}(\\mathcal{O}_V, \\mathcal{O}_W)_\\xi\n$$\nand the second by the integer\n$$\n\\sum (-1)^i\n\\text{length}_{\\mathcal{O}_{X \\times Y, \\Delta(\\xi)}}\n\\text{Tor}_i^{\\mathcal{O}_{X \\times Y}}(\n\\mathcal{O}_\\Delta, \\mathcal{O}_{V \\times W})_{\\Delta(\\xi)}\n$$\nHowever, by Lemma \\ref{lemma-tor-and-diagonal} we have\n$$\n\\text{Tor}_i^{\\mathcal{O}_X}(\\mathcal{O}_V, \\mathcal{O}_W)_\\xi \\cong\n\\text{Tor}_i^{\\mathcal{O}_{X \\times Y}}(\n\\mathcal{O}_\\Delta, \\mathcal{O}_{V \\times W})_{\\Delta(\\xi)}\n$$\nas $\\mathcal{O}_{X \\times X, \\Delta(\\xi)}$-modules. Thus equality\nof lengths (by Algebra, Lemma \\ref{algebra-lemma-length-independent}\nto be precise).\n\n\\medskip\\noindent\nPart (2) implies (3) because\n$\\text{pr}_{1, *} \\circ \\Delta_* = \\text{id}$ by\nLemma \\ref{lemma-compose-pushforward}.\n\\end{proof}\n\n\\begin{proposition}\n\\label{proposition-positivity}\n\\begin{reference}\nThis is one of the main results of \\cite{Serre_algebre_locale}.\n\\end{reference}\nLet $X$ be a nonsingular variety. Let $V \\subset X$ and\n$W \\subset Y$ be closed subvarieties which intersect properly.\nLet $Z \\subset V \\cap W$ be an irreducible component.\nThen $e(X, V \\cdot W, Z) > 0$.\n\\end{proposition}\n\n\\begin{proof}\nBy Lemma \\ref{lemma-reduction-diagonal} we have\n$$\ne(X, V \\cdot W, Z) = e(X \\times X, \\Delta \\cdot V \\times W, \\Delta(Z))\n$$\nSince $\\Delta : X \\to X \\times X$ is a regular immersion\n(see Lemma \\ref{lemma-diagonal-regular-immersion}), we see that\n$e(X \\times X, \\Delta \\cdot V \\times W, \\Delta(Z))$ is a positive\ninteger by Lemma \\ref{lemma-multiplicity-with-lci}.\n\\end{proof}\n\n\\noindent\nThe following is a key lemma in the development of the theory as\nis done in this chapter. Essentially, this lemma tells us that\nthe intersection numbers have a suitable additivity property.\n\n\\begin{lemma}\n\\label{lemma-tor-sheaf}\n\\begin{reference}\n\\cite[Chapter V]{Serre_algebre_locale}\n\\end{reference}\nLet $X$ be a nonsingular variety. Let $\\mathcal{F}$ and\n$\\mathcal{G}$ be coherent sheaves on $X$ with\n$\\dim(\\text{Supp}(\\mathcal{F})) \\leq r$,\n$\\dim(\\text{Supp}(\\mathcal{G})) \\leq s$, and\n$\\dim(\\text{Supp}(\\mathcal{F}) \\cap \\text{Supp}(\\mathcal{G}) )\n\\leq r + s - \\dim X$. In this case $[\\mathcal{F}]_r$ and $[\\mathcal{G}]_s$\nintersect properly and\n$$\n[\\mathcal{F}]_r \\cdot [\\mathcal{G}]_s =\n\\sum (-1)^p\n[\\text{Tor}_p^{\\mathcal{O}_X}(\\mathcal{F}, \\mathcal{G})]_{r + s - \\dim(X)}.\n$$\n\\end{lemma}\n\n\\begin{proof}\nThe statement that $[\\mathcal{F}]_r$ and $[\\mathcal{G}]_s$ intersect properly\nis immediate. Since we are proving an equality of cycles we may work\nlocally on $X$. (Observe that the formation of the intersection\nproduct of cycles, the formation of $\\text{Tor}$-sheaves, and\nforming the cycle associated to a coherent sheaf, each commute with\nrestriction to open subschemes.) Thus we may and do assume that $X$ is affine.\n\n\\medskip\\noindent\nDenote\n$$\nRHS(\\mathcal{F}, \\mathcal{G}) = [\\mathcal{F}]_r \\cdot [\\mathcal{G}]_s\n\\quad\\text{and}\\quad\nLHS(\\mathcal{F}, \\mathcal{G}) = \\sum (-1)^p\n[\\text{Tor}_p^{\\mathcal{O}_X}(\\mathcal{F}, \\mathcal{G})]_{r + s - \\dim(X)}\n$$\nConsider a short exact sequence\n$$\n0 \\to \\mathcal{F}_1 \\to \\mathcal{F}_2 \\to \\mathcal{F}_3 \\to 0\n$$\nof coherent sheaves on $X$ with\n$\\text{Supp}(\\mathcal{F}_i) \\subset \\text{Supp}(\\mathcal{F})$,\nthen both $LHS(\\mathcal{F}_i, \\mathcal{G})$ and\n$RHS(\\mathcal{F}_i, \\mathcal{G})$ are defined for $i = 1, 2, 3$\nand we have\n$$\nRHS(\\mathcal{F}_2, \\mathcal{G}) =\nRHS(\\mathcal{F}_1, \\mathcal{G}) + RHS(\\mathcal{F}_3, \\mathcal{G})\n$$\nand similarly for LHS. Namely, the support condition guarantees that\neverything is defined, the short exact sequence and additivity of lengths\ngives\n$$\n[\\mathcal{F}_2]_r = [\\mathcal{F}_1]_r  + [\\mathcal{F}_3]_r\n$$\n(Chow Homology, Lemma \\ref{chow-lemma-additivity-sheaf-cycle})\nwhich implies additivity for RHS. The long exact sequence of $\\text{Tor}$s\n$$\n\\ldots \\to \\text{Tor}_1(\\mathcal{F}_3, \\mathcal{G}) \\to\n\\text{Tor}_0(\\mathcal{F}_1, \\mathcal{G}) \\to\n\\text{Tor}_0(\\mathcal{F}_2, \\mathcal{G}) \\to\n\\text{Tor}_0(\\mathcal{F}_3, \\mathcal{G}) \\to 0\n$$\nand additivity of lengths as before implies additivity for LHS.\n\n\\medskip\\noindent\nBy Algebra, Lemma \\ref{algebra-lemma-filter-Noetherian-module}\nand the fact that $X$ is affine, we can find a filtration of $\\mathcal{F}$\nwhose graded pieces are structure sheaves of closed subvarieties of\n$\\text{Supp}(\\mathcal{F})$. The additivity shown in the previous paragraph,\nimplies that it suffices to prove $LHS = RHS$ with\n$\\mathcal{F}$ replaced by $\\mathcal{O}_V$ where\n$V \\subset \\text{Supp}(\\mathcal{F})$.\nBy symmetry we can do the same for $\\mathcal{G}$.\nThis reduces us to proving that\n$$\nLHS(\\mathcal{O}_V, \\mathcal{O}_W) = RHS(\\mathcal{O}_V, \\mathcal{O}_W)\n$$\nwhere $W \\subset \\text{Supp}(\\mathcal{G})$ is a closed subvariety.\nIf $\\dim(V) = r$ and $\\dim(W) = s$, then this equality is the\n{\\bf definition} of $V \\cdot W$. On the other hand, if\n$\\dim(V) < r$ or $\\dim(W) < s$, i.e., $[V]_r = 0$ or $[W]_s = 0$,\nthen we have to prove that $RHS(\\mathcal{O}_V, \\mathcal{O}_W) = 0$\n\\footnote{The reader can see that this is not a triviality by\ntaking $r = s = 1$ and $X$ a nonsingular surface and $V = W$\na closed point $x$ of $X$. In this case there are $3$ nonzero\n$\\text{Tor}$s of lengths $1, 2, 1$ at $x$.}.\n\n\\medskip\\noindent\nLet $Z \\subset V \\cap W$ be an irreducible component of dimension\n$r + s - \\dim(X)$. This is the maximal dimension of a component\nand it suffices to show that the coefficient of $Z$ in $RHS$ is zero.\nLet $\\xi \\in Z$ be the generic point. Write $A = \\mathcal{O}_{X, \\xi}$,\n$B = \\mathcal{O}_{X \\times X, \\Delta(\\xi)}$, and\n$C = \\mathcal{O}_{V \\times W, \\Delta(\\xi)}$.\nBy Lemma \\ref{lemma-tor-and-diagonal} we have\n$$\n\\text{coeff of }Z\\text{ in }\nRHS(\\mathcal{O}_V, \\mathcal{O}_W) = \n\\sum (-1)^i\n\\text{length}_B \\text{Tor}_i^B(A, C)\n$$\nSince $\\dim(V) < r$ or $\\dim(W) < s$ we have $\\dim(V \\times W) < r + s$\nwhich implies $\\dim(C) < \\dim(X)$ (small detail omitted). Moreover, the\nkernel $I$ of $B \\to A$ is generated by a regular sequence of\nlength $\\dim(X)$ (Lemma \\ref{lemma-diagonal-regular-immersion}).\nHence vanishing by Lemma \\ref{lemma-one-ideal-ci} because\nthe Hilbert function of $C$ with respect to $I$ has degree $\\dim(C) < n$\nby Algebra, Proposition \\ref{algebra-proposition-dimension}.\n\\end{proof}\n\n\\begin{remark}\n\\label{remark-Serre-conjectures}\nLet $(A, \\mathfrak m, \\kappa)$ be a regular local ring.\nLet $M$ and $N$ be nonzero finite $A$-modules such that $M \\otimes_A N$\nis supported in $\\{\\mathfrak m\\}$. Then\n$$\n\\chi(M, N) = \\sum (-1)^i \\text{length}_A \\text{Tor}_i^A(M, N)\n$$\nis finite. Let $r = \\dim(\\text{Supp}(M))$ and $s = \\dim(\\text{Supp}(N))$.\nIn \\cite{Serre_algebre_locale} it is shown that $r + s \\leq \\dim(A)$\nand the following conjectures are made:\n\\begin{enumerate}\n\\item if $r + s < \\dim(A)$, then $\\chi(M, N) = 0$, and\n\\item if $r + s = \\dim(A)$, then $\\chi(M, N) > 0$.\n\\end{enumerate}\nThe arguments that prove Lemma \\ref{lemma-tor-sheaf} and\nProposition \\ref{proposition-positivity} can be leveraged\n(as is done in Serre's text) to show that (1) and (2) are\ntrue if $A$ contains a field. Currently, conjecture (1) is known\nin general and it is known that $\\chi(M, N) \\geq 0$ in general (Gabber).\nPositivity is, as far as we know, still an open problem.\n\\end{remark}\n\n\n\n\\section{Associativity of intersections}\n\\label{section-associative}\n\n\\noindent\nIt is clear that proper intersections as defined above are commutative.\nUsing the key Lemma \\ref{lemma-tor-sheaf} we can prove that (proper)\nintersection products are associative.\n\n\\begin{lemma}\n\\label{lemma-associative}\nLet $X$ be a nonsingular variety. Let $U, V, W$ be closed\nsubvarieties. Assume that $U, V, W$ intersect properly pairwise\nand that $\\dim(U \\cap V \\cap W) \\leq \\dim(U) + \\dim(V) + \\dim(W) - 2\\dim(X)$.\nThen\n$$\nU \\cdot (V \\cdot W) = (U \\cdot V) \\cdot W\n$$\nas cycles on $X$.\n\\end{lemma}\n\n\\begin{proof}\nWe are going to use Lemma \\ref{lemma-tor-sheaf} without further mention.\nThis implies that\n\\begin{align*}\nV \\cdot W\n& =\n\\sum (-1)^i [\\text{Tor}_i(\\mathcal{O}_V, \\mathcal{O}_W)]_{b + c - n} \\\\\nU \\cdot (V \\cdot W)\n& =\n\\sum (-1)^{i + j}\n[\n\\text{Tor}_j(\\mathcal{O}_U, \\text{Tor}_i(\\mathcal{O}_V, \\mathcal{O}_W))\n]_{a + b + c - 2n} \\\\\nU \\cdot V\n& =\n\\sum (-1)^i [\\text{Tor}_i(\\mathcal{O}_U, \\mathcal{O}_V)]_{a + b - n} \\\\\n(U \\cdot V) \\cdot W\n& =\n\\sum (-1)^{i + j}\n[\n\\text{Tor}_j(\\text{Tor}_i(\\mathcal{O}_U, \\mathcal{O}_V), \\mathcal{O}_W))\n]_{a + b + c - 2n}\n\\end{align*}\nwhere $\\dim(U) = a$, $\\dim(V) = b$, $\\dim(W) = c$, $\\dim(X) = n$.\nThe assumptions in the lemma guarantee that the coherent sheaves\nin the formulae above satisfy the required bounds on dimensions\nof supports in order to make sense of these. Now consider the object\n$$\nK =\n\\mathcal{O}_U \\otimes^\\mathbf{L}_{\\mathcal{O}_X} \\mathcal{O}_V\n\\otimes^\\mathbf{L}_{\\mathcal{O}_X} \\mathcal{O}_W\n$$\nof the derived category $D_{\\textit{Coh}}(\\mathcal{O}_X)$.\nWe claim that the expressions obtained above for\n$U \\cdot (V \\cdot W)$ and $(U \\cdot V) \\cdot W$\nare equal to\n$$\n\\sum (-1)^k [H^k(K)]_{a + b + c - 2n}\n$$\nThis will prove the lemma. By symmetry it suffices to prove one\nof these equalities. To do this we represent $\\mathcal{O}_U$\nand $\\mathcal{O}_V \\otimes_{\\mathcal{O}_X}^\\mathbf{L} \\mathcal{O}_W$\nby K-flat complexes $M^\\bullet$ and $L^\\bullet$ and use the\nspectral sequence associated to the double complex\n$K^\\bullet \\otimes_{\\mathcal{O}_X} L^\\bullet$ in\nHomology, Section \\ref{homology-section-double-complex}.\nThis is a spectral sequence with $E_2$ page\n$$\nE_2^{p, q} =\n\\text{Tor}_{-p}(\\mathcal{O}_U, \\text{Tor}_{-q}(\\mathcal{O}_V, \\mathcal{O}_W))\n$$\nconverging to $H^{p + q}(K)$ (details omitted; compare with\nMore on Algebra, Example \\ref{more-algebra-example-tor}).\nSince lengths are additive in short\nexact sequences we see that the result is true.\n\\end{proof}\n\n\n\n\n\\section{Flat pullback and intersection products}\n\\label{section-flat-pullback-and-intersection-products}\n\n\\noindent\nShort discussion of the interaction between intersections and\nflat pullback.\n\n\\begin{lemma}\n\\label{lemma-flat-pull-back-and-intersections-sheaves}\nLet $f : X \\to Y$ be a flat morphism of nonsingular varieties. Set\n$e = \\dim(X) - \\dim(Y)$. Let $\\mathcal{F}$ and $\\mathcal{G}$ be coherent\nsheaves on $Y$ with $\\dim(\\text{Supp}(\\mathcal{F})) \\leq r$,\n$\\dim(\\text{Supp}(\\mathcal{G})) \\leq s$, and\n$\\dim(\\text{Supp}(\\mathcal{F}) \\cap \\text{Supp}(\\mathcal{G}) )\n\\leq r + s - \\dim(Y)$. In this case the cycles\n$[f^*\\mathcal{F}]_{r + e}$ and $[f^*\\mathcal{G}]_{s + e}$\nintersect properly and\n$$\nf^*([\\mathcal{F}]_r \\cdot [\\mathcal{G}]_s) =\n[f^*\\mathcal{F}]_{r + e} \\cdot [f^*\\mathcal{G}]_{s + e}\n$$\n\\end{lemma}\n\n\\begin{proof}\nThe statement that $[f^*\\mathcal{F}]_{r + e}$ and $[f^*\\mathcal{G}]_{s + e}$\nintersect properly is immediate from the assumption that $f$ has\nrelative dimension $e$. By\nLemmas \\ref{lemma-tor-sheaf} and \\ref{lemma-pullback}\nit suffices to show that\n$$\nf^*\\text{Tor}_i^{\\mathcal{O}_Y}(\\mathcal{F}, \\mathcal{G}) =\n\\text{Tor}_i^{\\mathcal{O}_X}(f^*\\mathcal{F}, f^*\\mathcal{G})\n$$\nas $\\mathcal{O}_X$-modules. This follows from\nCohomology, Lemma \\ref{cohomology-lemma-pullback-tensor-product}\nand the fact that $f^*$ is exact, so $Lf^*\\mathcal{F} = f^*\\mathcal{F}$\nand similarly for $\\mathcal{G}$.\n\\end{proof}\n\n\\begin{lemma}\n\\label{lemma-flat-pullback-and-intersections}\nLet $f : X \\to Y$ be a flat morphism of nonsingular varieties.\nLet $\\alpha$ be a $r$-cycle on $Y$ and $\\beta$ an $s$-cycle on $Y$.\nAssume that $\\alpha$ and $\\beta$ intersect properly. Then $f^*\\alpha$\nand $f^*\\beta$ intersect properly and\n$f^*( \\alpha \\cdot \\beta ) = f^*\\alpha \\cdot f^*\\beta$.\n\\end{lemma}\n\n\\begin{proof}\nBy linearity we may assume that $\\alpha = [V]$ and $\\beta = [W]$\nfor some closed subvarieties $V, W \\subset Y$ of dimension $r, s$.\nSay $f$ has relative dimension $e$. Then the lemma is a special case of\nLemma \\ref{lemma-flat-pull-back-and-intersections-sheaves}\nbecause $[V] = [\\mathcal{O}_V]_r$, $[W] = [\\mathcal{O}_W]_r$,\n$f^*[V] = [f^{-1}(V)]_{r + e} = [f^*\\mathcal{O}_V]_{r + e}$, and\n$f^*[W] = [f^{-1}(W)]_{s + e} = [f^*\\mathcal{O}_W]_{s + e}$.\n\\end{proof}\n\n\n\\section{Projection formula for flat proper morphisms}\n\\label{section-projection-formula-flat}\n\n\\noindent\nShort discussion of the projection formula for flat proper morphisms.\n\n\\begin{lemma}\n\\label{lemma-projection-formula-flat}\n\\begin{reference}\nSee \\cite[Chapter V, Section 7, formula (10)]{Serre_algebre_locale}\nfor a more general formula.\n\\end{reference}\nLet $f : X \\to Y$ be a flat proper morphism of nonsingular varieties.\nSet $e = \\dim(X) - \\dim(Y)$. Let $\\alpha$ be an $r$-cycle on $X$ and let\n$\\beta$ be a $s$-cycle on $Y$. Assume that $\\alpha$ and $f^*(\\beta)$ intersect\nproperly. Then $f_*(\\alpha)$ and $\\beta$ intersect properly and\n$$\nf_*(\\alpha) \\cdot \\beta = f_*( \\alpha \\cdot f^*\\beta)\n$$\n\\end{lemma}\n\n\\begin{proof}\nBy linearity we reduce to the case where $\\alpha = [V]$ and\n$\\beta = [W]$ for some closed subvariety $V \\subset X$ and\n$W \\subset Y$ of dimension $r$ and $s$. Then $f^{-1}(W)$ has\npure dimension $s + e$. We assume the cycles\n$[V]$ and $f^*[W]$ intersect properly. We will use without\nfurther mention the fact that $V \\cap f^{-1}(W) \\to f(V) \\cap W$\nis surjective.\n\n\\medskip\\noindent\nLet $a$ be the dimension of the generic fibre of $V \\to f(V)$.\nIf $a > 0$, then $f_*[V] = 0$. In particular $f_*\\alpha$ and $\\beta$\nintersect properly. To finish this case we have to show that\n$f_*([V] \\cdot f^*[W]) = 0$. However, since every fibre of\n$V \\to f(V)$ has dimension $\\geq a$ (see\nMorphisms, Lemma \\ref{morphisms-lemma-openness-bounded-dimension-fibres})\nwe conclude that every irreducible component $Z$ of $V \\cap f^{-1}(W)$\nhas fibres of dimension $\\geq a$ over $f(Z)$. This certainly\nimplies what we want.\n\n\\medskip\\noindent\nAssume that $V \\to f(V)$ is generically finite. Let $Z \\subset f(V) \\cap W$\nbe an irreducible component. Let $Z_i \\subset V \\cap f^{-1}(W)$,\n$i = 1, \\ldots, t$ be the irreducible components of $V \\cap f^{-1}(W)$\ndominating $Z$. By assumption each $Z_i$ has dimension\n$r + s + e - \\dim(X) = r + s - \\dim(Y)$. Hence\n$\\dim(Z) \\leq r + s - \\dim(Y)$. Thus we see that $f(V)$ and $W$\nintersect properly, $\\dim(Z) = r + s - \\dim(Y)$, and each\n$Z_i \\to Z$ is generically finite. In particular, it follows that\n$V \\to f(V)$ has finite fibre over the generic point $\\xi$ of $Z$.\nThus $V \\to Y$ is finite in an open neighbourhood of $\\xi$, see\nCohomology of Schemes, Lemma\n\\ref{coherent-lemma-proper-finite-fibre-finite-in-neighbourhood}.\nUsing a very general projection formula\\footnote{This can be avoided\nby working in an affine neighbourhood of $\\xi$ as above, choosing\nan affine open of $X$ containing the generic points of the $Z_i$, and\ntranslating the question into algebra. Doing this will produce a\nrelatively elementary proof of (\\ref{equation-stalks}).}\nfor derived tensor products, we get\n$$\nRf_*(\\mathcal{O}_V \\otimes_{\\mathcal{O}_X}^\\mathbf{L} Lf^*\\mathcal{O}_W) =\nRf_*\\mathcal{O}_V \\otimes_{\\mathcal{O}_Y}^\\mathbf{L} \\mathcal{O}_W\n$$\nsee Derived Categories of Schemes, Lemma\n\\ref{perfect-lemma-cohomology-base-change}.\nSince $f$ is flat, we see that $Lf^*\\mathcal{O}_W = f^*\\mathcal{O}_W$.\nSince $f|_V$ is finite in an open neighbourhood of $\\xi$ we have\n$$\n(Rf_*\\mathcal{F})_\\xi = (f_*\\mathcal{F})_\\xi\n$$\nfor any coherent sheaf on $X$ whose support is contained in $V$\n(see Cohomology of Schemes, Lemma\n\\ref{coherent-lemma-higher-direct-images-zero-finite-fibre}). Thus\nwe conclude that\n\\begin{equation}\n\\label{equation-stalks}\n\\left(\nf_*\\text{Tor}_i^{\\mathcal{O}_X}(\\mathcal{O}_V, f^*\\mathcal{O}_W)\n\\right)_\\xi =\n\\left(\\text{Tor}_i^{\\mathcal{O}_Y}(f_*\\mathcal{O}_V, \\mathcal{O}_W)\\right)_\\xi\n\\end{equation}\nfor all $i$. Since $f^*[W] = [f^*\\mathcal{O}_W]_{s + e}$ by\nLemma \\ref{lemma-pullback} we have\n$$\n[V] \\cdot f^*[W] =\n\\sum (-1)^i\n[\\text{Tor}_i^{\\mathcal{O}_X}(\\mathcal{O}_V,\nf^*\\mathcal{O}_W)]_{r + s - \\dim(Y)}\n$$\nby Lemma \\ref{lemma-tor-sheaf}. Applying\nLemma \\ref{lemma-push-coherent}\nwe find\n$$\nf_*([V] \\cdot f^*[W]) =\n\\sum (-1)^i\n[f_*\\text{Tor}_i^{\\mathcal{O}_X}(\\mathcal{O}_V,\nf^*\\mathcal{O}_W)]_{r + s - \\dim(Y)}\n$$\nSince $f_*[V] = [f_*\\mathcal{O}_V]_r$ by\nLemma \\ref{lemma-push-coherent} we have\n$$\n[f_*V] \\cdot [W] =\n\\sum (-1)^i\n[\\text{Tor}_i^{\\mathcal{O}_X}(f_*\\mathcal{O}_V,\n\\mathcal{O}_W)]_{r + s - \\dim(Y)}\n$$\nagain by Lemma \\ref{lemma-tor-sheaf}.\nComparing the formula for $f_*([V] \\cdot f^*[W])$ with\nthe formula for $f_*[V] \\cdot [W]$ and looking at the\nthe coefficient of $Z$ by taking lengths of stalks at $\\xi$, we see that\n(\\ref{equation-stalks}) finishes the proof.\n\\end{proof}\n\n\\begin{lemma}\n\\label{lemma-transfer}\nLet $X \\to P$ be a closed immersion of nonsingular varieties.\nLet $C' \\subset P \\times \\mathbf{P}^1$ be a closed subvariety of dimension\n$r + 1$. Assume\n\\begin{enumerate}\n\\item the fibre $C = C'_0$ has dimension $r$, i.e., $C' \\to \\mathbf{P}^1$\nis dominant,\n\\item $C'$ intersects $X \\times \\mathbf{P}^1$ properly,\n\\item $[C]_r$ intersects $X$ properly.\n\\end{enumerate}\nThen setting $\\alpha = [C]_r \\cdot X$ viewed as cycle on $X$ and\n$\\beta = C' \\cdot X \\times \\mathbf{P}^1$ viewed as cycle on\n$X \\times \\mathbf{P}^1$, we have\n$$\n\\alpha = \\text{pr}_{X, *}(\\beta \\cdot X \\times 0)\n$$\nas cycles on $X$ where $\\text{pr}_X : X \\times \\mathbf{P}^1 \\to X$ is the\nprojection.\n\\end{lemma}\n\n\\begin{proof}\nLet $\\text{pr} : P \\times \\mathbf{P}^1 \\to P$ be the projection.\nSince we are proving an equality of cycles it suffices to think of\n$\\alpha$, resp.\\ $\\beta$ as a cycle on $P$, resp.\\ $P \\times \\mathbf{P}^1$\nand prove the result for pushing forward by $\\text{pr}$.\nBecause $\\text{pr}^*X = X \\times \\mathbf{P}^1$ and\n$\\text{pr}$ defines an isomorphism of $C'_0$ onto $C$\nthe projection formula (Lemma \\ref{lemma-projection-formula-flat})\ngives\n$$\n\\text{pr}_*([C'_0]_r \\cdot X \\times \\mathbf{P}^1) = [C]_r \\cdot X = \\alpha\n$$\nOn the other hand, we have $[C'_0]_r = C' \\cdot P \\times 0$\nas cycles on $P \\times \\mathbf{P}^1$\nby Lemma \\ref{lemma-rational-equivalence-and-intersection}.\nHence\n$$\n[C'_0]_r \\cdot X \\times \\mathbf{P}^1 =\n(C' \\cdot P \\times 0) \\cdot X \\times \\mathbf{P}^1 =\n(C' \\cdot X \\times \\mathbf{P}^1) \\cdot P \\times 0\n$$\nby associativity (Lemma \\ref{lemma-associative}) and commutativity of the\nintersection product. It remains to show that the intersection product of\n$C' \\cdot X \\times \\mathbf{P}^1$ with $P \\times 0$ on\n$P \\times \\mathbf{P}^1$ is equal (as a cycle) to the intersection product of\n$\\beta$ with $X \\times 0$ on $X \\times \\mathbf{P}^1$. Write\n$C' \\cdot X \\times \\mathbf{P}^1 = \\sum n_k[E_k]$ and hence\n$\\beta = \\sum n_k[E_k]$ for some subvarieties\n$E_k \\subset X \\times \\mathbf{P}^1 \\subset P \\times \\mathbf{P}^1$.\nThen both intersections are equal to $\\sum m_k[E_{k, 0}]$ by\nLemma \\ref{lemma-rational-equivalence-and-intersection} applied twice.\nThis finishes the proof.\n\\end{proof}\n\n\n\n\n\\section{Projections}\n\\label{section-projection}\n\n\\noindent\nRecall that we are working over a fixed algebraically closed ground field\n$\\mathbf{C}$. If $V$ is a finite dimensional vector space over $\\mathbf{C}$\nthen we set\n$$\n\\mathbf{P}(V) = \\text{Proj}(\\text{Sym}(V))\n$$\nwhere $\\text{Sym}(V)$ is the symmetric algebra on $V$ over $\\mathbf{C}$.\nThe normalization is chosen such that\n$V = \\Gamma(\\mathbf{P}(V), \\mathcal{O}_{\\mathbf{P}(V)}(1))$.\nOf course we have $\\mathbf{P}(V) \\cong \\mathbf{P}^n_{\\mathbf{C}}$ if\n$\\dim(V) = n + 1$. We note that $\\mathbf{P}(V)$ is a nonsingular projective\nvariety.\n\n\\medskip\\noindent\nLet $p \\in \\mathbf{P}(V)$ be a closed point. The point $p$ corresponds to a\nsurjection $V \\to L_p$ of vector spaces where $\\dim(L_p) = 1$, see\nConstructions, Lemma \\ref{constructions-lemma-apply}.\nLet us denote $W_p = \\Ker(V \\to L_p)$.\n{\\it Projection from $p$} is the morphism\n$$\nr_p : \\mathbf{P}(V) \\setminus \\{p\\} \\longrightarrow \\mathbf{P}(W_p)\n$$\nof Constructions, Lemma \\ref{constructions-lemma-morphism-proj}.\nHere is a lemma to warm up.\n\n\\begin{lemma}\n\\label{lemma-projection-generically-finite}\nLet $V$ be a vector space of dimension $n + 1$.\nLet $X \\subset \\mathbf{P}(V)$ be a closed subscheme.\nIf $X \\not = \\mathbf{P}(V)$, then there is a nonempty Zariski open\n$U \\subset \\mathbf{P}(V)$\nsuch that for all closed points $p \\in U$ the restriction\nof the projection $r_p$ defines a finite morphism\n$r_p|_X : X \\to \\mathbf{P}(W_p)$.\n\\end{lemma}\n\n\\begin{proof}\nWe claim the lemma holds with $U = \\mathbf{P}(V) \\setminus X$. For a closed\npoint $p$ of $U$ we indeed obtain a morphism $r_p|_X : X \\to \\mathbf{P}(W_p)$.\nThis morphism is proper because $X$ is a proper scheme\n(Morphisms, Lemmas \\ref{morphisms-lemma-locally-projective-proper} and\n\\ref{morphisms-lemma-image-proper-scheme-closed}). On the other hand, the\nfibres of $r_p$ are affine lines as can be seen by a direct calculation.\nHence the fibres of $r_p|X$ are proper and affine, whence finite\n(Morphisms, Lemma \\ref{morphisms-lemma-finite-proper}).\nFinally, a proper morphism with finite fibres is finite\n(Cohomology of Schemes, Lemma \\ref{coherent-lemma-characterize-finite}).\n\\end{proof}\n\n\\begin{lemma}\n\\label{lemma-projection-generically-immersion}\nLet $V$ be a vector space of dimension $n + 1$.\nLet $X \\subset \\mathbf{P}(V)$ be a closed subvariety.\nLet $x \\in X$ be a nonsingular point.\n\\begin{enumerate}\n\\item If $\\dim(X) < n - 1$, then there is a nonempty Zariski open\n$U \\subset \\mathbf{P}(V)$ such that for all closed points $p \\in U$ the\nmorphism $r_p|_X : X \\to r_p(X)$ is an\nisomorphism over an open neighbourhood of $r_p(x)$.\n\\item If $\\dim(X) = n - 1$, then there is a nonempty Zariski open\n$U \\subset \\mathbf{P}(V)$ such that for all closed points $p \\in U$ the\nmorphism $r_p|_X : X \\to \\mathbf{P}(W_p)$ is \\'etale at $x$.\n\\end{enumerate}\n\\end{lemma}\n\n\\begin{proof}\nProof of (1). Note that if $x, y \\in X$ have the same image under\n$r_p$ then $p$ is on the line $\\overline{xy}$.\nConsider the finite type scheme\n$$\nT = \\{(y, p) \\mid\ny \\in X \\setminus \\{x\\},\\ p \\in \\mathbf{P}(V),\\ p \\in \\overline{xy}\\}\n$$\nand the morphisms $T \\to X$ and $T \\to \\mathbf{P}(V)$ given by\n$(y, p) \\mapsto y$ and $(y, p) \\mapsto p$.\nSince each fibre of $T \\to X$ is a line, we see that\nthe dimension of $T$ is $\\dim(X) + 1 < \\dim(\\mathbf{P}(V))$.\nHence $T \\to \\mathbf{P}(V)$ is not surjective. On the other hand,\nconsider the finite type scheme\n$$\nT' = \\{p \\mid\np \\in \\mathbf{P}(V) \\setminus \\{x\\},\n\\ \\overline{xp}\\text{ tangent to }X\\text{ at }x\\}\n$$\nThen the dimension of $T'$ is $\\dim(X) < \\dim(\\mathbf{P}(V))$.\nThus the morphism $T' \\to \\mathbf{P}(V)$ is not surjective either.\nLet $U \\subset \\mathbf{P}(V) \\setminus X$ be nonempty open and disjoint\nfrom these images; such a $U$ exists because the images of $T$ and $T'$\nin $\\mathbf{P}(V)$ are constructible by\nMorphisms, Lemma \\ref{morphisms-lemma-chevalley}.\nThen for $p \\in U$ closed the projection\n$r_p|_X : X \\to \\mathbf{P}(W_p)$ is injective on the\ntangent space at $x$ and $r_p^{-1}(\\{r_p(x)\\}) = \\{x\\}$.\nThis means that $r_p$ is unramified at $x$\n(Varieties, Lemma \\ref{varieties-lemma-injective-tangent-spaces-unramified}),\nfinite by Lemma \\ref{lemma-projection-generically-finite},\nand $r_p^{-1}(\\{r_p(x)\\}) = \\{x\\}$ thus \\'Etale Morphisms, Lemma\n\\ref{etale-lemma-finite-unramified-one-point} applies and\nthere is an open neighbourhood $R$ of $r_p(x)$\nin $\\mathbf{P}(W_p)$ such that $(r_p|_X)^{-1}(R) \\to R$ is a\nclosed immersion which proves (1).\n\n\\medskip\\noindent\nProof of (2). In this case we still conclude that the morphism\n$T' \\to \\mathbf{P}(V)$ is not surjective.\nArguing as above we conclude that for $U \\subset \\mathbf{P}(V)$\navoiding $X$ and the image of $T'$, the projection\n$r_p|_X : X \\to \\mathbf{P}(W_p)$ is \\'etale at $x$ and finite.\n\\end{proof}\n\n\\begin{lemma}\n\\label{lemma-projection-injective}\nLet $V$ be a vector space of dimension $n + 1$.\nLet $Y, Z \\subset \\mathbf{P}(V)$ be closed subvarieties.\nThere is a nonempty Zariski open $U \\subset \\mathbf{P}(V)$\nsuch that for all closed points $p \\in U$ we have\n$$\nY \\cap r_p^{-1}(r_p(Z)) = (Y \\cap Z) \\cup E\n$$\nwith $E \\subset Y$ closed and\n$\\dim(E) \\leq \\dim(Y) + \\dim(Z) + 1 - n$.\n\\end{lemma}\n\n\\begin{proof}\nSet $Y' = Y \\setminus Y \\cap Z$.\nLet $y \\in Y'$, $z \\in Z$ be closed points with $r_p(y) = r_p(z)$.\nThen $p$ is on the line $\\overline{yz}$ passing through $y$ and $z$.\nConsider the finite type scheme\n$$\nT = \\{(y, z, p) \\mid y \\in Y', z \\in Z, p \\in \\overline{yz}\\}\n$$\nand the morphism $T \\to \\mathbf{P}(V)$ given by $(y, z, p) \\mapsto p$.\nObserve that $T$ is irreducible and that $\\dim(T) = \\dim(Y) + \\dim(Z) + 1$.\nHence the general fibre of $T \\to \\mathbf{P}(V)$ has dimension at most\n$\\dim(Y) + \\dim(Z) + 1 - n$, more precisely, there exists a nonempty\nopen $U \\subset \\mathbf{P}(V) \\setminus (Y \\cup Z)$ over\nwhich the fibre has dimension at most $\\dim(Y) + \\dim(Z) + 1 - n$\n(Varieties, Lemma \\ref{varieties-lemma-dimension-fibres-locally-algebraic}).\nLet $p \\in U$ be a closed point and let $F \\subset T$ be the fibre\nof $T \\to \\mathbf{P}(V)$ over $p$. Then\n$$\n(Y \\cap r_p^{-1}(r_p(Z))) \\setminus (Y \\cap Z)\n$$\nis the image of $F \\to Y$, $(y, z, p) \\mapsto y$. Again by\nVarieties, Lemma \\ref{varieties-lemma-dimension-fibres-locally-algebraic}\nthe closure of the image of $F \\to Y$ has dimension at most\n$\\dim(Y) + \\dim(Z) + 1 - n$.\n\\end{proof}\n\n\\begin{lemma}\n\\label{lemma-find-lines}\nLet $V$ be a vector space. Let $B \\subset \\mathbf{P}(V)$\nbe a closed subvariety of codimension $\\geq 2$.\nLet $p \\in \\mathbf{P}(V)$ be a closed point, $p \\not \\in B$.\nThen there exists a line $\\ell \\subset \\mathbf{P}(V)$\nwith $\\ell \\cap B = \\emptyset$. Moreover, these lines\nsweep out an open subset of $\\mathbf{P}(V)$.\n\\end{lemma}\n\n\\begin{proof}\nConsider the image of $B$ under the projection\n$r_p : \\mathbf{P}(V) \\to \\mathbf{P}(W_p)$.\nSince $\\dim(W_p) = \\dim(V) - 1$, we see that $r_p(B)$\nhas codimension $\\geq 1$ in $\\mathbf{P}(W_p)$.\nFor any $q \\in \\mathbf{P}(V)$ with $r_p(q) \\not \\in r_p(B)$\nwe see that the line $\\ell = \\overline{pq}$ connecting $p$ and $q$ works.\n\\end{proof}\n\n\\begin{lemma}\n\\label{lemma-doubly-transitive}\nLet $V$ be a vector space. Let $G = \\text{PGL}(V)$.\nThen $G \\times \\mathbf{P}(V) \\to \\mathbf{P}(V)$ is\ndoubly transitive.\n\\end{lemma}\n\n\\begin{proof}\nOmitted. Hint: This follows from the fact that $\\text{GL}(V)$ acts doubly\ntransitive on pairs of linearly independent vectors.\n\\end{proof}\n\n\\begin{lemma}\n\\label{lemma-determinant}\nLet $k$ be a field. Let $n \\geq 1$ be an integer and let\n$x_{ij}, 1 \\leq i, j \\leq n$ be variables. Then\n$$\n\\det\n\\left(\n\\begin{matrix}\nx_{11} & x_{12} & \\ldots & x_{1n} \\\\\nx_{21} & \\ldots & \\ldots & \\ldots \\\\\n\\ldots & \\ldots & \\ldots & \\ldots \\\\\nx_{n1} & \\ldots & \\ldots & x_{nn}\n\\end{matrix}\n\\right)\n$$\nis an irreducible element of the polynomial ring $k[x_{ij}]$.\n\\end{lemma}\n\n\\begin{proof}\nLet $V$ be an $n$ dimensional vector space. Translating into geometry\nthe lemma signifies that the variety $C$ of non-invertible linear maps\n$V \\to V$ is irreducible. Let $W$ be a vector space of dimension $n - 1$.\nBy elementary linear algebra, the morphism\n$$\n\\Hom(W, V) \\times \\Hom(V, W) \\longrightarrow \\Hom(V, V),\\quad\n(\\psi, \\varphi) \\longmapsto \\psi \\circ \\varphi\n$$\nhas image $C$. Since the source is irreducible, so is the image.\n\\end{proof}\n\n\\noindent\nLet $V$ be a vector space of dimension $n + 1$. Set $E = \\text{End}(V)$.\nLet $E^\\vee = \\Hom(E, \\mathbf{C})$ be the dual vector space.\nWrite $\\mathbf{P} = \\mathbf{P}(E^\\vee)$.\nThere is a canonical linear map\n$$\nV \\longrightarrow V \\otimes_\\mathbf{C} E^\\vee = \\Hom(E, V)\n$$\nsending $v \\in V$ to the map $g \\mapsto g(v)$ in $\\Hom(E, V)$.\nRecall that we have a canonical map\n$E^\\vee \\to \\Gamma(\\mathbf{P}, \\mathcal{O}_\\mathbf{P}(1))$\nwhich is an isomorphism. Hence we obtain a canonical map\n$$\n\\psi : V \\otimes \\mathcal{O}_\\mathbf{P} \\to V \\otimes \\mathcal{O}_\\mathbf{P}(1)\n$$\nof sheaves of modules on $\\mathbf{P}$ which on global sections recovers\nthe given map. Recall that a projective bundle $\\mathbf{P}(\\mathcal{E})$\nis defined as the relative Proj of the symmetric algebra on $\\mathcal{E}$, see\nConstructions, Definition \\ref{constructions-definition-projective-bundle}.\nWe are going to study the rational\nmap between $\\mathbf{P}(V \\otimes \\mathcal{O}_\\mathbf{P}(1))$ and\n$\\mathbf{P}(V \\otimes \\mathcal{O}_\\mathbf{P})$ associated to $\\psi$. By\nConstructions, Lemma \\ref{constructions-lemma-relative-proj-base-change}\nwe have a canonical isomorphism\n$$\n\\mathbf{P}(V \\otimes \\mathcal{O}_\\mathbf{P}) = \\mathbf{P} \\times \\mathbf{P}(V)\n$$\nBy Constructions, Lemma \\ref{constructions-lemma-twisting-and-proj}\nwe see that\n$$\n\\mathbf{P}(V \\otimes \\mathcal{O}_\\mathbf{P}(1)) =\n\\mathbf{P}(V \\otimes \\mathcal{O}_\\mathbf{P}) = \\mathbf{P} \\times \\mathbf{P}(V)\n$$\nCombining this with\nConstructions, Lemma \\ref{constructions-lemma-morphism-relative-proj}\nwe obtain\n\\begin{equation}\n\\label{equation-r-psi}\n\\mathbf{P} \\times \\mathbf{P}(V) \\supset\nU(\\psi) \\xrightarrow{r_\\psi} \\mathbf{P} \\times \\mathbf{P}(V)\n\\end{equation}\nTo understand this better we work out what happens on fibres over\n$\\mathbf{P}$. Let $g \\in E$ be nonzero. This defines a nonzero map\n$E^\\vee \\to \\mathbf{C}$, hence a point $[g] \\in \\mathbf{P}$.\nOn the other hand, $g$ defines a $\\mathbf{C}$-linear map $g : V \\to V$.\nHence we obtain, by\nConstructions, Lemma \\ref{constructions-lemma-morphism-proj}\na map\n$$\n\\mathbf{P}(V) \\supset U(g) \\xrightarrow{r_g} \\mathbf{P}(V)\n$$\nWhat we will use below is that $U(g)$ is the fibre $U(\\psi)_{[g]}$ and\nthat $r_g$ is the fibre of $r_\\psi$ over the point $[g]$. Another observation\nwe will use is that the complement of $U(g)$ in $\\mathbf{P}(V)$ is\nthe image of the closed immersion\n$$\n\\mathbf{P}(\\Coker(g)) \\longrightarrow \\mathbf{P}(V)\n$$\nand the image of $r_g$ is the image of the closed immersion\n$$\n\\mathbf{P}(\\Im(g)) \\longrightarrow \\mathbf{P}(V)\n$$\n\n\\begin{lemma}\n\\label{lemma-make-family}\nWith notation as above. Let $X, Y$ be closed subvarieties of $\\mathbf{P}(V)$\nwhich intersect properly such that $X \\not = \\mathbf{P}(V)$. There exists\na line $\\ell \\subset \\mathbf{P}$ such that\n\\begin{enumerate}\n\\item $[\\text{id}_V] \\in \\ell$,\n\\item $X \\subset U_g$ for all $[g] \\in \\ell$,\n\\item $g(X)$ intersects $Y$ properly for all $[g] \\in \\ell$.\n\\end{enumerate}\n\\end{lemma}\n\n\\begin{proof}\nLet $B \\subset \\mathbf{P}$ be the set of ``bad'' points, i.e., those\npoints $[g]$ that violate either (2) or (3). Note that\n$[\\text{id}_V] \\not \\in B$ by assumption. Moreover, $B$ is closed.\nHence it suffices to prove that $\\dim(B) \\leq \\dim(\\mathbf{P}) - 2$\n(Lemma \\ref{lemma-find-lines}).\n\n\\medskip\\noindent\nFirst, consider the open $G = \\text{PGL}(V) \\subset \\mathbf{P}$\nconsisting of points $[g]$ such that $g : V \\to V$ is invertible.\nSince $G$ acts doubly transitively on $\\mathbf{P}(V)$\n(Lemma \\ref{lemma-doubly-transitive})\nwe see that\n$$\nT = \\{(x, y, [g]) \\mid x \\in X, y \\in Y, [g] \\in G, r_g(x) = y\\}\n$$\nis a locally trivial fibration over $X \\times Y$ with fibre equal\nto the stabilizer of a pair of points in $G$. Hence $T$ is a variety.\nObserve that the fibre of $T \\to G$ over $[g]$ is $r_g(X) \\cap Y$.\nThe morphism $T \\to G$ is surjective, because any translate of $X$\nintersects $Y$\n(Varieties, Lemma \\ref{varieties-lemma-intersection-in-projective-space}).\nSince the dimensions of fibres of a dominant morphism of varieties do\nnot jump in codimension $1$\n(Varieties, Lemma \\ref{varieties-lemma-dimension-fibres-locally-algebraic})\nwe conclude that $B \\cap G$ has codimension $\\geq 2$.\n\n\\medskip\\noindent\nNext we look at the complement $Z = \\mathbf{P} \\setminus G$.\nThis is an irreducible variety because the determinant is an\nirreducible polynomial (Lemma \\ref{lemma-determinant}).\nThus it suffices to prove that $B$ does not contain the\ngeneric point of $Z$. For a general point $[g] \\in Z$\nthe cokernel $V \\to \\Coker(g)$ has dimension $1$, hence\n$U(g)$ is the complement of a point. Since $X \\not = \\mathbf{P}(V)$\nwe see that for a general $[g] \\in Z$ we have $X \\subset U(g)$.\nMoreover, the morphism $r_g|_X : X \\to r_g(X)$ is finite, hence\n$\\dim(r_g(X)) = \\dim(X)$.\nOn the other hand, for such a $g$ the image of $r_g$ is the\nclosed subspace $H = \\mathbf{P}(\\Im(g)) \\subset \\mathbf{P}(V)$\nwhich has codimension $1$.\nFor general point of $Z$ we see that $H \\cap Y$ has dimension $1$\nless than $Y$ (compare with\nVarieties, Lemma \\ref{varieties-lemma-exact-sequence-induction}).\nThus we see that we have to show that $r_g(X)$ and $H \\cap Y$\nintersect properly in $H$. For a fixed choice of $H$, we can\nby postcomposing $g$ by an automorphism, move $r_g(X)$ by\nan arbitrary automorphism of $H = \\mathbf{P}(\\Im(g))$.\nThus we can argue as above to conclude that the intersection\nof $H \\cap Y$ with $r_g(X)$ is proper for general $g$ with given\n$H = \\mathbf{P}(\\Im(g))$. Some details omitted.\n\\end{proof}\n\n\n\n\n\n\n\n\\section{Moving Lemma}\n\\label{section-moving-lemma}\n\n\\noindent\nThe moving lemma states that given an $r$-cycle $\\alpha$ and a $s$ cycle\n$\\beta$ there exists $\\alpha'$, $\\alpha' \\sim_{rat} \\alpha$ such that\n$\\alpha'$ and $\\beta$ intersect properly (Lemma \\ref{lemma-moving-move}).\nSee \\cite{Samuel}, \\cite{ChevalleyI}, \\cite{ChevalleyII}.\nThe key to this is Lemma \\ref{lemma-moving}; the reader may find\nthis lemma in the form stated in\n\\cite[Example 11.4.1]{F} and find a proof in \\cite{Roberts}.\n\n\\begin{lemma}\n\\label{lemma-moving}\n\\begin{reference}\nSee \\cite{Roberts}.\n\\end{reference}\nLet $X \\subset \\mathbf{P}^N$ be a nonsingular closed subvariety.\nLet $n = \\dim(X)$ and $0 \\leq d, d' < n$. Let $Z \\subset X$ be a closed\nsubvariety of dimension $d$ and $T_i \\subset X$, $i \\in I$ be a finite\ncollection of closed subvarieties of dimension $d'$. Then there exists\na subvariety $C \\subset \\mathbf{P}^N$ such that $C$ intersects $X$\nproperly and such that\n$$\nC \\cdot X = Z + \\sum\\nolimits_{j \\in J} m_j Z_j\n$$\nwhere $Z_j \\subset X$ are irreducible of dimension $d$, distinct from $Z$, and\n$$\n\\dim(Z_j \\cap T_i) \\leq \\dim(Z \\cap T_i)\n$$\nwith strict inequality if $Z$ does not intersect $T_i$ properly in $X$.\n\\end{lemma}\n\n\\begin{proof}\nWrite $\\mathbf{P}^N = \\mathbf{P}(V_N)$ so $\\dim(V_N) = N + 1$ and set\n$X_N = X$. We are going to choose a sequence of projections from points\n\\begin{align*}\n& r_N : \n\\mathbf{P}(V_N) \\setminus \\{p_N\\} \\to \\mathbf{P}(V_{N - 1}), \\\\\n& r_{N - 1} :\n\\mathbf{P}(V_{N - 1}) \\setminus \\{p_{N - 1}\\} \\to \\mathbf{P}(V_{N - 2}), \\\\\n& \\ldots,\\\\\n& r_{n + 1} :\n\\mathbf{P}(V_{n + 1}) \\setminus \\{p_{n + 1}\\} \\to \\mathbf{P}(V_n)\n\\end{align*}\nas in Section \\ref{section-projection}. At each step we will choose\n$p_N, p_{N - 1}, \\ldots, p_{n + 1}$ in a suitable Zariski open set.\nPick a closed point $x \\in Z \\subset X$. For every $i$ pick\nclosed points $x_{it} \\in T_i \\cap Z$, at least one in each irreducible\ncomponent of $T_i \\cap Z$. Taking the composition we obtain\na morphism\n$$\n\\pi = (r_{n + 1} \\circ \\ldots \\circ r_N)|_X :\nX \\longrightarrow \\mathbf{P}(V_n)\n$$\nwhich has the following properties\n\\begin{enumerate}\n\\item $\\pi$ is finite,\n\\item $\\pi$ is \\'etale at $x$ and all $x_{it}$,\n\\item $\\pi|_Z : Z \\to \\pi(Z)$ is an isomorphism\nover an open neighbourhood of $\\pi(x_{it})$,\n\\item $T_i \\cap \\pi^{-1}(\\pi(Z)) = (T_i \\cap Z) \\cup E_i$ with\n$E_i \\subset T_i$ closed and\n$\\dim(E_i) \\leq d + d' + 1 - (n + 1) = d + d' - n$.\n\\end{enumerate}\nIt follows in a straightforward manner from\nLemmas \\ref{lemma-projection-generically-finite},\n\\ref{lemma-projection-generically-immersion}, and\n\\ref{lemma-projection-injective} and induction that we can do this;\nobserve that the last projection is from $\\mathbf{P}(V_{n + 1})$ and that\n$\\dim(V_{n + 1}) = n + 2$ which explains the inequality in (4).\n\n\\medskip\\noindent\nLet $C \\subset \\mathbf{P}(V_N)$ be the scheme theoretic closure of\n$(r_{n + 1} \\circ \\ldots \\circ r_N)^{-1}(\\pi(Z))$. Because $\\pi$\nis \\'etale at the point $x$ of $Z$, we see that the closed subscheme\n$C \\cap X$ contains $Z$ with multiplicity $1$ (local calculation omitted).\nHence by Lemma \\ref{lemma-transversal-subschemes} we conclude that\n$$\nC \\cdot X = [Z] + \\sum m_j[Z_j]\n$$\nfor some subvarieties $Z_j \\subset X$ of dimension $d$. Note that\n$$\nC \\cap X = \\pi^{-1}(\\pi(Z))\n$$\nset theoretically. Hence\n$T_i \\cap Z_j \\subset T_i \\cap \\pi^{-1}(\\pi(Z)) \\subset T_i \\cap Z \\cup E_i$.\nFor any irreducible component of $T_i \\cap Z$ contained in $E_i$ we\nhave the desired dimension bound. Finally, let $V$ be an irreducible\ncomponent of $T_i \\cap Z_j$ which is contained in $T_i \\cap Z$. To finish\nthe proof it suffices to show that $V$ does not contain any of the\npoints $x_{it}$, because then $\\dim(V) < \\dim(Z \\cap T_i)$.\nTo show this it suffices to show that $x_{it} \\not \\in Z_j$\nfor all $i, t, j$.\n\n\\medskip\\noindent\nSet $Z' = \\pi(Z)$ and $Z'' = \\pi^{-1}(Z')$, scheme theoretically. By\ncondition (3) we can find an open $U \\subset \\mathbf{P}(V_n)$ containing\n$\\pi(x_{it})$ such that $\\pi^{-1}(U) \\cap Z \\to U \\cap Z'$ is an isomorphism.\nIn particular, $Z \\to Z'$ is a local isomorphism at $x_{it}$.\nOn the other hand, $Z'' \\to Z'$ is \\'etale at $x_{it}$ by condition (2).\nHence the closed immersion $Z \\to Z''$ is \\'etale at $x_{it}$\n(Morphisms, Lemma \\ref{morphisms-lemma-etale-permanence}).\nThus $Z = Z''$ in a Zariski neighbourhood of $x_{it}$ which proves\nthe assertion.\n\\end{proof}\n\n\\noindent\nThe actual moving is done using the following lemma.\n\n\\begin{lemma}\n\\label{lemma-move}\nLet $C \\subset \\mathbf{P}^N$ be a closed subvariety.\nLet $X \\subset \\mathbf{P}^N$ be subvariety and let $T_i \\subset X$\nbe a finite collection of closed subvarieties.\nAssume that $C$ and $X$ intersect properly.\nThen there exists a closed subvariety\n$C' \\subset \\mathbf{P}^N \\times \\mathbf{P}^1$ such that\n\\begin{enumerate}\n\\item $C' \\to \\mathbf{P}^1$ is dominant,\n\\item $C'_0 = C$ scheme theoretically,\n\\item $C'$ and $X \\times \\mathbf{P}^1$ intersect properly,\n\\item $C'_\\infty$ properly intersects each of the given $T_i$.\n\\end{enumerate}\n\\end{lemma}\n\n\\begin{proof}\nWrite $\\mathbf{P}^N = \\mathbf{P}(V)$ so $\\dim(V) = N + 1$. Let\n$E = \\text{End}(V)$. Let $E^\\vee = \\Hom(E, \\mathbf{C})$. Set\n$\\mathbf{P} = \\mathbf{P}(E^\\vee)$ as in Lemma \\ref{lemma-make-family}.\nChoose a general line $\\ell \\subset \\mathbf{P}$ passing through $\\text{id}_V$.\nSet $C' \\subset \\ell \\times \\mathbf{P}(V)$ equal to the\nclosed subscheme having fibre $r_g(C)$ over $[g] \\in \\ell$.\nMore precisely, $C'$ is the image of\n$$\n\\ell \\times C \\subset \\mathbf{P} \\times \\mathbf{P}(V)\n$$\nunder the morphism (\\ref{equation-r-psi}). By Lemma \\ref{lemma-make-family}\nthis makes sense, i.e., $\\ell \\times C \\subset U(\\psi)$. The morphism\n$\\ell \\times C \\to C'$ is finite and $C'_{[g]} = r_g(C)$ set theoretically\nfor all $[g] \\in \\ell$. Parts (1) and (2) are clear with\n$0 = [\\text{id}_V] \\in \\ell$. Part (3) follows from the fact\nthat $r_g(C)$ and $X$ intersect properly for all $[g] \\in \\ell$.\nPart (4) follows from the fact that a general point $\\infty = [g] \\in \\ell$\nis a general point of $\\mathbf{P}$ and for such as point\n$r_g(C) \\cap T$ is proper for any closed subvariety $T$ of $\\mathbf{P}(V)$\n(see proof of Lemma \\ref{lemma-make-family}).\nSome details omitted.\n\\end{proof}\n\n\\begin{lemma}\n\\label{lemma-moving-move}\nLet $X$ be a nonsingular projective variety. Let $\\alpha$ be an\n$r$-cycle and $\\beta$ be an $s$-cycle on $X$. Then there exists\nan $r$-cycle $\\alpha'$ such that $\\alpha' \\sim_{rat} \\alpha$ and\nsuch that $\\alpha'$ and $\\beta$ intersect properly.\n\\end{lemma}\n\n\\begin{proof}\nWrite $\\beta = \\sum n_i[T_i]$ for some subvarieties $T_i \\subset X$\nof dimension $s$. By linearity we may assume that $\\alpha = [Z]$ for\nsome irreducible closed subvariety $Z \\subset X$ of dimension $r$.\nWe will prove the lemma by induction on the maximum $e$ of the integers\n$$\n\\dim(Z \\cap T_i)\n$$\nThe base case is $e = r + s - \\dim(X)$. In this case $Z$ intersects\n$\\beta$ properly and the lemma is trivial.\n\n\\medskip\\noindent\nInduction step. Assume that $e > r + s - \\dim(X)$. Choose an embedding\n$X \\subset \\mathbf{P}^N$ and apply Lemma \\ref{lemma-moving} to find a\nclosed subvariety $C \\subset \\mathbf{P}^N$ such that\n$C \\cdot X = [Z] + \\sum m_j[Z_j]$ and such that the induction\nhypothesis applies to each $Z_j$. Next, apply Lemma \\ref{lemma-move}\nto $C$, $X$, $T_i$ to find $C' \\subset \\mathbf{P}^N \\times \\mathbf{P}^1$.\nLet $\\gamma = C' \\cdot X \\times \\mathbf{P}^1$ viewed as a cycle\non $X \\times \\mathbf{P}^1$. By Lemma \\ref{lemma-transfer} we have\n$$\n[Z] + \\sum m_j[Z_j] = \\text{pr}_{X, *}(\\gamma \\cdot X \\times 0)\n$$\nOn the other hand the cycle\n$\\gamma_\\infty = \\text{pr}_{X, *}(\\gamma \\cdot X \\times \\infty)$\nis supported on $C'_\\infty \\cap X$ hence intersects $\\beta$ transversally.\nThus we see that $[Z] \\sim_{rat} - \\sum m_j[Z_j] + \\gamma_\\infty$\nby Lemma \\ref{lemma-rational-equivalence-and-intersection}. Since by\ninduction each $[Z_j]$ is rationally equivalent to a cycle which properly\nintersects $\\beta$ this finishes the proof.\n\\end{proof}\n\n\n\n\\section{Intersection products and rational equivalence}\n\\label{section-intersections-and-rational-equivalence}\n\n\\noindent\nWith definitions as above we show that the intersection product\nis well defined modulo rational equivalence. We first deal with a\nspecial case.\n\n\\begin{lemma}\n\\label{lemma-well-defined-special-case}\nLet $X$ be a nonsingular variety. Let\n$W \\subset X \\times \\mathbf{P}^1$ be an $(s + 1)$-dimensional subvariety\ndominating $\\mathbf{P}^1$. Let $W_a$, resp.\\ $W_b$ be the fibre of\n$W \\to \\mathbf{P}^1$ over $a$, resp.\\ $b$. Let $V$ be a $r$-dimensional\nsubvariety of $X$ such that $V$ intersects both $W_a$ and\n$W_b$ properly. Then $[V] \\cdot [W_a]_r \\sim_{rat} [V] \\cdot [W_b]_r$.\n\\end{lemma}\n\n\\begin{proof}\nWe have $[W_a]_r = \\text{pr}_{X,*}(W \\cdot X \\times a)$ and similarly for\n$[W_b]_r$, see Lemma \\ref{lemma-rational-equivalence-and-intersection}.\nThus we reduce to showing\n$$\nV \\cdot \\text{pr}_{X,*}( W \\cdot X \\times a) \\sim_{rat} V \\cdot\n\\text{pr}_{X,*}( W \\cdot X\\times b).\n$$\nApplying the projection formula\nLemma \\ref{lemma-projection-formula-flat} we get\n$$\nV \\cdot \\text{pr}_{X,*}( W \\cdot X\\times a) =\n\\text{pr}_{X,*}(V \\times \\mathbf{P}^1 \\cdot (W \\cdot X\\times a))\n$$\nand similarly for $b$. Thus we reduce to showing\n$$\n\\text{pr}_{X,*}(V \\times \\mathbf{P}^1 \\cdot (W \\cdot X\\times a))\n\\sim_{rat}\n\\text{pr}_{X,*}(V \\times \\mathbf{P}^1 \\cdot (W \\cdot X\\times b))\n$$\nIf $V \\times \\mathbf{P}^1$ intersects $W$ properly, then\nassociativity for the intersection multiplicities\n(Lemma \\ref{lemma-associative})\ngives $V \\times \\mathbf{P}^1 \\cdot (W \\cdot X\\times a) =\n(V \\times \\mathbf{P}^1 \\cdot W) \\cdot X \\times a$\nand similarly for $b$.  Thus we reduce to showing\n$$\n\\text{pr}_{X,*}((V \\times \\mathbf{P}^1 \\cdot W) \\cdot X\\times a)\n\\sim_{rat}\n\\text{pr}_{X,*}((V \\times \\mathbf{P}^1 \\cdot W) \\cdot X\\times b)\n$$\nwhich is true by Lemma \\ref{lemma-rational-equivalence-and-intersection}.\n\n\\medskip\\noindent\nThe argument above does not quite work. The obstruction is that\nwe do not know that $V \\times \\mathbf{P}^1$ and $W$ intersect properly.\nWe only know that $V$ and $W_a$ and $V$ and $W_b$ intersect properly.\nLet $Z_i$, $i \\in I$ be the irreducible components of\n$V \\times \\mathbf{P}^1 \\cap W$. Then we know that\n$\\dim(Z_i) \\geq r + 1 + s + 1 - n - 1 = r + s + 1 - n$ where $n = \\dim(X)$, see\nLemma \\ref{lemma-intersect-in-smooth}. Since we have assumed\nthat $V$ and $W_a$ intersect properly, we see that\n$\\dim(Z_{i, a}) = r + s - n$ or $Z_{i, a} = \\emptyset$.\nOn the other hand, if $Z_{i, a} \\not = \\emptyset$, then\n$\\dim(Z_{i, a}) \\geq \\dim(Z_i) - 1 = r + s - n$.\nIt follows that $\\dim(Z_i) = r + s + 1 - n$ if $Z_i$ meets $X \\times a$\nand in this case $Z_i \\to \\mathbf{P}^1$ is surjective.\nThus we may write $I = I' \\amalg I''$ where $I'$ is the set of $i \\in I$\nsuch that $Z_i \\to \\mathbf{P}^1$ is surjective and $I''$ is the set of\n$i \\in I$ such that $Z_i$ lies over a closed point $t_i \\in \\mathbf{P}^1$\nwith $t_i \\not = a$ and $t_i \\not = b$. Consider the cycle\n$$\n\\gamma = \\sum\\nolimits_{i \\in I'} e_i [Z_i]\n$$\nwhere we take\n$$\ne_i = \\sum\\nolimits_p (-1)^p\n\\text{length}_{\\mathcal{O}_{X \\times \\mathbf{P}^1, Z_i}}\n\\text{Tor}_p^{\\mathcal{O}_{X \\times \\mathbf{P}^1, Z_i}}(\n\\mathcal{O}_{V \\times \\mathbf{P}^1, Z_i}, \\mathcal{O}_{W, Z_i})\n$$\nWe will show that $\\gamma$ can be used as a replacement for\nthe intersection product of $V \\times \\mathbf{P}^1$ and $W$.\n\n\\medskip\\noindent\nWe will show this using associativity of intersection products in exactly\nthe same way as above. Let $U = \\mathbf{P}^1 \\setminus \\{t_i, i \\in I''\\}$.\nNote that $X \\times a$ and $X \\times b$ are contained in $X \\times U$.\nThe subvarieties\n$$\nV \\times U,\\quad W_U,\\quad X \\times a\\quad\\text{of}\\quad X \\times U\n$$\nintersect transversally pairwise by our choice of $U$ and moreover\n$\\dim(V \\times U \\cap W_U \\cap X \\times a) = \\dim(V \\cap W_a)$ has\nthe expected dimension. Thus we see that\n$$\nV \\times U \\cdot (W_U \\cdot X \\times a) =\n(V \\times U \\cdot W_U) \\cdot X \\times a\n$$\nas cycles on $X \\times U$ by Lemma \\ref{lemma-associative}.\nBy construction $\\gamma$ restricts to the cycle $V \\times U \\cdot W_U$\non $X \\times U$. Trivially,\n$V \\times \\mathbf{P}^1 \\cdot (W \\times X \\times a)$ restricts\nto $V \\times U \\cdot (W_U \\cdot X \\times a)$ on $X \\times U$.\nHence\n$$\nV \\times \\mathbf{P}^1 \\cdot (W \\cdot X \\times a) =\n\\gamma \\cdot X \\times a\n$$\nas cycles on $X \\times \\mathbf{P}^1$ (because both sides\nare contained in $X \\times U$ and are equal after restricting\nto $X \\times U$ by what was said before). Since we have the same for $b$\nwe conclude\n\\begin{align*}\nV \\cdot [W_a]\n& =\n\\text{pr}_{X,*}(V \\times \\mathbf{P}^1 \\cdot (W \\cdot X\\times a)) \\\\\n& =\n\\text{pr}_{X, *}(\\gamma \\cdot X \\times a) \\\\\n& \\sim_{rat} \n\\text{pr}_{X, *}(\\gamma \\cdot X \\times b) \\\\\n& =\n\\text{pr}_{X,*}(V \\times \\mathbf{P}^1 \\cdot (W \\cdot X\\times b)) \\\\\n& =\nV \\cdot [W_b]\n\\end{align*}\nThe first and the last equality by the first paragraph of the proof,\nthe second and penultimate equalities were shown in this paragraph, and\nthe middle equivalence is\nLemma \\ref{lemma-rational-equivalence-and-intersection}.\n\\end{proof}\n\n\\begin{theorem}\n\\label{theorem-well-defined}\nLet $X$ be a nonsingular projective variety. Let $\\alpha$, resp.\\ $\\beta$\nbe an $r$, resp.\\ $s$ cycle on $X$. Assume that $\\alpha$ and $\\beta$\nintersect properly so that $\\alpha \\cdot \\beta$ is defined. Finally,\nassume that $\\alpha \\sim_{rat} 0$. Then $\\alpha \\cdot \\beta \\sim_{rat} 0$.\n\\end{theorem}\n\n\\begin{proof}\nPick a closed immersion $X \\subset \\mathbf{P}^N$.\nBy linearity it suffices to prove the result when $\\beta = [Z]$ for some\n$s$-dimensional closed subvariety $Z \\subset X$ which intersects $\\alpha$\nproperly. The condition $\\alpha \\sim_{rat} 0$ means there\nare finitely many $(r + 1)$-dimensional closed subvarieties\n$W_i \\subset X \\times \\mathbf{P}^1$ such that\n$$\n\\alpha = \\sum [W_{i, a_i}]_r - [W_{i, b_i}]_r\n$$\nfor some pairs of points $a_i, b_i$ of $\\mathbf{P}^1$.\nLet $W_{i, a_i}^t$ and $W_{i, b_i}^t$ be the irreducible components\nof $W_{i, a_i}$ and $W_{i, b_i}$.\nWe will use induction on the maximum $d$ of the integers\n$$\n\\dim(Z \\cap W_{i, a_i}^t),\\quad \\dim(Z \\cap W_{i, b_i}^t)\n$$\nThe main problem in the rest of the proof is that although we know that $Z$\nintersects $\\alpha$ properly, it may not be the case that $Z$ intersects the\n``intermediate'' varieties $W_{i, a_i}^t$\nand $W_{i, b_i}^t$ properly, i.e., it may happen that $d >  r + s - \\dim(X)$.\n\n\\medskip\\noindent\nBase case: $d = r + s - \\dim(X)$. In this case all the intersections of\n$Z$ with the $W_{i, a_i}^t$ and $W_{i, b_i}^t$ are proper and the\ndesired result follows from Lemma \\ref{lemma-well-defined-special-case},\nbecause it applies to show that\n$[Z] \\cdot [W_{i, a_i}]_r \\sim_{rat} [Z] \\cdot [W_{i, b_i}]_r$ for\neach $i$.\n\n\\medskip\\noindent\nInduction step: $d >  r + s - \\dim(X)$. Apply\nLemma \\ref{lemma-moving} to $Z \\subset X$ and\nthe family of subvarieties $\\{W_{i, a_i}^t, W_{i, b_i}^t\\}$. Then we find a\nclosed subvariety $C \\subset \\mathbf{P}^N$ intersecting $X$\nproperly such that\n$$\nC \\cdot X = [Z] + \\sum m_j [Z_j]\n$$\nand such that\n$$\n\\dim(Z_j \\cap W_{i, a_i}^t) \\leq \\dim(Z \\cap W_{i, a_i}^t),\\quad\n\\dim(Z_j \\cap W_{i, b_i}^t) \\leq \\dim(Z \\cap W_{i, b_i}^t)\n$$\nwith strict inequality if the right hand side is $> r + s - \\dim(X)$.\nThis implies two things: (a) the induction hypothesis applies to\neach $Z_j$, and (b) $C \\cdot X$ and $\\alpha$ intersect properly (because\n$\\alpha$ is a linear combination of those $[W_{i, a_i}^t]$ and\n$[W_{i, a_i}^t]$ which intersect $Z$ properly).\nNext, pick $C' \\subset \\mathbf{P}^N \\times \\mathbf{P}^1$\nas in Lemma \\ref{lemma-move} with respect to $C$, $X$, and\n$W_{i, a_i}^t$, $W_{i, b_i}^t$.\nWrite $C' \\cdot X \\times \\mathbf{P}^1 = \\sum n_k [E_k]$ for\nsome subvarieties $E_k \\subset X \\times \\mathbf{P}^1$ of\ndimension $s + 1$. Note that $n_k > 0$ for all $k$ by\nProposition \\ref{proposition-positivity}.\nBy Lemma \\ref{lemma-transfer} we have\n$$\n[Z] + \\sum m_j [Z_j] = \\sum n_k[E_{k, 0}]_s\n$$\nSince $E_{k, 0} \\subset C \\cap X$ we see that $[E_{k, 0}]_s$ and $\\alpha$\nintersect properly. On the other hand, the cycle\n$$\n\\gamma = \\sum n_k[E_{k, \\infty}]_s\n$$\nis supported on $C'_\\infty \\cap X$ and hence\nproperly intersects each $W_{i, a_i}^t$, $W_{i, b_i}^t$.\nThus by the base case and linearity, we see that\n$$\n\\gamma \\cdot \\alpha \\sim_{rat} 0\n$$\nAs we have seen that $E_{k, 0}$ and $E_{k, \\infty}$\nintersect $\\alpha$ properly Lemma \\ref{lemma-well-defined-special-case}\napplied to $E_k \\subset X \\times \\mathbf{P}^1$ and $\\alpha$ gives\n$$\n[E_{k, 0}] \\cdot \\alpha \\sim_{rat} [E_{k, \\infty}] \\cdot \\alpha\n$$\nPutting everything together we have\n\\begin{align*}\n[Z] \\cdot \\alpha\n& =\n(\\sum n_k[E_{k, 0}]_r - \\sum m_j[Z_j]) \\cdot \\alpha \\\\\n& \\sim_{rat}\n\\sum n_k [E_{k, 0}] \\cdot \\alpha \\quad (\\text{by induction hypothesis})\\\\\n& \\sim_{rat}\n\\sum n_k [E_{k, \\infty}] \\cdot \\alpha \\quad (\\text{by the lemma})\\\\\n& =\n\\gamma \\cdot \\alpha \\\\\n& \\sim_{rat}\n0 \\quad (\\text{by base case})\n\\end{align*}\nThis finishes the proof.\n\\end{proof}\n\n\\begin{remark}\n\\label{remark-quasi-projective}\nLemma \\ref{lemma-moving-move} and\nTheorem \\ref{theorem-well-defined}\nalso hold for nonsingular quasi-projective varieties with\nthe same proof. The only change is that one needs to prove the following\nversion of the moving Lemma \\ref{lemma-moving}: Let  $X \\subset \\mathbf{P}^N$\nbe a closed subvariety. Let $n = \\dim(X)$ and $0 \\leq d, d' < n$. Let\n$X^{reg} \\subset X$ be the open subset of nonsingular points. Let\n$Z \\subset X^{reg}$ be a closed subvariety of dimension $d$ and\n$T_i \\subset X^{reg}$, $i \\in I$ be a finite collection of closed subvarieties\nof dimension $d'$. Then there exists a subvariety $C \\subset \\mathbf{P}^N$\nsuch that $C$ intersects $X$ properly and such that\n$$\n(C \\cdot X)|_{X^{reg}} = Z + \\sum\\nolimits_{j \\in J} m_j Z_j\n$$\nwhere $Z_j \\subset X^{reg}$ are irreducible of dimension $d$, distinct\nfrom $Z$, and\n$$\n\\dim(Z_j \\cap T_i) \\leq \\dim(Z \\cap T_i)\n$$\nwith strict inequality if $Z$ does not intersect $T_i$ properly in $X^{reg}$.\n\\end{remark}\n\n\n\n\\section{Chow rings}\n\\label{section-chow-rings}\n\n\\noindent\nLet $X$ be a nonsingular projective variety. We define the intersection\nproduct\n$$\nA_r(X) \\times A_s(X) \\longrightarrow A_{r + s - \\dim(X)}(X),\\quad\n(\\alpha, \\beta) \\longmapsto \\alpha \\cdot \\beta\n$$\nas follows. Let $\\alpha \\in Z_r(X)$ and $\\beta \\in Z_s(X)$.\nIf $\\alpha$ and $\\beta$ intersect properly, we use the\ndefinition given in Section \\ref{section-intersection-product}.\nIf not, then we choose $\\alpha \\sim_{rat} \\alpha'$ as in\nLemma \\ref{lemma-moving-move} and we set\n$$\n\\alpha \\cdot \\beta =\n\\text{class of }\\alpha' \\cdot \\beta \\in A_{r + s - \\dim(X)}(X)\n$$\nThis is well defined and passes through rational equivalence by\nTheorem \\ref{theorem-well-defined}. The intersection product\non $A_*(X)$ is commutative (this is clear), associative\n(Lemma \\ref{lemma-associative}) and has a unit $[X] \\in A_{\\dim(X)}(X)$.\n\n\\medskip\\noindent\nOften it is convenient to use $A^c(X) = A_{\\dim X - c}(X)$ to denote the\ngroup of codimension $c$ cycles modulo rational equivalence.\nThe intersection product defines a product\n$$\nA^k(X) \\times A^l(X) \\longrightarrow A^{k+l}(X)\n$$\nwhich is commutative, associative, and has a unit $1 = [X] \\in A^0(X)$.\n\n\n\\section{Pullback for a general morphism}\n\\label{section-general-pullback}\n\n\\noindent\nLet $f : X \\to Y$ be a morphism of nonsingular projective varieties.\nWe define\n$$\nf^* : A_k(Y) \\to A_{k+\\dim X - \\dim Y}(X)\n$$\nby the rule\n$$\nf^*(\\alpha) = pr_{X, *}(\\Gamma_f \\cdot pr_Y^*(\\alpha))\n$$\nwhere $\\Gamma_f \\subset X\\times Y$ is the graph of $f$. Note that in this\ngenerality, it is defined only on cycle classes and not on cycles. With the\nnotation $A^*$ introduced in Section \\ref{section-chow-rings}\nwe may think of pullback as a map\n$$\nf^* : A^*(Y) \\to A^*(X)\n$$\nin other words, it is a map of graded abelian groups.\n\n\\begin{lemma}\n\\label{lemma-pullback-and-intersection-product}\nLet $f : X \\to Y$ be a morphism of nonsingular projective varieties.\nThe pullback map on chow groups satisfies:\n\\begin{enumerate}\n\\item $f^* : A^*(Y) \\to A^*(X)$ is a ring map,\n\\item $(g \\circ f)^* = f^* \\circ g^*$ for a composable pair $f, g$,\n\\item the projection formula holds: $f_*(\\alpha) \\cdot \\beta =\nf_*( \\alpha \\cdot f^*\\beta)$, and\n\\item if $f$ is flat then it agrees with the previous definition.\n\\end{enumerate}\n\\end{lemma}\n\n\\begin{proof}\nAll of these follow readily from the results above.\n\n\\medskip\\noindent\nFor (1) it suffices to show that\n$\\text{pr}_{X,*}( \\Gamma_f \\cdot \\alpha \\cdot \\beta) =\n\\text{pr}_{X,*}(\\Gamma_f \\cdot \\alpha) \\cdot\n\\text{pr}_{X,*}(\\Gamma_f \\cdot \\beta)$\nfor cycles $\\alpha$, $\\beta$ on $X \\times Y$. If $\\alpha$ is a cycle on\n$X \\times Y$ which intersects $\\Gamma_f$ properly, then it is easy\nto see that\n$$\n\\Gamma_f \\cdot \\alpha =\n\\Gamma_f \\cdot \\text{pr}_X^*(\\text{pr}_{X,*}(\\Gamma_f \\cdot \\alpha))\n$$\nas cycles because $\\Gamma_f$ is a graph. Thus we get the first\nequality in\n\\begin{align*}\n\\text{pr}_{X,*}(\\Gamma_f \\cdot \\alpha \\cdot \\beta)\n& =\n\\text{pr}_{X,*}(\n\\Gamma_f \\cdot\n\\text{pr}_X^*(\\text{pr}_{X,*}(\\Gamma_f \\cdot \\alpha)) \\cdot \\beta) \\\\\n& =\n\\text{pr}_{X,*}(\\text{pr}_X^*(\\text{pr}_{X,*}(\\Gamma_f \\cdot \\alpha))\n\\cdot (\\Gamma_f \\cdot \\beta)) \\\\\n& =\n\\text{pr}_{X,*}(\\Gamma_f \\cdot \\alpha) \\cdot\n\\text{pr}_{X,*}(\\Gamma_f \\cdot \\beta)\n\\end{align*}\nthe last step by the projection formula in the flat case\n(Lemma \\ref{lemma-projection-formula-flat}).\n\n\\medskip\\noindent\nIf $g : Y \\to Z$ then property (2) follows formally from the observation that\n$$\n\\Gamma =\n\\text{pr}_{X \\times Y}^*\\Gamma_f \\cdot\n\\text{pr}_{Y \\times Z}^*\\Gamma_g\n$$\nin $Z_*(X \\times Y \\times Z)$ where $\\Gamma = \\{(x, f(x), g(f(x))\\}$\nand maps isomorphically to $\\Gamma_{g \\circ f}$ in $X \\times Z$.\nThe equality follows from the scheme theoretic equality and\nLemma \\ref{lemma-transversal}.\n\n\\medskip\\noindent\nFor (3) we use the projection formula for flat maps twice\n\\begin{align*}\nf_*(\\alpha \\cdot pr_{X, *}(\\Gamma_f \\cdot pr_Y^*(\\beta)))\n& =\nf_*(pr_{X, *}(pr_X^*\\alpha \\cdot \\Gamma_f \\cdot pr_Y^*(\\beta))) \\\\\n& =\npr_{Y, *}(pr_X^*\\alpha \\cdot \\Gamma_f \\cdot pr_Y^*(\\beta))) \\\\\n& =\npt_{Y, *}(pr_X^*\\alpha \\cdot \\Gamma_f) \\cdot \\beta \\\\\n& =\nf_*(\\alpha) \\cdot \\beta\n\\end{align*}\nwhere in the last equality we use the remark on graphs made above.\nThis proves (3).\n\n\\medskip\\noindent\nProperty (4) rests on identifying the intersection product\n$\\Gamma_f \\cdot pr_Y^*\\alpha$ in the case $f$ is flat. Namely, in this\ncase if $V \\subset Y$ is a closed subvariety, then every generic point\n$\\xi$ of the scheme $f^{-1}(V) \\cong \\Gamma_f \\cap pr_Y^{-1}(V)$\nlies over the generic point of $V$. Hence the local ring of\n$pr_Y^{-1}(V) = X \\times V$ at $\\xi$ is Cohen-Macaulay. Since\n$\\Gamma_f \\subset X \\times Y$ is a regular immersion (as a morphism of\nsmooth projective varieties) we find that\n$$\n\\Gamma_f \\cdot pr_Y^*[V] = [\\Gamma_f \\cap pr_Y^{-1}(V)]_d\n$$\nwith $d$ the dimension of $\\Gamma_f \\cap pr_Y^{-1}(V)$, see\nLemma \\ref{lemma-multiplicity-lci-CM}. Since $\\Gamma_f \\cap pr_Y^{-1}(V)$\nmaps isomorphically to $f^{-1}(V)$ we conclude.\n\\end{proof}\n\n\n\\section{Pullback of cycles}\n\\label{section-pullback-cycles}\n\n\\noindent\nSuppose that $X$ and $Y$ be nonsingular\nprojective varieties, and let $f : X \\to Y$ be a morphism.\nSuppose that $Z \\subset Y$ is a closed subvariety. Let $f^{-1}(Z)$\nbe the scheme theoretic inverse image:\n$$\n\\xymatrix{\nf^{-1}(Z) \\ar[r] \\ar[d] & Z \\ar[d] \\\\\nX \\ar[r] & Y\n}\n$$\nis a fibre product diagram of schemes. In particular $f^{-1}(Z) \\subset X$\nis a closed subscheme of $X$. In this case we always have\n$$\n\\dim f^{-1}(Z) \\geq \\dim Z + \\dim X - \\dim Y.\n$$\nIf equality holds in the formula above, then\n$f^*[Z] = [f^{-1}(Z)]_{\\dim Z + \\dim X - \\dim Y}$\nprovided that the scheme $Z$ is Cohen-Macaulay at the images\nof the generic points of $f^{-1}(Z)$. This follows by identifying\n$f^{-1}(Z)$ with the scheme theoretic intersection of $\\Gamma_f$\nand $X \\times Z$ and using Lemma \\ref{lemma-multiplicity-lci-CM}.\nDetails are similar to the proof of part (4) of\nLemma \\ref{lemma-pullback-and-intersection-product} above.\n\n\n\n\n\n\\input{chapters}\n\n\\bibliography{my}\n\\bibliographystyle{amsalpha}\n\n\\end{document}\n"

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