In [ ]:

    




    

In [1]:

    

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

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


    

In [2]:

    

nlp = spacy.load('en')
#nlp = spacy.load('en', parser=False, entity=False, add_vectors=False)


    

In [3]:

    

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 [4]:

    

def rule_based_annotation(doc):
    annotation = []
    for match in re.finditer('let \$(\S+( \S+){0,3})\$ be an? (\S+)', doc.text, re.IGNORECASE):
        annotation.append((match.span(1)[0],match.span(1)[1], 'VAR'))
        annotation.append((match.span(3)[0],match.span(3)[1], 'TYPE'))    
    return (doc.text, annotation)


    

In [5]:

    

annotated_data=[]

directory = os.fsencode('tex_files/')
for file in os.listdir(directory)[0:3]:
    filename = os.fsdecode(file)
    print("file: ", filename)
    doc = tex2doc(os.path.join(os.fsdecode(directory), filename))
    annotated_data.append(rule_based_annotation(doc))


    

    




file:  pione.tex
file:  intersection.tex
file:  spaces-simplicial.tex

In [6]:

    

print(len(annotated_data))
print(annotated_data[0])


    

    




3
("\\input{preamble}\n\n% OK, start here.\n%\n\\begin{document}\n\n\\title{Fundamental Groups of Schemes}\n\n\n\\maketitle\n\n\\phantomsection\n\\label{section-phantom}\n\n\\tableofcontents\n\n\\section{Introduction}\n\\label{section-introduction}\n\n\\noindent\nIn this chapter we discuss Grothendieck's fundamental group of a scheme\nand applications. A foundational reference is \\cite{SGA1}.\nA nice introduction is \\cite{Lenstra}.\nOther references \\cite{Murre-lectures} and \\cite{Grothendieck-Murre}.\n\n\n\n\n\n\n\n\n\n\n\\section{Schemes \\'etale over a point}\n\\label{section-schemes-etale-point}\n\n\\noindent\nIn this section we describe schemes \\'etale over the spectrum of a field.\nBefore we state the result we introduce the category of $G$-sets for a\ntopological group $G$.\n\n\\begin{definition}\n\\label{definition-G-set-continuous}\nLet $G$ be a topological group.\nA {\\it $G$-set}, sometime called a {\\it discrete $G$-set},\nis a set $X$ endowed with a left action $a : G \\times X \\to X$\nsuch that $a$ is continuous when $X$ is given the discrete topology and\n$G \\times X$ the product topology.\nA {\\it morphism of $G$-sets} $f : X \\to Y$ is simply any $G$-equivariant\nmap from $X$ to $Y$.\nThe category of $G$-sets is denoted {\\it $G\\textit{-Sets}$}.\n\\end{definition}\n\n\\noindent\nThe condition that $a : G \\times X \\to X$ is continuous signifies\nsimply that the stabilizer of any  $x \\in X$ is open in $G$.\nIf $G$ is an abstract group $G$ (i.e., a group but not a topological group)\nthen this agrees with our preceding definition (see for example\nSites, Example \\ref{sites-example-site-on-group})\nprovided we endow $G$ with the discrete topology.\n\n\\medskip\\noindent\nRecall that if $L/K$ is an infinite Galois extension then the\nGalois group $G = \\text{Gal}(L/K)$ comes endowed with a canonical\ntopology, see Fields, Section \\ref{fields-section-infinite-galois}.\n\n\\begin{lemma}\n\\label{lemma-sheaves-point}\nLet $K$ be a field. Let $K^{sep}$ a separable closure of $K$.\nConsider the profinite group $G = \\text{Gal}(K^{sep}/K)$.\nThe functor\n$$\n\\begin{matrix}\n\\text{schemes \\'etale over }K &\n\\longrightarrow &\nG\\textit{-Sets} \\\\\nX/K & \\longmapsto &\n\\Mor_{\\Spec(K)}(\\Spec(K^{sep}), X)\n\\end{matrix}\n$$\nis an equivalence of categories.\n\\end{lemma}\n\n\\begin{proof}\nA scheme $X$ over $K$ is \\'etale over $K$ if and only if\n$X \\cong \\coprod_{i\\in I} \\Spec(K_i)$ with\neach $K_i$ a finite separable extension of $K$\n(Morphisms, Lemma \\ref{morphisms-lemma-etale-over-field}).\nThe functor of the lemma associates to $X$ the $G$-set\n$$\n\\coprod\\nolimits_i \\Hom_K(K_i, K^{sep})\n$$\nwith its natural left $G$-action. Each element has an open stabilizer\nby definition of the topology on $G$. Conversely, any $G$-set $S$\nis a disjoint union of its orbits. Say $S = \\coprod S_i$. Pick $s_i \\in S_i$\nand denote $G_i \\subset G$ its open stabilizer. By Galois theory\n(Fields, Theorem \\ref{fields-theorem-inifinite-galois-theory})\nthe fields $(K^{sep})^{G_i}$ are finite separable field extensions of $K$, and\nhence the scheme\n$$\n\\coprod\\nolimits_i \\Spec((K^{sep})^{G_i})\n$$\nis \\'etale over $K$. This gives an inverse to the functor of the lemma.\nSome details omitted.\n\\end{proof}\n\n\\begin{remark}\n\\label{remark-covering-surjective}\nUnder the correspondence of Lemma \\ref{lemma-sheaves-point},\nthe coverings in the small \\'etale site\n$\\Spec(K)_\\etale$ of $K$ correspond to surjective families of\nmaps in $G\\textit{-Sets}$.\n\\end{remark}\n\n\n\n\n\n\n\n\n\n\\section{Galois categories}\n\\label{section-galois}\n\n\\noindent\nIn this section we discuss some of the material the reader can\nfind in \\cite[Expos\\'e V, Sections 4, 5, and 6]{SGA1}.\n\n\\medskip\\noindent\nLet $F : \\mathcal{C} \\to \\textit{Sets}$ be a functor.\nRecall that by our conventions categories have a set of objects and\nfor any pair of objects a set of morphisms. There is a canonical\ninjective map\n\\begin{equation}\n\\label{equation-embedding-product}\n\\text{Aut}(F)\n\\longrightarrow\n\\prod\\nolimits_{X \\in \\Ob(\\mathcal{C})} \\text{Aut}(F(X))\n\\end{equation}\nFor a set $E$ we endow $\\text{Aut}(E)$ with the compact open topology, see\nTopology, Example \\ref{topology-example-automorphisms-of-a-set}.\nOf course this is the discrete topology when $E$ is finite, which\nis the case of interest in this section\\footnote{When we discuss the\npro-\\'etale fundamental group the general case will be of interest.}.\nWe endow $\\text{Aut}(F)$ with the topology induced from the\nproduct topology on the right hand side of (\\ref{equation-embedding-product}).\nIn particular, the action maps\n$$\n\\text{Aut}(F) \\times F(X) \\longrightarrow F(X)\n$$\nare continuous when $F(X)$ is given the discrete topology because this\nis true for the action maps $\\text{Aut}(E) \\times E \\to E$ for any set $E$.\nThe universal property of our topology on $\\text{Aut}(F)$ is the following:\nsuppose that $G$ is a topological group and $G \\to \\text{Aut}(F)$\nis a group homomorphism such that the induced actions $G \\times F(X) \\to F(X)$\nare continuous for all $X \\in \\Ob(\\mathcal{C})$ where $F(X)$ has\nthe discrete topology. Then $G \\to \\text{Aut}(F)$ is continuous.\n\n\\medskip\\noindent\nThe following lemma tells us that the group of automorphisms of a functor\nto the category of finite sets is automatically a profinite group.\n\n\\begin{lemma}\n\\label{lemma-aut-inverse-limit}\nLet $\\mathcal{C}$ be a category and let $F : \\mathcal{C} \\to \\textit{Sets}$\nbe a functor. The map (\\ref{equation-embedding-product}) identifies\n$\\text{Aut}(F)$ with a closed subgroup of\n$\\prod_{X \\in \\Ob(\\mathcal{C})} \\text{Aut}(F(X))$.\nIn particular, if $F(X)$ is finite for all $X$, then\n$\\text{Aut}(F)$ is a profinite group.\n\\end{lemma}\n\n\\begin{proof}\nLet $\\xi = (\\gamma_X) \\in \\prod \\text{Aut}(F(X))$ be an element not in\n$\\text{Aut}(F)$. Then there exists a morphism $f : X \\to X'$ of $\\mathcal{C}$\nand an element $x \\in F(X)$ such that\n$F(f)(\\gamma_X(x)) \\not = \\gamma_{X'}(F(f)(x))$.\nConsider the open neighbourhood\n$U = \\{\\gamma \\in \\text{Aut}(F(X)) \\mid \\gamma(x) = \\gamma_X(x)\\}$\nof $\\gamma_X$ and the open neighbourhood\n$U' = \\{\\gamma' \\in \\text{Aut}(F(X')) \\mid \\gamma'(F(f)(x)) =\n\\gamma_{X'}(F(f)(x))\\}$.\nThen\n$U \\times U' \\times \\prod_{X'' \\not = X, X'} \\text{Aut}(F(X''))$\nis an open neighbourhood of $\\xi$ not meeting $\\text{Aut}(F)$.\nThe final statement is follows from the fact that\n$\\prod \\text{Aut}(F(X))$ is a profinite space if each $F(X)$ is finite.\n\\end{proof}\n\n\\begin{example}\n\\label{example-galois-category-G-sets}\nLet $G$ be a topological group. An important example will be the\nforgetful functor\n\\begin{equation}\n\\label{equation-forgetful}\n\\textit{Finite-}G\\textit{-Sets} \\longrightarrow \\textit{Sets}\n\\end{equation}\nwhere $\\textit{Finite-}G\\textit{-Sets}$ is the full subcategory of\n$G\\textit{-Sets}$ whose objects are the finite $G$-sets.\nThe category $G\\textit{-Sets}$ of $G$-sets is defined in\nDefinition \\ref{definition-G-set-continuous}.\n\\end{example}\n\n\\noindent\nLet $G$ be a topological group. The {\\it profinite completion} of $G$\nwill be the profinite group\n$$\nG^\\wedge =\n\\lim_{U \\subset G\\text{ open, normal, finite index}} G/U\n$$\nwith its profinite topology. Observe that the limit is cofiltered\nas a finite intersection of open, normal subgroups of finite index\nis another. The universal property of the profinite completion is\nthat any continuous map $G \\to H$ to a profinite group $H$ factors\ncanonically as $G \\to G^\\wedge \\to H$.\n\n\\begin{lemma}\n\\label{lemma-single-out-profinite}\nLet $G$ be a topological group. The automorphism group of the functor\n(\\ref{equation-forgetful}) endowed with its profinite topology from\nLemma \\ref{lemma-aut-inverse-limit} is the profinite completion of $G$.\n\\end{lemma}\n\n\\begin{proof}\nDenote $F_G$ the functor (\\ref{equation-forgetful}). Any morphism\n$X \\to Y$ in $\\textit{Finite-}G\\textit{-Sets}$ commutes with the action\nof $G$. Thus any $g \\in G$ defines an automorphism of $F_G$ and\nwe obtain a canonical homomorphism $G \\to \\text{Aut}(F_G)$ of groups.\nObserve that any finite $G$-set $X$ is a finite disjoint union of\n$G$-sets of the form $G/H_i$ with canonical $G$-action where\n$H_i \\subset G$ is an open subgroup of finite index. Then\n$U_i = \\bigcap gH_ig^{-1}$ is open, normal, and has finite index.\nMoreover $U_i$ acts trivially on $G/H_i$ hence\n$U = \\bigcap U_i$ acts trivially on $F_G(X)$. Hence the action\n$G \\times F_G(X) \\to F_G(X)$ is continuous. By the universal\nproperty of the topology on $\\text{Aut}(F_G)$ the map\n$G \\to \\text{Aut}(F_G)$ is continuous.\nBy Lemma \\ref{lemma-aut-inverse-limit} and the universal property\nof profinite completion there is an induced\ncontinuous group homomorphism\n$$\nG^\\wedge \\longrightarrow \\text{Aut}(F_G)\n$$\nMoreover, since $G/U$ acts faithfully on $G/U$ this map is\ninjective. If the image is dense, then the map is surjective and hence a\nhomeomorphism by Topology, Lemma \\ref{topology-lemma-bijective-map}.\n\n\\medskip\\noindent\nLet $\\gamma \\in \\text{Aut}(F_G)$ and let $X \\in \\Ob(\\mathcal{C})$.\nWe will show there is a $g \\in G$ such that $\\gamma$ and $g$\ninduce the same action on $F_G(X)$. This will finish the proof.\nAs before we see that $X$ is a finite disjoint union of $G/H_i$.\nWith $U_i$ and $U$ as above, the finite $G$-set $Y = G/U$\nsurjects onto $G/H_i$ for all $i$ and hence it suffices to\nfind $g \\in G$ such that $\\gamma$ and $g$ induce the same action\non $F_G(G/U) = G/U$. Let $e \\in G$ be the neutral element and\nsay that $\\gamma(eU) = g_0U$ for some $g_0 \\in G$. For any\n$g_1 \\in G$ the morphism\n$$\nR_{g_1} : G/U \\longrightarrow G/U,\\quad gU \\longmapsto gg_1U\n$$\nof $\\textit{Finite-}G\\textit{-Sets}$ commutes with the action of\n$\\gamma$. Hence\n$$\n\\gamma(g_1U) = \\gamma(R_{g_1}(eU)) = R_{g_1}(\\gamma(eU)) =\nR_{g_1}(g_0U) = g_0g_1U\n$$\nThus we see that $g = g_0$ works.\n\\end{proof}\n\n\\noindent\nRecall that an exact functor is one which commutes with all\nfinite limits and finite colimits. In particular such a functor\ncommutes with equalizers, coequalizers, fibred products,\npushouts, etc.\n\n\\begin{lemma}\n\\label{lemma-second-fundamental-functor}\nLet $G$ be a topological group. Let\n$F : \\textit{Finite-}G\\textit{-Sets} \\to \\textit{Sets}$\nbe an exact functor with $F(X)$ finite for all $X$.\nThen $F$ is isomorphic to the functor (\\ref{equation-forgetful}).\n\\end{lemma}\n\n\\begin{proof}\nLet $X$ be a nonempty object of $\\textit{Finite-}G\\textit{-Sets}$.\nThe diagram\n$$\n\\xymatrix{\nX \\ar[r] \\ar[d] & \\{*\\} \\ar[d] \\\\\n\\{*\\} \\ar[r] & \\{*\\}\n}\n$$\nis cocartesian. Hence we conclude that $F(X)$ is nonempty.\nLet $U \\subset G$ be an open, normal subgroup with finite index.\nObserve that\n$$\nG/U \\times G/U = \\coprod\\nolimits_{gU \\in G/U} G/U\n$$\nwhere the summand corresponding to $gU$ corresponds to the orbit of\n$(eU, gU)$ on the left hand side. Then we see that\n$$\nF(G/U) \\times F(G/U) = F(G/U \\times G/U) = \\coprod\\nolimits_{gU \\in G/U} F(G/U)\n$$\nHence $|F(G/U)| = |G/U|$ as $F(G/U)$ is nonempty. Thus we see that\n$$\n\\lim_{U \\subset G\\text{ open, normal, finite idex}} F(G/U)\n$$\nis nonempty (Categories, Lemma \\ref{categories-lemma-nonempty-limit}).\nPick $\\gamma = (\\gamma_U)$ an element in this limit.\nDenote $F_G$ the functor (\\ref{equation-forgetful}). We can identify\n$F_G$ with the functor\n$$\nX \\longmapsto \\colim_U \\Mor(G/U, X)\n$$\nwhere $f : G/U \\to X$ corresponds to $f(eU) \\in X = F_G(X)$\n(details omitted). Hence the element $\\gamma$ determines\na well defined map\n$$\nt : F_G \\longrightarrow F\n$$\nNamely, given $x \\in X$ choose $U$ and $f : G/U \\to X$ sending\n$eU$ to $x$ and then set $t_X(x) = F(f)(\\gamma_U)$.\nWe will show that $t$ induces a bijective map\n$t_{G/U} : F_G(G/U) \\to F(G/U)$ for any $U$.\nThis implies in a straightforward manner that $t$\nis an isomorphism (details omitted).\nSince $|F_G(G/U)| = |F(G/U)|$ it suffices to show\nthat $t_{G/U}$ is surjective. The image contains at least\none element, namely\n$t_{G/U}(eU) = F(\\text{id}_{G/U})(\\gamma_U) = \\gamma_U$.\nFor $g \\in G$ denote $R_g : G/U \\to G/U$ right multiplication.\nThen set of fixed points of $F(R_g) : F(G/U) \\to F(G/U)$\nis equal to $F(\\emptyset) = \\emptyset$ if $g \\not \\in U$ because $F$\ncommutes with equalizers. It follows that if\n$g_1, \\ldots, g_{|G/U|}$ is a system of representatives\nfor $G/U$, then the elements $F(R_{g_i})(\\gamma_U)$ are pairwise distinct\nand hence fill out $F(G/U)$. Then\n$$\nt_{G/U}(g_iU) = F(R_{g_i})(\\gamma_U)\n$$\nand the proof is complete.\n\\end{proof}\n\n\\begin{example}\n\\label{example-from-C-F-to-G-sets}\nLet $\\mathcal{C}$ be a category and let $F : \\mathcal{C} \\to \\textit{Sets}$\nbe a functor such that $F(X)$ is finite for all $X \\in \\Ob(\\mathcal{C})$.\nBy Lemma \\ref{lemma-aut-inverse-limit} we see that $G = \\text{Aut}(F)$\ncomes endowed with the structure of a profinite topological group in a\ncanonical manner. We obtain a functor\n\\begin{equation}\n\\label{equation-remember}\n\\mathcal{C} \\longrightarrow \\textit{Finite-}G\\textit{-Sets},\\quad\nX \\longmapsto F(X)\n\\end{equation}\nwhere $F(X)$ is endowed with the induced action of $G$. This action\nis continuous by our construction of the topology on $\\text{Aut}(F)$.\n\\end{example}\n\n\\noindent\nThe purpose of defining Galois categories is to single out those\npairs $(\\mathcal{C}, F)$ for which the functor (\\ref{equation-remember})\nis an equivalence. Our definition of a Galois category is as follows.\n\n\\begin{definition}\n\\label{definition-galois-category}\n\\begin{reference}\nDifferent from the definition in \\cite[Expos\\'e V, Definition 5.1]{SGA1}.\nCompare with \\cite[Definition 7.2.1]{BS}.\n\\end{reference}\nLet $\\mathcal{C}$ be a category and let $F : \\mathcal{C} \\to \\textit{Sets}$\nbe a functor. The pair $(\\mathcal{C}, F)$ is a {\\it Galois category} if\n\\begin{enumerate}\n\\item $\\mathcal{C}$ has finite limits and finite colimits,\n\\item\n\\label{item-connected-components}\nevery object of $\\mathcal{C}$ is a finite (possibly empty)\ncoproduct of connected objects,\n\\item $F(X)$ is finite for all $X \\in \\Ob(\\mathcal{C})$, and\n\\item $F$ reflects isomorphisms and is exact.\n\\end{enumerate}\nHere we say $X \\in \\Ob(\\mathcal{C})$ is connected if\nit is not initial and for any monomorphism $Y \\to X$\neither $Y$ is initial or $Y \\to X$ is an isomorphism.\n\\end{definition}\n\n\\noindent\n{\\bf Warning:} This definition is not the same (although eventually we'll\nsee it is equivalent) as the definition given in most references.\nNamely, in \\cite[Expos\\'e V, Definition 5.1]{SGA1} a Galois category is\ndefined to be a category equivalent to $\\textit{Finite-}G\\textit{-Sets}$\nfor some profinite group $G$. Then Grothendieck characterizes\nGalois categories by a list of axioms (G1) -- (G6) which are weaker\nthan our axioms above. The motivation for our choice is to stress the\nexistence of finite limits and finite colimits and exactness of the\nfunctor $F$. The price we'll pay for this later is that we'll have\nto work a bit harder to apply the results of this section.\n\n\\begin{lemma}\n\\label{lemma-epi-mono}\nLet $(\\mathcal{C}, F)$ be a Galois category. Let\n$X \\to Y \\in \\text{Arrows}(\\mathcal{C})$. Then\n\\begin{enumerate}\n\\item $F$ is faithful,\n\\item $X \\to Y$ is a monomorphism\n$\\Leftrightarrow F(X) \\to F(Y)$ is injective,\n\\item $X \\to Y$ is an epimorphism\n$\\Leftrightarrow F(X) \\to F(Y)$ is surjective,\n\\item an object $A$ of $\\mathcal{C}$ is initial if and only if\n$F(A) = \\emptyset$,\n\\item an object $Z$ of $\\mathcal{C}$ is final if and only if\n$F(Z)$ is a singleton,\n\\item if $X$ and $Y$ are connected, then $X \\to Y$ is an epimorphism,\n\\item\n\\label{item-one-element}\nif $X$ is connected and $a, b : X \\to Y$ are two morphisms\nthen $a = b$ as soon as $F(a)$ and $F(b)$ agree on one element of $F(X)$,\n\\item if $X = \\coprod_{i = 1, \\ldots, n} X_i$ and\n$Y = \\coprod_{j = 1, \\ldots, m} Y_j$ where $X_i$, $Y_j$ are connected,\nthen there is map $\\alpha : \\{1, \\ldots, n\\} \\to \\{1, \\ldots, m\\}$\nsuch that $X \\to Y$ comes from a collection of morphisms\n$X_i \\to Y_{\\alpha(i)}$.\n\\end{enumerate}\n\\end{lemma}\n\n\\begin{proof}\nProof of (1). Suppose $a, b : X \\to Y$ with $F(a) = F(b)$.\nLet $E$ be the equalizer of $a$ and $b$. Then $F(E) = F(X)$\nand we see that $E = X$ because $F$ reflects isomorphisms.\n\n\\medskip\\noindent\nProof of (2). This is true because $F$ turns the morphism $X \\to X \\times_Y X$\ninto the map $F(X) \\to F(X) \\times_{F(Y)} F(X)$ and $F$ reflects isomorphisms.\n\n\\medskip\\noindent\nProof of (3). This is true because $F$ turns the morphism $Y \\amalg_X Y \\to Y$\ninto the map $F(Y) \\amalg_{F(X)} F(Y) \\to F(Y)$ and $F$ reflects isomorphisms.\n\n\\medskip\\noindent\nProof of (4). There exists an initial object $A$ and certainly\n$F(A) = \\emptyset$. On the other hand, if $X$ is an object with\n$F(X) = \\emptyset$, then the unique map $A \\to X$ induces a bijection\n$F(A) \\to F(X)$ and hence $A \\to X$ is an isomorphism.\n\n\\medskip\\noindent\nProof of (5). There exists a final object $Z$ and certainly\n$F(Z)$ is a singleton. On the other hand, if $X$ is an object with\n$F(X)$ a singleton, then the unique map $X \\to Z$ induces a bijection\n$F(X) \\to F(Z)$ and hence $X \\to Z$ is an isomorphism.\n\n\\medskip\\noindent\nProof of (6). The equalizer $E$ of the two maps $Y \\to Y \\amalg_X Y$ is not\nan initial object of $\\mathcal{C}$ because $X \\to Y$ factors through $E$\nand $F(X) \\not = \\emptyset$. Hence $E = Y$ and we conclude.\n\n\\medskip\\noindent\nProof of (\\ref{item-one-element}).\nThe equalizer $E$ of $a$ and $b$ comes with a monomorphism\n$E \\to X$ and $F(E) \\subset F(X)$ is the set of elements where\n$F(a)$ and $F(b)$ agree. To finish use that either $E$ is initial\nor $E = X$.\n\n\\medskip\\noindent\nProof of (8). For each $i, j$ we see that $E_{ij} = X_i \\times_Y Y_j$\nis either initial or equal to $X_i$. Picking $s \\in F(X_i)$\nwe see that $E_{ij} = X_i$ if and only if $s$ maps to an element\nof $F(Y_j) \\subset F(Y)$, hence this happens for a unique $j = \\alpha(i)$.\n\\end{proof}\n\n\\noindent\nBy the lemma above we see that, given a connected object $X$ of a\nGalois category $(\\mathcal{C}, F)$, the automorphism group\n$\\text{Aut}(X)$ has order at most $|F(X)|$. Namely, given $s \\in F(X)$\nand $g \\in \\text{Aut}(X)$ we see that $g(s) = s$ if and only\nif $g = \\text{id}_X$ by (\\ref{item-one-element}).\nWe say $X$ is {\\it Galois} if equality holds.\nEquivalently, $X$ is Galois if it is connected and\n$\\text{Aut}(X)$ acts transitively on $F(X)$.\n\n\\begin{lemma}\n\\label{lemma-galois}\nLet $(\\mathcal{C}, F)$ be a Galois category. For any connected object $X$\nof $\\mathcal{C}$ there exists a Galois object $Y$ and a morphism $Y \\to X$.\n\\end{lemma}\n\n\\begin{proof}\nWe will use the results of Lemma \\ref{lemma-epi-mono} without further mention.\nLet $n = |F(X)|$. Consider $X^n$ endowed with its natural action of\n$S_n$. Let\n$$\nX^n = \\coprod\\nolimits_{t \\in T} Z_t\n$$\nbe the decomposition into connected objects. Pick a $t$ such that\n$F(Z_t)$ contains $(s_1, \\ldots, s_n)$ with $s_i$ pairwise distinct.\nIf $(s'_1, \\ldots, s'_n) \\in F(Z_t)$ is another element, then we\nclaim $s'_i$ are pairwise distinct as well. Namely, if not, say\n$s'_i = s'_j$, then $Z_t$ is the image of an connected component of\n$X^{n - 1}$ under the diagonal morphism\n$$\n\\Delta_{ij} : X^{n - 1} \\longrightarrow X^n\n$$\nSince morphisms of connected objects are epimorphisms and induce\nsurjections after applying $F$ it would follow that $s_i = s_j$\nwhich is not the case.\n\n\\medskip\\noindent\nLet $G \\subset S_n$ be the subgroup of elements with $g(Z_t) = Z_t$.\nLooking at the action of $S_n$ on\n$$\nF(X)^n = F(X^n) = \\coprod\\nolimits_{t' \\in T} F(Z_{t'})\n$$\nwe see that $G = \\{g \\in S_n \\mid g(s_1, \\ldots, s_n) \\in F(Z_t)\\}$.\nNow pick a second element $(s'_1, \\ldots, s'_n) \\in F(Z_t)$.\nAbove we have seen that $s'_i$ are pairwise distinct. Thus we can\nfind a $g \\in S_n$ with $g(s_1, \\ldots, s_n) = (s'_1, \\ldots, s'_n)$.\nIn other words, the action of $G$ on $F(Z_t)$ is transitive and\nthe proof is complete.\n\\end{proof}\n\n\\noindent\nHere is a key lemma.\n\n\\begin{lemma}\n\\label{lemma-tame}\n\\begin{reference}\nCompare with \\cite[Definition 7.2.4]{BS}.\n\\end{reference}\nLet $(\\mathcal{C}, F)$ be a Galois category. Let $G = \\text{Aut}(F)$\nbe as in Example \\ref{example-from-C-F-to-G-sets}. For any connected\n$X$ in $\\mathcal{C}$ the action of $G$ on $F(X)$ is transitive.\n\\end{lemma}\n\n\\begin{proof}\nWe will use the results of Lemma \\ref{lemma-epi-mono} without further mention.\nLet $I$ be the set of isomorphism classes of Galois objects in $\\mathcal{C}$.\nFor each $i \\in I$ let $X_i$ be a representative of the isomorphism class.\nChoose $\\gamma_i \\in F(X_i)$ for each $i \\in I$.\nWe define a partial ordering on $I$ by setting $i \\geq i'$ if\nand only if there is a morphism $f_{ii'} : X_i \\to X_{i'}$.\nGiven such a morphism we can post-compose by an automorphism\n$X_{i'} \\to X_{i'}$ to assure that $F(f_{ii'})(\\gamma_i) = \\gamma_{i'}$.\nWith this normalization the morphism $f_{ii'}$ is unique.\nObserve that $I$ is a directed partially ordered set:\n(Categories, Definition \\ref{categories-definition-directed-set})\nif $i_1, i_2 \\in I$ there exists a Galois object $Y$ and a morphism\n$Y \\to X_{i_1} \\times X_{i_2}$ by Lemma \\ref{lemma-galois} applied\nto a connected component of $X_{i_1} \\times X_{i_2}$.\nThen $Y \\cong X_i$ for some $i \\in I$ and $i \\geq i_1$, $i \\geq I_2$.\n\n\\medskip\\noindent\nWe claim that the functor $F$ is isomorphic to the functor $F'$\nwhich sends $X$ to\n$$\nF'(X) = \\colim_I \\Mor_\\mathcal{C}(X_i, X)\n$$\nvia the transformation of functors $t : F' \\to F$ defined as follows:\ngiven $f : X_i \\to X$ we set $t_X(f) = F(f)(\\gamma_i)$.\nUsing (\\ref{item-one-element}) we find that $t_X$ is injective.\nTo show surjectivity, let $\\gamma \\in F(X)$. Then we can immediately\nreduce to the case where $X$ is connected by the definition of\na Galois category. Then we may assume $X$ is Galois by\nLemma \\ref{lemma-galois}. In this case $X$ is isomorphic to $X_i$\nfor some $i$ and we can choose the isomorphism $X_i \\to X$ such\nthat $\\gamma_i$ maps to $\\gamma$ (by definition of Galois objects).\nWe conclude that $t$ is an isomorphism.\n\n\\medskip\\noindent\nSet $A_i = \\text{Aut}(X_i)$.\nWe claim that for $i \\geq i'$ there is a canonical map\n$h_{ii'} : A_i \\to A_{i'}$ such that for all $a \\in A_i$\nthe diagram\n$$\n\\xymatrix{\nX_i \\ar[d]_a \\ar[r]_{f_{ii'}} & X_{i'} \\ar[d]^{h_{ii'}(a)} \\\\\nX_i \\ar[r]^{f_{ii'}} & X_{i'}\n}\n$$\ncommutes. Namely, just let $h_{ii'}(a) = a' : X_{i'} \\to X_{i'}$\nbe the unique automorphism such that\n$F(a')(\\gamma_{i'}) = F(f_{ii'} \\circ a)(\\gamma_i)$.\nAs before this makes the diagram commute and moreover the choice\nis unique.\nIt follows that\n$h_{i'i''} \\circ h_{ii'} = h_{ii''}$\nif $i \\geq i' \\geq i''$.\nSince $F(X_i) \\to F(X_{i'})$ is surjective we see that\n$A_i \\to A_{i'}$ is surjective.\nTaking the inverse limit we obtain a group\n$$\nA = \\lim_I A_i\n$$\nThis is a profinite group since the automorphism groups are finite.\nThe map $A \\to A_i$ is surjective for all $i$ by\nCategories, Lemma \\ref{categories-lemma-nonempty-limit}.\n\n\\medskip\\noindent\nSince elements of $A$ act on the inverse system $X_i$ we get an action of\n$A$ (on the right) on $F'$ by pre-composing. In other words, we get\na homomorphism $A^{opp} \\to G$. Since $A \\to A_i$ is surjective we conclude\nthat $G$ acts transitively on $F(X_i)$ for all $i$. Since every connected\nobject is dominated by one of the $X_i$ we conclude the lemma is true.\n\\end{proof}\n\n\\begin{proposition}\n\\label{proposition-galois}\n\\begin{reference}\nThis is a weak version of \\cite[Expos\\'e V]{SGA1}.\nThe proof is borrowed from \\cite[Theorem 7.2.5]{BS}.\n\\end{reference}\nLet $(\\mathcal{C}, F)$ be a Galois category. Let $G = \\text{Aut}(F)$\nbe as in Example \\ref{example-from-C-F-to-G-sets}. The functor\n$F : \\mathcal{C} \\to \\textit{Finite-}G\\textit{-Sets}$\n(\\ref{equation-remember}) an equivalence.\n\\end{proposition}\n\n\\begin{proof}\nWe will use the results of Lemma \\ref{lemma-epi-mono} without further mention.\nIn particular we know the functor is faithful.\nBy Lemma \\ref{lemma-tame} we know that for any connected $X$ the\naction of $G$ on $F(X)$ is transitive. Hence $F$ preserves\nthe decomposition into connected components (existence of which is\nan axiom of a Galois category). Let $X$ and $Y$ be objects and let\n$s : F(X) \\to F(Y)$ be a map. Then the graph\n$\\Gamma_s \\subset F(X) \\times F(Y)$ of $s$\nis a union of connected components. Hence there exists a\nunion of connected components $Z$ of $X \\times Y$,\nwhich comes equipped with a monomorphism $Z \\to X \\times Y$,\nwith $F(Z) = \\Gamma_s$. Since $F(Z) \\to F(X)$ is bijective\nwe see that $Z \\to X$ is an isomorphism and we conclude\nthat $s = F(f)$ where $f : X \\cong Z \\to Y$ is the composition.\nHence $F$ is fully faithful.\n\n\\medskip\\noindent\nTo finish the proof we show that $F$ is essentially surjective.\nIt suffices to show that $G/H$ is in the essential image for\nany open subgroup $H \\subset G$ of finite index.\nBy definition of the topology on $G$ there exists a finite\ncollection of objects $X_i$ such that\n$$\n\\Ker(G \\longrightarrow \\prod\\nolimits_i \\text{Aut}(F(X_i)))\n$$\nis contained in $H$. We may assume $X_i$ is connected\nfor all $i$. We can choose a Galois object $Y$ mapping\nto a connected component of $\\prod X_i$ using\nLemma \\ref{lemma-galois}. Choose an isomorphism $F(Y) = G/U$\nin $G\\textit{-sets}$ for some open subgroup $U \\subset G$.\nAs $Y$ is Galois, the group\n$\\text{Aut}(Y) = \\text{Aut}_{G\\textit{-Sets}}(G/U)$ acts transitively\non $F(Y) = G/U$. This implies that $U$ is normal. Since\n$F(Y)$ surjects onto $F(X_i)$ for each $i$ we see that\n$U \\subset H$. Let $M \\subset \\text{Aut}(Y)$ be the finite subgroup\ncorresponding to\n$$\n(H/U)^{opp} \\subset (G/U)^{opp} = \\text{Aut}_{G\\textit{-Sets}}(G/U)\n= \\text{Aut}(Y).\n$$\nSet $X = Y/M$, i.e., $X$ is the coequalizer\nof the arrows $m : Y \\to Y$, $m \\in M$.\nSince $F$ is exact we see that $F(X) = G/H$ and the\nproof is complete.\n\\end{proof}\n\n\\begin{lemma}\n\\label{lemma-functoriality-galois}\nLet $(\\mathcal{C}, F)$ and $(\\mathcal{C}', F')$ be Galois categories.\nLet $H : \\mathcal{C} \\to \\mathcal{C}'$ be an exact functor.\nThere exists an isomorphism $t : F' \\circ H \\to F$.\nThe choice of $t$ determines a continuous homomorphism\n$h : G' = \\text{Aut}(F') \\to \\text{Aut}(F) = G$ and\na $2$-commutative diagram\n$$\n\\xymatrix{\n\\mathcal{C} \\ar[r]_H \\ar[d] & \\mathcal{C}' \\ar[d] \\\\\n\\textit{Finite-}G\\textit{-Sets} \\ar[r]^h &\n\\textit{Finite-}G'\\textit{-Sets}\n}\n$$\nThe map $h$ is independent of $t$ up\nto an inner automorphism of $G$.\nConversely, given a continuous homomorphism $h : G' \\to G$ there\nis an exact functor $H : \\mathcal{C} \\to \\mathcal{C}'$ and an\nisomorphism $t$ recovering $h$ as above.\n\\end{lemma}\n\n\\begin{proof}\nBy Proposition \\ref{proposition-galois} and\nLemma \\ref{lemma-single-out-profinite} we may assume\n$\\mathcal{C} = \\textit{Finite-}G\\textit{-Sets}$ and $F$ is the\nforgetful functor and similarly for $\\mathcal{C}'$. Thus the existence of\n$t$ follows from Lemma \\ref{lemma-second-fundamental-functor}. The map $h$\ncomes from transport of structure via $t$. The commutativity of the\ndiagram is obvious. Uniqueness of $h$ up to inner conjugation by\nan element of $G$ comes from the fact that the choice of $t$ is\nunique up to an element of $G$. The final statement is straightforward.\n\\end{proof}\n\n\n\n\n\n\\section{Functors and homomorphisms}\n\\label{section-translation}\n\n\\noindent\nLet $(\\mathcal{C}, F)$, $(\\mathcal{C}', F')$, $(\\mathcal{C}'', F'')$\nbe Galois categories. Set $G = \\text{Aut}(F)$, $G' = \\text{Aut}(F')$, and\n$G'' = \\text{Aut}(F'')$. Let $H : \\mathcal{C} \\to \\mathcal{C}'$\nand $H' : \\mathcal{C}' \\to \\mathcal{C}''$ be exact functors.\nLet $h : G' \\to G$ and $h' : G'' \\to G'$ be the corresponding\ncontinuous homomorphism as in Lemma \\ref{lemma-functoriality-galois}.\nIn this section we consider the corresponding $2$-commutative diagram\n\\begin{equation}\n\\label{equation-translation}\n\\vcenter{\n\\xymatrix{\n\\mathcal{C} \\ar[r]_H \\ar[d] &\n\\mathcal{C}' \\ar[r]_{H'} \\ar[d] &\n\\mathcal{C}'' \\ar[d] \\\\\n\\textit{Finite-}G\\textit{-Sets} \\ar[r]^h &\n\\textit{Finite-}G'\\textit{-Sets} \\ar[r]^{h'} &\n\\textit{Finite-}G''\\textit{-Sets}\n}\n}\n\\end{equation}\nand we relate exactness properties of the sequence\n$1 \\to G'' \\to G' \\to G \\to 1$ to properties of the functors $H$ and $H'$.\n\n\\begin{lemma}\n\\label{lemma-functoriality-galois-surjective}\nIn diagram (\\ref{equation-translation}) the following are equivalent\n\\begin{enumerate}\n\\item $h : G' \\to G$ is surjective,\n\\item $H : \\mathcal{C} \\to \\mathcal{C}'$ is fully faithful,\n\\item if $X \\in \\Ob(\\mathcal{C})$ is connected, then $H(X)$ is connected,\n\\item if $X \\in \\Ob(\\mathcal{C})$ is connected and there is\na morphism $*' \\to H(X)$ in $\\mathcal{C}'$, then\nthere is a morphism $* \\to X$, and\n\\item for any object $X$ of $\\mathcal{C}$ the map\n$\\Mor_\\mathcal{C}(*, X) \\to \\Mor_{\\mathcal{C}'}(*', H(X))$\nis bijective.\n\\end{enumerate}\nHere $*$ and $*'$ are final objects of $\\mathcal{C}$ and $\\mathcal{C}'$.\n\\end{lemma}\n\n\\begin{proof}\nThe implications (5) $\\Rightarrow$ (4) and (2) $\\Rightarrow$ (5) are clear.\n\n\\medskip\\noindent\nAssume (3). Let $X$ be a connected object of $\\mathcal{C}$ and let\n$*' \\to H(X)$ be a morphism. Since $H(X)$ is connected by (3)\nwe see that $*' \\to H(X)$ is an isomorphism. Hence the $G'$-set\ncorresponding to $H(X)$ has exactly one element, which means the\n$G$-set corresponding to $X$ has one element which means $X$ is\nisomorphic to the final object of $\\mathcal{C}$, in particular\nthere is a map $* \\to X$. In this way we see that (3) $\\Rightarrow$ (4).\n\n\\medskip\\noindent\nIf (1) is true, then the functor\n$\\textit{Finite-}G\\textit{-Sets} \\to \\textit{Finite-}G'\\textit{-Sets}$\nis fully faithful: in this case a map of $G$-sets commutes with the\naction of $G$ if and only if it commutes with the action of $G'$.\nThus (1) $\\Rightarrow$ (2).\n\n\\medskip\\noindent\nIf (1) is true, then for a $G$-set $X$ the $G$-orbits and $G'$-orbits\nagree. Thus (1) $\\Rightarrow$ (3).\n\n\\medskip\\noindent\nTo finish the proof it suffices to show that (4) implies (1).\nIf (1) is false, i.e., if $h$ is not surjective, then there is\nan open subgroup $U \\subset G$ containing $h(G')$ which is not\nequal to $G$. Then the finite $G$-set $M = G/U$ has a transitive\naction but $G'$ has a fixed point. The object $X$ of $\\mathcal{C}$\ncorresponding to $M$ would contradict (3). In this way we see that\n(3) $\\Rightarrow$ (1) and the proof is complete.\n\\end{proof}\n\n\\begin{lemma}\n\\label{lemma-composition-trivial}\nIn diagram (\\ref{equation-translation}) the following are equivalent\n\\begin{enumerate}\n\\item $h \\circ h'$ is trivial, and\n\\item the image of $H' \\circ H$ consists of objects isomorphic to finite\ncoproducts of final objects.\n\\end{enumerate}\n\\end{lemma}\n\n\\begin{proof}\nWe may replace $H$ and $H'$ by the canonical functors\n$\\textit{Finite-}G\\textit{-Sets} \\to \\textit{Finite-}G'\\textit{-Sets}\n\\to \\textit{Finite-}G''\\textit{-Sets}$ determined by $h$ and $h'$.\nThen we are saying that the action of $G''$ on every $G$-set is trivial\nif and only if the homomorphism $G'' \\to G$ is trivial. This is clear.\n\\end{proof}\n\n\\begin{lemma}\n\\label{lemma-functoriality-galois-ses}\nIn diagram (\\ref{equation-translation}) the following are equivalent\n\\begin{enumerate}\n\\item the sequence $G'' \\xrightarrow{h'} G' \\xrightarrow{h} G \\to 1$\nis exact in the following sense: $h$ is surjective, $h \\circ h'$ is trivial,\nand $\\Ker(h)$ is the smallest closed normal subgroup containing $\\Im(h')$,\n\\item $H$ is fully faithful and an object $X'$ of $\\mathcal{C}'$ is in\nthe essential image of $H$ if and only if $H'(X')$ is isomorphic to a\nfinite coproduct of final objects, and\n\\item $H$ is fully faithful, $H \\circ H'$ sends every object to a finite\ncoproduct of final objects, and for an object $X'$ of $\\mathcal{C}'$\nsuch that $H'(X')$ is a finite coproduct of final objects there exists\nan object $X$ of $\\mathcal{C}$ and an epimorphism $H(X) \\to X'$.\n\\end{enumerate}\n\\end{lemma}\n\n\\begin{proof}\nBy Lemmas \\ref{lemma-functoriality-galois-surjective} and\n\\ref{lemma-composition-trivial} we may assume that\n$H$ is fully faithful, $h$ is surjective, $H' \\circ H$ maps\nobjects to disjoint unions of the final object, and $h \\circ h'$\nis trivial. Let $N \\subset G'$ be the smallest closed normal\nsubgroup containing the image of $h'$. It is clear that\n$N \\subset \\Ker(h)$.\nWe may assume the functors $H$ and $H'$ are the canonical functors\n$\\textit{Finite-}G\\textit{-Sets} \\to \\textit{Finite-}G'\\textit{-Sets}\n\\to \\textit{Finite-}G''\\textit{-Sets}$ determined by $h$ and $h'$.\n\n\\medskip\\noindent\nSuppose that (2) holds. This means that for a finite $G'$-set $X'$\nsuch that $G''$ acts trivially, the action of $G'$ factors through $G$.\nApply this to $X' = G'/U'N$ where $U'$ is a small open subgroup of $G'$.\nThen we see that $\\Ker(h) \\subset U'N$ for all $U'$. Since $N$ is closed\nthis implies $\\Ker(h) \\subset N$, i.e., (1) holds.\n\n\\medskip\\noindent\nSuppose that (1) holds. This means that $N = \\Ker(h)$. Let $X'$ be a\nfinite $G'$-set such that $G''$ acts trivially. This means that\n$\\Ker(G' \\to \\text{Aut}(X'))$ is a closed normal subgroup containing\n$\\Im(h')$. Hence $N = \\Ker(h)$ is contained in it and the $G'$-action\non $X'$ factors through $G$, i.e., (2) holds.\n\n\\medskip\\noindent\nSuppose that (3) holds. This means that for a finite $G'$-set $X'$\nsuch that $G''$ acts trivially, there is a surjection of $G'$-sets\n$X \\to X'$ where $X$ is a $G$-set. Clearly this means the action of\n$G'$ on $X'$ factors through $G$, i.e., (2) holds.\n\n\\medskip\\noindent\nThe implication (2) $\\Rightarrow$ (3) is immediate. This finishes the proof.\n\\end{proof}\n\n\\begin{lemma}\n\\label{lemma-functoriality-galois-injective}\nIn diagram (\\ref{equation-translation}) the following are equivalent\n\\begin{enumerate}\n\\item $h'$ is injective, and\n\\item for every connected object $X''$ of $\\mathcal{C}''$\nthere exists an object $X'$ of $\\mathcal{C}'$ and a diagram\n$$\nX'' \\leftarrow Y'' \\rightarrow H(X')\n$$\nin $\\mathcal{C}''$ where $Y'' \\to X''$ is an epimorphism and\n$Y'' \\to H(X')$ is a monomorphism.\n\\end{enumerate}\n\\end{lemma}\n\n\\begin{proof}\nWe may replace $H'$ by the corresponding functor between the categories\nof finite $G'$-sets and finite $G''$-sets.\n\n\\medskip\\noindent\nAssume $h' : G'' \\to G'$ is injective. Let $H'' \\subset G''$\nbe an open subgroup. Since the topology on $G''$ is the induced\ntopology from $G'$ there exists an open subgroup $H' \\subset G'$\nsuch that $(h')^{-1}(H') \\subset H''$.\nThen the desired diagram is\n$$\nG''/H'' \\leftarrow G''/(h')^{-1}(H') \\rightarrow G'/H'\n$$\nConversely, assume (2) holds for the functor\n$\\textit{Finite-}G'\\textit{-Sets} \\to \\textit{Finite-}G''\\textit{-Sets}$.\nLet $g'' \\in \\Ker(h')$. Pick any open subgroup $H'' \\subset G''$.\nBy assumption there exists a finite $G'$-set $X'$ and a diagram\n$$\nG''/H'' \\leftarrow Y'' \\rightarrow X'\n$$\nof $G''$-sets with the left arrow surjective and the right arrow injective.\nSince $g''$ is in the kernel of $h'$ we see that $g''$ acts trivially on $X'$.\nHence $g''$ acts trivially on $Y''$ and hence trivially on $G''/H''$.\nThus $g'' \\in H''$. As this holds for all open subgroups we conclude\nthat $g''$ is the identity element as desired.\n\\end{proof}\n\n\\begin{lemma}\n\\label{lemma-functoriality-galois-normal}\nIn diagram (\\ref{equation-translation}) the following are equivalent\n\\begin{enumerate}\n\\item the image of $h'$ is normal, and\n\\item for every connected object $X'$ of $\\mathcal{C}'$ such that\nthere is a morphism from the final object of $\\mathcal{C}''$\nto $H'(X')$ we have that $H'(X')$ is isomorphic to a finite coproduct\nof final objects.\n\\end{enumerate}\n\\end{lemma}\n\n\\begin{proof}\nThis translates into the following statement for the continuous\ngroup homomorphism $h' : G'' \\to G'$: the image of $h'$ is normal\nif and only if every open subgroup $U' \\subset G'$ which\ncontains $h'(G'')$ also contains every conjugate of $h'(G'')$.\nThe result follows easily from this; some details omitted.\n\\end{proof}\n\n\n\n\n\n\n\\section{Finite \\'etale morphisms}\n\\label{section-finite-etale}\n\n\\noindent\nIn this section we prove enough basic results on finite \\'etale\nmorphisms to be able to construct the \\'etale fundamental group.\n\n\\medskip\\noindent\nLet $X$ be a scheme. We will use the notation $\\textit{F\\'Et}_X$\nto denote the category of schemes finite and \\'etale over $X$.\nThus\n\\begin{enumerate}\n\\item an object of $\\textit{F\\'Et}_X$ is a finite \\'etale morphism\n$Y \\to X$ with target $X$, and\n\\item a morphism in $\\textit{F\\'Et}_X$\nfrom $Y \\to X$ to $Y' \\to X$ is a morphism $Y \\to Y'$ making\nthe diagram\n$$\n\\xymatrix{\nY \\ar[rr] \\ar[rd] & &  Y' \\ar[ld] \\\\\n& X\n}\n$$\ncommute.\n\\end{enumerate}\nWe will often call an object of $\\textit{F\\'Et}_X$ a\n{\\it finite \\'etale cover} of $X$ (even if $Y$ is empty).\nIt turns out that there is a stack $p : \\textit{F\\'Et} \\to \\Sch$\nover the category of schemes whose fibre over $X$ is the category\n$\\textit{F\\'Et}_X$ just defined. See Examples of Stacks, Section\n\\ref{examples-stacks-section-finite-etale}.\n\n\\begin{example}\n\\label{example-finite-etale-geometric-point}\nLet $k$ be an algebraically closed field and $X = \\Spec(k)$. In this case\n$\\textit{F\\'Et}_X$ is equivalent to the category of finite sets. This works\nmore generally when $k$ is separably algebraically closed. The reason is\nthat a scheme \\'etale over $k$ is the disjoint union of spectra of\nfields finite separable over $k$, see\nMorphisms, Lemma \\ref{morphisms-lemma-etale-over-field}.\n\\end{example}\n\n\\begin{lemma}\n\\label{lemma-finite-etale-covers-limits-colimits}\nLet $X$ be a scheme. The category $\\textit{F\\'Et}_X$ has finite limits and\nfinite colimits and for any morphism $X' \\to X$ the base change functor\n$\\textit{F\\'Et}_X \\to \\textit{F\\'Et}_{X'}$ is exact.\n\\end{lemma}\n\n\\begin{proof}\nFinite limits and left exactness. By\nCategories, Lemma \\ref{categories-lemma-finite-limits-exist}\nit suffices to show that $\\textit{F\\'Et}_X$ has a final object\nand fibred products. This is clear because the category of\nall schemes over $X$ has a final object (namely $X$) and fibred products\nand fibred products of schemes finite \\'etale over $X$ are\nfinite \\'etale over $X$. Moreover, it is clear that base\nchange commutes with these operations and hence base change\nis left exact (Categories, Lemma\n\\ref{categories-lemma-characterize-left-exact}).\n\n\\medskip\\noindent\nFinite colimits and right exactness. By\nCategories, Lemma \\ref{categories-lemma-colimits-exist}\nit suffices to show that $\\textit{F\\'Et}_X$ has finite\ncoproducts and coequalizers. Finite coproducts are given\nby disjoint unions (the empty coproduct is the empty scheme).\nLet $a, b : Z \\to Y$ be two morphisms of $\\textit{F\\'Et}_X$.\nSince $Z \\to X$ and $Y \\to X$ are finite \\'etale we can write\n$Z = \\underline{\\Spec}(\\mathcal{C})$ and $Y = \\underline{\\Spec}(\\mathcal{B})$\nfor some finite locally free $\\mathcal{O}_X$-algebras $\\mathcal{C}$\nand $\\mathcal{B}$. The morphisms $a, b$ induce two maps\n$a^\\sharp, b^\\sharp : \\mathcal{B} \\to \\mathcal{C}$.\nLet $\\mathcal{A} = \\text{Eq}(a^\\sharp, b^\\sharp)$ be their\nequalizer. If\n$$\n\\underline{\\Spec}(\\mathcal{A}) \\longrightarrow X\n$$\nis finite \\'etale, then it is clear that this is the coequalizer\n(after all we can write any object of $\\textit{F\\'Et}_X$\nas the relative spectrum of a sheaf of $\\mathcal{O}_X$-algebras).\nThis we may do after replacing $X$ by the members of an \\'etale\ncovering (Descent, Lemmas \\ref{descent-lemma-descending-property-finite}\nand \\ref{descent-lemma-descending-property-separated}).\nThus by \\'Etale Morphisms, Lemma \\ref{etale-lemma-finite-etale-etale-local}\nwe may assume that\n$Y = \\coprod_{i = 1, \\ldots, n} X$ and $Z = \\coprod_{j = 1, \\ldots, m} X$.\nThen\n$$\n\\mathcal{C} = \\prod\\nolimits_{1 \\leq j \\leq m} \\mathcal{O}_X\n\\quad\\text{and}\\quad\n\\mathcal{B} = \\prod\\nolimits_{1 \\leq i \\leq n} \\mathcal{O}_X\n$$\nAfter a further replacement by the members of an open covering\nwe may assume that $a, b$ correspond to\nmaps $a_s, b_s : \\{1, \\ldots, m\\} \\to \\{1, \\ldots, n\\}$, i.e.,\nthe summand $X$ of $Z$ corresponding to the index $j$ maps into\nthe summand $X$ of $Y$ corresponding to the index $a_s(j)$, resp.\\ $b_s(j)$\nunder the morphism $a$, resp.\\ $b$.\nLet $\\{1, \\ldots, n\\} \\to T$ be the coequalizer of $a_s, b_s$.\nThen we see that\n$$\n\\mathcal{A} = \\prod\\nolimits_{t \\in T} \\mathcal{O}_X\n$$\nwhose spectrum is certainly finite \\'etale over $X$. We\nomit the verification that this is compatible with base change.\nThus base change is a right exact functor.\n\\end{proof}\n\n\\begin{remark}\n\\label{remark-colimits-commute-forgetful}\nLet $X$ be a scheme. Consider the natural functors\n$F_1 : \\textit{F\\'Et}_X \\to \\Sch$ and $F_2 : \\textit{F\\'Et}_X \\to \\Sch/X$.\nThen\n\\begin{enumerate}\n\\item The functors $F_1$ and $F_2$ commute with finite colimits.\n\\item The functor $F_2$ commutes with finite limits,\n\\item The functor $F_1$ commutes with connected finite limits, i.e.,\nwith equalizers and fibre products.\n\\end{enumerate}\nThe results on limits are immediate from the discussion in\nthe proof of Lemma \\ref{lemma-finite-etale-covers-limits-colimits}\nand Categories, Lemma \\ref{categories-lemma-connected-limit-over-X}.\nIt is clear that $F_1$ and $F_2$ commute with finite coproducts.\nBy the dual of Categories, Lemma\n\\ref{categories-lemma-characterize-left-exact}\nwe need to show that $F_1$ and $F_2$ commute with coequalizers.\nIn the proof of Lemma \\ref{lemma-finite-etale-covers-limits-colimits}\nwe saw that coequalizers in $\\textit{F\\'Et}_X$ look \\'etale locally\nlike this\n$$\n\\xymatrix{\n\\coprod_{j \\in J} U \\ar@<1ex>[r]^a \\ar@<-1ex>[r]_b &\n\\coprod_{i \\in I} U \\ar[r] &\n\\coprod_{t \\in \\text{Coeq}(a, b)} U\n}\n$$\nwhich is certainly a coequalizer in the category of schemes.\nHence the statement follows from the fact that being a coequalizer\nis fpqc local as formulate precisely in\nDescent, Lemma \\ref{descent-lemma-coequalizer-fpqc-local}.\n\\end{remark}\n\n\\begin{lemma}\n\\label{lemma-internal-hom-finite-etale}\nLet $X$ be a scheme. Given $U, V$ finite \\'etale over $X$ there\nexists a scheme $W$ finite \\'etale over $X$ such that\n$$\n\\Mor_X(X, W) = \\Mor_X(U, V)\n$$\nand such that the same remains true after any base change.\n\\end{lemma}\n\n\\begin{proof}\nBy More on Morphisms, Lemma\n\\ref{more-morphisms-lemma-hom-from-finite-locally-free-separated-lqf}\nthere exists a scheme $W$ representing $\\mathit{Mor}_X(U, V)$.\n(Use that an \\'etale morphism is locally quasi-finite by\nMorphisms, Lemmas \\ref{morphisms-lemma-etale-locally-quasi-finite}\nand that a finite morphism is separated.)\nThis scheme clearly satisfies the formula after any base change.\nTo finish the proof we have to show that $W \\to X$ is finite \\'etale.\nThis we may do after replacing $X$ by the members of an \\'etale\ncovering (Descent, Lemmas \\ref{descent-lemma-descending-property-finite}\nand \\ref{descent-lemma-descending-property-separated}).\nThus by \\'Etale Morphisms, Lemma \\ref{etale-lemma-finite-etale-etale-local}\nwe may assume that $U = \\coprod_{i = 1, \\ldots, n} X$\nand $V = \\coprod_{j = 1, \\ldots, m} X$.\nIn this case\n$W = \\coprod_{\\alpha : \\{1, \\ldots, n\\} \\to \\{1, \\ldots, m\\}} X$\nby inspection (details omitted) and the proof is complete.\n\\end{proof}\n\n\\noindent\nLet $X$ be a scheme. A {\\it geometric point} of $X$ is a morphism\n$\\Spec(k) \\to X$ where $k$ is algebraically closed. Such a point is\nusually denoted $\\overline{x}$, i.e., by an overlined small case letter.\nWe often use $\\overline{x}$ to denote the scheme $\\Spec(k)$ as well as\nthe morphism, and we use $\\kappa(\\overline{x})$\nto denote $k$. We say $\\overline{x}$ {\\it lies over} $x$\nto indicate that $x \\in X$ is the image of $\\overline{x}$.\nWe will discuss this further in\n\\'Etale Cohomology, Section \\ref{etale-cohomology-section-stalks}.\nGiven $\\overline{x}$ and an \\'etale morphism $U \\to X$ we can\nconsider\n$$\n|U_{\\overline{x}}| : \\text{the underlying set of points of the\nscheme }U_{\\overline{x}} = U \\times_X \\overline{x}\n$$\nSince $U_{\\overline{x}}$ as a scheme over $\\overline{x}$\nis a disjoint union of copies of $\\overline{x}$\n(Morphisms, Lemma \\ref{morphisms-lemma-etale-over-field})\nwe can also describe this set as\n$$\n|U_{\\overline{x}}| =\n\\left\\{\n\\begin{matrix}\n\\text{commutative} \\\\\n\\text{diagrams}\n\\end{matrix}\n\\vcenter{\n\\xymatrix{\n\\overline{x} \\ar[rd]_{\\overline{x}} \\ar[r]_{\\overline{u}} & U \\ar[d] \\\\\n& X\n}\n}\n\\right\\}\n$$\nThe assignment $U \\mapsto |U_{\\overline{x}}|$ is a functor\nwhich is often denoted $F_{\\overline{x}}$.\n\n\\begin{lemma}\n\\label{lemma-finite-etale-connected-galois-category}\nLet $X$ be a connected scheme. Let $\\overline{x}$ be a geometric point.\nThe functor\n$$\nF_{\\overline{x}} : \\textit{F\\'Et}_X \\longrightarrow \\textit{Sets},\\quad\nY \\longmapsto |Y_{\\overline{x}}|\n$$\ndefines a Galois category (Definition \\ref{definition-galois-category}).\n\\end{lemma}\n\n\\begin{proof}\nAfter identifying $\\textit{F\\'Et}_{\\overline{x}}$ with the category of\nfinite sets (Example \\ref{example-finite-etale-geometric-point})\nwe see that our functor $F_{\\overline{x}}$\nis nothing but the base change functor for the morphism $\\overline{x} \\to X$.\nThus we see that $\\textit{F\\'Et}_X$ has finite limits and finite colimits\nand that $F_{\\overline{x}}$ is exact by\nLemma \\ref{lemma-finite-etale-covers-limits-colimits}.\nWe will also use that finite limits in $\\textit{F\\'Et}_X$\nagree with the corresponding finite limits in the category\nof schemes over $X$, see Remark \\ref{remark-colimits-commute-forgetful}.\n\n\\medskip\\noindent\nIf $Y' \\to Y$ is a monomorphism in $\\textit{F\\'Et}_X$\nthen we see that $Y' \\to Y' \\times_Y Y'$ is an isomorphism, and\nhence $Y' \\to Y$ is a monomorphism of schemes. It follows that\n$Y' \\to Y$ is an open immersion\n(\\'Etale Morphisms, Theorem \\ref{etale-theorem-etale-radicial-open}). Since\n$Y'$ is finite over $X$ and $Y$ separated over $X$,\nthe morphism $Y' \\to Y$ is finite\n(Morphisms, Lemma \\ref{morphisms-lemma-finite-permanence}), hence closed\n(Morphisms, Lemma \\ref{morphisms-lemma-finite-proper}),\nhence it is the inclusion of an open and closed subscheme of $Y$.\nIt follows that $Y$ is a connected objects of the category\n$\\textit{F\\'Et}_X$ (as in Definition \\ref{definition-galois-category})\nif and only if $Y$ is connected as a scheme. Then it follows from\nTopology, Lemma \\ref{topology-lemma-finite-fibre-connected-components}\nthat $Y$ is a finite coproduct of its connected components\nboth as a scheme and in the sense of\nDefinition \\ref{definition-galois-category}.\n\n\\medskip\\noindent\nLet $Y \\to Z$ be a morphism in $\\textit{F\\'Et}_X$ which induces a\nbijection $F_{\\overline{x}}(Y) \\to F_{\\overline{x}}(Z)$. We have to\nshow that $Y \\to Z$ is an isomorphism. By the above we may assume\n$Z$ is connected. Since $Y \\to Z$ is finite \\'etale and hence finite\nlocally free it suffices to show that $Y \\to Z$ is finite locally\nfree of degree $1$. This is true in a neighbourhood of any point of\n$Z$ lying over $\\overline{x}$ and since $Z$ is connected and\nthe degree is locally constant we conclude.\n\\end{proof}\n\n\n\n\\section{Fundamental groups}\n\\label{section-fundamental-groups}\n\n\\noindent\nIn this section we define Grothendieck's algebraic fundamental group.\nThe following definition makes sense thanks to\nLemma \\ref{lemma-finite-etale-connected-galois-category}.\n\n\\begin{definition}\n\\label{definition-fundamental-group}\nLet $X$ be a connected scheme. Let $\\overline{x}$ be a geometric point\nof $X$. The {\\it fundamental group} of $X$ with\n{\\it base point} $\\overline{x}$ is the group\n$$\n\\pi_1(X, \\overline{x}) = \\text{Aut}(F_{\\overline{x}})\n$$\nof automorphisms of the fibre functor\n$F_{\\overline{x}} : \\textit{F\\'Et}_X \\to \\textit{Sets}$\nendowed with its canonical profinite topology from\nLemma \\ref{lemma-aut-inverse-limit}.\n\\end{definition}\n\n\\noindent\nCombining the above with the material from Section \\ref{section-galois}\nwe obtain the following theorem.\n\n\\begin{theorem}\n\\label{theorem-fundamental-group}\nLet $X$ be a connected scheme. Let $\\overline{x}$ be a geometric point\nof $X$.\n\\begin{enumerate}\n\\item The fibre functor $F_{\\overline{x}}$ defines an equivalence of\ncategories\n$$\n\\textit{F\\'Et}_X \\longrightarrow\n\\textit{Finite-}\\pi_1(X, \\overline{x})\\textit{-Sets}\n$$\n\\item Given a second geometric point $\\overline{x}'$ of $X$ there\nexists an isomorphism $t : F_{\\overline{x}} \\to F_{\\overline{x}'}$.\nThis gives an isomorphism $\\pi_1(X, \\overline{x}) \\to \\pi_1(X, \\overline{x}')$\ncompatible with the equivalences in (1). This isomorphism is\nindependent of $t$ up to inner conjugation.\n\\item Given a morphism $f : X \\to Y$ of connected schemes denote\n$\\overline{y} = f \\circ \\overline{x}$. There is a canonical\ncontinuous homomorphism\n$$\nf_* : \\pi_1(X, \\overline{x}) \\to \\pi_1(Y, \\overline{y})\n$$\nsuch that the diagram\n$$\n\\xymatrix{\n\\textit{F\\'Et}_Y \\ar[r]_{\\text{base change}} \\ar[d]_{F_{\\overline{y}}} &\n\\textit{F\\'Et}_X \\ar[d]^{F_{\\overline{x}}} \\\\\n\\textit{Finite-}\\pi_1(Y, \\overline{y})\\textit{-Sets} \\ar[r]^{f_*} &\n\\textit{Finite-}\\pi_1(X, \\overline{x})\\textit{-Sets}\n}\n$$\nis commutative.\n\\end{enumerate}\n\\end{theorem}\n\n\\begin{proof}\nPart (1) follows from Lemma \\ref{lemma-finite-etale-connected-galois-category}\nand Proposition \\ref{proposition-galois}.\nPart (2) is a special case of Lemma \\ref{lemma-functoriality-galois}.\nFor part (3) observe that the diagram\n$$\n\\xymatrix{\n\\textit{F\\'Et}_Y \\ar[r] \\ar[d]_{F_{\\overline{y}}} &\n\\textit{F\\'Et}_X \\ar[d]^{F_{\\overline{x}}} \\\\\n\\textit{Sets} \\ar@{=}[r] & \\textit{Sets}\n}\n$$\nis commutative (actually commutative, not just $2$-commutative) because\n$\\overline{y} = f \\circ \\overline{x}$. Hence\nwe can apply Lemma \\ref{lemma-functoriality-galois} with the implied\ntransformation of functors to get (3).\n\\end{proof}\n\n\\begin{remark}\n\\label{remark-finite-etale-under-galois}\nLet $X$ be a connected scheme with geometric point $\\overline{x}$.\nSince $F_{\\overline{x}} : \\textit{F\\'Et}_X \\to \\textit{Sets}$ is a\nGalois category (Lemma \\ref{lemma-finite-etale-connected-galois-category})\nthe material in Section \\ref{section-galois} applies.\nWe will say a finite \\'etale morphism $Y \\to X$ is a\n{\\it Galois cover} if $Y$ defines a Galois object of\n$\\textit{F\\'Et}_X$. Recall that this means that\n$Y$ is connected and that $G = \\text{Aut}(Y/X)$\nacts transitively (or equivalently simply transitively)\non $F_{\\overline{x}}(Y)$. For any finite \\'etale\nmorphism $f : Y \\to X$ with $Y$ connected, there is a Galois cover $Y' \\to X$\nwhich dominates $Y$ (Lemma \\ref{lemma-galois}).\nThe Galois objects of $\\textit{F\\'Et}_X$ correspond,\nvia the equivalence\n$F_{\\overline{x}} : \\textit{F\\'Et}_X \\to\n\\textit{Finite-}\\pi_1(X, \\overline{x})\\textit{-Sets}$\nof Theorem \\ref{theorem-fundamental-group},\nwith the finite $\\pi_1(X, \\overline{x})\\textit{-Sets}$\nof the form $G = \\pi_1(X, \\overline{x})/H$ where $H$ is a\nnormal open subgroup. Equivalently, if $G$ is a finite group\nand $\\pi_1(X, \\overline{x}) \\to G$ is a continuous surjection,\nthen $G$ viewed as a $\\pi_1(X, \\overline{x})$-set corresponds\nto a Galois covering.\n\\end{remark}\n\n\\begin{lemma}\n\\label{lemma-fundamental-group-Galois-group}\nLet $K$ be a field and set $X = \\Spec(K)$. Let $\\overline{K}$ be an\nalgebraic closure and denote $\\overline{x} : \\Spec(\\overline{K}) \\to X$\nthe corresponding geometric point. Let $K^{sep} \\subset \\overline{K}$\nbe the separable algebraic closure.\n\\begin{enumerate}\n\\item The functor of Lemma \\ref{lemma-sheaves-point} induces an equivalence\n$$\n\\textit{F\\'Et}_X \\longrightarrow\n\\textit{Finite-}\\text{Gal}(K^{sep}/K)\\textit{-Sets}.\n$$\ncompatible with $F_{\\overline{x}}$ and the functor\n$\\textit{Finite-}\\text{Gal}(K^{sep}/K)\\textit{-Sets} \\to \\textit{Sets}$.\n\\item This induces a canonical isomorphism\n$$\n\\text{Gal}(K^{sep}/K) \\longrightarrow \\pi_1(X, \\overline{x})\n$$\nof profinite topological groups.\n\\end{enumerate}\n\\end{lemma}\n\n\\begin{proof}\nThe functor of Lemma \\ref{lemma-sheaves-point} is the same as the functor\n$F_{\\overline{x}}$ because for any $Y$ \\'etale over $X$ we have\n$$\n\\Mor_X(\\Spec(\\overline{K}), Y) = \\Mor_X(\\Spec(K^{sep}), Y)\n$$\nNamely, as seen in the proof of Lemma \\ref{lemma-sheaves-point} we have\n$Y = \\coprod_{i \\in I} \\Spec(L_i)$ with $L_i/K$ finite separable over $K$.\nHence any $K$-algebra homomorphism $L_i \\to \\overline{K}$ factors\nthrough $K^{sep}$. Also, note that $F_{\\overline{x}}(Y)$ is finite\nif and only if $I$ is finite if and only if $Y \\to X$ is finite \\'etale.\nThis proves (1).\n\n\\medskip\\noindent\nPart (2) is a formal consequence of (1),\nLemma \\ref{lemma-functoriality-galois}, and\nLemma \\ref{lemma-single-out-profinite}.\n(Please also see the remark below.)\n\\end{proof}\n\n\\begin{remark}\n\\label{remark-variance}\nIn the situation of Lemma \\ref{lemma-fundamental-group-Galois-group}\nlet us give a more explicit construction of the isomorphism\n$\\text{Gal}(K^{sep}/K) \\to\n\\pi_1(X, \\overline{x}) = \\text{Aut}(F_{\\overline{x}})$.\nObserve that\n$\\text{Gal}(K^{sep}/K) = \\text{Aut}(\\overline{K}/K)$\nas $\\overline{K}$ is the perfection of $K^{sep}$.\nSince $F_{\\overline{x}}(Y) = \\Mor_X(\\Spec(\\overline{K}), Y)$\nwe may consider the map\n$$\n\\text{Aut}(\\overline{K}/K) \\times F_{\\overline{x}}(Y) \\to F_{\\overline{x}}(Y),\n\\quad\n(\\sigma, \\overline{y}) \\mapsto\n\\sigma \\cdot \\overline{y} = \\overline{y} \\circ \\Spec(\\sigma)\n$$\nThis is an action because\n$$\n\\sigma\\tau \\cdot \\overline{y} =\n\\overline{y} \\circ \\Spec(\\sigma\\tau) =\n\\overline{y} \\circ \\Spec(\\tau) \\circ \\Spec(\\sigma) =\n\\sigma \\cdot (\\tau \\cdot \\overline{y})\n$$\nThe action is functorial in $Y \\in \\textit{F\\'Et}_X$ and we\nobtain the desired map.\n\\end{remark}\n\n\n\n\n\n\n\n\\section{Topological invariance of the fundamental group}\n\\label{section-topological-invariance}\n\n\\noindent\nThe main result of this section is that a universal homeomorphism\nof connected schemes induces an isomorphism on fundamental groups.\nSee Proposition \\ref{proposition-universal-homeomorphism}.\n\n\\medskip\\noindent\nInstead of directly proving two schemes have the same fundamental\ngroup, we often prove that their categories of finite \\'etale\ncoverings are the same. This of course implies that\ntheir fundamental groups are equal provided they are connected.\n\n\\begin{lemma}\n\\label{lemma-what-equivalence-gives}\nLet $f : X \\to Y$ be a morphism of quasi-compact and quasi-separated schemes\nsuch that the base change functor $\\textit{F\\'Et}_Y \\to \\textit{F\\'Et}_X$\nis an equivalence of categories. In this case\n\\begin{enumerate}\n\\item $f$ induces a homeomorphism $\\pi_0(X) \\to \\pi_0(Y)$,\n\\item if $X$ or equivalently $Y$ is connected, then\n$\\pi_1(X, \\overline{x}) = \\pi_1(Y, \\overline{y})$.\n\\end{enumerate}\n\\end{lemma}\n\n\\begin{proof}\nLet $Y = Y_0 \\amalg Y_1$ be a decomposition into nonempty open and closed\nsubschemes. We claim that $f(X)$ meets both $Y_i$. Namely, if not,\nsay $f(X) \\subset Y_1$, then we can consider the finite \\'etale\nmorphism $V = Y_1 \\to Y$. This is not an\nisomorphism but $V \\times_Y X \\to X$ is an isomorphism, which is\na contradiction.\n\n\\medskip\\noindent\nSuppose that $X = X_0 \\amalg X_1$ is a decomposition into open and closed\nsubschemes. Consider the finite \\'etale morphism $U = X_1 \\to X$. Then\n$U = X \\times_Y V$ for some finite \\'etale morphism $V \\to Y$. The degree\nof the morphism $V \\to Y$ is locally constant, hence we obtain a decomposition\n$Y = \\coprod_{d \\geq 0} Y_d$ into open and closed subschemes\nsuch that $V \\to Y$ has degree $d$ over $Y_d$. Since\n$f^{-1}(Y_d) = \\emptyset$ for $d > 1$ we conclude that $Y_d = \\emptyset$\nfor $d > 1$ by the above. And we conclude that $f^{-1}(Y_i) = X_i$\nfor $i = 0, 1$.\n\n\\medskip\\noindent\nIt follows that $f^{-1}$ induces a bijection between the set of\nopen and closed subsets of $Y$ and the set of open and closed subsets of $X$.\nNote that $X$ and $Y$ are spectral spaces, see Properties, Lemma\n\\ref{properties-lemma-quasi-compact-quasi-separated-spectral}.\nBy Topology, Lemma \\ref{topology-lemma-connected-component-intersection}\nthe lattice of open and closed subsets of a spectral space\ndetermines the set of connected components.\nHence $\\pi_0(X) \\to \\pi_0(Y)$ is bijective. Since $\\pi_0(X)$ and\n$\\pi_0(Y)$ are profinite spaces\n(Topology, Lemma \\ref{topology-lemma-pi0-profinite})\nwe conclude that $\\pi_0(X) \\to \\pi_0(Y)$ is a homeomorphism by\nTopology, Lemma \\ref{topology-lemma-bijective-map}. This proves (1).\nPart (2) is immediate.\n\\end{proof}\n\n\\noindent\nThe following lemma tells us that the fundamental group of a henselian\npair is the fundamental group of the closed subset.\n\n\\begin{lemma}\n\\label{lemma-gabber}\nLet $(A, I)$ be a henselian pair. Set $X = \\Spec(A)$ and $Z = \\Spec(A/I)$.\nThe functor\n$$\n\\textit{F\\'Et}_X \\longrightarrow \\textit{F\\'Et}_Z,\\quad\nU \\longmapsto U \\times_X Z\n$$\nis an equivalence of categories.\n\\end{lemma}\n\n\\begin{proof}\nThis is a translation of\nMore on Algebra, Lemma \\ref{more-algebra-lemma-finite-etale-equivalence}.\n\\end{proof}\n\n\\noindent\nThe following lemma tells us that the fundamental group of a thickening\nis the same as the fundamental group of the original. We will use this\nin the proof of the strong proposition concerning universal homeomorphisms\nbelow.\n\n\\begin{lemma}\n\\label{lemma-thickening}\nLet $X \\subset X'$ be a thickening of schemes. The functor\n$$\n\\textit{F\\'Et}_{X'} \\longrightarrow \\textit{F\\'Et}_X,\\quad\nU' \\longmapsto U' \\times_{X'} X\n$$\nis an equivalence of categories.\n\\end{lemma}\n\n\\begin{proof}\nFor a discussion of thickenings see\nMore on Morphisms, Section \\ref{more-morphisms-section-thickenings}.\nLet $U' \\to X'$ be an \\'etale morphism such that $U = U' \\times_{X'} X \\to X$\nis finite \\'etale. Then $U' \\to X'$ is finite \\'etale as well.\nThis follows for example from More on Morphisms, Lemma\n\\ref{more-morphisms-lemma-properties-that-extend-over-thickenings}.\nNow, if $X \\subset X'$ is a finite order thickening then this remark\ncombined with \\'Etale Morphisms, Theorem\n\\ref{etale-theorem-remarkable-equivalence}\nproves the lemma. Below we will prove the lemma for general thickenings, but\nwe suggest the reader skip the proof.\n\n\\medskip\\noindent\nLet $X' = \\bigcup X_i'$ be an affine open covering. Set\n$X_i = X \\times_{X'} X_i'$, $X_{ij}' = X'_i \\cap X'_j$,\n$X_{ij} = X \\times_{X'} X_{ij}'$, $X_{ijk}' = X'_i \\cap X'_j \\cap X'_k$,\n$X_{ijk} = X \\times_{X'} X_{ijk}'$.\nSuppose that we can prove\nthe theorem for each of the thickenings\n$X_i \\subset X'_i$, $X_{ij} \\subset X_{ij}'$, and $X_{ijk} \\subset X_{ijk}'$.\nThen the result follows for $X \\subset X'$ by relative glueing of\nschemes, see\nConstructions, Section \\ref{constructions-section-relative-glueing}.\nObserve that the schemes $X_i'$, $X_{ij}'$, $X_{ijk}'$ are\neach separated as open subschemes of affine schemes. Repeating the\nargument one more time we reduce to the case where the schemes\n$X'_i$, $X_{ij}'$, $X_{ijk}'$ are affine.\n\n\\medskip\\noindent\nIn the affine case we have $X' = \\Spec(A')$ and $X = \\Spec(A'/I')$\nwhere $I'$ is a locally nilpotent ideal. Then $(A', I')$ is a\nhenselian pair (More on Algebra, Lemma\n\\ref{more-algebra-lemma-locally-nilpotent-henselian})\nand the result follows from Lemma \\ref{lemma-gabber} (which is\nmuch easier in this case).\n\\end{proof}\n\n\\noindent\nThe ``correct'' way to prove the following proposition would be to\ndeduce it from the invariance of the \\'etale site, see\n\\'Etale Cohomology, Theorem\n\\ref{etale-cohomology-theorem-topological-invariance}.\n\n\\begin{proposition}\n\\label{proposition-universal-homeomorphism}\nLet $f : X \\to Y$ be a universal homeomorphism of schemes. Then\n$$\n\\textit{F\\'Et}_Y \\longrightarrow \\textit{F\\'Et}_X,\\quad\nV \\longmapsto V \\times_Y X\n$$\nis an equivalence. Thus if $X$ and $Y$ are connected, then\n$f$ induces an isomorphism $\\pi_1(X, \\overline{x}) \\to \\pi_1(Y, \\overline{y})$\nof fundamental groups.\n\\end{proposition}\n\n\\begin{proof}\nRecall that a universal homeomorphism is the same thing as an\nintegral, universally injective, surjective morphism, see\nMorphisms, Lemma \\ref{morphisms-lemma-universal-homeomorphism}.\nIn particular, the diagonal $\\Delta : X \\to X \\times_Y X$ is a thickening\nby Morphisms, Lemma \\ref{morphisms-lemma-universally-injective}.\nThus by Lemma \\ref{lemma-thickening}\nwe see that given a finite \\'etale morphism $U \\to X$\nthere is a unique isomorphism\n$$\n\\varphi : U \\times_Y X \\to X \\times_Y U\n$$\nof schemes finite \\'etale over $X \\times_Y X$ which pulls back under\n$\\Delta$ to $\\text{id} : U \\to U$ over $X$.\nSince $X \\to X \\times_Y X \\times_Y X$\nis a thickening as well (it is bijective and a closed immersion)\nwe conclude that $(U, \\varphi)$ is a descent datum relative to $X/Y$.\nBy \\'Etale Morphisms, Proposition \\ref{etale-proposition-effective}\nwe conclude that $U = X \\times_Y V$ for some $V \\to Y$\nquasi-compact, separated, and \\'etale.\nWe omit the proof that $V \\to Y$ is finite (hints:\nthe morphism $U \\to V$ is surjective and $U \\to Y$ is integral).\nWe conclude that $\\textit{F\\'Et}_Y \\to \\textit{F\\'Et}_X$\nis essentially surjective.\n\n\\medskip\\noindent\nArguing in the same manner as above we see that given\n$V_1 \\to Y$ and $V_2 \\to Y$ in $\\textit{F\\'Et}_Y$ any\nmorphism $a : X \\times_Y V_1 \\to X \\times_Y V_2$ over $X$\nis compatible with the canonical descent data. Thus $a$\ndescends to a morphism $V_1 \\to V_2$ over $Y$ by\n\\'Etale Morphisms, Lemma \\ref{etale-lemma-fully-faithful-cases}.\n\\end{proof}\n\n\n\n\n\n\n\n\n\n\n\\section{Finite \\'etale covers of proper schemes}\n\\label{section-finite-etale-over-proper}\n\n\\noindent\nIn this section we show that the fundamental group of a connected proper\nscheme over a henselian local ring is the same as the fundamental\ngroup of its special fibre. We also show that the fundamental\ngroup of a connected proper scheme over an algebraically closed field $k$\ndoes not change if we replace $k$ by an algebraically closed extension.\nInstead of stating and proving the results in the connected case\nwe prove the results in general and we leave it to the reader to deduce\nthe result for fundamental groups using\nLemma \\ref{lemma-what-equivalence-gives}.\n\n\\begin{lemma}\n\\label{lemma-finite-etale-on-proper-over-henselian}\nLet $A$ be a henselian local ring. Let $X$ be a proper scheme over $A$\nwith closed fibre $X_0$. Then the functor\n$$\n\\textit{F\\'Et}_X \\to \\textit{F\\'Et}_{X_0},\\quad\nU \\longmapsto U_0 = U \\times_X X_0\n$$\nis an equivalence of categories.\n\\end{lemma}\n\n\\begin{proof}\nThe proof given here is an example of applying algebraization and\napproximation. We proceed in a number of stages.\n\n\\medskip\\noindent\nEssential surjectivity when $A$ is a complete local Noetherian ring.\nLet $X_n = X \\times_{\\Spec(A)} \\Spec(A/\\mathfrak m^{n + 1})$.\nBy \\'Etale Morphisms, Theorem \\ref{etale-theorem-remarkable-equivalence}\nthe inclusions\n$$\nX_0 \\to X_1 \\to X_2 \\to \\ldots\n$$\ninduce equivalence of categories between the category\nof schemes \\'etale over $X_0$ and the category of schemes\n\\'etale over $X_n$.\nMoreover, if $U_n \\to X_n$ corresponds to a finite \\'etale\nmorphism $U_0 \\to X_0$, then $U_n \\to X_n$ is finite too, for example\nby More on Morphisms, Lemma\n\\ref{more-morphisms-lemma-thicken-property-morphisms-cartesian}.\nIn this case the morphism $U_0 \\to \\Spec(A/\\mathfrak m)$\nis proper as $X_0$ is proper over $A/\\mathfrak m$. Thus we may apply\nGrothendieck's algebraization theorem\n(in the form of\nCohomology of Schemes, Lemma\n\\ref{coherent-lemma-algebraize-formal-scheme-finite-over-proper})\nto see that there is a finite morphism $U \\to X$ whose restriction\nto $X_0$ recovers $U_0$. By More on Morphisms, Lemma\n\\ref{more-morphisms-lemma-check-smoothness-on-infinitesimal-nbhds}\nwe see that $U \\to X$ is \\'etale at every point of $U_0$.\nHowever, since every point of $U$ specializes to a point of $U_0$\n(as $U$ is proper over $A$), we conclude that $U \\to X$ is \\'etale.\nIn this way we conclude the functor is essentially surjective.\n\n\\medskip\\noindent\nFully faithfulness when $A$ is a complete local Noetherian ring.\nLet $U \\to X$ and $V \\to X$ be finite \\'etale morphisms and\nlet $\\varphi_0 : U_0 \\to V_0$ be a morphism over $X_0$. Look at\nthe morphism\n$$\n\\Gamma_{\\varphi_0} : U_0 \\longrightarrow U_0 \\times_{X_0} V_0\n$$\nThis morphism is both finite \\'etale and a closed immersion.\nBy essential surjectivity applied to $X = U \\times_X V$ we find\na finite \\'etale morphism $W \\to U \\times_X V$ whose special\nfibre is isomorphic to $\\Gamma_{\\varphi_0}$. Consider the projection\n$W \\to U$. It is finite \\'etale and an isomorphism over $U_0$ by\nconstruction. By \\'Etale Morphisms, Lemma\n\\ref{etale-lemma-finite-etale-one-point}\n$W \\to U$ is an isomorphism in an open neighbourhood of $U_0$.\nThus it is an isomorphism and the composition $\\varphi : U \\cong W \\to V$\nis the desired lift of $\\varphi_0$.\n\n\\medskip\\noindent\nEssential surjectivity when $A$ is a henselian local Noetherian G-ring.\nLet $U_0 \\to X_0$ be a finite \\'etale morphism.\nLet $A^\\wedge$ be the completion of $A$ with respect to the maximal ideal.\nLet $X^\\wedge$ be the base change of $X$ to $A^\\wedge$.\nBy the result above there exists a finite \\'etale morphism\n$V \\to X^\\wedge$ whose special fibre is $U_0$.\nWrite $A^\\wedge = \\colim A_i$ with $A \\to A_i$ of finite type.\nBy Limits, Lemma \\ref{limits-lemma-descend-finite-presentation}\nthere exists an $i$ and a finitely presented morphism $U_i \\to X_{A_i}$\nwhose base change to $X^\\wedge$ is $V$. After increasing $i$\nwe may assume that $U_i \\to X_{A_i}$ is finite and \\'etale\n(Limits, Lemmas \\ref{limits-lemma-descend-finite-finite-presentation} and\n\\ref{limits-lemma-descend-etale}). Writing\n$$\nA_i = A[x_1, \\ldots, x_n]/(f_1, \\ldots, f_m)\n$$\nthe ring map $A_i \\to A^\\wedge$ can be reinterpreted as a solution\n$(a_1, \\ldots, a_n)$ in $A^\\wedge$ for the system of equations $f_j = 0$.\nBy Smoothing Ring Maps, Theorem \\ref{smoothing-theorem-approximation-property}\nwe can approximate this solution (to order $11$ for example) by a solution\n$(b_1, \\ldots, b_n)$ in $A$. Translating back we find an $A$-algebra map\n$A_i \\to A$ which gives the same closed point as the original map\n$A_i \\to A^\\wedge$ (as $11 > 1$). The base change $U \\to X$ of $V \\to X_{A_i}$\nby this ring map will therefore be a finite \\'etale morphism whose\nspecial fibre is isomorphic to $U_0$.\n\n\\medskip\\noindent\nFully faithfulness when $A$ is a henselian local Noetherian G-ring.\nThis can be deduced from essential surjectivity in exactly the same\nmanner as was done in the case that $A$ is complete Noetherian.\n\n\\medskip\\noindent\nGeneral case. Let $(A, \\mathfrak m)$ be a henselian local ring.\nSet $S = \\Spec(A)$ and denote $s \\in S$ the closed point. By Limits, Lemma\n\\ref{limits-lemma-proper-limit-of-proper-finite-presentation-noetherian}\nwe can write $X \\to \\Spec(A)$ as a cofiltered limit of\nproper morphisms $X_i \\to S_i$ with $S_i$ of finite type over $\\mathbf{Z}$.\nFor each $i$ let $s_i \\in S_i$ be the image of $s$.\nSince $S = \\lim S_i$ and $A = \\mathcal{O}_{S, s}$ we have\n$A = \\colim \\mathcal{O}_{S_i, s_i}$. The ring $A_i = \\mathcal{O}_{S_i, s_i}$\nis a Noetherian local G-ring (More on Algebra, Proposition\n\\ref{more-algebra-proposition-ubiquity-G-ring}).\nBy More on Algebra, Lemma \\ref{more-algebra-lemma-henselization-colimit}\nwe see that $A = \\colim A_i^h$. By\nMore on Algebra, Lemma \\ref{more-algebra-lemma-henselization-G-ring}\nthe rings $A_i^h$ are G-rings. Thus we see that $A = \\colim A_i^h$ and\n$$\nX = \\lim (X_i \\times_{S_i} \\Spec(A_i^h))\n$$\nas schemes. The category of schemes finite \\'etale over $X$ is the limit\nof the category of schemes finite \\'etale over\n$X_i \\times_{S_i} \\Spec(A_i^h)$ (by\nLimits, Lemmas\n\\ref{limits-lemma-descend-finite-presentation},\n\\ref{limits-lemma-descend-finite-finite-presentation}, and\n\\ref{limits-lemma-descend-etale})\nThe same thing is true for schemes finite \\'etale over\n$X_0 = \\lim (X_i \\times_{S_i} s_i)$.\nThus we formally deduce the result for $X / \\Spec(A)$\nfrom the result for the $(X_i \\times_{S_i} \\Spec(A_i^h)) / \\Spec(A_i^h)$\nwhich we dealt with above.\n\\end{proof}\n\n\\begin{lemma}\n\\label{lemma-finite-etale-invariant-over-proper}\nLet $k \\subset k'$ be an extension of algebraically closed fields.\nLet $X$ be a proper scheme over $k$. Then the functor\n$$\nU \\longmapsto U_{k'}\n$$\nis an equivalence of categories between schemes finite \\'etale over\n$X$ and schemes finite \\'etale over $X_{k'}$.\n\\end{lemma}\n\n\\begin{proof}\nLet us prove the functor is essentially surjective.\nLet $U' \\to X_{k'}$ be a finite \\'etale morphism.\nWrite $k' = \\colim A_i$ as a filtered colimit of finite type $k$-algebras.\nBy Limits, Lemma \\ref{limits-lemma-descend-finite-presentation}\nthere exists an $i$ and a finitely presented morphism $U_i \\to X_{A_i}$\nwhose base change to $X_{k'}$ is $U'$. After increasing $i$\nwe may assume that $U_i \\to X_{A_i}$ is finite and \\'etale\n(Limits, Lemmas \\ref{limits-lemma-descend-finite-finite-presentation} and\n\\ref{limits-lemma-descend-etale}).\nSince $k$ is algebraically closed we can find a\n$k$-valued point $t$ in $\\Spec(A_i)$. Let $U = (U_i)_t$ be the\nfibre of $U_i$ over $t$. Let $A_i^h$ be the\nhenselization of $(A_i)_{\\mathfrak m}$ where $\\mathfrak m$ is\nthe maximal ideal corresponding to the point $t$. By\nLemma \\ref{lemma-finite-etale-on-proper-over-henselian}\nwe see that $(U_i)_{A_i^h} = U \\times \\Spec(A_i^h)$ as schemes\nover $X_{A_i^h}$. Now since\n$A_i^h$ is algebraic over $A_i$ (see for example discussion in\nSmoothing Ring Maps, Example \\ref{smoothing-example-describe-henselian})\nand since $k'$ is algebraically closed\nwe can find a ring map $A_i^h \\to k'$ extending the given\ninclusion $A_i \\subset k'$. Hence we conclude that $U'$\nis isomorphic to the base change of $U$.\nThe proof of fully faithfulness is exactly the same.\n\\end{proof}\n\n\n\n\n\n\n\n\n\\section{Local connectedness}\n\\label{section-unibranch}\n\n\\noindent\nIn this section we ask when $\\pi_1(U) \\to \\pi_1(X)$ is surjective\nfor $U$ a dense open of a scheme $X$. We will see that this is the\ncase (roughly) when $U \\cap B$ is connected for any small\n``ball'' $B$ around a point $x \\in X \\setminus U$.\n\n\\begin{lemma}\n\\label{lemma-dense-faithful}\nLet $f : X \\to Y$ be a morphism of schemes. If $f(X)$ is dense in $Y$\nthen the base change functor $\\textit{F\\'Et}_Y \\to \\textit{F\\'Et}_X$\nis faithful.\n\\end{lemma}\n\n\\begin{proof}\nSince the category of finite \\'etale coverings has an\ninternal hom (Lemma \\ref{lemma-internal-hom-finite-etale})\nit suffices to prove the following: Given $W$ finite \\'etale over $Y$\nand a morphism $s : X \\to W$ over $X$ there is at most one section\n$t : Y \\to W$ such that $s = t \\circ f$. Consider two sections\n$t_1, t_2 : Y \\to W$ such that $s = t_1 \\circ f = t_2 \\circ f$.\nSince the equalizer of $t_1$ and $t_2$ is closed in $Y$\n(Schemes, Lemma \\ref{schemes-lemma-where-are-they-equal})\nand since $f(X)$ is dense in $Y$ we see that $t_1$ and $t_2$\nagree on $Y_{red}$. Then it follows that $t_1$ and $t_2$ have\nthe same image which is an open and closed subscheme of $W$ mapping\nisomorphically to $Y$\n(\\'Etale Morphisms, Proposition \\ref{etale-proposition-properties-sections})\nhence they are equal.\n\\end{proof}\n\n\\noindent\nThe condition in the following lemma that the punctured spectrum\nof the strict henselization is connected follows for example from\nthe assumption that the local ring is geometrically unibranch, see\nMore on Algebra, Lemma \\ref{more-algebra-lemma-geometrically-unibranch}.\nThere is a partial converse in\nProperties, Lemma \\ref{properties-lemma-geometrically-unibranch}.\n\n\\begin{lemma}\n\\label{lemma-same-etale-extensions}\nLet $(A, \\mathfrak m)$ be a local ring. Set $X = \\Spec(A)$\nand let $U = X \\setminus \\{\\mathfrak m\\}$. If the punctured spectrum\nof the strict henselization of $A$ is connected, then\n$$\n\\textit{F\\'Et}_X \\longrightarrow \\textit{F\\'Et}_U,\\quad\nY \\longmapsto Y \\times_X U\n$$\nis a fully faithful functor.\n\\end{lemma}\n\n\\begin{proof}\nAssume $A$ is strictly henselian. In this case any finite \\'etale\ncover $Y$ of $X$ is isomorphic to a finite disjoint union of\ncopies of $X$. Thus it suffices to prove that any morphism\n$U \\to U \\amalg \\ldots \\amalg U$ over $U$, extends uniquely to a morphism\n$X \\to X \\amalg \\ldots \\amalg X$ over $X$.\nIf $U$ is connected (in particular nonempty), then this is true.\n\n\\medskip\\noindent\nThe general case. Since the category of finite \\'etale coverings has an\ninternal hom (Lemma \\ref{lemma-internal-hom-finite-etale})\nit suffices to prove the following: Given $Y$ finite \\'etale over $X$\nany morphism $s : U \\to Y$ over $X$ extends to a morphism $t : X \\to Y$\nover $Y$. Let $A^{sh}$ be the strict henselization of $A$ and denote\n$X^{sh} = \\Spec(A^{sh})$, $U^{sh} = U \\times_X X^{sh}$,\n$Y^{sh} = Y \\times_X X^{sh}$. By the first paragraph and our assumption\non $A$, we can extend the base change $s^{sh} : U^{sh} \\to Y^{sh}$ of $s$ to\n$t^{sh} : X^{sh} \\to Y^{sh}$. Set $A' = A^{sh} \\otimes_A A^{sh}$.\nThen the two pullbacks $t'_1, t'_2$ of $t^{sh}$ to $X' = \\Spec(A')$\nare extensions of the pullback $s'$ of $s$ to $U' = U \\times_X X'$.\nAs $A \\to A'$ is flat we see that $U' \\subset X'$ is (topologically) dense\nby going down for $A \\to A'$\n(Algebra, Lemma \\ref{algebra-lemma-flat-going-down}). Thus\n$t'_1 = t'_2$ by Lemma \\ref{lemma-dense-faithful}.\nHence $t^{sh}$ descends to a morphism $t : X \\to Y$\nfor example by\nDescent, Lemma \\ref{descent-lemma-fpqc-universal-effective-epimorphisms}.\n\\end{proof}\n\n\\noindent\nIn view of Lemma \\ref{lemma-same-etale-extensions}\nit is interesting to know when the\npunctured spectrum of a ring (and of its strict henselization)\nis connected. The following famous lemma due to Hartshorne\ngives a sufficient condition.\n\n\\begin{lemma}\n\\label{lemma-depth-2-connected-punctured-spectrum}\n\\begin{reference}\n\\cite[Proposition 2.1]{Hartshorne-connectedness}\n\\end{reference}\n\\begin{slogan}\nHartshorne's connectedness\n\\end{slogan}\nLet $A$ be a Noetherian local ring of depth $\\geq 2$.\nThen the punctured spectra of $A$, $A^h$, and $A^{sh}$ are connected.\n\\end{lemma}\n\n\\begin{proof}\nLet $U$ be the punctured spectrum of $A$.\nIf $U$ is disconnected then we see that\n$\\Gamma(U, \\mathcal{O}_U)$ has a nontrivial idempotent.\nBut $A$, being local, does not have a nontrivial idempotent.\nHence $A \\to \\Gamma(U, \\mathcal{O}_U)$ is not an isomorphism.\nBy Local Cohomology, Lemma\n\\ref{local-cohomology-lemma-finiteness-pushforwards-and-H1-local}\nwe conclude that either $H^0_\\mathfrak m(A)$ or $H^1_\\mathfrak m(A)$\nis nonzero. Thus $\\text{depth}(A) \\leq 1$ by\nDualizing Complexes, Lemma \\ref{dualizing-lemma-depth}.\nTo see the result for $A^h$ and $A^{sh}$ use\nMore on Algebra, Lemma \\ref{more-algebra-lemma-henselization-depth}.\n\\end{proof}\n\n\\begin{lemma}\n\\label{lemma-quasi-compact-dense-open-connected-at-infinity-Noetherian}\nLet $X$ be a scheme. Let $U \\subset X$ be a dense open. Assume\n\\begin{enumerate}\n\\item the underlying topological space of $X$ is Noetherian, and\n\\item for every $x \\in X \\setminus U$ the punctured spectrum of the\nstrict henselization of $\\mathcal{O}_{X, x}$ is connected.\n\\end{enumerate}\nThen $\\textit{F\\'Et}_X \\to \\textit{F\\'et}_U$ is fully faithful.\n\\end{lemma}\n\n\\begin{proof}\nLet $Y_1, Y_2$ be finite \\'etale over $X$ and let\n$\\varphi : (Y_1)_U \\to (Y_2)_U$ be a morphism over $U$. We have to show that\n$\\varphi$ lifts uniquely to a morphism $Y_1 \\to Y_2$ over $X$.\nUniqueness follows from Lemma \\ref{lemma-dense-faithful}.\n\n\\medskip\\noindent\nLet $x \\in X \\setminus U$ be a generic point of an irreducible component\nof $X \\setminus U$. Set $V = U \\times_X \\Spec(\\mathcal{O}_{X, x})$.\nBy our choice of $x$ this is the punctured spectrum of\n$\\Spec(\\mathcal{O}_{X, x})$. By\nLemma \\ref{lemma-same-etale-extensions}\nwe can extend the morphism $\\varphi_V : (Y_1)_V \\to (Y_2)_V$\nuniquely to a morphism\n$(Y_1)_{\\Spec(\\mathcal{O}_{X, x})} \\to (Y_2)_{\\Spec(\\mathcal{O}_{X, x})}$.\nBy Limits, Lemma \\ref{limits-lemma-glueing-near-point}\nwe find an open $U \\subset U'$ containing $x$ and an extension\n$\\varphi' : (Y_1)_{U'} \\to (Y_2)_{U'}$ of $\\varphi$.\nSince the underlying topological space of $X$ is Noetherian\nthis finishes the proof by Noetherian induction on the complement\nof the open over which $\\varphi$ is defined.\n\\end{proof}\n\n\\begin{lemma}\n\\label{lemma-retrocompact-dense-open-connected-at-infinity-closed}\nLet $X$ be a scheme. Let $U \\subset X$ be a dense open. Assume\n\\begin{enumerate}\n\\item $U \\to X$ is quasi-compact,\n\\item every point of $X \\setminus U$ is closed, and\n\\item for every $x \\in X \\setminus U$ the punctured spectrum of the\nstrict henselization of $\\mathcal{O}_{X, x}$ is connected.\n\\end{enumerate}\nThen $\\textit{F\\'Et}_X \\to \\textit{F\\'et}_U$ is fully faithful.\n\\end{lemma}\n\n\\begin{proof}\nLet $Y_1, Y_2$ be finite \\'etale over $X$ and let\n$\\varphi : (Y_1)_U \\to (Y_2)_U$ be a morphism over $U$. We have to show that\n$\\varphi$ lifts uniquely to a morphism $Y_1 \\to Y_2$ over $X$.\nUniqueness follows from Lemma \\ref{lemma-dense-faithful}.\n\n\\medskip\\noindent\nLet $x \\in X \\setminus U$. Set $V = U \\times_X \\Spec(\\mathcal{O}_{X, x})$.\nSince every point of $X \\setminus U$ is closed $V$ is the punctured spectrum\nof $\\Spec(\\mathcal{O}_{X, x})$. By\nLemma \\ref{lemma-same-etale-extensions}\nwe can extend the morphism $\\varphi_V : (Y_1)_V \\to (Y_2)_V$\nuniquely to a morphism\n$(Y_1)_{\\Spec(\\mathcal{O}_{X, x})} \\to (Y_2)_{\\Spec(\\mathcal{O}_{X, x})}$.\nBy Limits, Lemma \\ref{limits-lemma-glueing-near-point}\n(this uses that $U$ is retrocompact in $X$)\nwe find an open $U \\subset U'_x$ containing $x$ and an extension\n$\\varphi'_x : (Y_1)_{U'_x} \\to (Y_2)_{U'_x}$ of $\\varphi$.\nNote that given two points $x, x' \\in X \\setminus U$ the\nmorphisms $\\varphi'_x$ and $\\varphi'_{x'}$ agree over\n$U'_x \\cap U'_{x'}$ as $U$ is dense in that open\n(Lemma \\ref{lemma-dense-faithful}). Thus we can extend $\\varphi$\nto $\\bigcup U'_x = X$ as desired.\n\\end{proof}\n\n\\begin{lemma}\n\\label{lemma-quasi-compact-dense-open-connected-at-infinity}\nLet $X$ be a scheme. Let $U \\subset X$ be a dense open. Assume\n\\begin{enumerate}\n\\item every quasi-compact open of $X$ has finitely many\nirreducible components,\n\\item for every $x \\in X \\setminus U$ the punctured spectrum of the\nstrict henselization of $\\mathcal{O}_{X, x}$ is connected.\n\\end{enumerate}\nThen $\\textit{F\\'Et}_X \\to \\textit{F\\'et}_U$ is fully faithful.\n\\end{lemma}\n\n\\begin{proof}\nLet $Y_1, Y_2$ be finite \\'etale over $X$ and let\n$\\varphi : (Y_1)_U \\to (Y_2)_U$ be a morphism over $U$. We have to show that\n$\\varphi$ lifts uniquely to a morphism $Y_1 \\to Y_2$ over $X$.\nUniqueness follows from Lemma \\ref{lemma-dense-faithful}.\nWe will prove existence by showing that we can enlarge $U$\nif $U \\not = X$ and using Zorn's lemma to finish the proof.\n\n\\medskip\\noindent\nLet $x \\in X \\setminus U$ be a generic point of an irreducible component\nof $X \\setminus U$. Set $V = U \\times_X \\Spec(\\mathcal{O}_{X, x})$.\nBy our choice of $x$ this is the punctured spectrum of\n$\\Spec(\\mathcal{O}_{X, x})$. By\nLemma \\ref{lemma-same-etale-extensions}\nwe can extend the morphism $\\varphi_V : (Y_1)_V \\to (Y_2)_V$\n(uniquely) to a morphism\n$(Y_1)_{\\Spec(\\mathcal{O}_{X, x})} \\to (Y_2)_{\\Spec(\\mathcal{O}_{X, x})}$.\nChoose an affine neighbourhood $W \\subset X$ of $x$.\nSince $U \\cap W$ is dense in $W$ it contains the generic points\n$\\eta_1, \\ldots, \\eta_n$ of $W$. Choose an affine open\n$W' \\subset W \\cap U$ containing $\\eta_1, \\ldots, \\eta_n$.\nSet $V' = W' \\times_X \\Spec(\\mathcal{O}_{X, x})$.\nBy Limits, Lemma \\ref{limits-lemma-glueing-near-point}\napplied to $x \\in W \\supset W'$\nwe find an open $W' \\subset W'' \\subset W$ with $x \\in W''$\nand a morphism $\\varphi'' : (Y_1)_{W''} \\to (Y_2)_{W''}$\nagreeing with $\\varphi$ over $W'$. Since $W'$ is dense in\n$W'' \\cap U$, we see by Lemma \\ref{lemma-dense-faithful}\nthat $\\varphi$ and $\\varphi''$ agree over $U \\cap W'$.\nThus $\\varphi$ and $\\varphi''$ glue to a morphism\n$\\varphi'$ over $U' = U \\cup W''$ agreeing with $\\varphi$ over $U$.\nObserve that $x \\in U'$ so that we've extended $\\varphi$\nto a strictly larger open.\n\n\\medskip\\noindent\nConsider the set $\\mathcal{S}$ of pairs $(U', \\varphi')$ where $U \\subset U'$\nand $\\varphi'$ is an extension of $\\varphi$. We endow $\\mathcal{S}$\nwith a partial ordering in the obvious manner. If $(U'_i, \\varphi'_i)$\nis a totally ordered subset, then it has a maximum $(U', \\varphi')$.\nJust take $U' = \\bigcup U'_i$ and let\n$\\varphi' : (Y_1)_{U'} \\to (Y_2)_{U'}$ be the morphism\nagreeing with $\\varphi'_i$ over $U'_i$. Thus Zorn's lemma applies\nand $\\mathcal{S}$ has a maximal element. By the argument above\nwe see that this maximal element is an extension of $\\varphi$\nover all of $X$.\n\\end{proof}\n\n\\begin{lemma}\n\\label{lemma-local-exact-sequence}\nLet $(A, \\mathfrak m)$ be a local ring. Set $X = \\Spec(A)$ and\n$U = X \\setminus \\{\\mathfrak m\\}$. Let $U^{sh}$ be the punctured spectrum\nof the strict henselization $A^{sh}$ of $A$.\nAssume $U$ is quasi-compact and $U^{sh}$ is connected. Then the sequence\n$$\n\\pi_1(U^{sh}, \\overline{u}) \\to \\pi_1(U, \\overline{u}) \\to\n\\pi_1(X, \\overline{u}) \\to 1\n$$\nis exact in the sense of Lemma \\ref{lemma-functoriality-galois-ses} part (1).\n\\end{lemma}\n\n\\begin{proof}\nThe map $\\pi_1(U) \\to \\pi_1(X)$ is surjective by\nLemmas \\ref{lemma-same-etale-extensions} and\n\\ref{lemma-functoriality-galois-surjective}.\n\n\\medskip\\noindent\nWrite $X^{sh} = \\Spec(A^{sh})$. Let $Y \\to X$ be a finite \\'etale morphism.\nThen $Y^{sh} = Y \\times_X X^{sh} \\to X^{sh}$ is a finite \\'etale morphism.\nSince $A^{sh}$ is strictly henselian we see that $Y^{sh}$ is isomorphic\nto a disjoint union of copies of $X^{sh}$. Thus the same is true for\n$Y \\times_X U^{sh}$. It follows that the composition\n$\\pi_1(U^{sh}) \\to \\pi_1(U) \\to \\pi_1(X)$ is trivial, see\nLemma \\ref{lemma-composition-trivial}.\n\n\\medskip\\noindent\nTo finish the proof, it suffices according to\nLemma \\ref{lemma-functoriality-galois-ses}\nto show the following: Given a finite \\'etale morphism\n$V \\to U$ such that $V \\times_U U^{sh}$ is a disjoint\nunion of copies of $U^{sh}$, we can find a finite \\'etale\nmorphism $Y \\to X$ with $V \\cong Y \\times_X U$ over $U$.\nThe assumption implies that there exists a finite \\'etale\nmorphism $Y^{sh} \\to X^{sh}$ and an isomorphism\n$V \\times_U U^{sh} \\cong Y^{sh} \\times_{X^{sh}} U^{sh}$.\nConsider the following diagram\n$$\n\\xymatrix{\nU \\ar[d] & U^{sh} \\ar[d] \\ar[l] &\nU^{sh} \\times_U U^{sh} \\ar[d] \\ar@<1ex>[l] \\ar@<-1ex>[l] &\nU^{sh} \\times_U U^{sh} \\times_U U^{sh}\n\\ar[d] \\ar@<1ex>[l] \\ar[l] \\ar@<-1ex>[l] \\\\\nX & X^{sh} \\ar[l] &\nX^{sh} \\times_X X^{sh} \\ar@<1ex>[l] \\ar@<-1ex>[l] &\nX^{sh} \\times_X X^{sh} \\times_X X^{sh} \\ar@<1ex>[l] \\ar[l] \\ar@<-1ex>[l]\n}\n$$\nSince $U \\subset X$ is quasi-compact by assumption, all the\ndownward arrows are quasi-compact open immersions.\nLet $\\xi \\in X^{sh} \\times_X X^{sh}$ be a point not\nin $U^{sh} \\times_U U^{sh}$. Then $\\xi$ lies over the closed\npoint $x^{sh}$ of $X^{sh}$.\nConsider the local ring homomorphism\n$$\nA^{sh} = \\mathcal{O}_{X^{sh}, x^{sh}} \\to\n\\mathcal{O}_{X^{sh} \\times_X X^{sh}, \\xi}\n$$\ndetermined by the first projection $X^{sh} \\times_X X^{sh}$.\nThis is a filtered colimit of local homomorphisms which are\nlocalizations \\'etale ring maps.\nSince $A^{sh}$ is strictly henselian, we conclude that it is an\nisomorphism. Since this holds for every $\\xi$ in the complement\nit follows there are no specializations among these points and\nhence every such $\\xi$ is a closed point (you can also prove\nthis directly). As the local ring at $\\xi$ is isomorphic\nto $A^{sh}$, it is strictly henselian and has connected punctured spectrum.\nSimilarly for points $\\xi$ of $X^{sh} \\times_X X^{sh} \\times_X X^{sh}$ not\nin $U^{sh} \\times_U U^{sh} \\times_U U^{sh}$. It follows from\nLemma \\ref{lemma-retrocompact-dense-open-connected-at-infinity-closed}\nthat pullback along the vertical arrows induce fully faithful functors on\nthe categories of finite \\'etale schemes. Thus the\ncanonical descent datum on $V \\times_U U^{sh}$ relative to\nthe fpqc covering $\\{U^{sh} \\to U\\}$ translates into a\ndescent datum for $Y^{sh}$ relative to the fpqc covering $\\{X^{sh} \\to X\\}$.\nSince $Y^{sh} \\to X^{sh}$ is finite hence affine, this descent datum is\neffective (Descent, Lemma \\ref{descent-lemma-affine}).\nThus we get an affine morphism $Y \\to X$ and an isomorphism\n$Y \\times_X X^{sh} \\to Y^{sh}$ compatible with descent data.\nBy fully faithfulness of descent data\n(as in Descent, Lemma \\ref{descent-lemma-refine-coverings-fully-faithful})\nwe get an isomorphism $V \\to U \\times_X Y$.\nFinally, $Y \\to X$ is finite \\'etale as $Y^{sh} \\to X^{sh}$ is, see\nDescent, Lemmas \\ref{descent-lemma-descending-property-etale} and\n\\ref{descent-lemma-descending-property-finite}.\n\\end{proof}\n\n\\noindent\nLet $X$ be an irreducible scheme. Let $\\eta \\in X$ be the geometric\npoint. The canonical morphism $\\eta \\to X$ induces a canonical map\n\\begin{equation}\n\\label{equation-inclusion-generic-point}\n\\text{Gal}(\\kappa(\\eta)^{sep}/\\kappa(\\eta)) = \\pi_1(\\eta, \\overline{\\eta})\n\\longrightarrow \\pi_1(X, \\overline{\\eta})\n\\end{equation}\nThe identification on the left hand side is\nLemma \\ref{lemma-fundamental-group-Galois-group}.\n\n\\begin{lemma}\n\\label{lemma-irreducible-geometrically-unibranch}\nLet $X$ be an irreducible, geometrically unibranch scheme.\nFor any nonempty open $U \\subset X$ the canonical map\n$$\n\\pi_1(U, \\overline{u}) \\longrightarrow \\pi_1(X, \\overline{u})\n$$\nis surjective. The map (\\ref{equation-inclusion-generic-point})\n$\\pi_1(\\eta, \\overline{\\eta}) \\to \\pi_1(X, \\overline{\\eta})$\nis surjective as well.\n\\end{lemma}\n\n\\begin{proof}\nBy Lemma \\ref{lemma-thickening} we may replace $X$ by its reduction.\nThus we may assume that $X$ is an integral scheme. By\nLemma \\ref{lemma-functoriality-galois-surjective}\nthe assertion of the lemma translates into the statement that\nthe functors $\\textit{F\\'Et}_X \\to \\textit{F\\'Et}_U$ and\n$\\textit{F\\'Et}_X \\to \\textit{F\\'Et}_\\eta$ are fully faithful.\n\n\\medskip\\noindent\nThe result for $\\textit{F\\'Et}_X \\to \\textit{F\\'Et}_U$ follows\nfrom Lemma \\ref{lemma-quasi-compact-dense-open-connected-at-infinity}\nand the fact that for a local ring $A$ which is\ngeometrically unibranch its strict henselization has an\nirreducible spectrum. See\nMore on Algebra, Lemma \\ref{more-algebra-lemma-geometrically-unibranch}.\n\n\\medskip\\noindent\nObserve that the residue field $\\kappa(\\eta) = \\mathcal{O}_{X, \\eta}$\nis the filtered colimit of $\\mathcal{O}_X(U)$ over $U \\subset X$\nnonempty open affine. Hence $\\textit{F\\'Et}_\\eta$ is the colimit of the\ncategories $\\textit{F\\'Et}_U$ over such $U$, see\nLimits, Lemmas \\ref{limits-lemma-descend-finite-presentation},\n\\ref{limits-lemma-descend-finite-finite-presentation}, and\n\\ref{limits-lemma-descend-etale}.\nA formal argument then shows that fully faithfulness for\n$\\textit{F\\'Et}_X \\to \\textit{F\\'Et}_\\eta$ follows from the\nfully faithfulness of the functors $\\textit{F\\'Et}_X \\to \\textit{F\\'Et}_U$.\n\\end{proof}\n\n\\begin{lemma}\n\\label{lemma-exact-sequence-finite-nr-closed-pts}\nLet $X$ be a scheme. Let $x_1, \\ldots, x_n \\in X$ be a finite\nnumber of closed points such that\n\\begin{enumerate}\n\\item $U = X \\setminus \\{x_1, \\ldots, x_n\\}$ is connected and is\na retrocompact open of $X$, and\n\\item for each $i$ the punctured spectrum $U_i^{sh}$ of the\nstrict henselization of $\\mathcal{O}_{X, x_i}$ is connected.\n\\end{enumerate}\nThen the map $\\pi_1(U) \\to \\pi_1(X)$ is surjective and the kernel\nis the smallest closed normal subgroup of $\\pi_1(U)$ containing\nthe image of $\\pi_1(U_i^{sh}) \\to \\pi_1(U)$ for $i = 1, \\ldots, n$.\n\\end{lemma}\n\n\\begin{proof}\nSurjectivity follows from\nLemmas \\ref{lemma-retrocompact-dense-open-connected-at-infinity-closed} and\n\\ref{lemma-functoriality-galois-surjective}.\nWe can consider the sequence of maps\n$$\n\\pi_1(U)  \\to \\ldots \\to\n\\pi_1(X \\setminus \\{x_1, x_2\\}) \\to \\pi_1(X \\setminus \\{x_1\\}) \\to \\pi_1(X)\n$$\nA group theory argument then shows it suffices to prove the statement on the\nkernel in the case $n = 1$ (details omitted). Write\n$x = x_1$, $U^{sh} = U_1^{sh}$,\nset $A = \\mathcal{O}_{X, x}$, and let $A^{sh}$ be the strict henselization.\nConsider the diagram\n$$\n\\xymatrix{\nU \\ar[d] &\n\\Spec(A) \\setminus \\{\\mathfrak m\\} \\ar[l] \\ar[d] &\nU^{sh} \\ar[d] \\ar[l] \\\\\nX & \\Spec(A) \\ar[l] & \\Spec(A^{sh}) \\ar[l]\n}\n$$\nBy Lemma \\ref{lemma-functoriality-galois-ses}\nwe have to show finite \\'etale morphisms\n$V \\to U$ which pull back to trivial coverings of $U^{sh}$\nextend to finite \\'etale schemes over $X$.\nBy Lemma \\ref{lemma-local-exact-sequence}\nwe know the corresponding statement\nfor finite \\'etale schemes over the punctured spectrum of $A$.\nHowever, by Limits, Lemma \\ref{limits-lemma-glueing-near-closed-point}\nschemes of finite presentation over $X$ are the same thing as\nschemes of finite presentation over $U$ and $A$ glued over\nthe punctured spectrum of $A$. This finishes the proof.\n\\end{proof}\n\n\n\n\n\n\n\n\n\n\\section{Fundamental groups of normal schemes}\n\\label{section-normal}\n\n\\noindent\nLet $X$ be an integral, geometrically unibranch scheme. In the previous section\nwe have seen that the fundamental group of $X$ is a quotient of the\nGalois group of the function field $K$ of $X$. Since the map is continuous\nthe kernel is a normal closed subgroup of the Galois group. Hence this kernel\ncorresponds to a Galois extension $M/K$ by Galois theory\n(Fields, Theorem \\ref{fields-theorem-inifinite-galois-theory}).\nIn this section we will determine $M$ when $X$ is a normal integral scheme.\n\n\\medskip\\noindent\nLet $X$ be an integral normal scheme with function field $K$.\nLet $K \\subset L$ be a finite extension. Consider the normalization\n$Y \\to X$ of $X$ in the morphism $\\Spec(L) \\to X$ as defined in\nMorphisms, Section \\ref{morphisms-section-normalization-X-in-Y}.\nWe will say (in this setting) that {\\it $X$ is unramified in $L$}\nif $Y \\to X$ is an unramified morphism of schemes. In\nLemma \\ref{lemma-unramified} we will elucidate this condition.\nObserve that the scheme theoretic fibre of $Y \\to X$ over $\\Spec(K)$\nis $\\Spec(L)$. Hence the field extension $L/K$ is separable if $X$ is\nunramified in $L$, see\nMorphisms, Lemmas \\ref{morphisms-lemma-unramified-over-field}.\n\n\\begin{lemma}\n\\label{lemma-unramified-in-L}\nIn the situation above the following are equivalent\n\\begin{enumerate}\n\\item $X$ is unramified in $L$,\n\\item $Y \\to X$ is \\'etale, and\n\\item $Y \\to X$ is finite \\'etale.\n\\end{enumerate}\n\\end{lemma}\n\n\\begin{proof}\nObserve that $Y \\to X$ is an integral morphism.\nIn each case the morphism $Y \\to X$ is locally of finite type\nby definition.\nHence we find that in each case the lemma is finite by\nMorphisms, Lemma \\ref{morphisms-lemma-finite-integral}.\nIn particular we see that (2) is equivalent to (3).\nAn \\'etale morphism is unramified, hence (2) implies (1).\n\n\\medskip\\noindent\nConversely, assume $Y \\to X$ is unramified. Let $x \\in X$.\nWe can choose an \\'etale neighbourhood $(U, u) \\to (X, x)$ such that\n$$\nY \\times_X U = \\coprod V_j \\longrightarrow U\n$$\nis a disjoint union of closed immersions, see\n\\'Etale Morphisms, Lemma \\ref{etale-lemma-finite-unramified-etale-local}.\nShrinking we may assume $U$ is quasi-compact.\nThen $U$ has finitely many irreducible components\n(Descent, Lemma \\ref{descent-lemma-locally-finite-nr-irred-local-fppf}).\nSince $U$ is normal\n(Descent, Lemma \\ref{descent-lemma-normal-local-smooth}) the\nirreducible components of $U$ are open and closed\n(Properties, Lemma \\ref{properties-lemma-normal-locally-finite-nr-irreducibles})\nand we may assume $U$ is irreducible. Then $U$ is an integral\nscheme whose generic point $\\xi$ maps to the generic point of $X$.\nOn the other hand, we know that $Y \\times_X U$\nis the normalization of $U$ in $\\Spec(L) \\times_X U$\nby More on Morphisms, Lemma\n\\ref{more-morphisms-lemma-normalization-smooth-localization}.\nEvery point of $\\Spec(L) \\times_X U$ maps to $\\xi$.\nThus every $V_j$ contains a point mapping to $\\xi$ by\nMorphisms, Lemma \\ref{morphisms-lemma-normalization-generic}.\nThus $V_j \\to U$ is an isomorphism as $U = \\overline{\\{\\xi\\}}$.\nThus $Y \\times_X U \\to U$ is \\'etale. By\nDescent, Lemma \\ref{descent-lemma-descending-property-etale}\nwe conclude that $Y \\to X$ is \\'etale over the\nimage of $U \\to X$ (an open neighbourhood of $x$).\n\\end{proof}\n\n\\begin{lemma}\n\\label{lemma-finite-etale-covering-normal-unramified}\nLet $X$ be a normal integral scheme with function field $K$.\nLet $Y \\to X$ be a finite \\'etale morphism. If $Y$ is connected,\nthen $Y$ is an integral normal scheme and $Y$ is the normalization\nof $X$ in the function field of $Y$.\n\\end{lemma}\n\n\\begin{proof}\nThe scheme $Y$ is normal by\nDescent, Lemma \\ref{descent-lemma-normal-local-smooth}.\nSince $Y \\to X$ is flat every generic point of $Y$ maps\nto the generic point of $X$ by\nMorphisms, Lemma \\ref{morphisms-lemma-generalizations-lift-flat}.\nSince $Y \\to X$ is finite we see that $Y$ has a finite number\nof irreducible components. Thus $Y$ is the disjoint union of\na finite number of integral normal schemes by\nProperties, Lemma \\ref{properties-lemma-normal-locally-finite-nr-irreducibles}.\nThus if $Y$ is connected, then $Y$ is an integral normal scheme.\n\n\\medskip\\noindent\nLet $L$ be the function field of $Y$ and let $Y' \\to X$ be the normalization\nof $X$ in $L$. By\nMorphisms, Lemma \\ref{morphisms-lemma-characterize-normalization}\nwe obtain a factorization $Y' \\to Y \\to X$ and $Y' \\to Y$ is\nthe normalization of $Y$ in $L$. Since $Y$ is normal it is clear\nthat $Y' = Y$ (this can also be deduced from\nMorphisms, Lemma \\ref{morphisms-lemma-finite-birational-over-normal}).\n\\end{proof}\n\n\\begin{proposition}\n\\label{proposition-normal}\nLet $X$ be a normal integral scheme with function field $K$.\nThen the canonical map (\\ref{equation-inclusion-generic-point})\n$$\n\\text{Gal}(K^{sep}/K) = \\pi_1(\\eta, \\overline{\\eta})\n\\longrightarrow \\pi_1(X, \\overline{\\eta})\n$$\nis identified with the quotient map\n$\\text{Gal}(K^{sep}/K) \\to \\text{Gal}(M/K)$ where $M \\subset K^{sep}$\nis the union of the finite subextensions $L$\nsuch that $X$ is unramified in $L$.\n\\end{proposition}\n\n\\begin{proof}\nThe normal scheme $X$ is geometrically unibranch\n(Properties, Lemma \\ref{properties-lemma-normal-geometrically-unibranch}).\nHence Lemma \\ref{lemma-irreducible-geometrically-unibranch} applies to $X$.\nThus $\\pi_1(\\eta, \\overline{\\eta}) \\to \\pi_1(X, \\overline{\\eta})$\nis surjective and top horizontal arrow of the commutative diagram\n$$\n\\xymatrix{\n\\textit{F\\'Et}_X \\ar[r] \\ar[d] \\ar[rd]_c & \\textit{F\\'Et}_\\eta \\ar[d] \\\\\n\\textit{Finite-}\\pi_1(X, \\overline{\\eta})\\textit{-sets} \\ar[r] &\n\\textit{Finite-}\\text{Gal}(K^{sep}/K)\\textit{-sets}\n}\n$$\nis fully faithful. The left vertical arrow is the equivalence of\nTheorem \\ref{theorem-fundamental-group}\nand the right vertical arrow is the equivalence of\nLemma \\ref{lemma-fundamental-group-Galois-group}. The lower\nhorizontal arrow is induced by the map of the proposition.\nBy Lemmas \\ref{lemma-unramified-in-L} and\n\\ref{lemma-finite-etale-covering-normal-unramified}\nwe see that the essential image of $c$\nconsists of $\\text{Gal}(K^{sep}/K)\\textit{-Sets}$ isomorphic\nto sets of the form\n$$\nS = \\Hom_K(\\prod\\nolimits_{i = 1, \\ldots, n} L_i, K^{sep}) =\n\\coprod\\nolimits_{i = 1, \\ldots, n} \\Hom_K(L_i, K^{sep})\n$$\nwith $L_i/K$ finite separable such that $X$ is unramified in $L_i$.\nThus if $M \\subset K^{sep}$ is as in the statement of the lemma,\nthen $\\text{Gal}(K^{sep}/M)$ is exactly the subgroup of\n$\\text{Gal}(K^{sep}/K)$ acting trivially on every object\nin the essential image of $c$. On the other hand, the essential image of $c$\nis exactly the category of $S$ such that the $\\text{Gal}(K^{sep}/K)$-action\nfactors through the surjection\n$\\text{Gal}(K^{sep}/K) \\to \\pi_1(X, \\overline{\\eta})$.\nWe conclude that $\\text{Gal}(K^{sep}/M)$ is the kernel.\nHence $\\text{Gal}(K^{sep}/M)$ is a normal subgroup, $M/K$ is Galois,\nand we have a short exact sequence\n$$\n1 \\to \\text{Gal}(K^{sep}/M) \\to\n\\text{Gal}(K^{sep}/K) \\to\n\\text{Gal}(M/K) \\to 1\n$$\nby Galois theory (Fields, Theorem\n\\ref{fields-theorem-inifinite-galois-theory} and\nLemma \\ref{fields-lemma-ses-infinite-galois}). The proof is done.\n\\end{proof}\n\n\\begin{lemma}\n\\label{lemma-local-exact-sequence-normal}\nLet $(A, \\mathfrak m)$ be a normal local ring.\nSet $X = \\Spec(A)$. Let $A^{sh}$ be the strict henselization of $A$.\nLet $K$ and $K^{sh}$ be the fraction fields of $A$ and $A^{sh}$.\nThen the sequence\n$$\n\\pi_1(\\Spec(K^{sh})) \\to \\pi_1(\\Spec(K)) \\to \\pi_1(X) \\to 1\n$$\nis exact in the sense of Lemma \\ref{lemma-functoriality-galois-ses} part (1).\n\\end{lemma}\n\n\\begin{proof}\nNote that $A^{sh}$ is a normal domain, see\nMore on Algebra, Lemma \\ref{more-algebra-lemma-henselization-normal}.\nThe map $\\pi_1(\\Spec(K)) \\to \\pi_1(X)$ is surjective by\nProposition \\ref{proposition-normal}.\n\n\\medskip\\noindent\nWrite $X^{sh} = \\Spec(A^{sh})$. Let $Y \\to X$ be a finite \\'etale morphism.\nThen $Y^{sh} = Y \\times_X X^{sh} \\to X^{sh}$ is a finite \\'etale morphism.\nSince $A^{sh}$ is strictly henselian we see that $Y^{sh}$ is isomorphic\nto a disjoint union of copies of $X^{sh}$. Thus the same is true for\n$Y \\times_X \\Spec(K^{sh})$. It follows that the composition\n$\\pi_1(\\Spec(K^{sh})) \\to \\pi_1(X)$ is trivial, see\nLemma \\ref{lemma-composition-trivial}.\n\n\\medskip\\noindent\nTo finish the proof, it suffices according to\nLemma \\ref{lemma-functoriality-galois-ses}\nto show the following: Given a finite \\'etale morphism\n$V \\to \\Spec(K)$ such that $V \\times_{\\Spec(K)} \\Spec(K^{sh})$\nis a disjoint union of copies of $\\Spec(K^{sh})$, we can find a\nfinite \\'etale morphism\n$Y \\to X$ with $V \\cong Y \\times_X \\Spec(K)$ over $\\Spec(K)$.\nWrite $V = \\Spec(L)$, so $L$ is a finite product of\nfinite separable extensions of $K$.\nLet $B \\subset L$ be the integral closure of $A$ in $L$.\nIf $A \\to B$ is \\'etale, then we can take $Y = \\Spec(B)$\nand the proof is complete. By\nAlgebra, Lemma \\ref{algebra-lemma-integral-closure-commutes-smooth}\n(and a limit argument we omit)\nwe see that $B \\otimes_A A^{sh}$ is the integral closure of\n$A^{sh}$ in $L^{sh} = L \\otimes_K K^{sh}$.\nOur assumption is that $L^{sh}$ is a product of copies of\n$K^{sh}$ and hence $B^{sh}$ is a product of copies of $A^{sh}$.\nThus $A^{sh} \\to B^{sh}$ is \\'etale. As $A \\to A^{sh}$ is\nfaithfully flat it follows that $A \\to B$ is \\'etale\n(Descent, Lemma \\ref{descent-lemma-descending-property-etale})\nas desired.\n\\end{proof}\n\n\n\n\n\n\n\n\\section{Group actions and integral closure}\n\\label{section-group-actions-integral}\n\n\\noindent\nIn this section we continue the discussion of\nMore on Algebra, Section \\ref{more-algebra-section-group-actions-integral}.\nRecall that a normal local ring is a domain by definition.\n\n\\begin{lemma}\n\\label{lemma-get-algebraic-closure}\nLet $A$ be a normal domain whose fraction field is separably algebraically\nclosed. Let $\\mathfrak p \\subset A$ be a nonzero prime ideal.\nThen the residue field $\\kappa(\\mathfrak p)$ is algebraically closed.\n\\end{lemma}\n\n\\begin{proof}\nAssume the lemma is not true to get a contradiction. Then there exists a\nmonic irreducible polynomial $P(T) \\in \\kappa(\\mathfrak p)[T]$ of\ndegree $d > 1$. After replacing $P$ by $a^d P(a^{-1}T)$ for suitable $a \\in A$\n(to clear denominators) we may assume that $P$ is the image of a\nmonic polynomial $Q$ in $A[T]$. Observe that $Q$ is irreducible in\n$f.f.(A)[T]$. Namely a factorization over $f.f.(A)$ leads to a factorization\nover $A$ by Algebra, Lemma \\ref{algebra-lemma-polynomials-divide}\nwhich we could reduce modulo $\\mathfrak p$ to get a factorization of $P$.\nAs $f.f.(A)$ is separably closed, $Q$ is not a separable polynomial\n(Fields, Definition \\ref{fields-definition-separable}).\nThen the characteristic of $f.f.(A)$ is $p > 0$ and $Q$ has\nvanishing linear term (Fields, Definition \\ref{fields-definition-separable}).\nHowever, then we can replace $Q$ by\n$Q + a T$ where $a \\in \\mathfrak p$ is nonzero to get a contradiction.\n\\end{proof}\n\n\\begin{lemma}\n\\label{lemma-normal-local-domain-separablly-closed-fraction-field}\nA normal local ring with separably closed fraction field is\nstrictly henselian.\n\\end{lemma}\n\n\\begin{proof}\nLet $(A, \\mathfrak m, \\kappa)$ be normal local with separably\nclosed fraction field $K$. If $A = K$, then we are done. If not,\nthen the residue field $\\kappa$ is algebraically closed\nby Lemma \\ref{lemma-get-algebraic-closure} and it suffices to\ncheck that $A$ is henselian.\nLet $f \\in A[T]$ be monic and let $a_0 \\in \\kappa$ be a root\nof multiplicity $1$ of the reduction $\\overline{f} \\in \\kappa[T]$.\nLet $f = \\prod f_i$ be the factorization in $K[T]$.\nBy Algebra, Lemma \\ref{algebra-lemma-polynomials-divide} we have\n$f_i \\in A[T]$. Thus $a_0$ is a root of $f_i$ for some $i$.\nAfter replacing $f$ by $f_i$ we may assume $f$ is irreducible.\nThen, since the derivative $f'$ cannot be zero in $A[T]$\nas $a_0$ is a single root, we conclude that $f$ is linear\ndue to the fact that $K$ is separably algebraically closed.\nThus $A$ is henselian, see\nAlgebra, Definition \\ref{algebra-definition-henselian}.\n\\end{proof}\n\n\\begin{lemma}\n\\label{lemma-inertia-base-change}\nLet $G$ be a finite group acting on a ring $R$. Let $R^G \\to A$ be a ring\nmap. Let $\\mathfrak q' \\subset A \\otimes_{R^G} R$ be a prime lying\nover the prime $\\mathfrak q \\subset R$. Then\n$$\nI_\\mathfrak q = \\{\\sigma \\in G \\mid\n\\sigma(\\mathfrak q) = \\mathfrak q\\text{ and }\n\\sigma \\bmod \\mathfrak q = \\text{id}_{\\kappa(\\mathfrak q)}\\}\n$$\nis equal to\n$$\nI_{\\mathfrak q'} = \\{\\sigma \\in G \\mid\n\\sigma(\\mathfrak q') = \\mathfrak q'\\text{ and }\n\\sigma \\bmod \\mathfrak q' = \\text{id}_{\\kappa(\\mathfrak q')}\\}\n$$\n\\end{lemma}\n\n\\begin{proof}\nSince $\\mathfrak q$ is the inverse image of $\\mathfrak q'$\nand since $\\kappa(\\mathfrak q) \\subset \\kappa(\\mathfrak q')$,\nwe get $I_{\\mathfrak q'} \\subset I_\\mathfrak q$.\nConversely, if $\\sigma \\in I_\\mathfrak q$, the $\\sigma$\nacts trivially on the fibre ring $A \\otimes_{R^G} \\kappa(\\mathfrak q)$.\nThus $\\sigma$ fixes all the primes lying over $\\mathfrak q$\nand induces the identity on their residue fields.\n\\end{proof}\n\n\\begin{lemma}\n\\label{lemma-inertia-invariants-etale}\nLet $G$ be a finite group acting on a ring $R$. Let $\\mathfrak q \\subset R$\nbe a prime. Set\n$$\nI = \\{\\sigma \\in G \\mid \\sigma(\\mathfrak q) = \\mathfrak q\n\\text{ and } \\sigma \\bmod \\mathfrak q = \\text{id}_\\mathfrak q\\}\n$$\nThen $R^G \\to R^I$ is \\'etale at $R^I \\cap \\mathfrak q$.\n\\end{lemma}\n\n\\begin{proof}\nThe strategy of the proof is to use \\'etale localization to\nreduce to the case where $R \\to R^I$ is a local isomorphism at\n$R^I \\cap \\mathfrak p$.\nLet $R^G \\to A$ be an \\'etale ring map. We claim that if the result\nholds for the action of $G$ on $A \\otimes_{R^G} R$ and some prime\n$\\mathfrak q'$ of $A \\otimes_{R^G} R$ lying over $\\mathfrak q$, then\nthe result is true.\n\n\\medskip\\noindent\nTo check this, note that since $R^G \\to A$ is flat we have\n$A = (A \\otimes_{R^G} R)^G$, see More on Algebra,\nLemma \\ref{more-algebra-lemma-base-change-invariants}.\nBy Lemma \\ref{lemma-inertia-base-change} the group $I$ does not change.\nThen a second application of More on Algebra,\nLemma \\ref{more-algebra-lemma-base-change-invariants}\nshows that $A \\otimes_{R^G} R^I = (A \\otimes_{R^G} R)^I$\n(because $R^I \\to A \\otimes_{R^G} R^I$ is flat).\nThus\n$$\n\\xymatrix{\n\\Spec((A \\otimes_{R^G} R)^I) \\ar[d] \\ar[r] & \\Spec(R^I) \\ar[d] \\\\\n\\Spec(A) \\ar[r] & \\Spec(R^G)\n}\n$$\nis cartesian and the horizontal arrows are \\'etale. Thus if the\nleft vertical arrow is \\'etale in some open neighbourhood $W$ of\n$(A \\otimes_{R^G} R)^I \\cap \\mathfrak q'$, then the right vertical\narrow is \\'etale at the points of the (open) image of $W$ in\n$\\Spec(R^I)$, see\nDescent, Lemma \\ref{descent-lemma-smooth-permanence}. In particular\nthe morphism $\\Spec(R^I) \\to \\Spec(R^G)$ is \\'etale at $R^I \\cap \\mathfrak q$.\n\n\\medskip\\noindent\nLet $\\mathfrak p = R^G \\cap \\mathfrak q$.\nBy More on Algebra, Lemma \\ref{more-algebra-lemma-one-orbit}\nthe fibre of $\\Spec(R) \\to \\Spec(R^G)$ over $\\mathfrak p$ is\nfinite. Moreover the residue field extensions at these points\nare algebraic, normal, with finite automorphism groups by\nMore on Algebra, Lemma \\ref{more-algebra-lemma-one-orbit-geometric}.\nThus we may apply\nMore on Morphisms,\nLemma \\ref{more-morphisms-lemma-etale-makes-integral-split}\nto the integral ring map $R^G \\to R$ and the prime $\\mathfrak p$.\nCombined with the claim above we reduce to the case where\n$R = A_1 \\times \\ldots \\times A_n$ with each $A_i$ having a single\nprime $\\mathfrak q_i$ lying over $\\mathfrak p$ such that the\nresidue field extensions $\\kappa(\\mathfrak q_i)/\\kappa(\\mathfrak p)$\nare purely inseparable. Of course $\\mathfrak q$ is one of\nthese primes, say $\\mathfrak q = \\mathfrak q_1$.\n\n\\medskip\\noindent\nIt may not be the case that $G$ permutes the factors $A_i$\n(this would be true if the spectrum of $A_i$ were connected,\nfor example if $R^G$ was local). This we can fix as follows;\nwe suggest the reader think this through for themselves, perhaps\nusing idempotents instead of topology.\nRecall that the product decomposition gives a corresponding\ndisjoint union decomposition of $\\Spec(R)$ by open and closed\nsubsets $U_i$. Since $G$ is finite, we can refine this covering\nby a finite disjoint union decomposition\n$\\Spec(R) = \\coprod_{j \\in J} W_j$ by open\nand closed subsets $W_j$, such that for all $j \\in J$ there exists\na $j' \\in J$ with $\\sigma(W_j) = W_{j'}$. The union of the\n$W_j$ not meeting $\\{\\mathfrak q_1, \\ldots, \\mathfrak q_n\\}$\nis a closed subset not meeting the fibre over $\\mathfrak p$\nhence maps to a closed subset of $\\Spec(R^G)$ not meeting\n$\\mathfrak p$ as $\\Spec(R) \\to \\Spec(R^G)$ is closed.\nHence after replacing $R^G$ by a principal localization\n(permissible by the claim) we may assume each $W_j$ meets\none of the points $\\mathfrak q_i$. Then we set $U_i = W_j$\nif $\\mathfrak q_i \\in W_j$. The corresponding product decomposition\n$R = A_1 \\times \\ldots \\times A_n$ is one\nwhere $G$ permutes the factors $A_i$.\n\n\\medskip\\noindent\nThus we may assume we have a product decomposition\n$R = A_1 \\times \\ldots \\times A_n$ compatible with $G$-action,\nwhere each $A_i$ has a single prime $\\mathfrak q_i$ lying\nover $\\mathfrak p$ and the field extensions\n$\\kappa(\\mathfrak q_i)/\\kappa(\\mathfrak p)$ are purely inseparable.\nWrite $A' = A_2 \\times \\ldots \\times A_n$ so that\n$$\nR = A_1 \\times A'\n$$\nSince $\\mathfrak q = \\mathfrak q_1$ we find that every\n$\\sigma \\in I$ preserves the product decomposition above.\nHence\n$$\nR^I = (A_1)^I \\times (A')^I\n$$\nObserve that $I = D = \\{\\sigma \\in G \\mid \\sigma(\\mathfrak q) = \\mathfrak q\\}$\nbecause $\\kappa(\\mathfrak q)/\\kappa(\\mathfrak p)$ is purely inseparable.\nSince the action of $G$ on primes over $\\mathfrak p$ is transitive\n(More on Algebra, Lemma \\ref{more-algebra-lemma-one-orbit})\nwe conclude that, the index of $I$ in $G$ is $n$ and we can write\n$G = eI \\amalg \\sigma_2I \\amalg \\ldots \\amalg \\sigma_nI$ so that\n$A_i = \\sigma_i(A_1)$ for $i = 2, \\ldots, n$. It follows that\n$$\nR^G = (A_1)^I.\n$$\nThus the map $R^G \\to R^I$ is \\'etale at $R^I \\cap \\mathfrak q$\nand the proof is complete.\n\\end{proof}\n\n\\noindent\nThe following lemma generalizes\nMore on Algebra, Lemma \\ref{more-algebra-lemma-inertial-invariants-unramified}.\n\n\\begin{lemma}\n\\label{lemma-inertial-invariants-unramified}\nLet $A$ be a normal domain with fraction field $K$.\nLet $L/K$ be a (possibly infinite) Galois extension.\nLet $G = \\text{Gal}(L/K)$ and let\n$B$ be the integral closure of $A$ in $L$.\nLet $\\mathfrak q \\subset B$. Set\n$$\nI = \\{\\sigma \\in G \\mid\n\\sigma(\\mathfrak q) = \\mathfrak q \\text{ and }\n\\sigma \\bmod \\mathfrak q = \\text{id}_{\\kappa(\\mathfrak q)}\\}\n$$\nThen $(B^I)_{B^I \\cap \\mathfrak q}$ is a filtered colimit\nof \\'etale $A$-algebras.\n\\end{lemma}\n\n\\begin{proof}\nWe can write $L$ as the filtered colimit of finite Galois extensions\nof $K$. Hence it suffices to prove this lemma in case $L/K$ is\na finite Galois extension, see\nAlgebra, Lemma \\ref{algebra-lemma-colimit-colimit-etale}.\nSince $A = B^G$ as $A$ is integrally\nclosed in $K = L^G$ the result follows from\nLemma \\ref{lemma-inertia-invariants-etale}.\n\\end{proof}\n\n\n\n\n\n\n\n\\section{Ramification theory}\n\\label{section-ramification}\n\n\\noindent\nIn this section we continue the discussion of\nMore on Algebra, Section \\ref{more-algebra-section-ramification}\nand we relate it to our discussion of the fundamental groups of schemes.\n\n\\medskip\\noindent\nLet $(A, \\mathfrak m, \\kappa)$ be a normal local ring with\nfraction field $K$. Choose a separable algebraic closure $K^{sep}$. Let\n$A^{sep}$ be the integral closure of $A$ in $K^{sep}$.\nChoose maximal ideal $\\mathfrak m^{sep} \\subset A^{sep}$.\nLet $A \\subset A^h \\subset A^{sh}$ be the henselization and strict\nhenselization. Observe that $A^h$ and $A^{sh}$ are normal rings as well\n(More on Algebra, Lemma \\ref{more-algebra-lemma-henselization-normal}).\nDenote $K^h$ and $K^{sh}$ their fraction fields.\nSince $(A^{sep})_{\\mathfrak m^{sep}}$ is strictly henselian by\nLemma \\ref{lemma-normal-local-domain-separablly-closed-fraction-field}\nwe can choose an $A$-algebra map $A^{sh} \\to (A^{sep})_{\\mathfrak m^{sep}}$.\nNamely, first choose a $\\kappa$-embedding\\footnote{This is possible\nbecause $\\kappa(\\mathfrak m^{sh})$ is a separable algebraic closure\nof $\\kappa$ and $\\kappa(\\mathfrak m^{sep})$ is an algebraic closure\nof $\\kappa$ by Lemma \\ref{lemma-get-algebraic-closure}.}\n$\\kappa(\\mathfrak m^{sh}) \\to \\kappa(\\mathfrak m^{sep})$ and\nthen extend (uniquely) to an $A$-algebra homomorphism by\nAlgebra, Lemma \\ref{algebra-lemma-strictly-henselian-functorial}.\nWe get the following diagram\n$$\n\\xymatrix{\nK^{sep} & K^{sh} \\ar[l] & K^h \\ar[l] & K \\ar[l] \\\\\n(A^{sep})_{\\mathfrak m^{sep}} \\ar[u] &\nA^{sh} \\ar[u] \\ar[l] &\nA^h \\ar[u] \\ar[l] &\nA \\ar[u] \\ar[l]\n}\n$$\nWe can take the fundamental groups of the spectra of these rings.\nOf course, since $K^{sep}$, $(A^{sep})_{\\mathfrak m^{sep}}$, and\n$A^{sh}$ are strictly henselian, for them we obtain trivial groups.\nThus the interesting part is the following\n\\begin{equation}\n\\label{equation-inertia-diagram-pione}\n\\vcenter{\n\\xymatrix{\n\\pi_1(U^{sh}) \\ar[r] \\ar[rd]_1 & \\pi_1(U^h) \\ar[d] \\ar[r] & \\pi_1(U) \\ar[d] \\\\\n& \\pi_1(X^h) \\ar[r] & \\pi_1(X)\n}\n}\n\\end{equation}\nHere $X^h$ and $X$ are the spectra of $A^h$ and $A$ and\n$U^{sh}$, $U^h$, $U$ are the spectra of $K^{sh}$, $K^h$, and $K$.\nThe label $1$ means that the map is trivial; this follows\nas it factors through the trivial group $\\pi_1(X^{sh})$.\nOn the other hand, the profinite group $G = \\text{Gal}(K^{sep}/K)$\nacts on $A^{sep}$ and we can make the following definitions\n$$\nD = \\{\\sigma \\in G \\mid \\sigma(\\mathfrak m^{sep}) = \\mathfrak m^{sep}\\}\n\\supset\nI = \\{\\sigma \\in D \\mid \\sigma \\bmod \\mathfrak m^{sep} =\n\\text{id}_{\\kappa(\\mathfrak m^{sep})}\\}\n$$\nThese groups are sometimes called the\n{\\it decomposition group} and the {\\it inertia group}\nespecially when $A$ is a discrete valuation ring.\n\n\\begin{lemma}\n\\label{lemma-identify-inertia}\nIn the situation described above, via the isomorphism\n$\\pi_1(U) = \\text{Gal}(K^{sep}/K)$ the diagram\n(\\ref{equation-inertia-diagram-pione})\ntranslates into the diagram\n$$\n\\xymatrix{\nI \\ar[r] \\ar[rd]_1 & D \\ar[d] \\ar[r] & \\text{Gal}(K^{sep}/K) \\ar[d] \\\\\n& \\text{Gal}(\\kappa(\\mathfrak m^{sh})/\\kappa) \\ar[r] & \\text{Gal}(M/K)\n}\n$$\nwhere $K^{sep}/M/K$ is the maximal subextension unramified\nwith respect to $A$. Moreover, the vertical arrows are surjective,\nthe kernel of the left vertical arrow is $I$ and the kernel of the\nright vertical arrow is\nthe smallest closed normal subgroup of $\\text{Gal}(K^{sep}/K)$\ncontaining $I$.\n\\end{lemma}\n\n\\begin{proof}\nBy construction the group $D$ acts on $(A^{sep})_{\\mathfrak m^{sep}}$\nover $A$. By the uniqueness of $A^{sh} \\to (A^{sep})_{\\mathfrak m^{sep}}$\ngiven the map on residue fields\n(Algebra, Lemma \\ref{algebra-lemma-strictly-henselian-functorial})\nwe see that the image of $A^{sh} \\to (A^{sep})_{\\mathfrak m^{sep}}$\nis contained in $((A^{sep})_{\\mathfrak m^{sep}})^I$.\nOn the other hand,\nLemma \\ref{lemma-inertial-invariants-unramified}\nshows that $((A^{sep})_{\\mathfrak m^{sep}})^I$\nis a filtered colimit of \\'etale extensions of $A$.\nSince $A^{sh}$ is the maximal such extension, we conclude\nthat $A^{sh} = ((A^{sep})_{\\mathfrak m^{sep}})^I$.\nHence $K^{sh} = (K^{sep})^I$.\n\n\\medskip\\noindent\nRecall that $I$ is the kernel of a surjective map\n$D \\to \\text{Aut}(\\kappa(\\mathfrak m^{sep})/\\kappa)$, see\nMore on Algebra, Lemma \\ref{more-algebra-lemma-one-orbit-geometric-galois}.\nWe have $\\text{Aut}(\\kappa(\\mathfrak m^{sep})/\\kappa) =\n\\text{Gal}(\\kappa(\\mathfrak m^{sh})/\\kappa)$\nas we have seen above that these fields are the algebraic\nand separable algebraic closures of $\\kappa$.\nOn the other hand, any automorphism of $A^{sh}$ over $A$\nis an automorphism of $A^{sh}$ over $A^h$ by the uniqueness\nin Algebra, Lemma \\ref{algebra-lemma-henselian-functorial}.\nFurthermore, $A^{sh}$ is the colimit of finite \\'etale\nextensions $A^h \\subset A'$ which correspond $1$-to-$1$\nwith finite separable extension $\\kappa'/\\kappa$, see\nAlgebra, Remark \\ref{algebra-remark-construct-sh-from-h}.\nThus\n$$\n\\text{Aut}(A^{sh}/A) = \\text{Aut}(A^{sh}/A^h) =\n\\text{Gal}(\\kappa(\\mathfrak m^{sh})/\\kappa)\n$$\nLet $\\kappa \\subset \\kappa'$ be a finite Galois extension with\nGalois group $G$. Let $A^h \\subset A'$ be the finite \\'etale extension\ncorresponding to $\\kappa \\subset \\kappa'$ by\nAlgebra, Lemma \\ref{algebra-lemma-henselian-cat-finite-etale}.\nThen it follows that\n$(A')^G = A^h$ by looking at fraction fields and degrees\n(small detail omitted). Taking the colimit we conclude that\n$(A^{sh})^{\\text{Gal}(\\kappa(\\mathfrak m^{sh})/\\kappa)} = A^h$.\nCombining all of the above, we find $A^h = ((A^{sep})_{\\mathfrak m^{sep}})^D$.\nHence $K^h = (K^{sep})^D$.\n\n\\medskip\\noindent\nSince $U$, $U^h$, $U^{sh}$ are the spectra of the fields\n$K$, $K^h$, $K^{sh}$ we see that the top lines of the diagrams\ncorrespond via\nLemma \\ref{lemma-fundamental-group-Galois-group}.\nBy Lemma \\ref{lemma-gabber} we have\n$\\pi_1(X^h) = \\text{Gal}(\\kappa(\\mathfrak m^{sh})/\\kappa)$.\nThe exactness of the sequence\n$1 \\to I \\to D \\to \\text{Gal}(\\kappa(\\mathfrak m^{sh})/\\kappa) \\to 1$\nwas pointed out above.\nBy Proposition \\ref{proposition-normal}\nwe see that $\\pi_1(X) = \\text{Gal}(M/K)$.\nFinally, the statement on the kernel of\n$\\text{Gal}(K^{sep}/K) \\to \\text{Gal}(M/K) = \\pi_1(X)$\nfollows from Lemma \\ref{lemma-local-exact-sequence-normal}.\nThis finishes the proof.\n\\end{proof}\n\n\\noindent\nLet $X$ be a normal integral scheme with function field $K$.\nLet $K^{sep}$ be a separable algebraic closure of $K$.\nLet $X^{sep} \\to X$ be the normalization of $X$ in $K^{sep}$.\nSince $G = \\text{Gal}(K^{sep}/K)$ acts on $K^{sep}$\nwe obtain a right action of $G$ on $X^{sep}$.\nFor $y \\in X^{sep}$ define\n$$\nD_y = \\{\\sigma \\in G \\mid \\sigma(y) = y\\} \\supset\nI_y = \\{\\sigma \\in D \\mid \\sigma \\bmod \\mathfrak m_y =\n\\text{id}_{\\kappa(y)} \\}\n$$\nsimilarly to the above. On the other hand, for $x \\in X$\nlet $\\mathcal{O}_{X, x}^{sh}$ be a strict henselization,\nlet $K_x^{sh}$ be the fraction field of $\\mathcal{O}_{X, x}^{sh}$\nand choose a $K$-embedding $K_x^{sh} \\to K^{sep}$.\n\n\\begin{lemma}\n\\label{lemma-normal-pione-quotient-inertia}\nLet $X$ be a normal integral scheme with function field $K$.\nWith notation as above, the following three subgroups of\n$\\text{Gal}(K^{sep}/K) = \\pi_1(\\Spec(K))$\nare equal\n\\begin{enumerate}\n\\item the kernel of the surjection\n$\\text{Gal}(K^{sep}/K) \\longrightarrow \\pi_1(X)$,\n\\item the smallest normal closed subgroup containing $I_y$\nfor all $y \\in X^{sep}$, and\n\\item the smallest normal closed subgroup containing\n$\\text{Gal}(K^{sep}/K_x^{sh})$ for all $x \\in  X$.\n\\end{enumerate}\n\\end{lemma}\n\n\\begin{proof}\nThe equivalence of (2) and (3) follows from\nLemma \\ref{lemma-identify-inertia}\nwhich tells us that $I_y$ is conjugate to $\\text{Gal}(K^{sep}/K_x^{sh})$\nif $y$ lies over $x$. By Lemma \\ref{lemma-local-exact-sequence-normal}\nwe see that $\\text{Gal}(K^{sep}/K_x^{sh})$ maps trivially to\n$\\pi_1(\\Spec(\\mathcal{O}_{X, x}))$ and therefore the subgroup\n$N \\subset G = \\text{Gal}(K^{sep}/K)$\nof (2) and (3) is contained in the kernel of\n$G \\longrightarrow \\pi_1(X)$.\n\n\\medskip\\noindent\nTo prove the other inclusion, since $N$ is normal, it suffices to prove:\ngiven $N \\subset U \\subset G$ with $U$ open normal,\nthe quotient map $G \\to G/U$ factors through $\\pi_1(X)$.\nIn other words, if $L/K$ is the Galois extension corresponding\nto $U$, then we have to show that $X$ is unramified in $L$\n(Section \\ref{section-normal}, especially\nProposition \\ref{proposition-normal}).\nIt suffices to do this when $X$ is affine (we do this\nso we can refer to algebra results in the rest of the proof).\nLet $Y \\to X$ be the normalization of $X$ in $L$.\nThe inclusion $L \\subset K^{sep}$ induces a morphism\n$\\pi : X^{sep} \\to Y$. For $y \\in X^{sep}$\nthe inertia group of $\\pi(y)$ in $\\text{Gal}(L/K)$\nis the image of $I_y$ in $\\text{Gal}(L/K)$; this follows\nfrom More on Algebra, Lemma\n\\ref{more-algebra-lemma-one-orbit-geometric-galois-compare}.\nSince $N \\subset U$ all these inertia groups are trivial.\nWe conclude that $Y \\to X$ is \\'etale by applying\nLemma \\ref{lemma-inertia-invariants-etale}.\n(Alternative: you can use Lemma \\ref{lemma-local-exact-sequence-normal}\nto see that the pullback of $Y$ to $\\Spec(\\mathcal{O}_{X, x})$ is\n\\'etale for all $x \\in X$ and then conclude from there\nwith a bit more work.)\n\\end{proof}\n\n\\begin{example}\n\\label{example-bigger-codim}\nLet $X$ be a normal integral Noetherian scheme with function field $K$.\nPurity of branch locus (see below) tells us that if $X$ is regular, then\nit suffices in Lemma \\ref{lemma-normal-pione-quotient-inertia}\nto consider the inertia groups $I = \\pi_1(\\Spec(K_x^{sh}))$\nfor points $x$ of codimension $1$ in $X$.\nIn general this is not enough however. Namely, let\n$Y = \\mathbf{A}_k^n = \\Spec(k[t_1, \\ldots, t_n])$\nwhere $k$ is a field not of characteristic $2$.\nLet $G = \\{\\pm 1\\}$ be the group of order $2$ acting on $Y$\nby multiplication on the coordinates. Set\n$$\nX = \\Spec(k[t_it_j, i, j \\in \\{1, \\ldots, n\\}])\n$$\nThe embedding $k[t_it_j] \\subset k[t_1, \\ldots, t_n]$\ndefines a degree $2$ morphism $Y \\to X$ which is unramified everywhere\nexcept over the maximal ideal $\\mathfrak m = (t_it_j)$\nwhich is a point of codimension $n$ in $X$.\n\\end{example}\n\n\\begin{lemma}\n\\label{lemma-unramified}\nLet $X$ be an integral normal scheme with function field $K$.\nLet $L/K$ be a finite extension. Let $Y \\to X$ be the normalization\nof $X$ in $L$. The following are equivalent\n\\begin{enumerate}\n\\item $X$ is unramified in $L$ as defined in Section \\ref{section-normal},\n\\item $Y \\to X$ is an unramified morphism of schemes,\n\\item $Y \\to X$ is an \\'etale morphism of schemes,\n\\item $Y \\to X$ is a finite \\'etale morphism of schemes,\n\\item for $x \\in X$ the projection\n$Y \\times_X \\Spec(\\mathcal{O}_{X, x}) \\to \\Spec(\\mathcal{O}_{X, x})$\nis unramified,\n\\item same as in (5) but with $\\mathcal{O}_{X, x}^h$,\n\\item same as in (5) but with $\\mathcal{O}_{X, x}^{sh}$,\n\\item for $x \\in X$ the scheme theoretic fibre $Y_x$\nis \\'etale over $x$ of degree $\\geq [L : K]$.\n\\end{enumerate}\nIf $L/K$ is Galois with Galois group $G$, then these are also\nequivalent to\n\\begin{enumerate}\n\\item[(9)] for $y \\in Y$ the group\n$I_y = \\{g \\in G \\mid g(y) = y\\text{ and }\ng \\bmod \\mathfrak m_y = \\text{id}_{\\kappa(y)}\\}$ is trivial.\n\\end{enumerate}\n\\end{lemma}\n\n\\begin{proof}\nThe equivalence of (1) and (2) is the definition of (1).\nThe equivalence of (2), (3), and (4) is Lemma \\ref{lemma-unramified-in-L}.\nIt is straightforward to prove that (4) $\\Rightarrow$ (5),\n(5) $\\Rightarrow$ (6), (6) $\\Rightarrow$ (7).\n\n\\medskip\\noindent\nAssume (7). Observe that $\\mathcal{O}_{X, x}^{sh}$ is a normal local domain\n(More on Algebra, Lemma \\ref{more-algebra-lemma-henselization-normal}).\nLet $L^{sh} = L \\otimes_K K_x^{sh}$ where $K_x^{sh}$ is the fraction field\nof $\\mathcal{O}_{X, x}^{sh}$. Then $L^{sh} = \\prod_{i = 1, \\ldots, n} L_i$\nwith $L_i/K_x^{sh}$ finite separable. By\nAlgebra, Lemma \\ref{algebra-lemma-integral-closure-commutes-smooth}\n(and a limit argument we omit)\nwe see that $Y \\times_X \\Spec(\\mathcal{O}_{X, x}^{sh})$\nis the integral closure of $\\Spec(\\mathcal{O}_{X, x}^{sh})$ in $L^{sh}$.\nHence by Lemma \\ref{lemma-unramified-in-L} (applied to the factors\n$L_i$ of $L^{sh}$) we see that\n$Y \\times_X \\Spec(\\mathcal{O}_{X, x}^{sh}) \\to \\Spec(\\mathcal{O}_{X, x}^{sh})$\nis finite \\'etale. Looking at the generic point we see that\nthe degree is equal to $[L : K]$ and hence we see that (8) is true.\n\n\\medskip\\noindent\nAssume (8). Assume that $x \\in X$ and that the scheme theoretic fibre $Y_x$\nis \\'etale over $x$ of degree $\\geq [L : K]$. Observe that this means\nthat $Y$ has $\\geq [L : K]$ geometric points lying over $x$.\nWe will show that $Y \\to X$ is finite \\'etale over a neighbourhood of $x$.\nThis will prove (1) holds.\nTo prove this we may assume $X = \\Spec(R)$, the point $x$ corresponds to\nthe prime $\\mathfrak p \\subset R$, and $Y = \\Spec(S)$. We apply\nMore on Morphisms,\nLemma \\ref{more-morphisms-lemma-etale-makes-integral-split} and we find an\n\\'etale neighbourhood $(U, u) \\to (X, x)$ such that\n$Y \\times_X U = V_1 \\amalg \\ldots \\amalg V_m$ such that $V_i$\nhas a unique point $v_i$ lying over $u$ with $\\kappa(v_i)/\\kappa(u)$\npurely inseparable. Shrinking $U$ if necessary we may assume $U$ is\na normal integral scheme with generic point $\\xi$ (use\nDescent, Lemmas \\ref{descent-lemma-locally-finite-nr-irred-local-fppf} and\n\\ref{descent-lemma-normal-local-smooth} and\nProperties, Lemma \\ref{properties-lemma-normal-locally-finite-nr-irreducibles}).\nBy our remark on geometric points we see that $m \\geq [L : K]$.\nOn the other hand, by More on Morphisms, Lemma\n\\ref{more-morphisms-lemma-normalization-smooth-localization}\nwe see that $\\coprod V_i \\to U$ is the normalization of $U$ in\n$\\Spec(L) \\times_X U$. As $K \\subset \\kappa(\\xi)$ is finite separable,\nwe can write $\\Spec(L) \\times_X U = \\Spec(\\prod_{i = 1, \\ldots, n} L_i)$\nwith $L_i/\\kappa(\\xi)$ finite and $[L : K] = \\sum [L_i : \\kappa(\\xi)]$.\nSince $V_j$ is nonempty for each $j$ and $m \\geq [L : K]$\nwe conclude that $m = n$ and $[L_i : \\kappa(\\xi)] = 1$\nfor all $i$. Then $V_j \\to U$ is an isomorphism in particular\n\\'etale, hence $Y \\times_X U \\to U$ is \\'etale. By\nDescent, Lemma \\ref{descent-lemma-descending-property-etale}\nwe conclude that $Y \\to X$ is \\'etale over the\nimage of $U \\to X$ (an open neighbourhood of $x$).\n\n\\medskip\\noindent\nAssume $L/K$ is Galois and (9) holds. Then $Y \\to X$ is \\'etale\nby Lemma \\ref{lemma-inertial-invariants-unramified}.\nWe omit the proof that (1) implies (9).\n\\end{proof}\n\n\\noindent\nIn the case of infinite Galois extensions of discrete valuation rings\nwe can say a tiny bit more. To do so we introduce the following notation.\nA subset $S \\subset \\mathbf{N}$ of integers is {\\it multiplicativity directed}\nif $1 \\in S$ and for $n, m \\in S$ there exists $k \\in S$ with\n$n | k$ and $m | k$. Define a partial ordering on $S$ by the rule\n$n \\geq_S m$ if and only if $m | n$. Given a field $\\kappa$ we obtain\nan inverse system of finite groups $\\{\\mu_n(\\kappa)\\}_{n \\in S}$\nwith transition maps\n$$\n\\mu_n(\\kappa) \\longrightarrow \\mu_m(\\kappa),\\quad\n\\zeta \\longmapsto \\zeta^{n/m}\n$$\nfor $n \\geq_S m$. Then we can form the profinite group\n$$\n\\lim_{n \\in S} \\mu_n(\\kappa)\n$$\nObserve that the limit is cofiltered (as $S$ is directed).\nThe construction is functorial in $\\kappa$. In particular\n$\\text{Aut}(\\kappa)$ acts on this profinite group.\nFor example, if $S = \\{1, n\\}$, then this gives $\\mu_n(\\kappa)$.\nIf $S = \\{1, \\ell, \\ell^2, \\ell^3, \\ldots\\}$ for some prime\n$\\ell$ different from the characteristic of $\\kappa$ this produces\n$\\lim_n \\mu_{\\ell^n}(\\kappa)$\nwhich is sometimes called the $\\ell$-adic Tate module of the multiplicative\ngroup of $\\kappa$ (compare with\nMore on Algebra, Example\n\\ref{more-algebra-example-spectral-sequence-principal}).\n\n\\begin{lemma}\n\\label{lemma-structure-decomposition}\nLet $A$ be a discrete valuation ring with fraction field $K$.\nLet $L/K$ be a (possibly infinite) Galois extension.\nLet $B$ be the integral closure of $A$ in $L$.\nLet $\\mathfrak m$ be a maximal ideal of $B$.\nLet $G = \\text{Gal}(L/K)$,\n$D = \\{\\sigma \\in G \\mid \\sigma(\\mathfrak m) = \\mathfrak m\\}$, and\n$I = \\{\\sigma \\in D \\mid \\sigma \\bmod \\mathfrak m =\n\\text{id}_{\\kappa(\\mathfrak m)}\\}$.\nThe decomposition group $D$ fits into a canonical exact sequence\n$$\n1 \\to I \\to D \\to \\text{Aut}(\\kappa(\\mathfrak m)/\\kappa_A) \\to 1\n$$\nThe inertia group $I$ fits into a canonical exact sequence\n$$\n1 \\to P \\to I \\to I_t \\to 1\n$$\nsuch that\n\\begin{enumerate}\n\\item $P$ is a normal subgroup of $D$,\n\\item $P$ is a pro-p-group if the characteristic of\n$\\kappa_A$ is $p > 1$ and $P = \\{1\\}$ if the characteristic of $\\kappa_A$\nis zero,\n\\item there is a multiplicatively directed $S \\subset \\mathbf{N}$\nsuch that $\\kappa(\\mathfrak m)$ contains a primitive $n$th root of unity\nfor each $n \\in S$ (elements of $S$ are prime to $p$),\n\\item there exists a canonical surjective map\n$$\n\\theta_{can} : I \\to \\lim_{n \\in S} \\mu_n(\\kappa(\\mathfrak m))\n$$\nwhose kernel is $P$, which satisfies\n$\\theta_{can}(\\tau \\sigma \\tau^{-1}) = \\tau(\\theta_{can}(\\sigma))$\nfor $\\tau \\in D$, $\\sigma \\in I$, and which induces an isomorphism\n$I_t \\to \\lim_{n \\in S} \\mu_n(\\kappa(\\mathfrak m))$.\n\\end{enumerate}\n\\end{lemma}\n\n\\begin{proof}\nThis is mostly a reformulation of the results on finite Galois extensions\nproved in More on Algebra, Section \\ref{more-algebra-section-ramification}.\nThe surjectivity of the map $D \\to \\text{Aut}(\\kappa(\\mathfrak m)/\\kappa)$ is\nMore on Algebra, Lemma \\ref{more-algebra-lemma-one-orbit-geometric-galois}.\nThis gives the first exact sequence.\n\n\\medskip\\noindent\nTo construct the second short exact sequence let $\\Lambda$ be the set\nof finite Galois subextensions, i.e., $\\lambda \\in \\Lambda$ corresponds\nto $L/L_\\lambda/K$. Set $G_\\lambda = \\text{Gal}(L_\\lambda/K)$.\nRecall that $G_\\lambda$ is an inverse system of finite groups with surjective\ntransition maps and that $G = \\lim_{\\lambda \\in \\Lambda} G_\\lambda$, see\nFields, Lemma \\ref{fields-lemma-infinite-galois-limit}.\nWe let $B_\\lambda$ be the integral closure of $A$ in $L_\\lambda$.\nThen we set $\\mathfrak m_\\lambda = \\mathfrak m \\cap B_\\lambda$\nand we denote $P_\\lambda, I_\\lambda, D_\\lambda$ the\nwild inertia, inertia, and decomposition group of\n$\\mathfrak m_\\lambda$, see More on Algebra, Lemma\n\\ref{more-algebra-lemma-galois-inertia}.\nFor $\\lambda \\geq \\lambda'$ the restriction defines\na commutative diagram\n$$\n\\xymatrix{\nP_\\lambda \\ar[d] \\ar[r] &\nI_\\lambda \\ar[d] \\ar[r] &\nD_\\lambda \\ar[d] \\ar[r] &\nG_\\lambda \\ar[d] \\\\\nP_{\\lambda'} \\ar[r] &\nI_{\\lambda'} \\ar[r] &\nD_{\\lambda'} \\ar[r] &\nG_{\\lambda'}\n}\n$$\nwith surjective vertical maps, see\nMore on Algebra, Lemma \\ref{more-algebra-lemma-compare-inertia}.\n\n\\medskip\\noindent\nFrom the definitions it follows immediately\nthat $I = \\lim I_\\lambda$ and $D = \\lim D_\\lambda$\nunder the isomorphism $G = \\lim G_\\lambda$ above.\nSince $L = \\colim L_\\lambda$ we have $B = \\colim B_\\lambda$\nand $\\kappa(\\mathfrak m) = \\colim \\kappa(\\mathfrak m_\\lambda)$.\nSince the transition maps of the system $D_\\lambda$\nare compatible with the maps\n$D_\\lambda \\to \\text{Aut}(\\kappa(\\mathfrak m_\\lambda)/\\kappa)$\n(see More on Algebra, Lemma \\ref{more-algebra-lemma-compare-inertia})\nwe see that the map $D \\to \\text{Aut}(\\kappa(\\mathfrak m)/\\kappa)$\nis the limit of the maps\n$D_\\lambda \\to \\text{Aut}(\\kappa(\\mathfrak m_\\lambda)/\\kappa)$.\n\n\\medskip\\noindent\nThere exist canonical maps\n$$\n\\theta_{\\lambda, can} :\nI_\\lambda\n\\longrightarrow\n\\mu_{n_\\lambda}(\\kappa(\\mathfrak m_\\lambda))\n$$\nwhere $n_\\lambda = |I_\\lambda|/|P_\\lambda|$, where\n$\\mu_{n_\\lambda}(\\kappa(\\mathfrak m_\\lambda))$ has\norder $n_\\lambda$, such that\n$\\theta_{\\lambda, can}(\\tau \\sigma \\tau^{-1}) =\n\\tau(\\theta_{\\lambda, can}(\\sigma))$ for\n$\\tau \\in D_\\lambda$ and $\\sigma \\in I_\\lambda$, and such that\nwe get commutative diagrams\n$$\n\\xymatrix{\nI_\\lambda \\ar[r]_-{\\theta_{\\lambda, can}} \\ar[d] &\n\\mu_{n_\\lambda}(\\kappa(\\mathfrak m_\\lambda))\n\\ar[d]^{(-)^{n_\\lambda/n_{\\lambda'}}} \\\\\nI_{\\lambda'} \\ar[r]^-{\\theta_{\\lambda', can}} &\n\\mu_{n_{\\lambda'}}(\\kappa(\\mathfrak m_{\\lambda'}))\n}\n$$\nsee\nMore on Algebra, Remark \\ref{more-algebra-remark-canonical-inertia-character}.\n\n\\medskip\\noindent\nLet $S \\subset \\mathbf{N}$ be the collection of integers $n_\\lambda$.\nSince $\\Lambda$ is directed, we see that $S$ is multiplicatively directed.\nBy the displayed commutative diagrams above we can take the limits of\nthe maps $\\theta_{\\lambda, can}$ to obtain\n$$\n\\theta_{can} : I \\to \\lim_{n \\in S} \\mu_n(\\kappa(\\mathfrak m)).\n$$\nThis map is continuous (small detail omitted). Since the transition maps\nof the system of $I_\\lambda$ are surjective\nand $\\Lambda$ is directed, the projections $I \\to I_\\lambda$\nare surjective. For every $\\lambda$ the diagram\n$$\n\\xymatrix{\nI \\ar[d] \\ar[r]_-{\\theta_{can}} &\n\\lim_{n \\in S} \\mu_n(\\kappa(\\mathfrak m)) \\ar[d] \\\\\nI_{\\lambda} \\ar[r]^-{\\theta_{\\lambda, can}} &\n\\mu_{n_\\lambda}(\\kappa(\\mathfrak m_\\lambda))\n}\n$$\ncommutes. Hence the image of $\\theta_{can}$ surjects onto the finite group\n$\\mu_{n_\\lambda}(\\kappa(\\mathfrak m)) =\n\\mu_{n_\\lambda}(\\kappa(\\mathfrak m_\\lambda))$ of order $n_\\lambda$\n(see above). It follows that the image of $\\theta_{can}$ is dense.\nOn the other hand $\\theta_{can}$ is continuous and the\nsource is a profinite group. Hence $\\theta_{can}$ is surjective\nby a topological argument.\n\n\\medskip\\noindent\nThe property $\\theta_{can}(\\tau \\sigma \\tau^{-1}) = \\tau(\\theta_{can}(\\sigma))$\nfor $\\tau \\in D$, $\\sigma \\in I$ follows from the corresponding properties\nof the maps $\\theta_{\\lambda, can}$ and the compatibility of the map\n$D \\to \\text{Aut}(\\kappa(\\mathfrak m))$ with the maps\n$D_\\lambda \\to \\text{Aut}(\\kappa(\\mathfrak m_\\lambda))$.\nSetting $P = \\Ker(\\theta_{can})$ this implies\nthat $P$ is a normal subgroup of $D$. Setting $I_t = I/P$\nwe obtain the isomorphism $I_t \\to \\lim_{n \\in S} \\mu_n(\\kappa(\\mathfrak m))$\nfrom the surjectivity of $\\theta_{can}$.\n\n\\medskip\\noindent\nTo finish the proof we show that $P = \\lim P_\\lambda$ which proves\nthat $P$ is a pro-p-group. Recall that the tame inertia group\n$I_{\\lambda, t} = I_\\lambda/P_\\lambda$ has order $n_\\lambda$.\nSince the transition maps $P_\\lambda \\to P_{\\lambda'}$ are surjective\nand $\\Lambda$ is directed, we obtain a short exact sequence\n$$\n1 \\to \\lim P_\\lambda \\to I \\to \\lim I_{\\lambda, t} \\to 1\n$$\n(details omitted). Since for each $\\lambda$ the map $\\theta_{\\lambda, can}$\ninduces an isomorphism\n$I_{\\lambda, t} \\cong \\mu_{n_\\lambda}(\\kappa(\\mathfrak m))$\nthe desired result follows.\n\\end{proof}\n\n\\begin{lemma}\n\\label{lemma-structure-decomposition-separable-closure}\nLet $A$ be a discrete valuation ring with fraction field $K$.\nLet $K^{sep}$ be a separable closure of $K$.\nLet $A^{sep}$ be the integral closure of $A$ in $K^{sep}$.\nLet $\\mathfrak m^{sep}$ be a maximal ideal of $A^{sep}$.\nLet $\\mathfrak m = \\mathfrak m^{sep} \\cap A$, let\n$\\kappa = A/\\mathfrak m$, and let\n$\\overline{\\kappa} = A^{sep}/\\mathfrak m^{sep}$.\nThen $\\overline{\\kappa}$ is an algebraic closure of $\\kappa$.\nLet $G = \\text{Gal}(K^{sep}/K)$,\n$D = \\{\\sigma \\in G \\mid \\sigma(\\mathfrak m^{sep}) = \\mathfrak m^{sep}\\}$, and\n$I = \\{\\sigma \\in D \\mid \\sigma \\bmod \\mathfrak m^{sep} =\n\\text{id}_{\\kappa(\\mathfrak m^{sep})}\\}$.\nThe decomposition group $D$ fits into a canonical exact sequence\n$$\n1 \\to I \\to D \\to \\text{Gal}(\\kappa^{sep}/\\kappa) \\to 1\n$$\nwhere $\\kappa^{sep} \\subset \\overline{\\kappa}$ is the separable\nclosure of $\\kappa$.\nThe inertia group $I$ fits into a canonical exact sequence\n$$\n1 \\to P \\to I \\to I_t \\to 1\n$$\nsuch that\n\\begin{enumerate}\n\\item $P$ is a normal subgroup of $D$,\n\\item $P$ is a pro-p-group if the characteristic of\n$\\kappa_A$ is $p > 1$ and $P = \\{1\\}$ if the characteristic of $\\kappa_A$\nis zero,\n\\item there exists a canonical surjective map\n$$\n\\theta_{can} : I \\to \\lim_{n\\text{ prime to }p} \\mu_n(\\kappa^{sep})\n$$\nwhose kernel is $P$, which satisfies\n$\\theta_{can}(\\tau \\sigma \\tau^{-1}) = \\tau(\\theta_{can}(\\sigma))$\nfor $\\tau \\in D$, $\\sigma \\in I$, and which induces an isomorphism\n$I_t \\to \\lim_{n\\text{ prime to }p} \\mu_n(\\kappa^{sep})$.\n\\end{enumerate}\n\\end{lemma}\n\n\\begin{proof}\nThe field $\\overline{\\kappa}$ is the algebraic closure of $\\kappa$ by\nLemma \\ref{lemma-get-algebraic-closure}.\nMost of the statements immediately follow from the corresponding\nparts of Lemma \\ref{lemma-structure-decomposition}. For example because\n$\\text{Aut}(\\overline{\\kappa}/\\kappa) = \\text{Gal}(\\kappa^{sep}/\\kappa)$\nwe obtain the first sequence.\nThen the only other assertion that needs a proof is the fact that\nwith $S$ as in Lemma \\ref{lemma-structure-decomposition} the\nlimit $\\lim_{n \\in S} \\mu_n(\\overline{\\kappa})$ is equal to\n$\\lim_{n\\text{ prime to }p} \\mu_n(\\kappa^{sep})$. To see this\nit suffices to show that every integer $n$ prime to $p$\ndivides an element of $S$.\nLet $\\pi \\in A$ be a uniformizer and consider the splitting\nfield $L$ of the polynomial $X^n - \\pi$. Since the polynomial\nis separable we see that $L$ is a finite Galois extension of $K$.\nChoose an embedding $L \\to K^{sep}$.\nObserve that if $B$ is the integral closure of $A$ in $L$,\nthen the ramification index of $A \\to B_{\\mathfrak m^{sep} \\cap B}$\nis divisible by $n$ (because $\\pi$ has an $n$th root in $B$; in fact\nthe ramification index equals $n$ but we do not need this).\nThen it follows from the construction of the $S$ in the proof of\nLemma \\ref{lemma-structure-decomposition}\nthat $n$ divides an element of $S$.\n\\end{proof}\n\n\n\n\n\n\n\n\n\\section{Geometric and arithmetic fundamental groups}\n\\label{section-galois-action}\n\n\\noindent\nIn this section we work out what happens when comparing the\nfundamental group of a scheme $X$ over a field $k$ with the\nfundamental group of $X_{\\overline{k}}$ where $\\overline{k}$\nis the algebraic closure of $k$.\n\n\\begin{lemma}\n\\label{lemma-limit}\nLet $I$ be a directed set. Let $X_i$ be an\ninverse system of quasi-compact and quasi-separated schemes\nover $I$ with affine transition morphisms.\nLet $X = \\lim X_i$ as in Limits, Section \\ref{limits-section-limits}.\nThen there is an equivalence of categories\n$$\n\\colim \\textit{F\\'Et}_{X_i} = \\textit{F\\'Et}_X\n$$\nIf $X_i$ is connected for all sufficiently large $i$ and $\\overline{x}$\nis a geometric point of $X$, then\n$$\n\\pi_1(X, \\overline{x}) = \\lim \\pi_1(X_i, \\overline{x})\n$$\n\\end{lemma}\n\n\\begin{proof}\nThe equivalence of categories follows from Limits, Lemmas\n\\ref{limits-lemma-descend-finite-presentation},\n\\ref{limits-lemma-descend-finite-finite-presentation}, and\n\\ref{limits-lemma-descend-etale}.\nThe second statement is formal given the statement on\ncategories.\n\\end{proof}\n\n\\begin{lemma}\n\\label{lemma-perfection}\nLet $k$ be a field with perfection $k^{perf}$. Let $X$ be a connected scheme\nover $k$. Then $X_{k^{perf}}$ is connected and\n$\\pi_1(X_{k^{perf}}) \\to \\pi_1(X)$ is an isomorphism.\n\\end{lemma}\n\n\\begin{proof}\nSpecial case of topological invariance of the fundamental group.\nSee Proposition \\ref{proposition-universal-homeomorphism}.\nTo see that $\\Spec(k^{perf}) \\to \\Spec(k)$ is a universal\nhomeomorphism you can use\nAlgebra, Lemma \\ref{algebra-lemma-radicial-integral-bijective}.\n\\end{proof}\n\n\\begin{lemma}\n\\label{lemma-ses-field}\nLet $k$ be a field with algebraic closure $\\overline{k}$.\nLet $X$ be a quasi-compact and quasi-separated scheme over $k$.\nIf the base change $X_{\\overline{k}}$ is connected, then\nthere is a short exact sequence\n$$\n1 \\to \\pi_1(X_{\\overline{k}}) \\to \\pi_1(X) \\to \\pi_1(\\Spec(k)) \\to 1\n$$\nof profinite topological groups.\n\\end{lemma}\n\n\\begin{proof}\nConnected objects of $\\textit{F\\'Et}_{\\Spec(k)}$ are of the form\n$\\Spec(k') \\to \\Spec(k)$ with $k'/k$ a finite separable extension.\nThen $X_{\\Spec{k'}}$ is connected, as the morphism\n$X_{\\overline{k}} \\to X_{\\Spec(k')}$ is surjective and\n$X_{\\overline{k}}$ is connected by assumption. Thus\n$\\pi_1(X) \\to \\pi_1(\\Spec(k))$ is surjective by\nLemma \\ref{lemma-functoriality-galois-surjective}.\n\n\\medskip\\noindent\nBefore we go on, note that we may assume that $k$ is a perfect field.\nNamely, we have $\\pi_1(X_{k^{perf}}) = \\pi_1(X)$ and\n$\\pi_1(\\Spec(k^{perf})) = \\pi_1(\\Spec(k))$ by Lemma \\ref{lemma-perfection}.\n\n\\medskip\\noindent\nIt is clear that the composition of the functors\n$\\textit{F\\'Et}_{\\Spec(k)} \\to \\textit{F\\'Et}_X \\to\n\\textit{F\\'Et}_{X_{\\overline{k}}}$ sends objects to disjoint unions\nof copies of $X_{\\Spec(\\overline{k})}$. Therefore the composition\n$\\pi_1(X_{\\overline{k}}) \\to \\pi_1(X) \\to \\pi_1(\\Spec(k))$\nis the trivial homomorphism by Lemma \\ref{lemma-composition-trivial}.\n\n\\medskip\\noindent\nLet $U \\to X$ be a finite \\'etale morphism with $U$ connected.\nObserve that $U \\times_X X_{\\overline{k}} = U_{\\overline{k}}$.\nSuppose that $U_{\\overline{k}} \\to X_{\\overline{k}}$\nhas a section $s : X_{\\overline{k}} \\to U_{\\overline{k}}$.\nThen $s(X_{\\overline{k}})$ is an open connected component of\n$U_{\\overline{k}}$. For $\\sigma \\in \\text{Gal}(\\overline{k}/k)$\ndenote $s^\\sigma$ the base change of $s$ by $\\Spec(\\sigma)$.\nSince $U_{\\overline{k}} \\to X_{\\overline{k}}$ is finite \\'etale\nit has only a finite number of sections. Thus\n$$\n\\overline{T} = \\bigcup s^\\sigma(X_{\\overline{k}})\n$$\nis a finite union and we see that $\\overline{T}$ is a\n$\\text{Gal}(\\overline{k}/k)$-stable open and closed subset.\nBy Varieties, Lemma \\ref{varieties-lemma-closed-fixed-by-Galois}\nwe see that $\\overline{T}$ is the inverse image of a closed\nsubset $T \\subset U$. Since $U_{\\overline{k}} \\to U$ is open\n(Morphisms, Lemma \\ref{morphisms-lemma-scheme-over-field-universally-open})\nwe conclude that $T$ is open as well. As $U$ is connected we\nsee that $T = U$. Hence $U_{\\overline{k}}$ is a (finite) disjoint\nunion of copies of $X_{\\overline{k}}$. By\nLemma \\ref{lemma-functoriality-galois-normal} we conclude that the image of\n$\\pi_1(X_{\\overline{k}}) \\to \\pi_1(X)$ is normal.\n\n\\medskip\\noindent\nLet $V \\to X_{\\overline{k}}$ be a finite \\'etale cover. Recall that\n$\\overline{k}$ is the union of finite separable extensions of $k$.\nBy Lemma \\ref{lemma-limit} we find a finite separable extension $k'/k$\nand a finite \\'etale morphism $U \\to X_{k'}$ such that\n$V = X_{\\overline{k}} \\times_{X_{k'}} U =\nU \\times_{\\Spec(k')} \\Spec(\\overline{k})$.\nThen the composition $U \\to X_{k'} \\to X$ is finite \\'etale\nand $U \\times_{\\Spec(k)} \\Spec(\\overline{k})$\ncontains $V = U \\times_{\\Spec(k')} \\Spec(\\overline{k})$\nas an open and closed subscheme. (Because $\\Spec(\\overline{k})$\nis an open and closed subscheme of\n$\\Spec(k') \\times_{\\Spec(k)} \\Spec(\\overline{k})$ via\nthe multiplication map $k' \\otimes_k \\overline{k} \\to \\overline{k}$.) By\nLemma \\ref{lemma-functoriality-galois-injective}\nwe conclude that $\\pi_1(X_{\\overline{k}}) \\to \\pi_1(X)$ is injective.\n\n\\medskip\\noindent\nFinally, we have to show that for any finite \\'etale morphism\n$U \\to X$ such that $U_{\\overline{k}}$ is a disjoint union\nof copies of $X_{\\overline{k}}$ there is a finite \\'etale\nmorphism $V \\to \\Spec(k)$ and a surjection $V \\times_{\\Spec(k)} X \\to U$.\nSee Lemma \\ref{lemma-functoriality-galois-ses}.\nArguing as above using Lemma \\ref{lemma-limit}\nwe find a finite separable extension $k'/k$\nsuch that there is an isomorphism\n$U_{k'} \\cong \\coprod_{i = 1, \\ldots, n} X_{k'}$.\nThus setting $V = \\coprod_{i = 1, \\ldots, n} \\Spec(k')$\nwe conclude.\n\\end{proof}\n\n\n\n\n\n\n\\section{Homotopy exact sequence}\n\\label{section-homotopy-exact-sequence}\n\n\\noindent\nIn this section we discuss the following result.\nLet $f : X \\to S$ be a flat proper morphism of\nfinite presentation whose\ngeometric fibres are connected and reduced.\nAssume $S$ is connected and let $\\overline{s}$\nbe a geometric point of $S$. Then there is an exact\nsequence\n$$\n\\pi_1(X_{\\overline{s}}) \\to \\pi_1(X) \\to \\pi_1(S) \\to 1\n$$\nof fundamental groups. See\nProposition \\ref{proposition-first-homotopy-sequence}.\n\n\\begin{lemma}\n\\label{lemma-stein-factorization-etale}\n\\begin{reference}\n\\cite[Expose X, Proposition 1.2, p. 262]{SGA1}.\n\\end{reference}\nLet $f : X \\to S$ be a proper morphism of schemes.\nLet $X \\to S' \\to S$ be the Stein factorization of $f$, see\nMore on Morphisms, Theorem\n\\ref{more-morphisms-theorem-stein-factorization-general}.\nIf $f$ is of finite presentation, flat, with geometrically\nreduced fibres, then $S' \\to S$ is finite \\'etale.\n\\end{lemma}\n\n\\begin{proof}\nLet $s \\in S$. Set $n$ be the number of connected components of\nthe geometric fibre $X_{\\overline{s}}$. Note that $n < \\infty$ as the geometric\nfibre of $X \\to S$ at $s$ is a proper scheme over a field, hence Noetherian,\nhence has a finite number of connected components.\nBy More on Morphisms, Lemma\n\\ref{more-morphisms-lemma-stein-universally-closed-residue-fields}\nthere are finitely many points $s'_1, \\ldots, s'_m \\in S'$ lying over $s$\nand for each $i$ the extension $\\kappa(s'_i)/\\kappa(s)$ is finite.\nMore on Morphisms,\nLemma \\ref{more-morphisms-lemma-etale-makes-integral-split}\ntells us that after replacing $S$ by an \\'etale neighbourhood\nof $s$ we may assume $S' = V_1 \\amalg \\ldots \\amalg V_m$ as a scheme\nwith $s'_i \\in V_i$ and $\\kappa(s'_i)/\\kappa(s)$ purely inseparable.\nIn this case the schemes $X_{s_i'}$ are geometrically connected\nover $\\kappa(s)$, hence $m = n$.\nThe schemes $X_i = (f')^{-1}(V_i)$, $i = 1, \\ldots, n$\nare proper, flat, of finite presentation, with geometrically\nreduced fibres over $S$. It suffices to prove the lemma\nfor each of the morphisms $X_i \\to S$. This reduces us to the case where\n$X_{\\overline{s}}$ is connected.\n\n\\medskip\\noindent\nAssume that $X_{\\overline{s}}$ is connected. By\nMore on Morphisms, Lemma \\ref{more-morphisms-lemma-proper-flat-geom-red}\nwe see that $X \\to S$ has geometrically connected\nfibres in a neighbourhood of $s$. Thus\nwe may assume the fibres of $X \\to S$ are geometrically connected.\nThen $f_*\\mathcal{O}_X = \\mathcal{O}_S$ by\nDerived Categories of Schemes, Lemma\n\\ref{perfect-lemma-proper-flat-geom-red-connected}\nwhich finishes the proof.\n\\end{proof}\n\n\\begin{proposition}\n\\label{proposition-first-homotopy-sequence}\nLet $f : X \\to S$ be a flat proper morphism of finite presentation whose\ngeometric fibres are connected and reduced. Assume $S$ is connected and\nlet $\\overline{s}$ be a geometric point of $S$. Then there is an exact\nsequence\n$$\n\\pi_1(X_{\\overline{s}}) \\to \\pi_1(X) \\to \\pi_1(S) \\to 1\n$$\nof fundamental groups.\n\\end{proposition}\n\n\\begin{proof}\nLet $Y \\to X$ be a finite \\'etale morphism. Consider the Stein factorization\n$$\n\\xymatrix{\nY \\ar[d] \\ar[r] & X \\ar[d] \\\\\nT \\ar[r] & S\n}\n$$\nof $Y \\to S$. By Lemma \\ref{lemma-stein-factorization-etale}\nthe morphism $T \\to S$ is finite \\'etale. In this way we obtain\na functor $\\textit{F\\'Et}_X \\to \\textit{F\\'Et}_S$.\nFor any finite \\'etale morphism $U \\to S$ a morphism\n$Y \\to U \\times_S X$ over $X$ is the same thing as a morphism\n$Y \\to U$ over $S$ and such a morphism factors uniquely through\nthe Stein factorization, i.e., corresponds to a unique\nmorphism $T \\to U$\n(by the construction of the Stein factorization as a relative\nnormalization in More on Morphisms, Lemma\n\\ref{more-morphisms-lemma-stein-universally-closed}\nand factorization by\nMorphisms, Lemma \\ref{morphisms-lemma-characterize-normalization}).\nThus we see that the functors\n$\\textit{F\\'Et}_X \\to \\textit{F\\'Et}_S$ and\n$\\textit{F\\'Et}_S \\to \\textit{F\\'Et}_X$ are adjoints.\nNote that the Stein factorization of $U \\times_S X \\to S$ is\n$U$, because the fibres of $U \\times_S X \\to U$ are geometrically connected.\n\n\\medskip\\noindent\nBy the discussion above and\nCategories, Lemma \\ref{categories-lemma-adjoint-fully-faithful}\nwe conclude that\n$\\textit{F\\'Et}_S \\to \\textit{F\\'Et}_X$\nis fully faithful, i.e., $\\pi_1(X) \\to \\pi_1(S)$ is surjective\n(Lemma \\ref{lemma-functoriality-galois-surjective}).\n\n\\medskip\\noindent\nIt is immediate that the composition\n$\\textit{F\\'Et}_S \\to \\textit{F\\'Et}_X \\to \\textit{F\\'Et}_{X_{\\overline{s}}}$\nsends any $U$ to a disjoint union of copies of $X_{\\overline{s}}$.\nHence $\\pi_1(X_{\\overline{s}}) \\to \\pi_1(X) \\to \\pi_1(S)$ is trivial\nby Lemma \\ref{lemma-composition-trivial}.\n\n\\medskip\\noindent\nLet $Y \\to X$ be a finite \\'etale morphism with $Y$ connected such that\n$Y \\times_X X_{\\overline{s}}$ contains a connected component $Z$\nisomorphic to $X_{\\overline{s}}$. Consider the Stein factorization $T$\nas above. Let $\\overline{t} \\in T_{\\overline{s}}$ be the point corresponding\nto the fibre $Z$. Observe that $T$ is connected (as the image of a connected\nscheme) and by the surjectivity above $T \\times_S X$ is connected.\nNow consider the factorization\n$$\n\\pi : Y \\longrightarrow T \\times_S X\n$$\nLet $\\overline{x} \\in X_{\\overline{s}}$ be any closed point. Note that\n$\\kappa(\\overline{t}) = \\kappa(\\overline{s}) = \\kappa(\\overline{x})$\nis an algebraically closed field.\nThen the fibre of $\\pi$ over $(\\overline{t}, \\overline{x})$ consists\nof a unique point, namely the unique point $\\overline{z} \\in Z$\ncorresponding to $\\overline{x} \\in X_{\\overline{s}}$ via the\nisomorphism $Z \\to X_{\\overline{s}}$. We conclude that the finite\n\\'etale morphism $\\pi$ has degree $1$ in a neighbourhood of\n$(\\overline{t}, \\overline{x})$. Since $T \\times_S X$ is connected\nit has degree $1$ everywhere and we find tat $Y \\cong T \\times_S X$.\nThus $Y \\times_X X_{\\overline{s}}$ splits completely.\nCombining all of the above we see that\nLemmas \\ref{lemma-functoriality-galois-ses} and\n\\ref{lemma-functoriality-galois-normal}\nboth apply and the proof is complete.\n\\end{proof}\n\n\n\n\n\\section{Specialization maps}\n\\label{section-specialization-map}\n\n\\noindent\nIn this section we construct specialization maps.\nLet $f : X \\to S$ be a proper morphism of schemes\nwith geometrically connected fibres.\nLet $s' \\leadsto s$ be a specialization of points in $S$.\nLet $\\overline{s}$ and $\\overline{s}'$ be geometric points\nlying over $s$ and $s'$. Then there is a specialization map\n$$\nsp : \\pi_1(X_{\\overline{s}'}) \\longrightarrow \\pi_1(X_{\\overline{s}})\n$$\nThe construction of this map is as follows. Let $A$ be the\nstrict henselization of $\\mathcal{O}_{S, s}$ with respect to\n$\\kappa(s) \\subset \\kappa(s)^{sep} \\subset \\kappa(\\overline{s})$, see\nAlgebra, Definition \\ref{algebra-definition-henselization}.\nSince $s' \\leadsto s$ the point $s'$ corresponds to a point of\n$\\Spec(\\mathcal{O}_{S, s})$ and hence there is at least one point\n(and potentially many points)\nof $\\Spec(A)$ over $s'$ whose residue field is a separable algebraic\nextension of $\\kappa(s')$.\nSince $\\kappa(\\overline{s}')$ is algebraically closed we can choose\na morphism $\\varphi : \\overline{s}' \\to \\Spec(A)$ giving rise to a commutative\ndiagram\n$$\n\\xymatrix{\n\\overline{s}' \\ar[r]_-\\varphi \\ar[rd] &\n\\Spec(A) \\ar[d] &\n\\overline{s} \\ar[l] \\ar[ld] \\\\\n& S\n}\n$$\nThe specialization map is the composition\n$$\n\\pi_1(X_{\\overline{s}'}) \\longrightarrow\n\\pi_1(X_A) =\n\\pi_1(X_{\\kappa(s)^{sep}}) =\n\\pi_1(X_{\\overline{s}})\n$$\nwhere the first equality is\nLemma \\ref{lemma-finite-etale-on-proper-over-henselian}\nand the second follows from\nLemmas \\ref{lemma-perfection} and\n\\ref{lemma-finite-etale-invariant-over-proper}.\nBy construction the specialization map fits into a commutative\ndiagram\n$$\n\\xymatrix{\n\\pi_1(X_{\\overline{s}'}) \\ar[rr]_{sp} \\ar[rd] & &\n\\pi_1(X_{\\overline{s}}) \\ar[ld] \\\\\n& \\pi_1(X)\n}\n$$\nprovided that $X$ is connected. The specialization map depends on the\nchoice of $\\varphi : \\overline{s}' \\to \\Spec(A)$ above and we will\nwrite $sp_\\varphi$ if we want to indicate this.\n\n\\begin{lemma}\n\\label{lemma-specialization-map-base-change}\nConsider a commutative diagram\n$$\n\\xymatrix{\nY \\ar[d]_g \\ar[r] & X \\ar[d]^f \\\\\nT \\ar[r] & S\n}\n$$\nof schemes where $f$ and $g$ are proper with geometrically connected\nfibres. Let $t' \\leadsto t$ be a specialization of points in $T$\nand consider a specialization map\n$sp : \\pi_1(Y_{\\overline{t}'}) \\to \\pi_1(Y_{\\overline{t}})$ as above.\nThen there is a commutative diagram\n$$\n\\xymatrix{\n\\pi_1(Y_{\\overline{t}'}) \\ar[r]_{sp} \\ar[d] & \\pi_1(Y_{\\overline{t}}) \\ar[d] \\\\\n\\pi_1(X_{\\overline{s}'}) \\ar[r]^{sp} & \\pi_1(X_{\\overline{s}})\n}\n$$\nof specialization maps where $\\overline{s}$ and $\\overline{s}'$\nare the images of $\\overline{t}$ and $\\overline{t}'$.\n\\end{lemma}\n\n\\begin{proof}\nLet $B$ be the strict henselization of $\\mathcal{O}_{T, t}$ with respect to\n$\\kappa(t) \\subset \\kappa(t)^{sep} \\subset \\kappa(\\overline{t})$.\nPick $\\psi : \\overline{t}' \\to \\Spec(B)$ lifting $\\overline{t}' \\to T$\nas in the construction of the specialization map.\nLet $s$ and $s'$ denote the images of $t$ and $t'$ in $S$.\nLet $A$ be the strict henselization of $\\mathcal{O}_{S, s}$\nwith respect to\n$\\kappa(s) \\subset \\kappa(s)^{sep} \\subset \\kappa(\\overline{s})$.\nSince $\\kappa(\\overline{s}) = \\kappa(\\overline{t})$,\nby the functoriality of strict henselization\n(Algebra, Lemma \\ref{algebra-lemma-strictly-henselian-functorial})\nwe obtain a ring map $A \\to B$ fitting into the commutative diagram\n$$\n\\xymatrix{\n\\overline{t}' \\ar[r]_-\\psi \\ar[d] & \\Spec(B) \\ar[d] \\ar[r] & T \\ar[d] \\\\\n\\overline{s}' \\ar[r]^-\\varphi & \\Spec(A) \\ar[r] & S\n}\n$$\nHere the morphism $\\varphi : \\overline{s}' \\to \\Spec(A)$ is simply taken\nto be the composition $\\overline{t}' \\to \\Spec(B) \\to \\Spec(A)$.\nApplying base change we obtain a commutative diagram\n$$\n\\xymatrix{\nY_{\\overline{t}'} \\ar[r] \\ar[d] & Y_B \\ar[d] \\\\\nX_{\\overline{s}'} \\ar[r] & X_A\n}\n$$\nand from the construction of the specialization map the commutativity\nof this diagram implies the commutativity of the diagram of the lemma.\n\\end{proof}\n\n\\begin{lemma}\n\\label{lemma-specialization-map-composition}\nLet $f : X \\to S$ be a proper morphism with geometrically connected fibres.\nLet $s'' \\leadsto s' \\leadsto s$ be specializations of points of $S$.\nA composition of specialization maps\n$\\pi_1(X_{\\overline{s}''}) \\to \\pi_1(X_{\\overline{s}'}) \\to\n\\pi_1(X_{\\overline{s}})$ is a specialization map\n$\\pi_1(X_{\\overline{s}''}) \\to \\pi_1(X_{\\overline{s}})$.\n\\end{lemma}\n\n\\begin{proof}\nLet $\\mathcal{O}_{S, s} \\to A$ be the strict henselization\nconstructed using $\\kappa(s) \\to \\kappa(\\overline{s})$.\nLet $A \\to \\kappa(\\overline{s}')$ be the map used to construct\nthe first specialization map. Let $\\mathcal{O}_{S, s'} \\to A'$\nbe the strict henselization constructed using\n$\\kappa(s') \\subset \\kappa(\\overline{s}')$.\nBy functoriality of strict henselization, there is a map\n$A \\to A'$ such that the composition with $A' \\to \\kappa(\\overline{s}')$\nis the given map\n(Algebra, Lemma \\ref{algebra-lemma-map-into-henselian-colimit}).\nNext, let $A' \\to \\kappa(\\overline{s}'')$ be the map used to\nconstruct the second specialization map. Then it is clear that\nthe composition of the first and second specialization maps\nis the specialization map\n$\\pi_1(X_{\\overline{s}''}) \\to \\pi_1(X_{\\overline{s}})$\nconstructed using $A \\to A' \\to \\kappa(\\overline{s}'')$.\n\\end{proof}\n\n\\noindent\nLet $X \\to S$ be a proper morphism with geometrically connected fibres.\nLet $R$ be a strictly henselian valuation ring with algebraically\nclosed fraction field and let $\\Spec(R) \\to S$\nbe a morphism. Let $\\eta, s \\in \\Spec(R)$ be the generic and closed point.\nThen we can consider the specialization map\n$$\nsp_R : \\pi_1(X_\\eta) \\to \\pi_1(X_s)\n$$\nfor the base change $X_R/\\Spec(R)$. Note that this makes sense as both\n$\\eta$ and $s$ have algebraically closed residue fields.\n\n\\begin{lemma}\n\\label{lemma-specialization-map-valuation-ring}\nLet $f : X \\to S$ be a proper morphism with geometrically connected fibres.\nLet $s' \\leadsto s$ be a specialization of points of $S$ and let\n$sp : \\pi_1(X_{\\overline{s}'}) \\to \\pi_1(X_{\\overline{s}})$\nbe a specialization map. Then there exists a strictly henselian\nvaluation ring $R$ over $S$ with algebraically closed fraction field\nsuch that $sp$ is isomorphic to $sp_R$ defined above.\n\\end{lemma}\n\n\\begin{proof}\nLet $\\mathcal{O}_{S, s} \\to A$ be the strict henselization\nconstructed using $\\kappa(s) \\to \\kappa(\\overline{s})$.\nLet $A \\to \\kappa(\\overline{s}')$ be the map used to construct $sp$.\nLet $R \\subset \\kappa(\\overline{s}')$ be a valuation ring with\nfraction field $\\kappa(\\overline{s}')$ dominating the image of $A$.\nSee Algebra, Lemma \\ref{algebra-lemma-dominate}.\nObserve that $R$ is strictly henselian for example by\nLemma \\ref{lemma-normal-local-domain-separablly-closed-fraction-field}\nand Algebra, Lemma \\ref{algebra-lemma-valuation-ring-normal}.\nThen the lemma is clear.\n\\end{proof}\n\n\\noindent\nLet $X \\to S$ be a proper morphism with geometrically connected fibres.\nLet $R$ be a strictly henselian discrete valuation ring and let\n$\\Spec(R) \\to S$ be a morphism. Let $\\eta, s \\in \\Spec(R)$ be the\ngeneric and closed point. Then we can consider the specialization map\n$$\nsp_R : \\pi_1(X_{\\overline{\\eta}}) \\to \\pi_1(X_s)\n$$\nfor the base change $X_R/\\Spec(R)$. Note that this makes sense as $s$\nhas algebraically closed residue field.\n\n\\begin{lemma}\n\\label{lemma-specialization-map-discrete-valuation-ring}\nLet $f : X \\to S$ be a proper morphism with geometrically connected fibres.\nLet $s' \\leadsto s$ be a specialization of points of $S$ and let\n$sp : \\pi_1(X_{\\overline{s}'}) \\to \\pi_1(X_{\\overline{s}})$\nbe a specialization map. If $S$ is Noetherian, then\nthere exists a strictly henselian\ndiscrete valuation ring $R$ over $S$ such that $sp$ is isomorphic to $sp_R$\ndefined above.\n\\end{lemma}\n\n\\begin{proof}\nLet $\\mathcal{O}_{S, s} \\to A$ be the strict henselization\nconstructed using $\\kappa(s) \\to \\kappa(\\overline{s})$.\nLet $A \\to \\kappa(\\overline{s}')$ be the map used to construct $sp$.\nLet $R \\subset \\kappa(\\overline{s}')$ be a discrete valuation ring\ndominating the image of $A$, see Algebra, Lemma \\ref{algebra-lemma-exists-dvr}.\nChoose a diagram of fields\n$$\n\\xymatrix{\n\\kappa(\\overline{s}) \\ar[r] & k \\\\\nA/\\mathfrak m_A \\ar[r] \\ar[u] & R/\\mathfrak m_R \\ar[u]\n}\n$$\nwith $k$ algebraically closed. Let $R^{sh}$ be the strict\nhenselization of $R$ constructed using $R \\to k$. Then\n$R^{sh}$ is a discrete valuation ring by\nMore on Algebra, Lemma \\ref{more-algebra-lemma-henselization-dvr}.\nDenote $\\eta, o$ the generic and closed point of $\\Spec(R^{sh})$.\nSince the diagram of schemes\n$$\n\\xymatrix{\n\\overline{\\eta} \\ar[d] \\ar[r] & \\Spec(R^{sh}) \\ar[d] &\n\\Spec(k) \\ar[d] \\ar[l] \\\\\n\\overline{s}' \\ar[r] & \\Spec(A) & \\overline{s} \\ar[l]\n}\n$$\ncommutes, we obtain a commutative diagram\n$$\n\\xymatrix{\n\\pi_1(X_{\\overline{\\eta}}) \\ar[d] \\ar[r]_{sp_{R^{sh}}} & \\pi_1(X_o) \\ar[d] \\\\\n\\pi_1(X_{\\overline{s}'}) \\ar[r]^{sp} & X_{\\overline{s}}\n}\n$$\nof specialization maps by the construction of these maps.\nSince the vertical arrows are isomorphisms\n(Lemma \\ref{lemma-finite-etale-invariant-over-proper}), this proves the lemma.\n\\end{proof}\n\n\n\n\n\n\n\n\n\n\\section{Finite \\'etale covers of punctured spectra, I}\n\\label{section-pi1-punctured-spec}\n\n\\noindent\nWe first prove some results \\'a la Lefschetz.\n\n\\begin{situation}\n\\label{situation-local-lefschetz}\nLet $(A, \\mathfrak m)$ be a Noetherian local ring and $f \\in \\mathfrak m$.\nWe set $X = \\Spec(A)$ and $X_0 = \\Spec(A/fA)$ and we\nlet $U = X \\setminus \\{\\mathfrak m\\}$ and\n$U_0 = X_0 \\setminus \\{\\mathfrak m\\}$ be the punctured spectrum of\n$A$ and $A/fA$.\n\\end{situation}\n\n\\noindent\nRecall that for a scheme $X$ the category of schemes finite\n\\'etale over $X$ is denoted $\\textit{F\\'Et}_X$, see\nSection \\ref{section-finite-etale}.\nIn Situation \\ref{situation-local-lefschetz}\nwe will study the base change functors\n$$\n\\xymatrix{\n\\textit{F\\'Et}_X \\ar[d] \\ar[r] & \\textit{F\\'Et}_U \\ar[d] \\\\\n\\textit{F\\'Et}_{X_0} \\ar[r] & \\textit{F\\'Et}_{U_0}\n}\n$$\nIn many case the right vertical arrow is faithful.\n\n\\begin{lemma}\n\\label{lemma-faithful}\nIn Situation \\ref{situation-local-lefschetz}.\nAssume one of the following holds\n\\begin{enumerate}\n\\item $\\dim(A/\\mathfrak p) \\geq 2$ for every minimal prime\n$\\mathfrak p \\subset A$ with $f \\not \\in \\mathfrak p$, or\n\\item every connected component of $U$ meets $U_0$.\n\\end{enumerate}\nThen\n$$\n\\textit{F\\'Et}_U \\longrightarrow \\textit{F\\'Et}_{U_0},\\quad\nV \\longmapsto V_0 = V \\times_U U_0\n$$\nis a faithful functor.\n\\end{lemma}\n\n\\begin{proof}\nLet $a, b : V \\to W$ be two morphisms of schemes finite \\'etale over $U$\nwhose restriction to $U_0$ are the same. Assumption (1)\nmeans that every irreducible component of $U$ meets $U_0$, see\nAlgebra, Lemma \\ref{algebra-lemma-one-equation}.\nThe image of any irreducible component of $V$ is an\nirreducible component of $U$ and hence meets $U_0$.\nHence $V_0$ meets every connected component of $V$ and\nwe conclude that $a = b$ by \\'Etale Morphisms, Proposition\n\\ref{etale-proposition-equality}.\nIn case (2) the argument is the same using that the image\nof a connected component of $V$ is a connected component of $U$.\n\\end{proof}\n\n\\noindent\nBefore we prove something more interesting, we need a couple of lemmas.\n\n\\begin{lemma}\n\\label{lemma-fill-in-missing}\nIn Situation \\ref{situation-local-lefschetz}. Let $V \\to U$ be a finite\nmorphism.  Let $A^\\wedge$ be the $\\mathfrak m$-adic completion of $A$,\nlet $X' = \\Spec(A^\\wedge)$ and let $U'$ and $V'$ be the base changes of\n$U$ and $V$ to $X'$. If $Y' \\to X'$ is a finite morphism such that\n$V' = Y' \\times_{X'} U'$, then there exists a finite morphism $Y \\to X$\nsuch that $V = Y \\times_X U$ and $Y' = Y \\times_X X'$.\n\\end{lemma}\n\n\\begin{proof}\nThis is a straightforward application of\nMore on Algebra, Proposition \\ref{more-algebra-proposition-equivalence}.\nNamely, choose generators $f_1, \\ldots, f_t$ of $\\mathfrak m$.\nFor each $i$ write $V \\times_U D(f_i) = \\Spec(B_i)$.\nFor $1 \\leq i, j \\leq n$ we obtain an isomorphism\n$\\alpha_{ij} : (B_i)_{f_j} \\to (B_j)_{f_i}$ of $A_{f_if_j}$-algebras\nbecause the spectrum of both represent $V \\times_U D(f_if_j)$.\nWrite $Y' = \\Spec(B')$. Since $V \\times_U U' = Y \\times_{X'} U'$\nwe get isomorphisms $\\alpha_i : B'_{f_i} \\to B_i \\otimes_A A^\\wedge$.\nA straightforward argument shows that $(B', B_i, \\alpha_i, \\alpha_{ij})$\nis an object of $\\text{Glue}(A \\to A^\\wedge, f_1, \\ldots, f_t)$, see\nMore on Algebra, Remark \\ref{more-algebra-remark-glueing-data}.\nApplying the proposition cited above (and using\nMore on Algebra, Remark \\ref{more-algebra-remark-formal-glueing-algebras}\nto obtain the algebra structure) we find an $A$-algebra $B$ such that\n$\\text{Can}(B)$ is isomorphic to $(B', B_i, \\alpha_i, \\alpha_{ij})$.\nSetting $Y = \\Spec(B)$ we see that $Y \\to X$ is a morphism\nwhich comes equipped with compatible isomorphisms\n$V \\cong Y \\times_X U$ and $Y' = Y \\times_X X'$ as desired.\n\\end{proof}\n\n\\begin{lemma}\n\\label{lemma-fully-faithful-henselian-completion}\nIn Situation \\ref{situation-local-lefschetz} assume $A$ is henselian\nor more generally that $(A, (f))$ is a henselian pair.\nLet $A^\\wedge$ be the $\\mathfrak m$-adic completion of $A$,\nlet $X' = \\Spec(A^\\wedge)$ and let $U'$ and $U'_0$ be the base changes of\n$U$ and $U_0$ to $X'$. If $\\textit{F\\'Et}_{U'} \\to \\textit{F\\'Et}_{U'_0}$\nis fully faithful, then $\\textit{F\\'Et}_U \\to \\textit{F\\'Et}_{U_0}$\nis fully faithful.\n\\end{lemma}\n\n\\begin{proof}\nAssume $\\textit{F\\'Et}_{U'} \\longrightarrow \\textit{F\\'Et}_{U'_0}$\nis a fully faithful. Since $X' \\to X$ is faithfully flat, it is\nimmediate that the functor $V \\to V_0 = V \\times_U U_0$ is faithful.\nSince the category of finite \\'etale coverings has an internal hom\n(Lemma \\ref{lemma-internal-hom-finite-etale})\nit suffices to prove the following: Given $V$ finite \\'etale over $U$\nwe have\n$$\n\\Mor_U(U, V) = \\Mor_{U_0}(U_0, V_0)\n$$\nThe we assume we have a morphism $s_0 : U_0 \\to V_0$ over $U_0$ and we will\nproduce a morphism $s : U \\to V$ over $U$.\n\n\\medskip\\noindent\nBy our assumption there does exist a morphism $s' : U' \\to V'$\nwhose restriction to $V'_0$ is the base change $s'_0$ of $s_0$.\nSince $V' \\to U'$ is finite \\'etale this means that $V' = s'(U') \\amalg W'$\nfor some $W' \\to U'$ finite and \\'etale.\nChoose a finite morphism $Z' \\to X'$ such that $W' = Z' \\times_{X'} U'$.\nThis is possible by Zariski's main theorem in the form stated in\nMore on Morphisms, Lemma\n\\ref{more-morphisms-lemma-quasi-finite-separated-pass-through-finite}\n(small detail omitted).\nThen\n$$\nV' = s'(U') \\amalg W' \\longrightarrow X' \\amalg Z' = Y'\n$$\nis an open immersion such that $V' = Y' \\times_{X'} U'$.\nBy Lemma \\ref{lemma-fill-in-missing} we can find $Y \\to X$ finite\nsuch that $V = Y \\times_X U$ and $Y' = Y \\times_X X'$.\nWrite $Y = \\Spec(B)$ so that $Y' = \\Spec(B \\otimes_A A^\\wedge)$.\nThen $B \\otimes_A A^\\wedge$ has an idempotent $e'$\ncorresponding to the open and closed subscheme $X'$ of $Y' = X' \\amalg Z'$.\n\n\\medskip\\noindent\nThe case $A$ is henselian (slightly easier). The image $\\overline{e}$\nof $e'$ in $B \\otimes_A \\kappa(\\mathfrak m) = B/\\mathfrak mB$ lifts to an\nidempotent $e$ of $B$ as $A$ is henselian (because $B$ is a product of\nlocal rings by Algebra, Lemma \\ref{algebra-lemma-characterize-henselian}).\nThen we see that $e$ maps to $e'$ by uniqueness of lifts of idempotents\n(using that $B \\otimes_A A^\\wedge$ is a product of local rings).\nLet $Y_1 \\subset Y$ be the open and closed subscheme corresponding to $e$.\nThen $Y_1 \\times_X X' = s'(X')$ which implies that $Y_1 \\to X$ is\nan isomorphism (by faithfully flat descent) and gives the desired section.\n\n\\medskip\\noindent\nThe case where $(A, (f))$ is a henselian pair. Here we use that $s'$ is\na lift of $s'_0$. Namely, let $Y_{0, 1} \\subset Y_0 = Y \\times_X X_0$\nbe the closure of $s_0(U_0) \\subset V_0 = Y_0 \\times_{X_0} U_0$.\nAs $X' \\to X$ is flat, the base change $Y'_{0, 1} \\subset Y'_0$\nis the closure of $s'_0(U'_0)$ which is equal to $X'_0 \\subset Y'_0$\n(see Morphisms, Lemma\n\\ref{morphisms-lemma-flat-base-change-scheme-theoretic-image}).\nSince $Y'_0 \\to Y_0$ is submersive\n(Morphisms, Lemma \\ref{morphisms-lemma-fpqc-quotient-topology})\nwe conclude that $Y_{0, 1}$ is open and closed in $Y_0$.\nLet $e_0 \\in B/fB$ be the corresponding idempotent.\nBy More on Algebra, Lemma\n\\ref{more-algebra-lemma-characterize-henselian-pair}\nwe can lift $e_0$ to an idempotent $e \\in B$.\nThen we conclude as before.\n\\end{proof}\n\n\\noindent\nThe following lemma will be superseded by\nLemma \\ref{lemma-fully-faithful-minimal} below.\n\n\\begin{lemma}\n\\label{lemma-fully-faithful}\nIn Situation \\ref{situation-local-lefschetz}.\nAssume $f$ is a nonzerodivisor, that $A$ has depth $\\geq 3$, and that\n$A$ is henselian or more generally $(A, (f))$ is a henselian pair. Then\n$$\n\\textit{F\\'Et}_U \\longrightarrow \\textit{F\\'Et}_{U_0},\\quad\nV \\longmapsto V_0 = V \\times_U U_0\n$$\nis a fully faithful functor.\n\\end{lemma}\n\n\\begin{proof}\nBy Lemma \\ref{lemma-fully-faithful-henselian-completion} we may assume $A$\nis a complete local Noetherian ring. The functor is faithful by\nLemma \\ref{lemma-faithful} (to see the assumption of that lemma holds, apply\nAlgebra, Lemma \\ref{algebra-lemma-depth-dim-associated-primes}).\nSince the category of finite \\'etale coverings has an internal hom\n(Lemma \\ref{lemma-internal-hom-finite-etale})\nit suffices to prove the following: Given $V$ finite \\'etale over $U$ we have\n$$\n\\Mor_U(U, V) = \\Mor_{U_0}(U_0, V_0)\n$$\nIf we have a morphism $U_0 \\to V_0$ over $U_0$, then we obtain an\ndecomposition $V_0 = U_0 \\amalg V'_0$ into open and closed subschemes.\nWe will show that this implies the same thing for $V$ thereby\nfinishing the proof.\n\n\\medskip\\noindent\nFor $n \\geq 1$ let $U_n$ be the punctured spectrum of $A/f^{n + 1}A$\nand let $V_n \\to U_n$ be the base change of $V \\to U$. By\n\\'Etale Morphisms, Theorem \\ref{etale-theorem-remarkable-equivalence}\nwe conclude that there is a unique decomposition\n$V_n = U_n \\amalg V'_n$\ninto open and closed subschemes whose base change to $U_0$ recovers\nthe given decomposition.\n\n\\medskip\\noindent\nSince $A$ has depth $\\geq 3$ and $f$ is a nonzerodivisor, we see\nthat $A/fA$ has depth $\\geq 2$\n(Algebra, Lemma \\ref{algebra-lemma-depth-drops-by-one}).\nThis implies the\nvanishing of $H^0_\\mathfrak m(A/fA)$ and $H^1_\\mathfrak m(A/fA)$, see\nDualizing Complexes, Lemma \\ref{dualizing-lemma-depth}.\nThis in turn\ntells us that $A/fA \\to \\Gamma(U_0, \\mathcal{O}_{U_0})$ is an isomorphism, see\nLocal Cohomology, Lemma\n\\ref{local-cohomology-lemma-finiteness-pushforwards-and-H1-local}.\nAs $f$ is a nonzerodivisor we obtain short exact sequences\n$$\n0 \\to A/fA \\xrightarrow{f^n} A/f^{n + 1}A \\to A/f^n A \\to 0\n$$\nInduction on $n$ shows that\n$H^0_\\mathfrak m(A/f^{n + 1}A) = H^1_\\mathfrak m(A/f^{n + 1}A) = 0$\nfor all $n$. Hence the same reasoning shows that\n$A/f^{n + 1}A \\to \\Gamma(U_n, \\mathcal{O}_{U_n})$\nis an isomorphism.\nCombined with the decompositions above this determines a map\n$$\n\\Gamma(V, \\mathcal{O}_V) \\to\n\\lim \\Gamma(V_n, \\mathcal{O}_{V_n}) \\to\n\\lim \\Gamma(U_n, \\mathcal{O}_{U_n}) = A\n$$\nSince $V \\to U$ is affine, this $A$-algebra map corresponds to\na section $U \\to V$ as desired.\n\\end{proof}\n\n\\noindent\nIn the following lemma we prove fully faithfulness under very weak assumptions.\nNote that the assumptions do not imply that $U$ is a connected scheme, but\nthe conclusion guarantees that $U$ and $U_0$ have the same number of\nconnected components.\n\n\\begin{lemma}\n\\label{lemma-fully-faithful-minimal}\n\\begin{reference}\n\\cite[Corollary 1.11]{Bhatt-local}\n\\end{reference}\nIn Situation \\ref{situation-local-lefschetz}. Assume\n\\begin{enumerate}\n\\item $f$ is a nonzerodivisor,\n\\item $H^1_\\mathfrak m(A)$ is finite,\n\\item $H^2_\\mathfrak m(A)$ is annihilated by a power of $f$, and\n\\item $A$ is henselian or more generally $(A, (f))$ is a henselian pair.\n\\end{enumerate}\nThen\n$$\n\\textit{F\\'Et}_U \\longrightarrow \\textit{F\\'Et}_{U_0},\\quad\nV \\longmapsto V_0 = V \\times_U U_0\n$$\nis a fully faithful functor.\n\\end{lemma}\n\n\\begin{proof}\nBy Lemma \\ref{lemma-fully-faithful-henselian-completion}\nwe may assume that $A$ is a Noetherian complete local ring.\n(The assumptions carry over; use\nDualizing Complexes, Lemma \\ref{dualizing-lemma-torsion-change-rings}.)\n\n\\medskip\\noindent\nAssume $A$ is complete in addition to the other conditions.\nWe will show that given $\\pi : V \\to U$ finite \\'etale, the set\nof connected components of $V$ agrees with the set of connected\ncomponents of $V_0$. This will prove the lemma because the\ncategory of finite \\'etale covers has internal hom\n(Lemma \\ref{lemma-internal-hom-finite-etale})\nand images of sections are connected components\n(\\'Etale Morphisms, Proposition \\ref{etale-proposition-properties-sections}).\nSome details omitted.\n\n\\medskip\\noindent\nSet $\\mathcal{B} = \\pi_*\\mathcal{O}_V$. This is a finite locally free\n$\\mathcal{O}_U$-algebra. Thus\n$\\text{Ass}(\\mathcal{B}) = \\text{Ass}(\\mathcal{O}_U)$.\nAssumption (2) means that $H^0(U, \\mathcal{O}_U)$ is a finite\n$A$-module and equivalently that $j_*\\mathcal{O}_U$ is coherent\n(Local Cohomology, Lemma\n\\ref{local-cohomology-lemma-finiteness-pushforwards-and-H1-local}).\nBy Local Cohomology, Proposition \\ref{local-cohomology-proposition-kollar}\nand the agreement of $\\text{Ass}$\nwe see that the same holds for $\\mathcal{B}$ and we conclude\nthat $B = \\Gamma(U, \\mathcal{B}) = \\Gamma(V, \\mathcal{O}_V)$\nis a finite $A$-algebra.\n\n\\medskip\\noindent\nNext, using that $H^2_\\mathfrak m(A) = H^1(U, \\mathcal{O}_U)$\nis annihilated by $f^n$ for some $n$ we see that\n$H^1(U, \\mathcal{B}) = H^1(V, \\mathcal{O}_V)$\nis annihilated by $f^m$ for some $m$, see\nLocal Cohomology, Lemma \\ref{local-cohomology-lemma-annihilate-Hp}.\n\n\\medskip\\noindent\nAt this point we apply Local Cohomology, Lemma\n\\ref{local-cohomology-lemma-formal-functions-principal} to\nthe scheme $V$ over $\\Spec(A)$ and the sheaf $\\mathcal{O}_V$\nwith $p = 0$. Since $f$ is a nonzerodivisor in $A$ the $f$-power torsion\nsubsheaf of $\\mathcal{O}_V$ is zero. The first short exact sequence\nof the lemma collapses to become\n$$\nH^0 = \\lim H^0(V, \\mathcal{O}_V/f^n\\mathcal{O}_V) =\n\\lim H^0(V_n, \\mathcal{O}_{V_n})\n$$\nwhere $V_n \\subset V$ is the closed subscheme cut out by $f^{n + 1}$.\nSince $H^1(V, \\mathcal{O}_V)$ is annihilated by a power\nof $f$ we see that the Tate module $T_f(H^1(V, \\mathcal{O}_V))$ is zero.\nOn the other hand, since $A$ is complete and\n$B = H^0(V, \\mathcal{O}_V)$ is a finite $A$-module\nit is complete (Algebra, Lemma \\ref{algebra-lemma-completion-tensor})\nhence derived complete\n(More on Algebra,\nProposition \\ref{more-algebra-proposition-derived-complete-modules})\nand hence equal to its derived $f$-adic completion.\nThus we see that $H^0 = B$.\nSince\n$$\nV_0 \\subset V_1 \\subset V_2 \\subset \\ldots\n$$\nare nilpotent thickenings the connected components of these schemes\nagree. Correspondingly the maps\n$$\n\\ldots \\to\nH^0(V_2, \\mathcal{O}_{V_2}) \\to\nH^0(V_1, \\mathcal{O}_{V_1}) \\to\nH^0(V_0, \\mathcal{O}_{V_0})\n$$\ninduce bijections between idempotents. Hence the map\n$B \\to H^0(V_0, \\mathcal{O}_{V_0})$ induces a bijection between\nidempotents and we conclude.\n\\end{proof}\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\\section{Purity in local case, I}\n\\label{section-local-purity}\n\n\\noindent\nLet $(A, \\mathfrak m)$ be a Noetherian local ring. Set $X = \\Spec(A)$\nand let $U = X \\setminus \\{\\mathfrak m\\}$ be the punctured spectrum.\nWe say {\\it purity holds for $(A, \\mathfrak m)$} if the restriction functor\n$$\n\\textit{F\\'Et}_X \\longrightarrow \\textit{F\\'Et}_U\n$$\nis essentially surjective. In this section we try to understand how the\nquestion changes when one passes from $X$ to a hypersurface $X_0$ in $X$,\nin other words, we study a kind of local Lefschetz property for the\nfundamental groups of punctured spectra.\nThese results will be useful to proceed by induction on dimension\nin the proofs of our main results on local purity, namely,\nLemma \\ref{lemma-local-purity} and\nProposition \\ref{proposition-purity-complete-intersection}.\n\n\\begin{lemma}\n\\label{lemma-sections-over-punctured-spec}\nLet $(A, \\mathfrak m)$ be a Noetherian local ring. Set $X = \\Spec(A)$\nand let $U = X \\setminus \\{\\mathfrak m\\}$.\nLet $\\pi : Y \\to X$ be a finite morphism such that\n$\\text{depth}(\\mathcal{O}_{Y, y}) \\geq 2$ for all closed points\n$y \\in Y$.\nThen $Y$ is the spectrum of $B = \\mathcal{O}_Y(\\pi^{-1}(U))$.\n\\end{lemma}\n\n\\begin{proof}\nSet $V = \\pi^{-1}(U)$ and denote $\\pi' : V \\to U$ the restriction of $\\pi$.\nConsider the $\\mathcal{O}_X$-module map\n$$\n\\pi_*\\mathcal{O}_Y \\longrightarrow j_*\\pi'_*\\mathcal{O}_V\n$$\nwhere $j : U \\to X$ is the inclusion morphism. We claim\nDivisors, Lemma \\ref{divisors-lemma-check-isomorphism-via-depth-and-ass}\napplies to this map. If so, then $B = \\Gamma(Y, \\mathcal{O}_Y)$\nand we see that the lemma holds. Let $x \\in X$.\nIf $x \\in U$, then the map is an\nisomorphism on stalks as $V = Y \\times_X U$.\nIf $x$ is the closed point, then\n$x \\not \\in \\text{Ass}(j_*\\pi_*\\mathcal{O}_V)$\n(Divisors, Lemmas \\ref{divisors-lemma-weakass-pushforward} and\n\\ref{divisors-lemma-weakly-ass-support}).\nThus it suffices to show that\n$\\text{depth}((\\pi_*\\mathcal{O}_Y)_x) \\geq 2$.\nLet $y_1, \\ldots, y_n \\in Y$ be the points mapping to $x$.\nBy Algebra, Lemma \\ref{algebra-lemma-depth-goes-down-finite}\nit suffices to show that\n$\\text{depth}(\\mathcal{O}_{Y, y_i}) \\geq 2$ for $i = 1, \\ldots, n$.\nSince this is the assumption of the lemma the proof is complete.\n\\end{proof}\n\n\\begin{lemma}\n\\label{lemma-reformulate-purity}\nLet $(A, \\mathfrak m)$ be a Noetherian local ring. Set $X = \\Spec(A)$\nand let $U = X \\setminus \\{\\mathfrak m\\}$.\nLet $V$ be finite \\'etale\nover $U$. Assume $A$ has depth $\\geq 2$. The following are equivalent\n\\begin{enumerate}\n\\item $V = Y \\times_X U$ for some $Y \\to X$ finite \\'etale,\n\\item $B = \\Gamma(V, \\mathcal{O}_V)$ is finite \\'etale over $A$.\n\\end{enumerate}\n\\end{lemma}\n\n\\begin{proof}\nDenote $\\pi : V \\to U$ the given finite \\'etale morphism.\nAssume $Y$ as in (1) exists. Let $x \\in X$ be the point\ncorresponding to $\\mathfrak m$.\nLet $y \\in Y$ be a point mapping to $x$. We claim that\n$\\text{depth}(\\mathcal{O}_{Y, y}) \\geq 2$.\nThis is true because $Y \\to X$ is \\'etale and hence\n$A = \\mathcal{O}_{X, x}$ and $\\mathcal{O}_{Y, y}$ have\nthe same depth (Algebra, Lemma \\ref{algebra-lemma-apply-grothendieck}).\nHence Lemma \\ref{lemma-sections-over-punctured-spec}\napplies and $Y = \\Spec(B)$.\n\n\\medskip\\noindent\nThe implication (2) $\\Rightarrow$ (1) is easier and the\ndetails are omitted.\n\\end{proof}\n\n\\begin{lemma}\n\\label{lemma-reformulate-purity-normal}\nLet $(A, \\mathfrak m)$ be a Noetherian local ring. Set $X = \\Spec(A)$\nand let $U = X \\setminus \\{\\mathfrak m\\}$. Assume $A$ is normal\nof dimension $\\geq 2$. The functor\n$$\n\\textit{F\\'Et}_U \\longrightarrow\n\\left\\{\n\\begin{matrix}\n\\text{finite normal }A\\text{-algebras }B\\text{ such} \\\\\n\\text{that }\\Spec(B) \\to X\\text{ is \\'etale over }U\n\\end{matrix}\n\\right\\},\n\\quad\nV \\longmapsto \\Gamma(V, \\mathcal{O}_V)\n$$\nis an equivalence. Moreover, $V = Y \\times_X U$ for some $Y \\to X$\nfinite \\'etale if and only if $B = \\Gamma(V, \\mathcal{O}_V)$\nis finite \\'etale over $A$.\n\\end{lemma}\n\n\\begin{proof}\nObserve that $\\text{depth}(A) \\geq 2$ because $A$ is normal\n(Serre's criterion for normality, Algebra, Lemma\n\\ref{algebra-lemma-criterion-normal}).\nThus the final statement follows from Lemma \\ref{lemma-reformulate-purity}.\nGiven $\\pi : V \\to U$ finite \\'etale, set $B = \\Gamma(V, \\mathcal{O}_V)$.\nIf we can show that $B$ is normal and finite over $A$, then\nwe obtain the displayed functor. Since there is an obvious\nquasi-inverse functor, this is also all that we have to show.\n\n\\medskip\\noindent\nSince $A$ is normal, the scheme $V$ is normal\n(Descent, Lemma \\ref{descent-lemma-normal-local-smooth}).\nHence $V$ is a finite disjoint union of integral schemes\n(Properties, Lemma \\ref{properties-lemma-normal-Noetherian}).\nThus we may assume $V$ is integral.\nIn this case the function field $L$ of $V$\n(Morphisms, Section \\ref{morphisms-section-rational-maps})\nis a finite separable extension of $f.f.(A)$\n(because we get it by looking at the generic fibre\nof $V \\to U$ and using Morphisms, Lemma\n\\ref{morphisms-lemma-etale-over-field}).\nBy Algebra, Lemma\n\\ref{algebra-lemma-Noetherian-normal-domain-finite-separable-extension}\nthe integral closure $B' \\subset L$ of $A$ in $L$ is finite over $A$.\nBy More on Algebra, Lemma \\ref{more-algebra-lemma-integral-closure-reflexive}\nwe see that $B'$ is a reflexive $A$-module, which in turn implies\nthat $\\text{depth}_A(B') \\geq 2$ by\nMore on Algebra, Lemma \\ref{more-algebra-lemma-reflexive-over-normal}.\n\n\\medskip\\noindent\nLet $f \\in \\mathfrak m$. Then $B_f = \\Gamma(V \\times_U D(f), \\mathcal{O}_V)$\n(Properties, Lemma \\ref{properties-lemma-invert-f-sections}).\nHence $B'_f = B_f$ because $B_f$ is normal (see above),\nfinite over $A_f$ with fraction field $L$.\nIt follows that $V = \\Spec(B') \\times_X U$.\nThen we conclude that $B = B'$ from\nLemma \\ref{lemma-sections-over-punctured-spec}\napplied to $\\Spec(B') \\to X$.\nThis lemma applies because the localizations $B'_{\\mathfrak m'}$\nof $B'$ at maximal ideals $\\mathfrak m' \\subset B'$ lying over\n$\\mathfrak m$ have depth $\\geq 2$ by\nAlgebra, Lemma \\ref{algebra-lemma-depth-goes-down-finite}\nand the remark on depth in the preceding paragraph.\n\\end{proof}\n\n\\begin{lemma}\n\\label{lemma-purity-and-completion}\nLet $(A, \\mathfrak m)$ be a Noetherian local ring. Set $X = \\Spec(A)$\nand let $U = X \\setminus \\{\\mathfrak m\\}$.\nLet $V$ be finite \\'etale over $U$.\nLet $A^\\wedge$ be the $\\mathfrak m$-adic completion of $A$,\nlet $X' = \\Spec(A^\\wedge)$ and let $U'$ and $V'$ be the base changes of\n$U$ and $V$ to $X'$. The following are equivalent\n\\begin{enumerate}\n\\item $V = Y \\times_X U$ for some $Y \\to X$ finite \\'etale, and\n\\item $V' = Y' \\times_{X'} U'$ for some $Y' \\to X'$ finite \\'etale.\n\\end{enumerate}\n\\end{lemma}\n\n\\begin{proof}\nThe implication (1) $\\Rightarrow$ (2) follows from taking the base change\nof a solution $Y \\to X$. Let $Y' \\to X'$ be as in (2).\nBy Lemma \\ref{lemma-fill-in-missing} we can find $Y \\to X$ finite\nsuch that $V = Y \\times_X U$ and $Y' = Y \\times_X X'$.\nBy descent we see that $Y \\to X$ is finite \\'etale\n(Algebra, Lemmas \\ref{algebra-lemma-descend-properties-modules} and\n\\ref{algebra-lemma-etale}). This finishes the proof.\n\\end{proof}\n\n\\noindent\nThe following lemma will be superseded by\nLemma \\ref{lemma-lift-purity-general}.\n\n\\begin{lemma}\n\\label{lemma-lift-purity}\nIn Situation \\ref{situation-local-lefschetz}.\nLet $V$ be finite \\'etale over $U$. Assume\n\\begin{enumerate}\n\\item $f$ is a nonzerodivisor,\n\\item $A$ has depth $\\geq 3$,\n\\item $V_0 = V \\times_U U_0$ is equal to $Y_0 \\times_{X_0} U_0$\nfor some $Y_0 \\to X_0$ finite \\'etale.\n\\end{enumerate}\nThen $V = Y \\times_X U$ for some $Y \\to X$ finite \\'etale.\n\\end{lemma}\n\n\\begin{proof}\nWe reduce to the complete case by Lemma \\ref{lemma-purity-and-completion}.\nAlternatively you can use Lemma \\ref{lemma-reformulate-purity},\ncohomology and base change\n(Cohomology of Schemes, Lemma\n\\ref{coherent-lemma-flat-base-change-cohomology}), and descent\n(Algebra, Lemmas \\ref{algebra-lemma-descend-properties-modules} and\n\\ref{algebra-lemma-etale}).\n\n\\medskip\\noindent\nIn the complete case we can lift $Y_0 \\to X_0$ to a finite\n\\'etale morphism $Y \\to X$ by\nMore on Algebra, Lemma \\ref{more-algebra-lemma-finite-etale-equivalence};\nobserve that $(A, fA)$ is a henselian pair by\nMore on Algebra, Lemma \\ref{more-algebra-lemma-complete-henselian}.\nThen we can use Lemma \\ref{lemma-fully-faithful}\nto see that $V$ is isomorphic to $Y \\times_X U$ and\nthe proof is complete.\n\\end{proof}\n\n\\noindent\nThe point of the following lemma is that the assumptions do not force\n$A$ to have depth $\\geq 3$. For example if $A$ is a complete normal\nlocal domain of dimension $\\geq 3$ and $f \\in \\mathfrak m$ is nonzero,\nthen the assumptions are satisfied.\n\n\\begin{lemma}\n\\label{lemma-lift-purity-general}\nIn Situation \\ref{situation-local-lefschetz}.\nLet $V$ be finite \\'etale over $U$. Assume\n\\begin{enumerate}\n\\item $f$ is a nonzerodivisor,\n\\item $H^1_\\mathfrak m(A)$ is a finite $A$-module,\n\\item a power of $f$ annihilates $H^2_\\mathfrak m(A)$,\n\\item $V_0 = V \\times_U U_0$ is equal to $Y_0 \\times_{X_0} U_0$\nfor some $Y_0 \\to X_0$ finite \\'etale.\n\\end{enumerate}\nThen $V = Y \\times_X U$ for some $Y \\to X$ finite \\'etale.\n\\end{lemma}\n\n\\begin{proof}\nWe reduce to the complete case using Lemma \\ref{lemma-purity-and-completion}.\n(The assumptions carry over; use Dualizing Complexes, Lemma\n\\ref{dualizing-lemma-torsion-change-rings}.)\n\n\\medskip\\noindent\nIn the complete case we can lift $Y_0 \\to X_0$ to a finite \\'etale\nmorphism $Y \\to X$ by\nMore on Algebra, Lemma \\ref{more-algebra-lemma-finite-etale-equivalence};\nobserve that $(A, fA)$ is a henselian pair by\nMore on Algebra, Lemma \\ref{more-algebra-lemma-complete-henselian}.\nThen we can use Lemma \\ref{lemma-fully-faithful-minimal}\nto see that $V$ is isomorphic to $Y \\times_X U$ and\nthe proof is complete.\n\\end{proof}\n\n\n\n\n\n\n\n\n\n\n\\section{Purity of branch locus}\n\\label{section-purity}\n\n\\noindent\nWe will use the discriminant of a finite locally free morphism. See\nDiscriminants, Section \\ref{discriminant-section-discriminant}.\n\n\\begin{lemma}\n\\label{lemma-find-point-codim-1}\nLet $(A, \\mathfrak m)$ be a Noetherian local ring with $\\dim(A) \\geq 1$.\nLet $f \\in \\mathfrak m$. Then there exist a $\\mathfrak p \\in V(f)$ with\n$\\dim(A_\\mathfrak p) = 1$.\n\\end{lemma}\n\n\\begin{proof}\nBy induction on $\\dim(A)$. If $\\dim(A) = 1$, then $\\mathfrak p = \\mathfrak m$\nworks. If $\\dim(A) > 1$, then let $Z \\subset \\Spec(A)$ be an irreducible\ncomponent of dimension $> 1$. Then $V(f) \\cap Z$ has dimension $> 0$\n(Algebra, Lemma \\ref{algebra-lemma-one-equation}). Pick a prime\n$\\mathfrak q \\in V(f) \\cap Z$, $\\mathfrak q \\not = \\mathfrak m$\ncorresponding to a closed point of the punctured spectrum of $A$;\nthis is possible by\nProperties, Lemma \\ref{properties-lemma-complement-closed-point-Jacobson}.\nThen $\\mathfrak q$ is not the generic point of $Z$. Hence\n$0 < \\dim(A_\\mathfrak q) < \\dim(A)$ and $f \\in \\mathfrak q A_\\mathfrak q$.\nBy induction on the dimension we can find\n$f \\in \\mathfrak p \\subset A_\\mathfrak q$ with\n$\\dim((A_\\mathfrak q)_\\mathfrak p) = 1$.\nThen $\\mathfrak p \\cap A$ works.\n\\end{proof}\n\n\\begin{lemma}\n\\label{lemma-ramification-quasi-finite-flat}\nLet $f : X \\to Y$ be a morphism of locally Noetherian schemes.\nLet $x \\in X$. Assume\n\\begin{enumerate}\n\\item $f$ is flat,\n\\item $f$ is quasi-finite at $x$,\n\\item $x$ is not a generic point of an irreducible component of $X$,\n\\item for specializations $x' \\leadsto x$ with\n$\\dim(\\mathcal{O}_{X, x'}) = 1$ our $f$ is unramified at $x'$.\n\\end{enumerate}\nThen $f$ is \\'etale at $x$.\n\\end{lemma}\n\n\\begin{proof}\nObserve that the set of points where $f$ is unramified is the same as\nthe set of points where $f$ is \\'etale and that this set is open.\nSee Morphisms, Definitions \\ref{morphisms-definition-unramified}\nand \\ref{morphisms-definition-etale} and\nLemma \\ref{morphisms-lemma-flat-unramified-etale}.\nTo check $f$ is \\'etale at $x$ we may work \\'etale\nlocally on the base and on the\ntarget (Descent, Lemmas \\ref{descent-lemma-descending-property-etale} and\n\\ref{descent-lemma-etale-etale-local-source}).\nThus we can apply More on Morphisms, Lemma\n\\ref{more-morphisms-lemma-etale-makes-quasi-finite-finite-at-point}\nand assume that $f : X \\to Y$ is finite and that $x$ is the unique\npoint of $X$ lying over $y = f(x)$.\nThen it follows that $f$ is finite locally free\n(Morphisms, Lemma \\ref{morphisms-lemma-finite-flat}).\n\n\\medskip\\noindent\nAssume $f$ is finite locally free and that $x$ is the unique point of\n$X$ lying over $y = f(x)$. By\nDiscriminants, Lemma \\ref{discriminant-lemma-discriminant}\nwe find a locally principal closed subscheme $D_\\pi \\subset Y$\nsuch that $y' \\in D_\\pi$ if and only if there exists an $x' \\in X$\nwith $f(x') = y'$ and $f$ ramified at $x'$. Thus we have to prove\nthat $y \\not \\in D_\\pi$. Assume $y \\in D_\\pi$ to get a contradiction.\n\n\\medskip\\noindent\nBy condition (3) we have $\\dim(\\mathcal{O}_{X, x}) \\geq 1$.\nWe have $\\dim(\\mathcal{O}_{X, x}) = \\dim(\\mathcal{O}_{Y, y})$ by\nAlgebra, Lemma \\ref{algebra-lemma-dimension-base-fibre-equals-total}.\nBy Lemma \\ref{lemma-find-point-codim-1}\nwe can find $y' \\in D_\\pi$ specializing to $y$\nwith $\\dim(\\mathcal{O}_{Y, y'}) = 1$.\nChoose $x' \\in X$ with $f(x') = y'$ where $f$ is ramified. Since $f$\nis finite it is closed, and hence $x' \\leadsto x$.\nWe have $\\dim(\\mathcal{O}_{X, x'}) = \\dim(\\mathcal{O}_{Y, y'}) = 1$\nas before. This contradicts property (4).\n\\end{proof}\n\n\\begin{lemma}\n\\label{lemma-local-purity}\nLet $(A, \\mathfrak m)$ be a regular local ring of dimension $d \\geq 2$.\nSet $X = \\Spec(A)$ and $U = X \\setminus \\{\\mathfrak m\\}$. Then\n\\begin{enumerate}\n\\item the functor $\\textit{F\\'Et}_X \\to \\textit{F\\'Et}_U$\nis essentially surjective,\n\\item any finite $A \\to B$ with $B$ normal which\ninduces a finite \\'etale morphism on punctured spectra is \\'etale.\n\\end{enumerate}\n\\end{lemma}\n\n\\begin{proof}\nRecall that a regular local ring is normal by\nAlgebra, Lemma \\ref{algebra-lemma-regular-normal}.\nHence (1) and (2) are equivalent by\nLemma \\ref{lemma-reformulate-purity-normal}.\nWe prove the lemma by induction on $d$.\n\n\\medskip\\noindent\nThe case $d = 2$. In this case $A \\to B$ is flat.\nNamely, we have going down for $A \\to B$ by\nAlgebra, Proposition \\ref{algebra-proposition-going-down-normal-integral}.\nThen $\\dim(B_{\\mathfrak m'}) = 2$ for all maximal ideals\n$\\mathfrak m' \\subset B$ by\nAlgebra, Lemma \\ref{algebra-lemma-dimension-base-fibre-equals-total}.\nThen $B_{\\mathfrak m'}$ is Cohen-Macaulay by\nAlgebra, Lemma \\ref{algebra-lemma-criterion-normal}.\nHence and this is the important step\nAlgebra, Lemma \\ref{algebra-lemma-CM-over-regular-flat}\napplies to show $A \\to B_{\\mathfrak m'}$ is flat.\nThen Algebra, Lemma \\ref{algebra-lemma-flat-localization}\nshows $A \\to B$ is flat. Thus we can apply\nLemma \\ref{lemma-ramification-quasi-finite-flat}\n(or you can directly argue using the easier\nDiscriminants, Lemma \\ref{discriminant-lemma-discriminant})\nto see that $A \\to B$ is \\'etale.\n\n\\medskip\\noindent\nThe case $d \\geq 3$. Let $V \\to U$ be finite \\'etale.\nLet $f \\in \\mathfrak m_A$, $f \\not \\in \\mathfrak m_A^2$.\nThen $A/fA$ is a regular local ring of dimension $d - 1 \\geq 2$, see\nAlgebra, Lemma \\ref{algebra-lemma-regular-ring-CM}.\nLet $U_0$ be the punctured spectrum of $A/fA$ and let\n$V_0 = V \\times_U U_0$.\nBy Lemma \\ref{lemma-lift-purity} (or the more general\nLemma \\ref{lemma-lift-purity-general})\nit suffices to show that $V_0$ is in the essential\nimage of $\\textit{F\\'Et}_{\\Spec(A/fA)} \\to \\textit{F\\'Et}_{U_0}$.\nThis follows from the induction hypothesis.\n\\end{proof}\n\n\\begin{lemma}[Purity of branch locus]\n\\label{lemma-purity}\n\\begin{reference}\n\\cite{Nagata-Purity} and \\cite[Exp. X, Thm. 3.1]{SGA1}\n\\end{reference}\n\\begin{history}\nThis result was first stated and proved by Zariski in\ngeometric form in \\cite{Zariski-Purity}.\nThe generalization to nonperfect ground fields by Nagata\nwas published as the next article in the same volume of the\nProceedings of the National Academy of Sciences of the United States of America\nin \\cite{Nagata-Remarks-Purity}. In the following year Nagata\nproved the result for Noetherian local rings in \\cite{Nagata-Purity}.\nHis proof uses a result of Chow which is a Bertini theorem for\ncomplete local domains, see \\cite{Chow-Bertini};\nthe history of Bertini's theorems is discussed in\nKleiman's historical article \\cite{Kleiman-Bertini}.\nA few years later a completely different proof was found by\nAuslander, see \\cite{Auslander-Purity}.\n\\end{history}\nLet $f : X \\to Y$ be a morphism of locally Noetherian schemes.\nLet $x \\in X$ and set $y = f(x)$. Assume\n\\begin{enumerate}\n\\item $\\mathcal{O}_{X, x}$ is normal,\n\\item $\\mathcal{O}_{Y, y}$ is regular,\n\\item $f$ is quasi-finite at $x$,\n\\item $\\dim(\\mathcal{O}_{X, x}) = \\dim(\\mathcal{O}_{Y, y}) \\geq 1$\n\\item for specializations $x' \\leadsto x$ with\n$\\dim(\\mathcal{O}_{X, x'}) = 1$ our $f$ is unramified at $x'$.\n\\end{enumerate}\nThen $f$ is \\'etale at $x$.\n\\end{lemma}\n\n\\begin{proof}\nWe will prove the lemma by induction on\n$d = \\dim(\\mathcal{O}_{X, x}) = \\dim(\\mathcal{O}_{Y, y})$.\n\n\\medskip\\noindent\nAn uninteresting case is when $d = 1$.\nIn that case we are assuming that $f$ is unramified at $x$\nand that $\\mathcal{O}_{Y, y}$ is a discrete valuation ring\n(Algebra, Lemma \\ref{algebra-lemma-characterize-dvr}).\nThen $\\mathcal{O}_{X, x}$ is flat over $\\mathcal{O}_{Y, y}$\n(otherwise the map would not be quasi-finite at $x$)\nand we see that $f$ is flat at $x$. Since flat $+$\nunramified is \\'etale we conclude (some details omitted).\n\n\\medskip\\noindent\nThe case $d \\geq 2$. We will use induction on $d$ to reduce\nto the case discussed in Lemma \\ref{lemma-local-purity}.\nTo check $f$ is \\'etale at $x$ we may work \\'etale locally\non the base and on the target\n(Descent, Lemmas \\ref{descent-lemma-descending-property-etale} and\n\\ref{descent-lemma-etale-etale-local-source}).\nThus we can apply More on Morphisms, Lemma\n\\ref{more-morphisms-lemma-etale-makes-quasi-finite-finite-at-point}\nand assume that $f : X \\to Y$ is finite and that $x$ is the unique\npoint of $X$ lying over $y$. Here we use that \\'etale extensions of\nlocal rings do not change dimension, normality, and regularity, see\nMore on Algebra, Section \\ref{more-algebra-section-permanence-etale}\nand\n\\'Etale Morphisms, Section \\ref{etale-section-properties-permanence}.\n\n\\medskip\\noindent\nNext, we can base change by $\\Spec(\\mathcal{O}_{Y, y})$\nand assume that $Y$ is the spectrum of a regular local ring.\nIt follows that $X = \\Spec(\\mathcal{O}_{X, x})$ as\nevery point of $X$ necessarily specializes to $x$.\n\n\\medskip\\noindent\nThe ring map $\\mathcal{O}_{Y, y} \\to \\mathcal{O}_{X, x}$ is\nfinite and necessarily injective (by equality of dimensions).\nWe conclude we have going down for\n$\\mathcal{O}_{Y, y} \\to \\mathcal{O}_{X, x}$ by\nAlgebra, Proposition \\ref{algebra-proposition-going-down-normal-integral}\n(and the fact that a regular ring is a normal ring by\nAlgebra, Lemma \\ref{algebra-lemma-regular-normal}).\nPick $x' \\in X$, $x' \\not = x$ with image $y' = f(x')$.\nThen $\\mathcal{O}_{X, x'}$ is normal as a localization\nof a normal domain. Similarly, $\\mathcal{O}_{Y, y'}$ is\nregular (see Algebra, Lemma\n\\ref{algebra-lemma-localization-of-regular-local-is-regular}).\nWe have $\\dim(\\mathcal{O}_{X, x'}) = \\dim(\\mathcal{O}_{Y, y'})$ by\nAlgebra, Lemma \\ref{algebra-lemma-dimension-base-fibre-equals-total}\n(we checked going down above).\nOf course these dimensions are strictly less than $d$ as $x' \\not = x$\nand by induction on $d$ we conclude that $f$ is \\'etale at $x'$.\n\n\\medskip\\noindent\nThus we arrive at the following situation: We have a finite\nlocal homomorphism $A \\to B$ of Noetherian local rings\nof dimension $d \\geq 2$, with $A$ regular, $B$ normal, which\ninduces a finite \\'etale morphism $V \\to U$ on punctured spectra.\nOur goal is to show that $A \\to B$ is \\'etale.\nThis follows from Lemma \\ref{lemma-local-purity}\nand the proof is complete.\n\\end{proof}\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\\section{Finite \\'etale covers of punctured spectra, II}\n\\label{section-pi1-punctured-spec-II}\n\n\\noindent\nIn this section we prove some variants of the material discussed\nin Section \\ref{section-pi1-punctured-spec}. Suppose\nwe have a Noetherian local ring $(A, \\mathfrak m)$ and $f \\in \\mathfrak m$.\nWe set $X = \\Spec(A)$ and $X_0 = \\Spec(A/fA)$ and we\nlet $U = X \\setminus \\{\\mathfrak m\\}$ and\n$U_0 = X_0 \\setminus \\{\\mathfrak m\\}$ be the punctured spectrum of\n$A$ and $A/fA$. All of this is exactly as in\nSituation \\ref{situation-local-lefschetz}.\nThe difference is that we will consider the functor\n$$\n\\colim_{U_0 \\subset U' \\subset U} \\textit{F\\'Et}_{U'}\n\\longrightarrow \\textit{F\\'Et}_{U_0},\\quad\nV' \\longmapsto V_0 = V' \\times_{U'} U_0\n$$\nIn other words, we will not try to lift finite \\'etale coverings\nof $U_0$ to all of $U$, but just to some open neighbourhood\n$U'$ of $U_0$ in $U$.\n\n\\begin{lemma}\n\\label{lemma-faithful-general}\nIn Situation \\ref{situation-local-lefschetz}.\nLet $U' \\subset U$ be open and contain $U_0$.\nAssume $\\dim(A/\\mathfrak p) \\geq 2$ for every minimal prime\n$\\mathfrak p \\subset A$ corresponding to a point of $U'$. Then\n$$\n\\textit{F\\'Et}_{U'} \\longrightarrow \\textit{F\\'Et}_{U_0},\\quad\nV' \\longmapsto V_0 = V' \\times_{U'} U_0\n$$\nis a faithful functor. Moreover, there exists a $U'$ satisfying\nthese assumptions.\n\\end{lemma}\n\n\\begin{proof}\nLet $a, b : V' \\to W'$ be two morphisms of schemes finite \\'etale\nover $U'$ whose restriction to $U_0$ are the same. By\nAlgebra, Lemma \\ref{algebra-lemma-one-equation}\nwe see that $V(\\mathfrak p)$ meets $U_0$ for\nevery prime $\\mathfrak p$ of $A$ with $\\dim(A/\\mathfrak p) \\geq 2$.\nThe assumption therefore implies that every\nirreducible component of $U'$ meets $U_0$.\nThe image of any irreducible component of $V'$ is an\nirreducible component of $U'$ and hence meets $U_0$.\nHence $V_0$ meets every connected component of $V'$ and\nwe conclude that $a = b$ by \\'Etale Morphisms, Proposition\n\\ref{etale-proposition-equality}.\nTo see the existence of such a $U'$ note that if\n$\\mathfrak p \\subset A$ is a prime with $\\dim(A/\\mathfrak p) = 1$\nthen $\\mathfrak p$ corresponds to a closed point of $U$.\n\\end{proof}\n\n\\begin{lemma}\n\\label{lemma-fully-faithful-general-better}\nIn Situation \\ref{situation-local-lefschetz} assume\n\\begin{enumerate}\n\\item $A$ has a dualizing complex and is $f$-adically complete,\n\\item $f$ is a nonzerodivisor,\n\\item for $x \\in X \\setminus X_0$ whose closure $\\overline{\\{x\\}}$\nin $X$ meets $U_0$ we have $\\text{depth}(\\mathcal{O}_{X, x}) \\geq 1$\nor $\\text{depth}(\\mathcal{O}_{X, x}) + \\dim(\\overline{\\{x\\}}) > 2$.\n\\end{enumerate}\nLet $V'$, $W'$ be finite \\'etale over an open $U' \\subset U$\nwhich contains $U_0$. Let\n$\\varphi_0 : V' \\times_{U'} U_0 \\to W' \\times_{U'} U_0$\nbe a morphism over $U_0$.\nThen there exists an open $U'' \\subset U'$\ncontaining $U_0$ and a morphism\n$\\varphi : V' \\times_{U'} U'' \\to W' \\times_{U'} U''$\nlifting $\\varphi_0$.\n\\end{lemma}\n\n\\begin{proof}\nSince the category of finite \\'etale coverings has an internal hom\n(Lemma \\ref{lemma-internal-hom-finite-etale})\nit suffices to prove the following: Given $V'$ finite \\'etale over $U'$\nany section $U_0 \\to V' \\times_{U'} U_0$ extends to a section of $V'$\nover some open $U'' \\subset U'$ containing $U_0$.\nGiven our section we obtain a decomposition\n$V' \\times_{U'} U_0 = U_0 \\amalg R_0$ into open and closed subschemes.\nWe will show that this implies the same thing for $V' \\times_{U'} U''$\nfor some $U'' \\subset U'$ open containing $U_0$ thereby\nfinishing the proof.\n\n\\medskip\\noindent\nFor $n \\geq 1$ let $U_n$ be the punctured spectrum of $A/f^{n + 1}A$. By\n\\'Etale Morphisms, Theorem \\ref{etale-theorem-remarkable-equivalence}\nwe conclude that there is a unique decomposition\n$V' \\times_{U'} U_n = U_n \\amalg R_n$\ninto open and closed subschemes whose base change to $U_0$ recovers\nthe given decomposition.\n\n\\medskip\\noindent\nVia the inclusions $U_n \\to V' \\times_{U'} U_n \\to V'$\nwe obtain an $A$-algebra map\n$$\n\\Gamma(V', \\mathcal{O}_{V'}) \\to B = \\lim H^0(U_n, \\mathcal{O}_{U_n})\n$$\nBy Local Cohomology, Theorem\n\\ref{local-cohomology-theorem-algebraization-formal-sections}\napplied with $s = 1$ and $\\mathcal{F} = \\mathcal{O}_U$ we see that\n$B = H^0(U'', \\mathcal{O}_{U''})$ for some open $U'' \\subset U$\ncontaining $U_0$. Since $V \\to U$ is affine, this $A$-algebra map\ncorresponds to a morphism $U'' \\to V'$ over $U'$ as desired.\n\\end{proof}\n\n\\begin{lemma}\n\\label{lemma-fully-faithful-general}\nIn Situation \\ref{situation-local-lefschetz} assume\n\\begin{enumerate}\n\\item $A$ is $f$-adically complete,\n\\item $f$ is a nonzerodivisor, and\n\\item $H^1_\\mathfrak m(A/fA)$ is a finite $A$-module.\n\\end{enumerate}\nLet $V'$, $W'$ be finite \\'etale over an open $U' \\subset U$\nwhich contains $U_0$. Let\n$\\varphi_0 : V' \\times_{U'} U_0 \\to W' \\times_{U'} U_0$\nbe a morphism over $U_0$.\nThen there exists an open $U'' \\subset U'$\ncontaining $U_0$ and a morphism\n$\\varphi : V' \\times_{U'} U'' \\to W' \\times_{U'} U''$\nlifting $\\varphi_0$.\n\\end{lemma}\n\n\\begin{proof}\nThis lemma is a variant of\nLemma \\ref{lemma-fully-faithful-general-better}\nand if $A$ is a complete local ring, then it follows from that lemma.\nWe suggest the reader skip the proof.\n\n\\medskip\\noindent\nSince the category of finite \\'etale coverings has an internal hom\n(Lemma \\ref{lemma-internal-hom-finite-etale})\nit suffices to prove the following: Given $V'$ finite \\'etale over $U'$\nany section $U_0 \\to V' \\times_{U'} U_0$ extends to a section of $V'$\nover some open $U'' \\subset U'$ containing $U_0$.\nGiven our section we obtain a decomposition\n$V' \\times_{U'} U_0 = U_0 \\amalg R_0$ into open and closed subschemes.\nWe will show that this implies the same thing for $V' \\times_{U'} U''$\nfor some $U'' \\subset U'$ open containing $U_0$ thereby\nfinishing the proof.\n\n\\medskip\\noindent\nFor $n \\geq 1$ let $U_n$ be the punctured spectrum of $A/f^{n + 1}A$. By\n\\'Etale Morphisms, Theorem \\ref{etale-theorem-remarkable-equivalence}\nwe conclude that there is a unique decomposition\n$V' \\times_{U'} U_n = U_n \\amalg R_n$\ninto open and closed subschemes whose base change to $U_0$ recovers\nthe given decomposition.\n\n\\medskip\\noindent\nThe finiteness of $H^1_\\mathfrak m(A/fA)$ tells us that\n$B_0 = \\Gamma(U_0, \\mathcal{O}_{U_0})$ is a finite $A$-module, see\nLocal Cohomology, Lemma\n\\ref{local-cohomology-lemma-finiteness-pushforwards-and-H1-local}.\nSet $B_n = \\Gamma(U_n, \\mathcal{O}_{U_n})$.\nAs $f$ is a nonzerodivisor we have exact sequences\n$$\n0 \\to A/f^nA \\xrightarrow{f} A/f^{n + 1}A \\to A/fA \\to 0\n$$\nand hence short exact sequences $0 \\to \\mathcal{O}_{U_n} \\to\n\\mathcal{O}_{U_{n + 1}} \\to \\mathcal{O}_{U_0} \\to 0$.\nThus we may apply Local Cohomology, Lemma\n\\ref{local-cohomology-lemma-limit-finite}\nto the inverse system $\\mathcal{O}_{U_n}$ on $U$.\nWe find that $B = \\lim B_n$ is a finite $A$-algebra, such that\n$f$ is a nonzerodivisor on $B$,\nand such that $B/fB \\subset B_0$.\nVia the inclusions $U_n \\to V' \\times_{U'} U_n \\to V'$\nwe obtain an $A$-algebra map $\\Gamma(V', \\mathcal{O}_{V'}) \\to B$.\nSince $V \\to U$ is affine, this $A$-algebra map corresponds to\na morphism\n$$\n\\Spec(B) \\times_{\\Spec(A)} U' \\longrightarrow V'\n$$\nover $U'$.\n\n\\medskip\\noindent\nLet $\\mathfrak q \\in U_0$ be a prime. The kernel and cokernel of\n$A/fA \\to B_0$ have support contained in $\\{\\mathfrak m\\}$ (see above).\nHence the same is true for the map $A/fA \\to B/fB$.\nThen $A_\\mathfrak q \\to B_\\mathfrak q$ is finite and\ninduces an isomorphism $(A/fA)_\\mathfrak q \\to (B/fB)_\\mathfrak q$.\nSince $f$ is a nonzerodivisor on $B$ it follows that\n$A_\\mathfrak q \\to B_\\mathfrak q$ is an isomorphism.\nUsing finiteness again we find $g \\in A$, $g \\not \\in \\mathfrak q$\nsuch that $A_g \\to B_g$ is an isomorphism.\nIt follows that $\\Spec(B) \\to \\Spec(A)$ is an\nisomorphism over an open $U'' \\subset U'$ which produces\nthe desired section by the above.\n\\end{proof}\n\n\\begin{lemma}\n\\label{lemma-essentially-surjective-general-better}\nIn Situation \\ref{situation-local-lefschetz} assume\n\\begin{enumerate}\n\\item $A$ has a dualizing complex and is $f$-adically complete,\n\\item $f$ is a nonzerodivisor,\n\\item $A$ is $f$-adically complete,\n\\item if $\\mathfrak p \\in V(f) \\setminus \\{\\mathfrak m\\}$, then\n$\\text{depth}((A/f)_\\mathfrak p) + \\dim(A/\\mathfrak p) > 1$, and\n\\item if $\\mathfrak p \\not \\in V(f)$ and\n$V(\\mathfrak p) \\cap V(f) \\not = \\{\\mathfrak m\\}$, then\n$\\text{depth}(A_\\mathfrak p) + \\dim(A/\\mathfrak p) > 3$.\n\\end{enumerate}\nFor any finite \\'etale morphism $V_0 \\to U_0$ there exists an open\n$U' \\subset U$ containing $U_0$ and a finite \\'etale morphism\n$V' \\to U'$ whose base change to $U_0$ is $V_0 \\to U_0$.\n\\end{lemma}\n\n\\begin{proof}\nFor $n \\geq 1$ let $U_n$ be the punctured spectrum of $A/f^{n + 1}A$.\nBy \\'Etale Morphisms, Theorem \\ref{etale-theorem-remarkable-equivalence}\nwe conclude that there is a unique finite \\'etale morphism\n$\\pi_n : V_n \\to U_n$ whose base change to $U_0$ recovers $V_0 \\to U_0$.\nConsider the sheaves $\\mathcal{F}_n = \\pi_{n, *}\\mathcal{O}_{V_n}$.\nWe may and do view $\\mathcal{F}_n$ as an $\\mathcal{O}_U$-module on $U$\nwich is locally isomorphic to\n$(\\mathcal{O}_U/f^{n + 1}\\mathcal{O}_U)^{\\oplus r}$. By\nLocal Cohomology, Lemma \\ref{local-cohomology-lemma-algebraization-principal}\nthere exists a coherent $\\mathcal{O}_U$-module $\\mathcal{F}$\nand a compatible system of isomorphisms\n$$\n\\mathcal{F}/f^{n + 1}\\mathcal{F} \\to \\mathcal{F}_n\n$$\nof $\\mathcal{O}_U$-modules. If $x \\in U_0$, then the $f$-adic\ncompletion of the stalk $\\mathcal{F}_x$ is isomorphic to\na finite free module over the $f$-adic completion of $\\mathcal{O}_{U, u}$.\nHence $\\mathcal{F}$ is finite locally free in an open neighbourhood\n$U'$ of $U_0$.\n\n\\medskip\\noindent\nTo construct an algebra structure on $\\mathcal{F}$ consider the coherent\n$\\mathcal{O}_U$-module\n$$\n\\mathcal{H} = \\SheafHom_{\\mathcal{O}_U}(\n\\mathcal{F}\\otimes_{\\mathcal{O}_U} \\mathcal{F}, \\mathcal{F})\n$$\nObserve that $\\mathcal{H}|_{U'}$ is finite locally free. The multiplication\nmaps\n$\\mathcal{F}_n \\otimes_{\\mathcal{O}_U} \\mathcal{F}_n \\to \\mathcal{F}_n$\ncoming from the fact that $\\mathcal{F}_n = \\pi_{n, *}\\mathcal{O}_{V_n}$\nare sheaves of algebras defines an an element in\n$$\n\\lim \\Gamma(U, \\mathcal{H}/f^{n + 1}\\mathcal{H})\n$$\nBy Local Cohomology, Theorem\n\\ref{local-cohomology-theorem-algebraization-formal-sections}\nthis comes from a section $\\mu \\in \\Gamma(U', \\mathcal{F})$\nafter possibly shrinking $U'$. After possibly shrinking furter\nwe may assume $\\mu$ defines a commutative $\\mathcal{O}_{U'}$-algebra\nstructure on $\\mathcal{F}$ compatible with the given algebra\nstructures on $\\mathcal{F}_n$.\nSetting\n$$\nV' = \\underline{\\Spec}_{U'}((\\mathcal{F}|_{U'}, \\mu))\n$$\nwe obtain a finite locally free scheme over $U'$ whose restriction\nto $U_n$ is isomorphic to $V_n$. It follows that $V' \\to U'$\nis \\'etale at all points lying over $U_0$, see\nMore on Morphisms, Lemma\n\\ref{more-morphisms-lemma-check-smoothness-on-infinitesimal-nbhds}.\nThis finishes the proof.\n\\end{proof}\n\n\\begin{lemma}\n\\label{lemma-essentially-surjective-general}\nIn Situation \\ref{situation-local-lefschetz} assume\n\\begin{enumerate}\n\\item $A$ is $f$-adically complete,\n\\item $f$ is a nonzerodivisor,\n\\item $H^1_\\mathfrak m(A/fA)$ and $H^2_\\mathfrak m(A/fA)$\nare finite $A$-modules.\n\\end{enumerate}\nFor any finite \\'etale morphism $V_0 \\to U_0$ there exists an open\n$U' \\subset U$ containing $U_0$ and a finite \\'etale morphism\n$V' \\to U'$ whose base change to $U_0$ is $V_0 \\to U_0$.\n\\end{lemma}\n\n\\begin{proof}\nThis lemma is a variant of\nLemma \\ref{lemma-essentially-surjective-general-better}\nand if $A$ is a complete local ring, then it follows from that lemma.\nWe suggest the reader skip the proof.\n\n\\medskip\\noindent\nFor $n \\geq 1$ let $U_n$ be the punctured spectrum of $A/f^{n + 1}A$.\nBy \\'Etale Morphisms, Theorem \\ref{etale-theorem-remarkable-equivalence}\nwe conclude that there is a unique finite \\'etale morphism\n$\\pi_n : V_n \\to U_n$ whose base change to $U_0$ recovers $V_0 \\to U_0$.\nConsider the sheaves $\\mathcal{F}_n = \\pi_{n, *}\\mathcal{O}_{V_n}$.\nWe may view $\\mathcal{F}_n$ as an $\\mathcal{O}_U$-module on $U$.\nAs $f$ is a nonzerodivisor we obtain short exact sequences\n$$\n0 \\to A/f^nA \\xrightarrow{f} A/f^{n + 1}A \\to A/fA \\to 0\n$$\nand because $V_n \\to U_n$ is finite locally free we have corresponding\nshort exact sequences\n$0 \\to \\mathcal{F}_n \\to \\mathcal{F}_{n + 1} \\to \\mathcal{F}_0 \\to 0$.\n\n\\medskip\\noindent\nWe will use Local Cohomology, Lemma\n\\ref{local-cohomology-lemma-finiteness-pushforwards-and-H1-local}\nwithout further mention.\nOur assumptions imply that $H^0(U, \\mathcal{O}_{U_0})$ and\n$H^1(U, \\mathcal{O}_{U_0})$ are finite $A$-modules.\nHence the same thing is true for $\\mathcal{F}_0$, see\nLocal Cohomology, Lemma\n\\ref{local-cohomology-lemma-finiteness-for-finite-locally-free}.\nThus $H^0(U, \\mathcal{F}_0)$ is a finite $A$-module\nand $H^1(U, \\mathcal{F}_0)$ has finite length\n(as a finite $A$-module which is $\\mathfrak m$-power torsion).\nThus Local Cohomology, Lemmas \\ref{local-cohomology-lemma-limit-finite} and\n\\ref{local-cohomology-lemma-ML} apply to the system above. Set\n$$\nB_n = \\Gamma(V_n, \\mathcal{O}_{V_n}) = \\Gamma(U, \\mathcal{F}_n)\n$$\nWe conclude that the system $(B_n)$ satisfies the Mittag-Leffler condition,\nthat $B = \\lim B_n$ is a finite $A$-algebra, that $f$ is a nonzerodivisor\non $B$ and that $B/fB \\subset B_0$. To finish the proof,\nwe will show that the finite morphism\n$\\Spec(B) \\to \\Spec(A)$ (a) becomes isomorphic to $V_0 \\to U_0$\nafter base change to $U_0$ and (b) is \\'etale at all points lying\nover $U_0$.\n\n\\medskip\\noindent\nLet $\\mathfrak q \\in U_0$ be a prime. By the Mittag-Leffler\ncondition, we know that $B/fB \\subset B_0$ is the image of\n$B_{n + 1} \\to B_0$ for some $n$. Since the cokernel of $B_{n + 1} \\to B_0$\nis contained in $H^1(U, \\mathcal{F}_n)$ which is $\\mathfrak m$-power\ntorsion, we conclude that $B/fB \\to B_0$ becomes an isomorphism\nafter localizing at $\\mathfrak q$. This proves (a).\nThus $A_\\mathfrak q \\to B_\\mathfrak q$\nis finite and $(A/fA)_\\mathfrak q \\to (B/fB)_\\mathfrak q$ is \\'etale.\nSince $f$ is a nonzerodivisor on $B$ it follows that\n$A_\\mathfrak q \\to B_\\mathfrak q$ is flat\n(Algebra, Lemma \\ref{algebra-lemma-variant-local-criterion-flatness}).\nThus $A \\to B$ is \\'etale at all primes lying over $\\mathfrak q$\n(for example by Algebra, Lemma \\ref{algebra-lemma-characterize-etale})\nwhich proves (b).\n\\end{proof}\n\n\\begin{remark}\n\\label{remark-combine}\nLet $(A, \\mathfrak m)$ be a complete local ring and $f \\in \\mathfrak m$\na nonzerodivisor. Let $U$, resp.\\ $U_0$ be the punctured spectrum of\n$A$, resp.\\ $A/fA$. Assume\n\\begin{enumerate}\n\\item if $\\mathfrak p \\in V(f) \\setminus \\{\\mathfrak m\\}$, then\n$\\text{depth}((A/f)_\\mathfrak p) + \\dim(A/\\mathfrak p) > 1$, and\n\\item if $\\mathfrak p \\not \\in V(f)$ and\n$V(\\mathfrak p) \\cap V(f) \\not = \\{\\mathfrak m\\}$, then\n$\\text{depth}(A_\\mathfrak p) + \\dim(A/\\mathfrak p) > 3$.\n\\end{enumerate}\nCombining Lemmas \\ref{lemma-faithful-general},\n\\ref{lemma-fully-faithful-general-better}, and\n\\ref{lemma-essentially-surjective-general-better}\nwe see that the category\n$$\n\\colim\\nolimits_{U' \\subset U\\text{ open, }U_0 \\subset U}\n\\text{ category of schemes finite \\'etale over }U'\n$$\nis equivalent to the category of schemes finite \\'etale over $U_0$.\nFor example it suffices if every irreducible component of $\\Spec(A)$\nhas dimension $\\geq 4$ and $A$ is $(S_2)$.\nFor example, if $A$ is a normal domain of dimension $\\geq 4$!\n\\end{remark}\n\n\n\n\n\n\n\n\\section{Purity in local case, II}\n\\label{section-local-purity-II}\n\n\\noindent\nThis section is the continuation of Section \\ref{section-local-purity}.\nIn the next lemma we say {\\it purity holds} for a Noetherian local ring\n$(A, \\mathfrak m)$ if the restriction functor\n$\\textit{F\\'Et}_X \\to \\textit{F\\'Et}_U$ is essentially\nsurjective where $X = \\Spec(A)$ and $U = X \\setminus \\{\\mathfrak m\\}$\nis the punctured spectrum.\n\n\\begin{lemma}\n\\label{lemma-purity-inherited-by-hypersurface-better}\nLet $(A, \\mathfrak m)$ be a Noetherian local ring.\nLet $f \\in \\mathfrak m$. Assume\n\\begin{enumerate}\n\\item $A$ has a dualizing complex and is $f$-adically complete,\n\\item $f$ is a nonzerodivisor,\n\\item if $\\mathfrak p \\in V(f) \\setminus \\{\\mathfrak m\\}$, then\n$\\text{depth}((A/f)_\\mathfrak p) + \\dim(A/\\mathfrak p) > 1$, and\n\\item if $\\mathfrak p \\not \\in V(f)$ and\n$V(\\mathfrak p) \\cap V(f) \\not = \\{\\mathfrak m\\}$, then\n$\\text{depth}(A_\\mathfrak p) + \\dim(A/\\mathfrak p) > 3$,\n\\item for every maximal ideal $\\mathfrak p \\subset A_f$\npurity holds for $(A_f)_\\mathfrak p$, and\n\\item purity holds for $A$.\n\\end{enumerate}\nThen purity holds for $A/fA$.\n\\end{lemma}\n\n\\begin{proof}\nDenote $X = \\Spec(A)$ and $U = X \\setminus \\{\\mathfrak m\\}$\nthe punctured spectrum. Similarly we have $X_0 = \\Spec(A/fA)$\nand $U_0 = X_0 \\setminus \\{\\mathfrak m\\}$.\nLet $V_0 \\to U_0$ be a finite \\'etale morphism. By\nLemma \\ref{lemma-essentially-surjective-general-better}\nthere exists an open $U' \\subset U$ containing $U_0$ and\na finite \\'etale morphism $V' \\to U$ whose base change to $U_0$\nis isomorphic to $V_0 \\to U_0$. Since $U' \\supset U_0$\nwe see that $U \\setminus U'$ consists of points corresponding\nto prime ideals $\\mathfrak p_1, \\ldots, \\mathfrak p_n$ as in (4).\nBy assumption we can find finite \\'etale morphisms\n$V'_i \\to \\Spec(A_{\\mathfrak p_i})$ agreeing with\n$V' \\to U'$ over $U' \\times_U \\Spec(A_{\\mathfrak p_i})$.\nBy Limits, Lemma \\ref{limits-lemma-glueing-near-closed-point}\napplied $n$ times we see that $V' \\to U'$ extends to a finite \\'etale\nmorphism $V \\to U$. By assumption (5) we find that $V \\to U$ extends\nto a finite \\'etale morphism $Y \\to X$. Then the restriction of\n$Y$ to $X_0$ is the desired extension of $V_0 \\to U_0$.\n\\end{proof}\n\n\\begin{lemma}\n\\label{lemma-purity-inherited-by-hypersurface}\nLet $(A, \\mathfrak m)$ be a Noetherian local ring.\nLet $f \\in \\mathfrak m$. Assume\n\\begin{enumerate}\n\\item $A$ is $f$-adically complete,\n\\item $f$ is a nonzerodivisor,\n\\item $H^1_\\mathfrak m(A/fA)$ and $H^2_\\mathfrak m(A/fA)$ are finite\n$A$-modules,\n\\item for every maximal ideal $\\mathfrak p \\subset A_f$\npurity holds for $(A_f)_\\mathfrak p$,\n\\item purity holds for $A$.\n\\end{enumerate}\nThen purity holds for $A/fA$.\n\\end{lemma}\n\n\\begin{proof}\nThe proof is identical to the proof of\nLemma \\ref{lemma-purity-inherited-by-hypersurface-better}\nusing\nLemma \\ref{lemma-essentially-surjective-general}\nin stead of\nLemma \\ref{lemma-essentially-surjective-general-better}.\n\\end{proof}\n\n\\noindent\nNow we can bootstrap the earlier results to prove that\npurity holds for complete intersections of dimension $\\geq 3$.\nRecall that a Noetherian local ring is called a complete\nintersection if its completion is the quotient of a\nregular local ring by the ideal generated by a regular sequence.\nSee the discussion in Divided Power Algebra, Section \\ref{dpa-section-lci}.\n\n\\begin{proposition}\n\\label{proposition-purity-complete-intersection}\nLet $(A, \\mathfrak m)$ be a Noetherian local ring. If $A$ is a\ncomplete intersection of dimension $\\geq 3$, then purity\nholds for $A$ in the sense that any finite \\'etale cover of\nthe punctured spectrum extends.\n\\end{proposition}\n\n\\begin{proof}\nBy Lemma \\ref{lemma-purity-and-completion} we may assume that $A$ is\na complete local ring. By assumption we can write\n$A = B/(f_1, \\ldots, f_r)$ where $B$ is a complete regular local\nring and $f_1, \\ldots, f_r$ is a regular sequence.\nWe will finish the proof by induction on $r$.\nThe base case is $r = 0$ which follows from\nLemma \\ref{lemma-local-purity} which applies to\nregular rings of dimension $\\geq 2$.\n\n\\medskip\\noindent\nAssume that $A = B/(f_1, \\ldots, f_r)$ and that the proposition\nholds for $r - 1$. Set $A' = B/(f_1, \\ldots, f_{r - 1})$ and apply\nLemma \\ref{lemma-purity-inherited-by-hypersurface} to $f_r \\in A'$.\nThis is permissible:\ncondition (1) holds as $f_1, \\ldots, f_r$ is a regular sequence,\ncondition (2) holds as $B$ and hence $A'$ is complete,\ncondition (3) holds as $A = A'/f_r A'$ is Cohen-Macaulay of dimension\n$\\dim(A) \\geq 3$, see Dualizing Complexes, Lemma \\ref{dualizing-lemma-depth},\ncondition (4) holds by induction hypothesis as\n$\\dim((A'_{f_r})_\\mathfrak p) \\geq 3$ for a maximal\nprime $\\mathfrak p$ of $A'_{f_r}$ and as\n$(A'_{f_r})_\\mathfrak p = B_\\mathfrak q/(f_1, \\ldots, f_{r - 1})$\nfor some $\\mathfrak q \\subset B$,\ncondition (5) holds by induction hypothesis.\n\\end{proof}\n\n\n\n\n\n\n\\section{Specialization maps in the smooth proper case}\n\\label{section-specialization-smooth-proper}\n\n\\noindent\nIn this section we discuss the following result.\nLet $f : X \\to S$ be a proper smooth morphism of schemes.\nLet $s \\leadsto s'$ be a specialization of points in $S$.\nThen the specialization map\n$$\nsp : \\pi_1(X_{\\overline{s}}) \\longrightarrow \\pi_1(X_{\\overline{s}'})\n$$\nof Section \\ref{section-specialization-map}\nis surjective and\n\\begin{enumerate}\n\\item if the characteristic of $\\kappa(s')$ is zero, then it is\nan isomorphism, or\n\\item if the characteristic of $\\kappa(s')$ is $p > 0$, then it\ninduces an isomorphism on maximal prime-to-$p$ quotients.\n\\end{enumerate}\n\n\\begin{lemma}\n\\label{lemma-specialization-map-surjective}\nLet $f : X \\to S$ be a flat proper morphism with geometrically\nconnected fibres. Let $s' \\leadsto s$ be a specialization.\nIf $X_s$ is geometrically reduced, then the specialization\nmap $sp : \\pi_1(X_{\\overline{s}'}) \\to \\pi_1(X_{\\overline{s}})$\nis surjective.\n\\end{lemma}\n\n\\begin{proof}\nSince $X_s$ is geometrically reduced, we may assume all\nfibres are geometrically reduced after possibly shrinking $S$, see\nMore on Morphisms, Lemma \\ref{more-morphisms-lemma-geometrically-reduced-open}.\nLet $\\mathcal{O}_{S, s} \\to A \\to \\kappa(\\overline{s}')$ be as\nin the construction of the specialization map, see\nSection \\ref{section-specialization-map}.\nThus it suffices to show that\n$$\n\\pi_1(X_{\\overline{s}'}) \\to \\pi_1(X_A)\n$$\nis surjective. This follows from\nProposition \\ref{proposition-first-homotopy-sequence}\nand $\\pi_1(\\Spec(A)) = \\{1\\}$.\n\\end{proof}\n\n\\begin{proposition}\n\\label{proposition-specialization-map-isomorphism}\nLet $f : X \\to S$ be a smooth proper morphism with geometrically\nconnected fibres. Let $s' \\leadsto s$ be a specialization.\nIf the characteristic to $\\kappa(s)$ is zero, then the specialization\nmap\n$$\nsp : \\pi_1(X_{\\overline{s}'}) \\to \\pi_1(X_{\\overline{s}})\n$$\nis an isomorphism.\n\\end{proposition}\n\n\\begin{proof}\nThe map is surjective by\nLemma \\ref{lemma-specialization-map-surjective}.\nThus we have to show it is injective.\n\n\\medskip\\noindent\nWe may assume $S$ is affine. Then $S$ is a cofiltered limit of affine\nschemes of finite type over $\\mathbf{Z}$.\nHence we can assume $X \\to S$ is the\nbase change of $X_0 \\to S_0$ where $S_0$ is the spectrum of a finite\ntype $\\mathbf{Z}$-algebra and $X_0 \\to S_0$ is smooth and proper.\nSee Limits, Lemma \\ref{limits-lemma-descend-finite-presentation},\n\\ref{limits-lemma-descend-smooth}, and\n\\ref{limits-lemma-eventually-proper}. By\nLemma \\ref{lemma-specialization-map-base-change}\nwe reduce to the case where the base is Noetherian.\n\n\\medskip\\noindent\nApplying Lemma \\ref{lemma-specialization-map-discrete-valuation-ring}\nwe reduce to the case where the base $S$ is the spectrum of a\nstrictly henselian discrete valuation ring $A$ and we are\nlooking at the specialization map over $A$.\nLet $K$ be the fraction field of $A$.\nChoose an algebraic closure $\\overline{K}$ which\ncorresponds to a geometric generic point $\\overline{\\eta}$ of $\\Spec(A)$.\nFor $\\overline{K}/L/K$ finite separable, let $B \\subset L$ be the\nintegral closure of $A$ in $L$. This is a discrete\nvaluation ring by\nMore on Algebra, Remark \\ref{more-algebra-remark-finite-separable-extension}.\n\n\\medskip\\noindent\nLet $X \\to \\Spec(A)$ be as in the previous paragraph.\nTo show injectivity of the specialization map\nit suffices to prove that every finite\n\\'etale cover $V$ of $X_{\\overline{\\eta}}$ is the base\nchange of a finite \\'etale cover $Y \\to X$.\nNamely, then $\\pi_1(X_{\\overline{\\eta}}) \\to \\pi_1(X) = \\pi_1(X_s)$\nis injective by Lemma \\ref{lemma-functoriality-galois-injective}.\n\n\\medskip\\noindent\nGiven $V$ we can first descend $V$ to $V' \\to X_{K^{sep}}$ by\nLemma \\ref{lemma-perfection} and then to\n$V'' \\to X_L$ by Lemma \\ref{lemma-limit}.\nLet $Z \\to X_B$ be the normalization of $X_B$ in $V''$.\nObserve that $Z$ is normal and that $Z_L = V''$ as schemes\nover $X_L$. Hence $Z \\to X_B$ is finite \\'etale over\nthe generic fibre. The problem is that we do not know that\n$Z \\to X_B$ is everywhere \\'etale. Since $X \\to \\Spec(A)$\nhas geometrically connected smooth fibres, we see that\nthe special fibre $X_s$ is geometrically irreducible.\nHence the special fibre of $X_B \\to \\Spec(B)$ is irreducible;\nlet $\\xi_B$ be its generic point. Let\n$\\xi_1, \\ldots, \\xi_r$ be the points of $Z$ mapping to\n$\\xi_B$. Our first (and it will turn out only) problem\nis now that the extensions\n$$\n\\mathcal{O}_{X_B, \\xi_B} \\subset \\mathcal{O}_{Z, \\xi_i}\n$$\nof discrete valuation rings may be ramified. Let $e_i$ be\nthe ramification index of this extension. Note that since the\ncharacteristic of $\\kappa(s)$ is zero, the ramification is tame!\n\n\\medskip\\noindent\nTo get rid of the ramification we are going to choose a further finite\nseparable extension $K^{sep}/L'/L/K$ such that the ramification\nindex $e$ of the induced extensions $B'/B$ is divisible by $e_i$.\nConsider the normalized base change $Z'$ of $Z$ with respect to\n$\\Spec(B') \\to \\Spec(B)$, see discussion in\nMore on Morphisms, Section \\ref{more-morphisms-section-reduced-fibre-theorem}.\nLet $\\xi_{i, j}$ be the points of $Z'$ mapping to $\\xi_{B'}$\nand to $\\xi_i$ in $Z$. Then the local rings\n$$\n\\mathcal{O}_{Z', \\xi_{i, j}}\n$$\nare the localizations of the integral closure of $\\mathcal{O}_{Z, \\xi_i}$ in\n$L' \\otimes_L f.f.(\\mathcal{O}_{Z, \\xi_i})$. Hence Abhyankar's lemma\n(More on Algebra, Lemma \\ref{more-algebra-lemma-abhyankar})\ntells us that\n$$\n\\mathcal{O}_{X_{B'}, \\xi_{B'}} \\subset \\mathcal{O}_{Z', \\xi_{i, j}}\n$$\nis unramified. We conclude that the morphism $Z' \\to X_{B'}$\nis \\'etale away from codimension $1$. Hence by purity of\nbranch locus (Lemma \\ref{lemma-purity})\nwe see that $Z' \\to X_{B'}$ is finite \\'etale!\n\n\\medskip\\noindent\nHowever, since the residue field extension induced by $A \\to B'$\nis trivial (as the residue field of $A$ is algebraically closed\nbeing separably closed of characteristic zero)\nwe conclude that $Z'$ is the base change of a finite \\'etale\ncover $Y \\to X$ by applying\nLemma \\ref{lemma-finite-etale-on-proper-over-henselian}\ntwice (first to get $Y$ over $A$, then to prove that\nthe pullback to $B$ is isomorphic to $Z'$).\nThis finishes the proof.\n\\end{proof}\n\n\\noindent\nLet $G$ be a profinite group. Let $p$ be a prime number.\nThe {\\it maximal prime-to-$p$ quotient} is by definition\n$$\nG' = \\lim_{U \\subset G\\text{ open, normal, index prime to }p} G/U\n$$\nIf $X$ is a connected scheme and $p$ is given, then the maximal\nprime-to-$p$ quotient of $\\pi_1(X)$ is denoted $\\pi'_1(X)$.\n\n\\begin{theorem}\n\\label{theorem-specialization-map-isomorphism-prime-to-p}\nLet $f : X \\to S$ be a smooth proper morphism with geometrically\nconnected fibres. Let $s' \\leadsto s$ be a specialization.\nIf the characteristic of $\\kappa(s)$ is $p$, then the specialization\nmap\n$$\nsp : \\pi_1(X_{\\overline{s}'}) \\to \\pi_1(X_{\\overline{s}})\n$$\nis surjective and induces an isomorphism\n$$\n\\pi'_1(X_{\\overline{s}'}) \\cong \\pi'_1(X_{\\overline{s}})\n$$\nof the maximal prime-to-p quotients\n\\end{theorem}\n\n\\begin{proof}\nThis is proved in exactly the same manner as\nProposition \\ref{proposition-specialization-map-isomorphism}\nwith the following differences\n\\begin{enumerate}\n\\item Given $X/A$ we no longer show that the functor\n$\\textit{F\\'Et}_X \\to \\textit{F\\'Et}_{X_{\\overline{\\eta}}}$\nis essentially surjective. We show only that Galois objects\nwhose Galois group has order prime to $p$ are in the essential\nimage. This will be enough to conclude the injectivity of\n$\\pi'_1(X_{\\overline{s}'}) \\to \\pi'_1(X_{\\overline{s}})$ by\nexactly the same argument.\n\\item The extensions\n$\\mathcal{O}_{X_B, \\xi_B} \\subset \\mathcal{O}_{Z, \\xi_i}$\nare tamely ramified as the associated extension of fraction\nfields is Galois with group of order prime to $p$. See\nMore on Algebra, Lemma \\ref{more-algebra-lemma-galois-conclusion}.\n\\item The extension $\\kappa_A \\subset \\kappa_B$ is no longer\nnecessarily trivial, but it is purely inseparable.\nHence the morphism $X_{\\kappa_B} \\to X_{\\kappa_A}$\nis a universal homeomorphism and induces an isomorphism\nof fundamental groups by Proposition \\ref{proposition-universal-homeomorphism}.\n\\end{enumerate}\n\\end{proof}\n\n\n\n\n\n\n\n\\section{Tame ramification}\n\\label{section-tame}\n\n\\noindent\nLet $X \\to Y$ be a finite \\'etale morphism of schemes of finite type\nover $\\mathbf{Z}$. There are many ways to define what it means for $f$\nto be tamely ramified at $\\infty$. The article \\cite{Kerz-Schmidt}\ndiscusses to what extent these notions agree.\n\n\\medskip\\noindent\nIn this section we discuss a different more elementary question which\nprecedes the notion of tameness at infinity. Namely, given a scheme\n$X$ and a dense open $U \\subset X$ when is a finite  morphism $f : Y \\to X$\ntamely ramified relative to $D = X \\setminus U$? We will use the definition\nas given in \\cite{Grothendieck-Murre} but only in the case that $D$ is\na divisor with normal crossings.\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\\input{chapters}\n\n\\bibliography{my}\n\\bibliographystyle{amsalpha}\n\n\\end{document}\n", [(795, 796, 'VAR'), (803, 814, 'TYPE'), (1864, 1865, 'VAR'), (1872, 1878, 'TYPE'), (5202, 5213, 'VAR'), (5220, 5228, 'TYPE'), (6343, 6344, 'VAR'), (6351, 6362, 'TYPE'), (6799, 6800, 'VAR'), (6807, 6818, 'TYPE'), (7326, 7327, 'VAR'), (7334, 7345, 'TYPE'), (9888, 9889, 'VAR'), (9896, 9907, 'TYPE'), (10125, 10126, 'VAR'), (10133, 10141, 'TYPE'), (10337, 10348, 'VAR'), (10356, 10361, 'TYPE'), (12245, 12256, 'VAR'), (12263, 12271, 'TYPE'), (13294, 13305, 'VAR'), (13312, 13320, 'TYPE'), (14678, 14694, 'VAR'), (14701, 14707, 'TYPE'), (18043, 18059, 'VAR'), (18066, 18072, 'TYPE'), (19686, 19702, 'VAR'), (19709, 19715, 'TYPE'), (20091, 20094, 'VAR'), (20101, 20115, 'TYPE'), (23151, 23167, 'VAR'), (23174, 23180, 'TYPE'), (28595, 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'TYPE'), (177205, 177223, 'VAR'), (177231, 177242, 'TYPE'), (180259, 180275, 'VAR'), (180282, 180289, 'TYPE'), (193893, 193912, 'VAR'), (193919, 193925, 'TYPE'), (200249, 200268, 'VAR'), (200275, 200281, 'TYPE'), (201109, 201125, 'VAR'), (201132, 201140, 'TYPE'), (202629, 202645, 'VAR'), (202652, 202662, 'TYPE'), (203472, 203483, 'VAR'), (203490, 203496, 'TYPE'), (204436, 204452, 'VAR'), (204459, 204469, 'TYPE'), (205563, 205579, 'VAR'), (205586, 205596, 'TYPE'), (207247, 207260, 'VAR'), (207267, 207281, 'TYPE'), (207850, 207863, 'VAR'), (207870, 207885, 'TYPE'), (208776, 208789, 'VAR'), (208796, 208811, 'TYPE'), (213260, 213261, 'VAR'), (213268, 213277, 'TYPE'), (213290, 213291, 'VAR'), (213298, 213303, 'TYPE'), (213728, 213741, 'VAR'), (213748, 213763, 'TYPE'), (215266, 215273, 'VAR'), (215280, 215286, 'TYPE')])

In [7]:

    

random.shuffle(annotated_data)
train_data = annotated_data[:-1]
test_data = annotated_data[-1:] #we hold out one tex file for testing


    

In [8]:

    

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 [9]:

    

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


    

In [10]:

    

#first test on a simple sentence
doc = nlp.make_doc('Let $S$ be a scheme.')  
nlp.tagger(doc)
ner(doc)
for word in doc:
    print(word.text, word.ent_type_)


    

    




Let 
$ 
S$ 
be 
a 
scheme TYPE
. TYPE

In [11]:

    

#then test on the hold out tex file; there are no 'VAR' tags detected, maybe something to do with dollar sign tokenization...
doc = nlp.make_doc(test_data[0][0])  
nlp.tagger(doc)
ner(doc)
for word in doc
    print(word.text, "\t" + word.ent_type_)


    

    




\input{preamble 	
} 	


 	
% 	
OK 	
, 	
start 	
here 	
. 	

 	
% 	

 	
\begin{document 	
} 	


 	
\title{Simplicial 	
Spaces 	
} 	



 	
\maketitle 	


 	
\phantomsection 	

 	
\label{section 	
- 	
phantom 	
} 	


 	
\tableofcontents 	


 	
\section{Introduction 	
} 	

 	
\label{section 	
- 	
introduction 	
} 	


 	
\noindent 	

 	
This 	
chapter 	
develops 	
some 	
theory 	
concerning 	
simplicial 	
topological 	
spaces 	
, 	

 	
simplicial 	
ringed 	
spaces 	
, 	
simplicial 	
schemes 	
, 	
and 	
simplicial 	
algebraic 	
spaces 	
. 	

 	
The 	
theory 	
of 	
simplicial 	
spaces 	
sometimes 	
allows 	
one 	
to 	
prove 	
local 	
to 	
global 	

 	
principles 	
which 	
appear 	
difficult 	
to 	
prove 	
in 	
other 	
ways 	
. 	

 	
Some 	
example 	
applications 	
can 	
be 	
found 	
in 	
the 	
papers 	

 	
\cite{faltings_finiteness 	
} 	
, 	
\cite{Kiehl 	
} 	
, 	
and 	
\cite{HodgeIII}. 	


 	
\medskip\noindent 	

 	
We 	
assume 	
throughout 	
that 	
the 	
reader 	
is 	
familiar 	
with 	
the 	
basic 	
concepts 	

 	
and 	
results 	
of 	
the 	
chapter 	
Simplicial 	
Methods 	
, 	
see 	

 	
Simplicial 	
, 	
Section 	
\ref{simplicial 	
- 	
section 	
- 	
introduction}. 	

 	
In 	
particular 	
, 	
we 	
continue 	
to 	
write 	
$ 	
X$ 	
and 	
not 	
$ 	
X_\bullet$ 	

 	
for 	
a 	
simplicial 	
object 	
. 	









 	
\section{Simplicial 	
topological 	
spaces 	
} 	

 	
\label{section 	
- 	
simplicial 	
- 	
top 	
} 	


 	
\noindent 	

 	
A 	
{ 	
\it 	
simplicial 	
space 	
} 	
is 	
a 	
simplicial 	
object 	
in 	
the 	
category 	
of 	

 	
topological 	
spaces 	
where 	
morphisms 	
are 	
continuous 	
maps 	
of 	
topological 	

 	
spaces 	
. 	
( 	
We 	
will 	
use 	
`` 	
simplicial 	
algebraic 	
space 	
'' 	
to 	
refer 	
to 	
simplicial 	

 	
objects 	
in 	
the 	
category 	
of 	
algebraic 	
spaces 	
. 	
) 	

 	
We 	
may 	
picture 	
a 	
simplicial 	
space 	
$ 	
X$ 	
as 	
follows 	

 	
$ 	
$ 	

 	
\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 	
] 	

 	
} 	

 	
$ 	
$ 	

 	
Here 	
there 	
are 	
two 	
morphisms 	
$ 	
d^1_0 	
, 	
d^1_1 	
: 	
X_1 	
\to 	
X_0 	
$ 	

 	
and 	
a 	
single 	
morphism 	
$ 	
s^0_0 	
: 	
X_0 	
\to 	
X_1 	
$ 	
, 	
etc 	
. 	

 	
It 	
is 	
important 	
to 	
keep 	
in 	
mind 	
that 	
$ 	
d^n_i 	
: 	
X_n 	
\to 	
X_{n 	
- 	
1}$ 	

 	
should 	
be 	
thought 	
of 	
as 	
a 	
`` 	
projection 	
forgetting 	
the 	

 	
$ 	
i$th 	
coordinate 	
'' 	
and 	
$ 	
s^n_j 	
: 	
X_n 	
\to 	
X_{n 	
+ 	
1}$ 	
as 	
the 	
diagonal 	

 	
map 	
repeating 	
the 	
$ 	
j$th 	
coordinate 	
. 	


 	
\medskip\noindent 	

 	
Let 	
$ 	
X$ 	
be 	
a 	
simplicial 	TYPE
space 	
. 	
We 	
associate 	
a 	
site 	

 	
$ 	
X_{Zar}$\footnote{This 	
notation 	
is 	
similar 	
to 	
the 	
notation 	
in 	

 	
Sites 	
, 	
Example 	
\ref{sites 	
- 	
example 	
- 	
site 	
- 	
topological 	
} 	

 	
and 	

 	
Topologies 	
, 	
Definition 	
\ref{topologies 	
- 	
definition 	
- 	
big 	
- 	
small 	
- 	
Zariski}. 	
} 	

 	
to 	
$ 	
X$ 	
as 	
follows 	
. 	

 	
\begin{enumerate 	
} 	

 	
\item 	
An 	
object 	
of 	
$ 	
X_{Zar}$ 	
is 	
an 	
open 	
$ 	
U$ 	
of 	
$ 	
X_n$ 	
for 	
some 	
$ 	
n$ 	
, 	

 	
\item 	
a 	
morphism 	
$ 	
U 	
\to 	
V$ 	
of 	
$ 	
X_{Zar}$ 	
is 	
given 	
by 	
a 	

 	
$ 	
\varphi 	
: 	
[ 	
m 	
] 	
\to 	
[ 	
n]$ 	
where 	
$ 	
n 	
, 	
m$ 	
are 	
such 	
that 	

 	
$ 	
U 	
\subset 	
X_n$ 	
, 	
$ 	
V 	
\subset 	
X_m$ 	
and 	
$ 	
\varphi$ 	
is 	
such 	
that 	

 	
$ 	
X(\varphi)(U 	
) 	
\subset 	
V$ 	
, 	
and 	

 	
\item 	
a 	
covering 	
$ 	
\{U_i 	
\to 	
U\}$ 	
in 	
$ 	
X_{Zar}$ 	
means 	

 	
that 	
$ 	
U 	
, 	
U_i 	
\subset 	
X_n$ 	
are 	
open 	
, 	
the 	
maps 	
$ 	
U_i 	
\to 	
U$ 	
are 	

 	
given 	
by 	
$ 	
\text{id 	
} 	
: 	
[ 	
n 	
] 	
\to 	
[ 	
n]$ 	
, 	
and 	
$ 	
U 	
= 	
\bigcup 	
U_i$. 	

 	
\end{enumerate 	
} 	

 	
Note 	
that 	
in 	
particular 	
, 	
if 	
$ 	
U 	
\to 	
V$ 	
is 	
a 	
morphism 	
of 	
$ 	
X_{Zar}$ 	

 	
given 	
by 	
$ 	
\varphi$ 	
, 	
then 	
$ 	
X(\varphi 	
) 	
: 	
X_n 	
\to 	
X_m$ 	
does 	
in 	
fact 	

 	
induce 	
a 	
continuous 	
map 	
$ 	
U 	
\to 	
V$ 	
of 	
topological 	
spaces 	
. 	


 	
\noindent 	

 	
It 	
is 	
clear 	
that 	
the 	
above 	
is 	
a 	
special 	
case 	
of 	
a 	
construction 	
that 	

 	
associates 	
to 	
any 	
diagram 	
of 	
topological 	
spaces 	
a 	
site 	
. 	
We 	
formulate 	

 	
the 	
obligatory 	
lemma 	
. 	


 	
\begin{lemma 	
} 	

 	
\label{lemma 	
- 	
simplicial 	
- 	
site 	
} 	

 	
Let 	
$ 	
X$ 	
be 	
a 	
simplicial 	TYPE
space 	
. 	
Then 	
$ 	
X_{Zar}$ 	

 	
as 	
defined 	
above 	
is 	
a 	
site 	
. 	

 	
\end{lemma 	
} 	


 	
\begin{proof 	
} 	

 	
Omitted 	
. 	

 	
\end{proof 	
} 	


 	
\noindent 	

 	
Let 	
$ 	
X$ 	
be 	
a 	
simplicial 	TYPE
space 	
. 	
Let 	
$ 	
\mathcal{F}$ 	
be 	
a 	
sheaf 	TYPE
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 	
} 	
) 	
. 	

 	
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 	TYPE
space 	
. 	
There 	
is 	
an 	
equivalence 	
of 	

 	
categories 	
between 	

 	
\begin{enumerate 	
} 	

 	
\item 	
$ 	
\Sh(X_{Zar})$ 	
, 	
and 	

 	
\item 	
category 	
of 	
systems 	
$ 	
( 	
\mathcal{F}_n 	
, 	
\mathcal{F}(\varphi))$ 	

 	
described 	
above 	
. 	

 	
\end{enumerate 	
} 	

 	
\end{lemma 	
} 	


 	
\begin{proof 	
} 	

 	
See 	
discussion 	
above 	
. 	

 	
\end{proof 	
} 	


 	
\begin{lemma 	
} 	

 	
\label{lemma 	
- 	
simplicial 	
- 	
space 	
- 	
site 	
- 	
functorial 	
} 	

 	
Let 	
$ 	
f 	
: 	
Y 	
\to 	
X$ 	
be 	
a 	
morphism 	
of 	
simplicial 	
spaces 	
. 	

 	
Then 	
the 	
functor 	
$ 	
u 	
: 	
X_{Zar 	
} 	
\to 	
Y_{Zar}$ 	

 	
which 	
associates 	
to 	
the 	
open 	
$ 	
U 	
\subset 	
X_n$ 	
the 	
open 	

 	
$ 	
f_n^{-1}(U 	
) 	
\subset 	
Y_n$ 	
defines 	
a 	
morphism 	
of 	
sites 	

 	
$ 	
f_{Zar 	
} 	
: 	
Y_{Zar 	
} 	
\to 	
X_{Zar}$. 	

 	
\end{lemma 	
} 	


 	
\begin{proof 	
} 	

 	
It 	
is 	
clear 	
that 	
$ 	
u$ 	
is 	
a 	
continuous 	
functor 	
. 	
Hence 	
we 	
obtain 	
functors 	

 	
$ 	
f_{Zar 	
, 	
* 	
} 	
= 	
u^s$ 	
and 	
$ 	
f_{Zar}^{-1 	
} 	
= 	
u_s$ 	
, 	
see 	

 	
Sites 	
, 	
Section 	
\ref{sites 	
- 	
section 	
- 	
morphism 	
- 	
sites}. 	

 	
To 	
see 	
that 	
we 	
obtain 	
a 	
morphism 	
of 	
sites 	
we 	
have 	
to 	
show 	

 	
that 	
$ 	
u_s$ 	
is 	
exact 	
. 	
We 	
will 	
use 	

 	
Sites 	
, 	
Lemma 	
\ref{sites 	
- 	
lemma 	
- 	
directed 	
- 	
morphism 	
} 	
to 	
see 	
this 	
. 	

 	
Let 	
$ 	
V 	
\subset 	
Y_n$ 	
be 	
an 	
open 	TYPE
subset 	
. 	
The 	
category 	

 	
$ 	
\mathcal{I}_V^u$ 	
( 	
see 	
Sites 	
, 	
Section 	
\ref{sites 	
- 	
section 	
- 	
functoriality 	
- 	
PSh 	
} 	
) 	

 	
consists 	
of 	
pairs 	
$ 	
( 	
U 	
, 	
\varphi)$ 	
where 	

 	
$ 	
\varphi 	
: 	
[ 	
m 	
] 	
\to 	
[ 	
n]$ 	
and 	
$ 	
U 	
\subset 	
X_m$ 	
open 	
such 	
that 	

 	
$ 	
Y(\varphi)(V 	
) 	
\subset 	
f_m^{-1}(U)$. 	
Moreover 	
, 	
a 	
morphism 	

 	
$ 	
( 	
U 	
, 	
\varphi 	
) 	
\to 	
( 	
U 	
' 	
, 	
\varphi')$ 	
is 	
given 	
by 	
a 	

 	
$ 	
\psi 	
: 	
[ 	
m 	
' 	
] 	
\to 	
[ 	
m]$ 	
such 	
that 	
$ 	
X(\psi)(U 	
) 	
\subset 	
U'$ 	

 	
and 	
$ 	
\varphi 	
\circ 	
\psi 	
= 	
\varphi'$. 	

 	
It 	
is 	
our 	
task 	
to 	
show 	
that 	
$ 	
\mathcal{I}_V^u$ 	
is 	
cofiltered 	
. 	


 	
\medskip\noindent 	

 	
We 	
verify 	
the 	
conditions 	
of 	

 	
Categories 	
, 	
Definition 	
\ref{categories 	
- 	
definition 	
- 	
codirected}. 	

 	
Condition 	
( 	
1 	
) 	
holds 	
because 	
$ 	
( 	
X_n 	
, 	
\text{id}_{[n]})$ 	
is 	
an 	
object 	
. 	

 	
Let 	
$ 	
( 	
U 	
, 	
\varphi)$ 	
be 	
an 	
object 	
. 	
The 	
condition 	

 	
$ 	
Y(\varphi)(V 	
) 	
\subset 	
f_m^{-1}(U)$ 	
is 	
equivalent 	
to 	

 	
$ 	
V 	
\subset 	
f_n^{-1}(X(\varphi)^{-1}(U))$. 	
Hence 	
we 	
obtain 	
a 	
morphism 	

 	
$ 	
( 	
X(\varphi)^{-1}(U 	
) 	
, 	
\text{id}_{[n 	
] 	
} 	
) 	
\to 	
( 	
U 	
, 	
\varphi)$ 	
given 	

 	
by 	
setting 	
$ 	
\psi 	
= 	
\varphi$. 	
Moreover 	
, 	
given 	
a 	
pair 	
of 	
objects 	

 	
of 	
the 	
form 	
$ 	
( 	
U 	
, 	
\text{id}_{[n]})$ 	
and 	
$ 	
( 	
U 	
' 	
, 	
\text{id}_{[n]})$ 	

 	
we 	
see 	
there 	
exists 	
an 	
object 	
, 	
namely 	
$ 	
( 	
U 	
\cap 	
U 	
' 	
, 	
\text{id}_{[n]})$ 	
, 	

 	
which 	
maps 	
to 	
both 	
of 	
them 	
. 	
Thus 	
condition 	
( 	
2 	
) 	
holds 	
. 	

 	
To 	
verify 	
condition 	
( 	
3 	
) 	
suppose 	
given 	
two 	
morphisms 	

 	
$ 	
a 	
, 	
a 	
' 	
: 	
( 	
U 	
, 	
\varphi 	
) 	
\to 	
( 	
U 	
' 	
, 	
\varphi')$ 	
given 	
by 	
$ 	
\psi 	
, 	
\psi 	
' 	
: 	
[ 	
m 	
' 	
] 	
\to 	
[ 	
m]$. 	

 	
Then 	
precomposing 	
with 	
the 	
morphism 	

 	
$ 	
( 	
X(\varphi)^{-1}(U 	
) 	
, 	
\text{id}_{[n 	
] 	
} 	
) 	
\to 	
( 	
U 	
, 	
\varphi)$ 	
given 	

 	
by 	
$ 	
\varphi$ 	
equalizes 	
$ 	
a 	
, 	
a'$ 	
because 	

 	
$ 	
\varphi 	
\circ 	
\psi 	
= 	
\varphi 	
' 	
= 	
\varphi 	
\circ 	
\psi'$. 	

 	
This 	
finishes 	
the 	
proof 	
. 	

 	
\end{proof 	
} 	


 	
\begin{lemma 	
} 	

 	
\label{lemma 	
- 	
describe 	
- 	
functoriality 	
} 	

 	
Let 	
$ 	
f 	
: 	
Y 	
\to 	
X$ 	
be 	
a 	
morphism 	
of 	
simplicial 	
spaces 	
. 	
In 	
terms 	
of 	
the 	

 	
description 	
of 	
sheaves 	
in 	

 	
Lemma 	
\ref{lemma 	
- 	
describe 	
- 	
sheaves 	
- 	
simplicial 	
- 	
site 	
} 	
the 	

 	
morphism 	
$ 	
f_{Zar}$ 	
of 	
Lemma 	
\ref{lemma 	
- 	
simplicial 	
- 	
space 	
- 	
site 	
- 	
functorial 	
} 	

 	
can 	
be 	
described 	
as 	
follows 	
. 	

 	
\begin{enumerate 	
} 	

 	
\item 	
If 	
$ 	
\mathcal{G}$ 	
is 	
a 	
sheaf 	
on 	
$ 	
Y$ 	
, 	
then 	

 	
$ 	
( 	
f_{Zar 	
, 	
* 	
} 	
\mathcal{G})_n 	
= 	
f_{n 	
, 	
* 	
} 	
\mathcal{G}_n$. 	

 	
\item 	
If 	
$ 	
\mathcal{F}$ 	
is 	
a 	
sheaf 	
on 	
$ 	
X$ 	
, 	
then 	

 	
$ 	
( 	
f_{Zar}^{-1}\mathcal{F})_n 	
= 	
f_n^{-1}\mathcal{F}_n$. 	

 	
\end{enumerate 	
} 	

 	
\end{lemma 	
} 	


 	
\begin{proof 	
} 	

 	
The 	
first 	
part 	
is 	
immediate 	
from 	
the 	
definitions 	
. 	
For 	
the 	
second 	
part 	
, 	
note 	

 	
that 	
in 	
the 	
proof 	
of 	

 	
Lemma 	
\ref{lemma 	
- 	
simplicial 	
- 	
space 	
- 	
site 	
- 	
functorial 	
} 	

 	
we 	
have 	
shown 	
that 	
for 	
a 	
$ 	
V 	
\subset 	
Y_n$ 	
open 	
the 	
category 	

 	
$ 	
( 	
\mathcal{I}_V^u)^{opp}$ 	
contains 	
as 	
a 	
cofinal 	
subcategory 	

 	
the 	
category 	
of 	
opens 	
$ 	
U 	
\subset 	
X_n$ 	
with 	
$ 	
f_n^{-1}(U 	
) 	
\supset 	
V$ 	

 	
and 	
morphisms 	
given 	
by 	
inclusions 	
. 	
Hence 	
we 	
see 	
that 	
the 	
restriction 	

 	
of 	
$ 	
u_p\mathcal{F}$ 	
to 	
opens 	
of 	
$ 	
Y_n$ 	
is 	
the 	
presheaf 	

 	
$ 	
f_{n 	
, 	
p}\mathcal{F}_n$ 	
as 	
defined 	
in 	

 	
Sheaves 	
, 	
Lemma 	
\ref{sheaves 	
- 	
lemma 	
- 	
pullback 	
- 	
presheaves}. 	

 	
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 	TYPE
space 	
. 	
In 	

 	
Sites 	
, 	
Example 	
\ref{sites 	
- 	
example 	
- 	
site 	
- 	
topological 	
} 	

 	
we 	
denoted 	
$ 	
X_{Zar}$ 	
the 	
site 	
consisting 	
of 	
opens 	
of 	
$ 	
X$ 	

 	
with 	
inclusions 	
as 	
morphisms 	
and 	
coverings 	
given 	
by 	
open 	
coverings 	
. 	

 	
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 	TYPE
space 	
. 	
The 	
functor 	

 	
$ 	
X_{n 	
, 	
Zar 	
} 	
\to 	
X_{Zar}$ 	
, 	
$ 	
U 	
\mapsto 	
U$ 	
is 	
continuous 	

 	
and 	
cocontinuous 	
. 	
The 	
associated 	
morphism 	
of 	

 	
topoi 	
$ 	
g_n 	
: 	
\Sh(X_n 	
) 	
\to 	
\Sh(X_{Zar})$ 	
satisfies 	

 	
\begin{enumerate 	
} 	

 	
\item 	
$ 	
g_n^{-1}$ 	
associates 	
to 	
the 	
sheaf 	
$ 	
\mathcal{F}$ 	
on 	
$ 	
X$ 	

 	
the 	
sheaf 	
$ 	
\mathcal{F}_n$ 	
on 	
$ 	
X_n$ 	
, 	

 	
\item 	
$ 	
g_n^{-1 	
} 	
: 	
\Sh(X_{Zar 	
} 	
) 	
\to 	
\Sh(X_n)$ 	
has 	
a 	
left 	
adjoint 	
$ 	
g^{Sh}_{n!}$ 	
, 	

 	
\item 	
$ 	
g^{Sh}_{n!}$ 	
commutes 	
with 	
finite 	
connected 	
limits 	
, 	

 	
\item 	
$ 	
g_n^{-1 	
} 	
: 	
\textit{Ab}(X_{Zar 	
} 	
) 	
\to 	
\textit{Ab}(X_n)$ 	

 	
has 	
a 	
left 	
adjoint 	
$ 	
g_{n!}$ 	
, 	
and 	

 	
\item 	
$ 	
g_{n!}$ 	
is 	
exact 	
. 	

 	
\end{enumerate 	
} 	

 	
\end{lemma 	
} 	


 	
\begin{proof 	
} 	

 	
Besides 	
the 	
properties 	
of 	
our 	
functor 	
mentioned 	
in 	
the 	
statement 	
, 	

 	
the 	
category 	
$ 	
X_{n 	
, 	
Zar}$ 	
has 	
fibre 	
products 	
and 	
equalizers 	

 	
and 	
the 	
functor 	
commutes 	
with 	
them 	
( 	
beware 	
that 	
$ 	
X_{Zar}$ 	
does 	
not 	

 	
have 	
all 	
fibre 	
products 	
) 	
. 	
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 	TYPE
space 	
. 	
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 	
. 	
Then 	

 	
$ 	
$ 	

 	
\xymatrix 	
{ 	

 	
\Sh(Y_n 	
) 	
\ar[d 	
] 	
\ar[r]_{f_n 	
} 	
& 	
\Sh(X_n 	
) 	
\ar[d 	
] 	
\\ 	

 	
\Sh(Y_{Zar 	
} 	
) 	
\ar[r]^{f_{Zar 	
} 	
} 	
& 	
\Sh(X_{Zar 	
} 	
) 	

 	
} 	

 	
$ 	
$ 	

 	
is 	
a 	
commutative 	
diagram 	
of 	
topoi 	
. 	

 	
\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 	TYPE
space 	
and 	
let 	
$ 	
a 	
: 	
Y 	
\to 	
X$ 	
be 	
an 	
augmentation 	TYPE

 	
( 	
Simplicial 	
, 	
Definition 	
\ref{simplicial 	
- 	
definition 	
- 	
augmentation 	
} 	
) 	
. 	

 	
Let 	
$ 	
a_n 	
: 	
Y_n 	
\to 	
X$ 	
be 	
the 	
corresponding 	
morphisms 	
of 	
topological 	
spaces 	
. 	

 	
There 	
is 	
a 	
canonical 	
morphism 	
of 	
topoi 	

 	
$ 	
$ 	

 	
a 	
: 	
\Sh(Y_{Zar 	
} 	
) 	
\to 	
\Sh(X 	
) 	

 	
$ 	
$ 	

 	
with 	
the 	
following 	
properties 	
: 	

 	
\begin{enumerate 	
} 	

 	
\item 	
$ 	
a^{-1}\mathcal{F}$ 	
is 	
the 	
sheaf 	
restricting 	
to 	
$ 	
a_n^{-1}\mathcal{F}$ 	

 	
on 	
$ 	
Y_n$ 	
, 	

 	
\item 	
$ 	
a_m 	
\circ 	
Y(\varphi 	
) 	
= 	
a_n$ 	
for 	
all 	
$ 	
\varphi 	
: 	
[ 	
m 	
] 	
\to 	
[ 	
n]$ 	
, 	

 	
\item 	
$ 	
a 	
\circ 	
g_n 	
= 	
a_n$ 	
as 	
morphisms 	
of 	
topoi 	
with 	

 	
$ 	
g_n$ 	
as 	
in 	
Lemma 	
\ref{lemma 	
- 	
restriction 	
- 	
to 	
- 	
components 	
} 	
, 	

 	
\item 	
$ 	
a_*\mathcal{G}$ 	
for 	
$ 	
\mathcal{G 	
} 	
\in 	
\Sh(Y_{Zar})$ 	

 	
is 	
the 	
equalizer 	
of 	
the 	
two 	
maps 	

 	
$ 	
a_{0 	
, 	
* 	
} 	
\mathcal{G}_0 	
\to 	
a_{1 	
, 	
* 	
} 	
\mathcal{G}_1$. 	

 	
\end{enumerate 	
} 	

 	
\end{lemma 	
} 	


 	
\begin{proof 	
} 	

 	
Part 	
( 	
2 	
) 	
holds 	
for 	
augmentations 	
of 	
simplicial 	
objects 	
in 	
any 	
category 	
. 	

 	
Thus 	
$ 	
Y(\varphi)^{-1 	
} 	
a_m^{-1 	
} 	
\mathcal{F 	
} 	
= 	
a_n^{-1}\mathcal{F}$ 	

 	
which 	
defines 	
an 	
$ 	
Y(\varphi)$-map 	
from 	
$ 	
a_m^{-1}\mathcal{F}$ 	

 	
to 	
$ 	
a_n^{-1}\mathcal{F}$. 	

 	
Thus 	
we 	
can 	
use 	
( 	
1 	
) 	
as 	
the 	
definition 	
of 	
$ 	
a^{-1}\mathcal{F}$ 	
( 	
using 	

 	
Lemma 	
\ref{lemma 	
- 	
describe 	
- 	
sheaves 	
- 	
simplicial 	
- 	
site 	
} 	
) 	
and 	

 	
( 	
4 	
) 	
as 	
the 	
definition 	
of 	
$ 	
a_*$. 	
If 	
this 	
defines 	
a 	
morphism 	
of 	
topoi 	

 	
then 	
part 	
( 	
3 	
) 	
follows 	
because 	
we 	
'll 	
have 	
$ 	
g_n^{-1 	
} 	
\circ 	
a^{-1 	
} 	
= 	
a_n^{-1}$ 	

 	
by 	
construction 	
. 	
To 	
check 	
$ 	
a$ 	
is 	
a 	
morphism 	
of 	
topoi 	
we 	
have 	
to 	
show 	

 	
that 	
$ 	
a^{-1}$ 	
is 	
left 	
adjoint 	
to 	
$ 	
a_*$ 	
and 	
we 	
have 	
to 	
show 	
that 	

 	
$ 	
a^{-1}$ 	
is 	
exact 	
. 	
The 	
last 	
fact 	
is 	
immediate 	
from 	
the 	
exactness 	
of 	

 	
the 	
functors 	
$ 	
a_n^{-1}$. 	


 	
\medskip\noindent 	

 	
Let 	
$ 	
\mathcal{F}$ 	
be 	
an 	
object 	TYPE
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]$. 	
Then 	
we 	
can 	
look 	
at 	

 	
the 	
composition 	

 	
$ 	
$ 	

 	
\mathcal{F 	
} 	
\xrightarrow{\alpha 	
} 	
a_*\mathcal{G 	
} 	

 	
\to 	
a_{0 	
, 	
* 	
} 	
\mathcal{G}_0 	
\xrightarrow{\mathcal{G}(\varphi 	
) 	
} 	

 	
a_{n 	
, 	
* 	
} 	
\mathcal{G}_n 	

 	
$ 	
$ 	

 	
These 	
are 	
adjoint 	
to 	
maps 	
$ 	
a_n^{-1}\mathcal{F 	
} 	
\to 	
\mathcal{G}_n$ 	

 	
which 	
define 	
a 	
morphism 	
of 	
sheaves 	
$ 	
a^{-1}\mathcal{F 	
} 	
\to 	
\mathcal{G}$. 	

 	
We 	
omit 	
the 	
proof 	
that 	
the 	
constructions 	
given 	
above 	
define 	

 	
mutually 	
inverse 	
bijections 	

 	
$ 	
$ 	

 	
\Mor_{\Sh(Y_{Zar})}(a^{-1}\mathcal{F 	
} 	
, 	
\mathcal{G 	
} 	
) 	
= 	

 	
\Mor_{\Sh(X)}(\mathcal{F 	
} 	
, 	
a_*\mathcal{G 	
} 	
) 	

 	
$ 	
$ 	

 	
This 	
finishes 	
the 	
proof 	
. 	
An 	
interesting 	
observation 	
is 	
here 	
that 	

 	
this 	
morphism 	
of 	
topoi 	
does 	
not 	
correspond 	
to 	
any 	
obvious 	
geometric 	

 	
functor 	
between 	
the 	
sites 	
defining 	
the 	
topoi 	
. 	

 	
\end{proof 	
} 	


 	
\begin{lemma 	
} 	

 	
\label{lemma 	
- 	
simplicial 	
- 	
resolution 	
- 	
Z 	
} 	

 	
Let 	
$ 	
X$ 	
be 	
a 	
simplicial 	TYPE
topological 	
space 	
. 	
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 	TYPE
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 	
] 	
) 	
] 	

 	
$ 	
$ 	

 	
In 	
other 	
words 	
, 	
this 	
is 	
the 	
complex 	
associated 	
to 	
the 	

 	
free 	
abelian 	
group 	
on 	
the 	
simplicial 	
set 	
$ 	
\Delta[m]$ 	
, 	
see 	

 	
Simplicial 	
, 	
Example 	
\ref{simplicial 	
- 	
example 	
- 	
simplex 	
- 	
simplicial 	
- 	
set}. 	

 	
Since 	
$ 	
\Delta[m]$ 	
is 	
homotopy 	
equivalent 	
to 	
$ 	
\Delta[0]$ 	
, 	
see 	

 	
Simplicial 	
, 	
Example 	
\ref{simplicial 	
- 	
example 	
- 	
simplex 	
- 	
contractible 	
} 	
, 	

 	
and 	
since 	
`` 	
taking 	
free 	
abelian 	
groups 	
'' 	
is 	
a 	
functor 	
, 	

 	
we 	
see 	
that 	
the 	
complex 	
above 	
is 	
homotopy 	
equivalent 	
to 	

 	
the 	
free 	
abelian 	
group 	
on 	
$ 	
\Delta[0]$ 	

 	
( 	
Simplicial 	
, 	
Remark 	
\ref{simplicial 	
- 	
remark 	
- 	
homotopy 	
- 	
better 	
} 	
and 	

 	
Lemma 	
\ref{simplicial 	
- 	
lemma 	
- 	
homotopy 	
- 	
equivalence 	
- 	
s 	
- 	
N 	
} 	
) 	
. 	

 	
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 	TYPE
topological 	
space 	
. 	
Let 	
$ 	
\mathcal{F}$ 	
be 	
an 	
abelian 	TYPE

 	
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 	TYPE
resolution 	
. 	

 	
Consider 	
the 	
double 	
complex 	
with 	
terms 	

 	
$ 	
$ 	

 	
A^{p 	
, 	
q 	
} 	
= 	
\mathcal{I}^q(X_p 	
) 	

 	
$ 	
$ 	

 	
and 	
first 	
differential 	
given 	
by 	
the 	
alternating 	
sum 	
along 	
the 	
maps 	

 	
$ 	
d^{p 	
+ 	
1}_i$-maps 	
$ 	
\mathcal{I}_p^q 	
\to 	
\mathcal{I}_{p 	
+ 	
1}^q$ 	
, 	
see 	

 	
Lemma 	
\ref{lemma 	
- 	
describe 	
- 	
sheaves 	
- 	
simplicial 	
- 	
site}. 	
Note 	
that 	

 	
$ 	
$ 	

 	
A^{p 	
, 	
q 	
} 	
= 	
\Gamma(X_p 	
, 	
\mathcal{I}_p^q 	
) 	
= 	

 	
\Mor_{\textit{PSh}}(h_{X_p 	
} 	
, 	
\mathcal{I}^q 	
) 	
= 	

 	
\Mor_{\textit{PAb}}(\mathbf{Z}_{X_p 	
} 	
, 	
\mathcal{I}^q 	
) 	

 	
$ 	
$ 	

 	
Hence 	
it 	
follows 	
from 	
Lemma 	
\ref{lemma 	
- 	
simplicial 	
- 	
resolution 	
- 	
Z 	
} 	
and 	

 	
Cohomology 	
on 	
Sites 	
, 	
Lemma 	

 	
\ref{sites 	
- 	
cohomology 	
- 	
lemma 	
- 	
injective 	
- 	
abelian 	
- 	
sheaf 	
- 	
injective 	
- 	
presheaf 	
} 	

 	
that 	
the 	
rows 	
of 	
the 	
double 	
complex 	
are 	
exact 	
in 	
positive 	
degrees 	
and 	

 	
evaluate 	
to 	
$ 	
\Gamma(X_{Zar 	
} 	
, 	
\mathcal{I}^q)$ 	
in 	
degree 	
$ 	
0$. 	

 	
On 	
the 	
other 	
hand 	
, 	
since 	
restriction 	
is 	
exact 	

 	
( 	
Lemma 	
\ref{lemma 	
- 	
restriction 	
- 	
to 	
- 	
components 	
} 	
) 	

 	
the 	
map 	

 	
$ 	
$ 	

 	
\mathcal{F}_p 	
\to 	
\mathcal{I}_p^\bullet 	

 	
$ 	
$ 	

 	
is 	
a 	
resolution 	
. 	
The 	
sheaves 	
$ 	
\mathcal{I}_p^q$ 	
are 	
injective 	

 	
abelian 	
sheaves 	
on 	
$ 	
X_p$ 	

 	
( 	
Lemma 	
\ref{lemma 	
- 	
restriction 	
- 	
injective 	
- 	
to 	
- 	
component 	
} 	
) 	
. 	

 	
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 	TYPE
space 	
and 	
let 	
$ 	
a 	
: 	
X 	
\to 	
Y$ 	

 	
be 	
an 	
augmentation 	TYPE
. 	
Let 	
$ 	
\mathcal{F}$ 	
be 	
an 	
abelian 	TYPE
sheaf 	

 	
on 	
$ 	
X_{Zar}$. 	
Then 	
$ 	
R^na_*\mathcal{F}$ 	
is 	
the 	
sheaf 	
associated 	

 	
to 	
the 	
presheaf 	

 	
$ 	
$ 	

 	
V 	
\longmapsto 	
H^n((X 	
\times_Y 	
V)_{Zar 	
} 	
, 	
\mathcal{F}|_{(X 	
\times_Y 	
V)_{Zar 	
} 	
} 	
) 	

 	
$ 	
$ 	

 	
\end{lemma 	
} 	


 	
\begin{proof 	
} 	

 	
This 	
is 	
the 	
analogue 	
of 	

 	
Cohomology 	
, 	
Lemma 	
\ref{cohomology 	
- 	
lemma 	
- 	
describe 	
- 	
higher 	
- 	
direct 	
- 	
images 	
} 	
or 	
of 	

 	
Cohomology 	
on 	
Sites 	
, 	
Lemma 	
\ref{sites 	
- 	
cohomology 	
- 	
lemma 	
- 	
higher 	
- 	
direct 	
- 	
images 	
} 	

 	
and 	
we 	
strongly 	
encourge 	
the 	
reader 	
to 	
skip 	
the 	
proof 	
. 	

 	
Choosing 	
an 	
injective 	
resolution 	
of 	
$ 	
\mathcal{F}$ 	
on 	

 	
$ 	
X_{Zar}$ 	
and 	
using 	
the 	
definitions 	
we 	
see 	
that 	
it 	
suffices 	
to 	
show 	
: 	

 	
( 	
1 	
) 	
the 	
restriction 	
of 	
an 	
injective 	
abelian 	

 	
sheaf 	
on 	
$ 	
X_{Zar}$ 	
to 	
$ 	
( 	
X 	
\times_Y 	
V)_{Zar}$ 	
is 	
an 	
injective 	
abelian 	
sheaf 	
and 	

 	
( 	
2 	
) 	
$ 	
a_*\mathcal{F}$ 	
is 	
equal 	
to 	
the 	
rule 	

 	
$ 	
$ 	

 	
V 	
\longmapsto 	
H^0((X 	
\times_Y 	
V)_{Zar 	
} 	
, 	
\mathcal{F}|_{(X 	
\times_Y 	
V)_{Zar 	
} 	
} 	
) 	

 	
$ 	
$ 	

 	
Part 	
( 	
2 	
) 	
follows 	
from 	
the 	
following 	
facts 	

 	
\begin{enumerate 	
} 	

 	
\item[(2a 	
) 	
] 	
$ 	
a_*\mathcal{F}$ 	
is 	
the 	
equalizer 	
of 	
the 	
two 	
maps 	

 	
$ 	
a_{0 	
, 	
* 	
} 	
\mathcal{F}_0 	
\to 	
a_{1 	
, 	
* 	
} 	
\mathcal{F}_1 	
$ 	

 	
by 	
Lemma 	
\ref{lemma 	
- 	
augmentation 	
} 	
, 	

 	
\item[(2b 	
) 	
] 	
$ 	
a_{0 	
, 	
* 	
} 	
\mathcal{F}_0(V 	
) 	
= 	

 	
H^0(a_0^{-1}(V 	
) 	
, 	
\mathcal{F}_0)$ 	
and 	

 	
$ 	
a_{1 	
, 	
* 	
} 	
\mathcal{F}_1(V 	
) 	
= 	
H^0(a_1^{-1}(V 	
) 	
, 	
\mathcal{F}_1)$ 	
, 	

 	
\item[(2c 	
) 	
] 	
$ 	
X_0 	
\times_Y 	
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 	TYPE
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 	TYPE
space 	
. 	
Let 	
$ 	
X_\bullet$ 	
be 	
the 	
constant 	

 	
simplicial 	
topological 	
space 	
with 	
value 	
$ 	
X$. 	
The 	
functor 	

 	
$ 	
$ 	

 	
X_{\bullet 	
, 	
Zar 	
} 	
\longrightarrow 	
X_{Zar},\quad 	

 	
U 	
\longmapsto 	
U 	

 	
$ 	
$ 	

 	
is 	
continuous 	
and 	
cocontinuous 	
and 	
defines 	
a 	
morphism 	
of 	

 	
topoi 	
$ 	
g 	
: 	
\Sh(X_{\bullet 	
, 	
Zar 	
} 	
) 	
\to 	
\Sh(X)$ 	
as 	
well 	
as 	
a 	
left 	
adjoint 	

 	
$ 	
g_!$ 	
to 	
$ 	
g^{-1}$. 	
We 	
have 	

 	
\begin{enumerate 	
} 	

 	
\item 	
$ 	
g^{-1}$ 	
associates 	
to 	
a 	
sheaf 	
on 	
$ 	
X$ 	
the 	
constant 	
cosimplicial 	

 	
sheaf 	
on 	
$ 	
X$ 	
, 	

 	
\item 	
$ 	
g_!$ 	
associates 	
to 	
a 	
sheaf 	
$ 	
\mathcal{F}$ 	
on 	
$ 	
X_{\bullet 	
, 	
Zar}$ 	
the 	

 	
sheaf 	
$ 	
\mathcal{F}_0 	
$ 	
, 	
and 	

 	
\item 	
$ 	
g_*$ 	
associates 	
to 	
a 	
sheaf 	
$ 	
\mathcal{F}$ 	
on 	
$ 	
X_{\bullet 	
, 	
Zar}$ 	
the 	

 	
equalizer 	
of 	
the 	
two 	
maps 	
$ 	
\mathcal{F}_0 	
\to 	
\mathcal{F}_1$. 	

 	
\end{enumerate 	
} 	

 	
\end{lemma 	
} 	


 	
\begin{proof 	
} 	

 	
The 	
statements 	
about 	
the 	
functor 	
are 	
straightforward 	
to 	
verify 	
. 	

 	
The 	
existence 	
of 	
$ 	
g$ 	
and 	
$ 	
g_!$ 	
follow 	
from 	

 	
Sites 	
, 	
Lemmas 	
\ref{sites 	
- 	
lemma 	
- 	
cocontinuous 	
- 	
morphism 	
- 	
topoi 	
} 	
and 	

 	
\ref{sites 	
- 	
lemma 	
- 	
when 	
- 	
shriek}. 	
The 	
description 	
of 	

 	
$ 	
g^{-1}$ 	
is 	
immediate 	
from 	
Sites 	
, 	
Lemma 	
\ref{sites 	
- 	
lemma 	
- 	
when 	
- 	
shriek}. 	

 	
The 	
description 	
of 	
$ 	
g_*$ 	
and 	
$ 	
g_!$ 	
follows 	
as 	
the 	
functors 	
given 	
are 	

 	
right 	
and 	
left 	
adjoint 	
to 	
$ 	
g^{-1}$. 	

 	
\end{proof 	
} 	









 	
\section{Simplicial 	
sites 	
and 	
topoi 	
} 	

 	
\label{section 	
- 	
simplicial 	
- 	
sites 	
} 	


 	
\noindent 	

 	
It 	
seems 	
natural 	
to 	
define 	
a 	
{ 	
\it 	
simplicial 	
site 	
} 	
as 	
a 	
simplicial 	

 	
object 	
in 	
the 	
( 	
big 	
) 	
category 	
whose 	
objects 	
are 	
sites 	

 	
and 	
whose 	
morphisms 	
are 	
morphisms 	
of 	
sites 	
. 	

 	
See 	
Sites 	
, 	
Definitions 	
\ref{sites 	
- 	
definition 	
- 	
site 	
} 	
and 	

 	
\ref{sites 	
- 	
definition 	
- 	
morphism 	
- 	
sites 	
} 	

 	
with 	
composition 	
of 	
morphisms 	
as 	
in 	

 	
Sites 	
, 	
Lemma 	
\ref{sites 	
- 	
lemma 	
- 	
composition 	
- 	
morphisms 	
- 	
sites}. 	

 	
But 	
here 	
are 	
some 	
variants 	
one 	
might 	
want 	
to 	
consider 	
: 	

 	
( 	
a 	
) 	
we 	
could 	
work 	
with 	
cocontinuous 	
functors 	

 	
( 	
see 	
Sites 	
, 	
Sections 	
\ref{sites 	
- 	
section 	
- 	
cocontinuous 	
- 	
functors 	
} 	
and 	

 	
\ref{sites 	
- 	
section 	
- 	
cocontinuous 	
- 	
morphism 	
- 	
topoi 	
} 	
) 	
between 	
sites 	
instead 	
, 	

 	
( 	
b 	
) 	
we 	
could 	
work 	
in 	
a 	
suitable 	
$ 	
2$-category 	
of 	
sites 	
where 	
one 	
introduces 	

 	
the 	
notion 	
of 	
a 	
$ 	
2$-morphism 	
between 	
morphisms 	
of 	
sites 	
, 	

 	
( 	
c 	
) 	
we 	
could 	
work 	
in 	
a 	
$ 	
2$-category 	
constructed 	
out 	
of 	
cocontinuous 	

 	
functors 	
. 	
Instead 	
of 	
picking 	
one 	
of 	
these 	
variants 	
as 	
a 	
definition 	

 	
we 	
will 	
simply 	
develop 	
theory 	
as 	
needed 	
. 	


 	
\medskip\noindent 	

 	
Certainly 	
a 	
{ 	
\it 	
simplicial 	
topos 	
} 	
should 	
probably 	
be 	
defined 	
as 	
a 	

 	
pseudo 	
- 	
functor 	
from 	
$ 	
\Delta^{opp}$ 	
into 	
the 	
$ 	
2$-category 	
of 	
topoi 	
. 	

 	
See 	
Categories 	
, 	
Definition 	
\ref{categories 	
- 	
definition 	
- 	
functor 	
- 	
into-2-category 	
} 	

 	
and 	
Sites 	
, 	
Section 	
\ref{sites 	
- 	
section 	
- 	
topoi 	
} 	
and 	

 	
\ref{sites 	
- 	
section-2-category}. 	
We 	
will 	
try 	
to 	
avoid 	
working 	
with 	
such 	

 	
a 	
beast 	
if 	
possible 	
. 	


 	
\medskip\noindent 	

 	
{ 	
\bf 	
Case 	
A. 	
} 	

 	
Let 	
$ 	
\mathcal{C}$ 	
be 	
a 	
simplicial 	TYPE
object 	
in 	
the 	
category 	
whose 	
objects 	

 	
are 	
sites 	
and 	
whose 	
morphisms 	
are 	
morphisms 	
of 	
sites 	
. 	
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 	TYPE
object 	
in 	
the 	
category 	
of 	
sites 	
. 	

 	
With 	
notation 	
as 	
above 	
we 	
construct 	
a 	
site 	
$ 	
\mathcal{C}_{total}$ 	
as 	
follows 	
. 	

 	
\begin{enumerate 	
} 	

 	
\item 	
An 	
object 	
of 	
$ 	
\mathcal{C}_{total}$ 	
is 	
an 	
object 	
$ 	
U$ 	
of 	

 	
$ 	
\mathcal{C}_n$ 	
for 	
some 	
$ 	
n$ 	
, 	

 	
\item 	
a 	
morphism 	
$ 	
( 	
\varphi 	
, 	
f 	
) 	
: 	
U 	
\to 	
V$ 	
of 	
$ 	
\mathcal{C}_{total}$ 	

 	
is 	
given 	
by 	
a 	
map 	
$ 	
\varphi 	
: 	
[ 	
m 	
] 	
\to 	
[ 	
n]$ 	
with 	

 	
$ 	
U 	
\in 	
\Ob(\mathcal{C}_n)$ 	
, 	
$ 	
V 	
\in 	
\Ob(\mathcal{C}_m)$ 	

 	
and 	
a 	
morphism 	
$ 	
f 	
: 	
U 	
\to 	
u_\varphi(V)$ 	
of 	
$ 	
\mathcal{C}_n$ 	
, 	
and 	

 	
\item 	
a 	
covering 	
$ 	
\{(\text{id 	
} 	
, 	
f_i 	
) 	
: 	
  	
U_i 	
\to 	
U\}$ 	
in 	
$ 	
\mathcal{C}_{total}$ 	

 	
is 	
given 	
by 	
an 	
$ 	
n$ 	
and 	
a 	
covering 	
$ 	
\{f_i 	
: 	
U_i 	
\to 	
U\}$ 	

 	
of 	
$ 	
\mathcal{C}_n$. 	

 	
\end{enumerate 	
} 	

 	
\end{lemma 	
} 	


 	
\begin{proof 	
} 	

 	
Composition 	
of 	
$ 	
( 	
\varphi 	
, 	
f 	
) 	
: 	
U 	
\to 	
V$ 	
with 	
$ 	
( 	
\psi 	
, 	
g 	
) 	
: 	
V 	
\to 	
W$ 	

 	
is 	
given 	
by 	
$ 	
( 	
\varphi 	
\circ 	
\psi 	
, 	
u_\varphi(g 	
) 	
\circ 	
f)$. 	

 	
This 	
uses 	
that 	
$ 	
u_\varphi 	
\circ 	
u_\psi 	
= 	
u_{\varphi 	
\circ 	
\psi}$. 	


 	
\medskip\noindent 	

 	
Let 	
$ 	
\{(\text{id 	
} 	
, 	
f_i 	
) 	
: 	
  	
U_i 	
\to 	
U\}$ 	
be 	
a 	
covering 	TYPE
as 	
in 	
( 	
3 	
) 	

 	
and 	
let 	
$ 	
( 	
\varphi 	
, 	
g 	
) 	
: 	
W 	
\to 	
U$ 	
be 	
a 	
morphism 	TYPE
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}$. 	
This 	
makes 	
sense 	
as 	
by 	
our 	

 	
definition 	
of 	
morphisms 	
of 	
sites 	
, 	
the 	
required 	
fibre 	
products 	

 	
in 	
$ 	
\mathcal{C}_m$ 	
exist 	
since 	
$ 	
u_\varphi$ 	
transforms 	
coverings 	
into 	

 	
coverings 	
. 	
The 	
same 	
reasoning 	
implies 	
the 	
claim 	
( 	
details 	
omitted 	
) 	
. 	

 	
Thus 	
we 	
see 	
that 	
the 	
collection 	
of 	
coverings 	
is 	
stable 	
under 	
base 	

 	
change 	
. 	
The 	
other 	
axioms 	
of 	
a 	
site 	
are 	
immediate 	
. 	

 	
\end{proof 	
} 	


 	
\noindent 	

 	
{ 	
\bf 	
Case 	
B. 	
} 	

 	
Let 	
$ 	
\mathcal{C}$ 	
be 	
a 	
simplicial 	TYPE
object 	
in 	
the 	
category 	
whose 	
objects 	
are 	

 	
sites 	
and 	
whose 	
morphisms 	
are 	
cocontinuous 	
functors 	
. 	
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 	TYPE
object 	
in 	
the 	
category 	
whose 	
objects 	
are 	

 	
sites 	
and 	
whose 	
morphisms 	
are 	
cocontinuous 	
functors 	
. 	
With 	
notation 	
as 	
above 	
, 	

 	
assume 	
the 	
functors 	
$ 	
u_\varphi 	
: 	
\mathcal{C}_n 	
\to 	
\mathcal{C}_m$ 	

 	
have 	
property 	
$ 	
P$ 	
of 	
Sites 	
, 	
Remark 	
\ref{sites 	
- 	
remark 	
- 	
cartesian 	
- 	
cocontinuous}. 	

 	
Then 	
we 	
can 	
construct 	
a 	
site 	
$ 	
\mathcal{C}_{total}$ 	
as 	
follows 	
. 	

 	
\begin{enumerate 	
} 	

 	
\item 	
An 	
object 	
of 	
$ 	
\mathcal{C}_{total}$ 	
is 	
an 	
object 	
$ 	
U$ 	
of 	

 	
$ 	
\mathcal{C}_n$ 	
for 	
some 	
$ 	
n$ 	
, 	

 	
\item 	
a 	
morphism 	
$ 	
( 	
\varphi 	
, 	
f 	
) 	
: 	
U 	
\to 	
V$ 	
of 	
$ 	
\mathcal{C}_{total}$ 	

 	
is 	
given 	
by 	
a 	
map 	
$ 	
\varphi 	
: 	
[ 	
m 	
] 	
\to 	
[ 	
n]$ 	
with 	

 	
$ 	
U 	
\in 	
\Ob(\mathcal{C}_n)$ 	
, 	
$ 	
V 	
\in 	
\Ob(\mathcal{C}_m)$ 	

 	
and 	
a 	
morphism 	
$ 	
f 	
: 	
u_\varphi(U 	
) 	
\to 	
V$ 	
of 	
$ 	
\mathcal{C}_m$ 	
, 	
and 	

 	
\item 	
a 	
covering 	
$ 	
\{(\text{id 	
} 	
, 	
f_i 	
) 	
: 	
  	
U_i 	
\to 	
U\}$ 	
in 	
$ 	
\mathcal{C}_{total}$ 	

 	
is 	
given 	
by 	
an 	
$ 	
n$ 	
and 	
a 	
covering 	
$ 	
\{f_i 	
: 	
U_i 	
\to 	
U\}$ 	

 	
of 	
$ 	
\mathcal{C}_n$. 	

 	
\end{enumerate 	
} 	

 	
\end{lemma 	
} 	


 	
\begin{proof 	
} 	

 	
Composition 	
of 	
$ 	
( 	
\varphi 	
, 	
f 	
) 	
: 	
U 	
\to 	
V$ 	
with 	
$ 	
( 	
\psi 	
, 	
g 	
) 	
: 	
V 	
\to 	
W$ 	

 	
is 	
given 	
by 	
$ 	
( 	
\varphi 	
\circ 	
\psi 	
, 	
g 	
\circ 	
u_\psi(f))$. 	

 	
This 	
uses 	
that 	
$ 	
u_\psi 	
\circ 	
u_\varphi 	
= 	
u_{\varphi 	
\circ 	
\psi}$. 	


 	
\medskip\noindent 	

 	
Let 	
$ 	
\{(\text{id 	
} 	
, 	
f_i 	
) 	
: 	
  	
U_i 	
\to 	
U\}$ 	
be 	
a 	
covering 	TYPE
as 	
in 	
( 	
3 	
) 	

 	
and 	
let 	
$ 	
( 	
\varphi 	
, 	
g 	
) 	
: 	
W 	
\to 	
U$ 	
be 	
a 	
morphism 	TYPE
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 	
) 	
. 	

 	
Since 	
the 	
family 	
$ 	
\{W 	
\times_{g 	
, 	
U 	
, 	
f_i 	
} 	
U_i 	
\to 	
W\}$ 	
is 	
a 	

 	
covering 	
of 	
$ 	
\mathcal{C}_m$ 	
by 	
property 	
$ 	
P$ 	
we 	
see 	
that 	

 	
the 	
collection 	
of 	
coverings 	
is 	
stable 	
under 	
base 	

 	
change 	
. 	
The 	
other 	
axioms 	
of 	
a 	
site 	
are 	
immediate 	
. 	

 	
\end{proof 	
} 	


 	
\begin{situation 	
} 	

 	
\label{situation 	
- 	
simplicial 	
- 	
site 	
} 	

 	
Here 	
we 	
have 	
one 	
of 	
the 	
following 	
two 	
cases 	
: 	

 	
\begin{enumerate 	
} 	

 	
\item[(A 	
) 	
] 	
$ 	
\mathcal{C}$ 	
is 	
a 	
simplicial 	
object 	
in 	
the 	
category 	
whose 	

 	
objects 	
are 	
sites 	
and 	
whose 	
morphisms 	
are 	
morphisms 	
of 	
sites 	
. 	
For 	
every 	

 	
morphism 	
$ 	
\varphi 	
: 	
[ 	
m 	
] 	
\to 	
[ 	
n]$ 	
of 	
$ 	
\Delta$ 	
we 	
have 	
a 	
morphism 	
of 	
sites 	

 	
$ 	
f_\varphi 	
: 	
\mathcal{C}_n 	
\to 	
\mathcal{C}_m$ 	
given 	
by 	
a 	
continuous 	

 	
functor 	
$ 	
u_\varphi 	
: 	
\mathcal{C}_m 	
\to 	
\mathcal{C}_n$. 	

 	
\item[(B 	
) 	
] 	
$ 	
\mathcal{C}$ 	
is 	
a 	
simplicial 	
object 	
in 	
the 	
category 	
whose 	

 	
objects 	
are 	
sites 	
and 	
whose 	
morphisms 	
are 	
cocontinuous 	
functors 	
having 	

 	
property 	
$ 	
P$ 	
of 	
Sites 	
, 	
Remark 	
\ref{sites 	
- 	
remark 	
- 	
cartesian 	
- 	
cocontinuous}. 	

 	
For 	
every 	
morphism 	
$ 	
\varphi 	
: 	
[ 	
m 	
] 	
\to 	
[ 	
n]$ 	
of 	
$ 	
\Delta$ 	
we 	
have 	
a 	
cocontinuous 	

 	
functor 	
$ 	
u_\varphi 	
: 	
\mathcal{C}_n 	
\to 	
\mathcal{C}_m$ 	
which 	
induces 	
a 	

 	
morphism 	
of 	
topoi 	
$ 	
f_\varphi 	
: 	
\Sh(\mathcal{C}_n 	
) 	
\to 	
\Sh(\mathcal{C}_m)$. 	

 	
\end{enumerate 	
} 	

 	
As 	
usual 	
we 	
will 	
denote 	
$ 	
f_\varphi^{-1}$ 	
and 	
$ 	
f_{\varphi 	
, 	
* 	
} 	
$ 	
the 	

 	
pullback 	
and 	
pushforward 	
. 	
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 	
) 	
. 	

 	
\end{situation 	
} 	


 	
\noindent 	

 	
Let 	
$ 	
\mathcal{C}$ 	
be 	
as 	
in 	
Situation 	
\ref{situation 	
- 	
simplicial 	
- 	
site}. 	

 	
Let 	
$ 	
\mathcal{F}$ 	
be 	
a 	
sheaf 	TYPE
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 	
] 	
\to 	
[ 	
n]$ 	
the 	
restriction 	
maps 	
of 	
$ 	
\mathcal{F}$ 	

 	
along 	
the 	
morphisms 	
$ 	
( 	
\varphi 	
, 	
f 	
) 	
: 	
U 	
\to 	
V$ 	
with 	

 	
$ 	
U 	
\in 	
\Ob(\mathcal{C}_n)$ 	
and 	
$ 	
V 	
\in 	
\Ob(\mathcal{C}_m)$ 	

 	
define 	
an 	
element 	
$ 	
\mathcal{F}(\varphi)$ 	
of 	

 	
$ 	
$ 	

 	
\Mor_{\Sh(\mathcal{C}_m)}(\mathcal{F}_m 	
, 	
f_{\varphi 	
, 	
* 	
} 	
\mathcal{F}_n 	
) 	
= 	

 	
\Mor_{\Sh(\mathcal{C}_n)}(f_\varphi^{-1}\mathcal{F}_m 	
, 	
\mathcal{F}_n 	
) 	

 	
$ 	
$ 	

 	
Moreover 	
, 	
given 	
$ 	
\varphi 	
: 	
[ 	
m 	
] 	
\to 	
[ 	
n]$ 	
and 	
$ 	
\psi 	
: 	
[ 	
l 	
] 	
\to 	
[ 	
m]$ 	

 	
the 	
diagrams 	

 	
$ 	
$ 	

 	
\vcenter 	
{ 	

 	
\xymatrix 	
{ 	

 	
\mathcal{F}_l 	
\ar[rr]_{\mathcal{F}(\varphi 	
\circ 	
\psi 	
) 	
} 	

 	
\ar[rd]_{\mathcal{F}(\psi 	
) 	
} 	

 	
& 	
& 	
f_{\varphi 	
\circ 	
\psi 	
, 	
* 	
} 	
\mathcal{F}_n 	
\\ 	

 	
& 	
f_{\psi 	
, 	
* 	
} 	
\mathcal{F}_m 	
\ar[ur]_{f_{\psi 	
, 	
* 	
} 	
\mathcal{F}(\varphi 	
) 	
} 	

 	
} 	

 	
} 	

 	
\quad\text{and}\quad 	

 	
\vcenter 	
{ 	

 	
\xymatrix 	
{ 	

 	
f_{\varphi 	
\circ 	
\psi}^{-1}\mathcal{F}_l 	

 	
\ar[rr]_{\mathcal{F}(\varphi 	
\circ 	
\psi 	
) 	
} 	

 	
\ar[rd]_{f_\varphi^{-1}\mathcal{F}(\psi 	
) 	
} 	

 	
& 	
& 	
\mathcal{F}_n 	
\\ 	

 	
& 	
f_\varphi^{-1}\mathcal{F}_m 	
\ar[ur]_{\mathcal{F}(\varphi 	
) 	
} 	

 	
} 	

 	
} 	

 	
$ 	
$ 	

 	
commute 	
. 	
Clearly 	
, 	
the 	
converse 	
statement 	
is 	
true 	
as 	
well 	
: 	
if 	
we 	
have 	
a 	
system 	

 	
$ 	
( 	
\{\mathcal{F}_n\}_{n 	
\geq 	
0 	
} 	
, 	

 	
\{\mathcal{F}(\varphi)\}_{\varphi 	
\in 	
\text{Arrows}(\Delta)})$ 	

 	
satisfying 	
the 	
commutativity 	
constraints 	
above 	
, 	

 	
then 	
we 	
obtain 	
a 	
sheaf 	
on 	
$ 	
\mathcal{C}_{total}$. 	


 	
\begin{lemma 	
} 	

 	
\label{lemma 	
- 	
describe 	
- 	
sheaves 	
- 	
simplicial 	
- 	
site 	
- 	
site 	
} 	

 	
In 	
Situation 	
\ref{situation 	
- 	
simplicial 	
- 	
site 	
} 	
there 	
is 	
an 	
equivalence 	
of 	

 	
categories 	
between 	

 	
\begin{enumerate 	
} 	

 	
\item 	
$ 	
\Sh(\mathcal{C}_{total})$ 	
, 	
and 	

 	
\item 	
the 	
category 	
of 	
systems 	
$ 	
( 	
\mathcal{F}_n 	
, 	
\mathcal{F}(\varphi))$ 	

 	
described 	
above 	
. 	

 	
\end{enumerate 	
} 	

 	
In 	
particular 	
, 	
the 	
topos 	
$ 	
\Sh(\mathcal{C}_{total})$ 	
only 	
depends 	
on 	

 	
the 	
topoi 	
$ 	
\Sh(\mathcal{C}_n)$ 	
and 	
the 	
morphisms 	
of 	
topoi 	
$ 	
f_\varphi$. 	

 	
\end{lemma 	
} 	


 	
\begin{proof 	
} 	

 	
See 	
discussion 	
above 	
. 	

 	
\end{proof 	
} 	


 	
\begin{lemma 	
} 	

 	
\label{lemma 	
- 	
restriction 	
- 	
to 	
- 	
components 	
- 	
site 	
} 	

 	
In 	
Situation 	
\ref{situation 	
- 	
simplicial 	
- 	
site 	
} 	
the 	
functor 	

 	
$ 	
\mathcal{C}_n 	
\to 	
\mathcal{C}_{total}$ 	
, 	
$ 	
U 	
\mapsto 	
U$ 	
is 	
continuous 	

 	
and 	
cocontinuous 	
. 	
The 	
associated 	
morphism 	
of 	

 	
topoi 	
$ 	
g_n 	
: 	
\Sh(\mathcal{C}_n 	
) 	
\to 	
\Sh(\mathcal{C}_{total})$ 	
satisfies 	

 	
\begin{enumerate 	
} 	

 	
\item 	
$ 	
g_n^{-1}$ 	
associates 	
to 	
the 	
sheaf 	
$ 	
\mathcal{F}$ 	
on 	
$ 	
\mathcal{C}_{total}$ 	

 	
the 	
sheaf 	
$ 	
\mathcal{F}_n$ 	
on 	
$ 	
\mathcal{C}_n$ 	
, 	

 	
\item 	
$ 	
g_n^{-1 	
} 	
: 	
\Sh(\mathcal{C}_{total 	
} 	
) 	
\to 	
\Sh(\mathcal{C}_n)$ 	

 	
has 	
a 	
left 	
adjoint 	
$ 	
g^{Sh}_{n!}$ 	
, 	

 	
\item 	
for 	
$ 	
\mathcal{G}$ 	
in 	
$ 	
\Sh(\mathcal{C}_n)$ 	
the 	
restriction 	
of 	

 	
$ 	
g_{n!}^{Sh}\mathcal{G}$ 	
to 	
$ 	
\mathcal{C}_m$ 	
is 	

 	
$ 	
\coprod\nolimits_{\varphi 	
: 	
[ 	
n 	
] 	
\to 	
[ 	
m 	
] 	
} 	
f_\varphi^{-1}\mathcal{G}$ 	
, 	

 	
\item 	
$ 	
g_{n!}^{Sh}$ 	
commutes 	
with 	
finite 	
connected 	
limits 	
, 	

 	
\item 	
$ 	
g_n^{-1 	
} 	
: 	
\textit{Ab}(\mathcal{C}_{total 	
} 	
) 	
\to 	

 	
\textit{Ab}(\mathcal{C}_n)$ 	
has 	
a 	
left 	
adjoint 	
$ 	
g_{n!}$ 	
, 	

 	
\item 	
for 	
$ 	
\mathcal{G}$ 	
in 	
$ 	
\textit{Ab}(\mathcal{C}_n)$ 	
the 	
restriction 	
of 	

 	
$ 	
g_{n!}\mathcal{G}$ 	
to 	
$ 	
\mathcal{C}_m$ 	
is 	

 	
$ 	
\bigoplus\nolimits_{\varphi 	
: 	
[ 	
n 	
] 	
\to 	
[ 	
m 	
] 	
} 	
f_\varphi^{-1}\mathcal{G}$ 	
, 	
and 	

 	
\item 	
$ 	
g_{n!}$ 	
is 	
exact 	
. 	

 	
\end{enumerate 	
} 	

 	
\end{lemma 	
} 	


 	
\begin{proof 	
} 	

 	
Case 	
A. 	
If 	
$ 	
\{U_i 	
\to 	
U\}_{i 	
\in 	
I}$ 	
is 	
a 	
covering 	
in 	
$ 	
\mathcal{C}_n$ 	

 	
then 	
the 	
image 	
$ 	
\{U_i 	
\to 	
U\}_{i 	
\in 	
I}$ 	
is 	
a 	
covering 	
in 	
$ 	
\mathcal{C}_{total}$ 	

 	
by 	
definition 	
( 	
Lemma 	
\ref{lemma 	
- 	
simplicial 	
- 	
site 	
- 	
site 	
} 	
) 	
. 	
For 	
a 	
morphism 	

 	
$ 	
V 	
\to 	
U$ 	
of 	
$ 	
\mathcal{C}_n$ 	
, 	
the 	
fibre 	
product 	

 	
$ 	
V 	
\times_U 	
U_i$ 	
in 	
$ 	
\mathcal{C}_n$ 	
is 	
also 	
the 	

 	
the 	
fibre 	
product 	
in 	
$ 	
\mathcal{C}_{total}$ 	
( 	
by 	
the 	
claim 	
in 	
the 	

 	
proof 	
of 	
Lemma 	
\ref{lemma 	
- 	
simplicial 	
- 	
site 	
- 	
site 	
} 	
) 	
. 	

 	
Therefore 	
our 	
functor 	
is 	
continuous 	
. 	
On 	
the 	
other 	
hand 	
, 	
our 	
functor 	

 	
defines 	
a 	
bijection 	
between 	
coverings 	
of 	
$ 	
U$ 	
in 	
$ 	
\mathcal{C}_n$ 	

 	
and 	
coverings 	
of 	
$ 	
U$ 	
in 	
$ 	
\mathcal{C}_{total}$. 	
Therefore 	
it 	
is 	

 	
certainly 	
the 	
case 	
that 	
our 	
functor 	
is 	
cocontinuous 	
. 	


 	
\medskip\noindent 	

 	
Case 	
B. 	
If 	
$ 	
\{U_i 	
\to 	
U\}_{i 	
\in 	
I}$ 	
is 	
a 	
covering 	
in 	
$ 	
\mathcal{C}_n$ 	

 	
then 	
the 	
image 	
$ 	
\{U_i 	
\to 	
U\}_{i 	
\in 	
I}$ 	
is 	
a 	
covering 	
in 	
$ 	
\mathcal{C}_{total}$ 	

 	
by 	
definition 	
( 	
Lemma 	
\ref{lemma 	
- 	
simplicial 	
- 	
cocontinuous 	
- 	
site 	
} 	
) 	
. 	
For 	
a 	
morphism 	

 	
$ 	
V 	
\to 	
U$ 	
of 	
$ 	
\mathcal{C}_n$ 	
, 	
the 	
fibre 	
product 	

 	
$ 	
V 	
\times_U 	
U_i$ 	
in 	
$ 	
\mathcal{C}_n$ 	
is 	
also 	
the 	

 	
the 	
fibre 	
product 	
in 	
$ 	
\mathcal{C}_{total}$ 	
( 	
by 	
the 	
claim 	
in 	
the 	

 	
proof 	
of 	
Lemma 	
\ref{lemma 	
- 	
simplicial 	
- 	
cocontinuous 	
- 	
site 	
} 	
) 	
. 	

 	
Therefore 	
our 	
functor 	
is 	
continuous 	
. 	
On 	
the 	
other 	
hand 	
, 	
our 	
functor 	

 	
defines 	
a 	
bijection 	
between 	
coverings 	
of 	
$ 	
U$ 	
in 	
$ 	
\mathcal{C}_n$ 	

 	
and 	
coverings 	
of 	
$ 	
U$ 	
in 	
$ 	
\mathcal{C}_{total}$. 	
Therefore 	
it 	
is 	

 	
certainly 	
the 	
case 	
that 	
our 	
functor 	
is 	
cocontinuous 	
. 	


 	
\medskip\noindent 	

 	
At 	
this 	
point 	
part 	
( 	
1 	
) 	
and 	
the 	
existence 	
of 	
$ 	
g^{Sh}_{n!}$ 	
and 	
$ 	
g_{n!}$ 	

 	
in 	
cases 	
A 	
and 	
B 	
follows 	
from 	

 	
Sites 	
, 	
Lemmas 	
\ref{sites 	
- 	
lemma 	
- 	
cocontinuous 	
- 	
morphism 	
- 	
topoi 	
} 	
and 	

 	
\ref{sites 	
- 	
lemma 	
- 	
when 	
- 	
shriek 	
} 	

 	
and 	

 	
Modules 	
on 	
Sites 	
, 	
Lemmas 	
\ref{sites 	
- 	
modules 	
- 	
lemma 	
- 	
g 	
- 	
shriek 	
- 	
adjoint 	
} 	
and 	

 	
\ref{sites 	
- 	
modules 	
- 	
lemma 	
- 	
back 	
- 	
and 	
- 	
forth}. 	


 	
\medskip\noindent 	

 	
Proof 	
of 	
( 	
3 	
) 	
. 	
Let 	
$ 	
\mathcal{G}$ 	
be 	
a 	
sheaf 	TYPE
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 	
. 	

 	
Given 	
a 	
map 	
$ 	
\psi 	
: 	
[ 	
m 	
] 	
\to 	
[ 	
m']$ 	
the 	
map 	

 	
$ 	
\mathcal{H}(\psi 	
) 	
: 	
f_\psi^{-1}\mathcal{H}_m 	
\to 	
\mathcal{H}_{m'}$ 	

 	
is 	
given 	
on 	
components 	
by 	
the 	
identifications 	

 	
$ 	
$ 	

 	
f_\psi^{-1 	
} 	
f_\varphi^{-1 	
} 	
\mathcal{G 	
} 	
\to 	

 	
f_{\psi 	
\circ 	
\varphi}^{-1}\mathcal{G 	
} 	

 	
$ 	
$ 	

 	
Observe 	
that 	
given 	
a 	
map 	
$ 	
\alpha 	
: 	
\mathcal{H 	
} 	
\to 	
\mathcal{F}$ 	

 	
of 	
sheaves 	
on 	
$ 	
\mathcal{C}_{total}$ 	
we 	
obtain 	
a 	
map 	

 	
$ 	
\mathcal{G 	
} 	
\to 	
\mathcal{F}_n$ 	

 	
corresponding 	
to 	
the 	
restriction 	
of 	
$ 	
\alpha_n$ 	
to 	
the 	
component 	

 	
$ 	
\mathcal{G}$ 	
in 	
$ 	
\mathcal{H}_n$. 	
Conversely 	
, 	
given 	
a 	
map 	

 	
$ 	
\beta 	
: 	
\mathcal{G 	
} 	
\to 	
\mathcal{F}_n$ 	
of 	
sheaves 	
on 	
$ 	
\mathcal{C}_n$ 	

 	
we 	
can 	
define 	

 	
$ 	
\alpha 	
: 	
\mathcal{H 	
} 	
\to 	
\mathcal{F}$ 	
by 	
letting 	
$ 	
\alpha_m$ 	

 	
be 	
the 	
map 	
which 	
on 	
components 	

 	
$ 	
$ 	

 	
f_\varphi^{-1}\mathcal{G 	
} 	
\to 	
\mathcal{F}_m 	

 	
$ 	
$ 	

 	
uses 	
the 	
maps 	
adjoint 	
to 	
$ 	
\mathcal{F}(\varphi 	
) 	
\circ 	
f_\varphi^{-1}\beta$. 	

 	
We 	
omit 	
the 	
arguments 	
showing 	
these 	
two 	
constructions 	
give 	

 	
mutually 	
inverse 	
maps 	

 	
$ 	
$ 	

 	
\Mor_{\Sh(\mathcal{C}_n)}(\mathcal{G 	
} 	
, 	
\mathcal{F}_n 	
) 	
= 	

 	
\Mor_{\Sh(\mathcal{C}_{total})}(\mathcal{H 	
} 	
, 	
\mathcal{F 	
} 	
) 	

 	
$ 	
$ 	

 	
Thus 	
$ 	
\mathcal{H 	
} 	
= 	
g^{Sh}_{n!}\mathcal{G}$ 	
as 	
desired 	
. 	


 	
\medskip\noindent 	

 	
Proof 	
of 	
( 	
4 	
) 	
. 	
If 	
$ 	
\mathcal{G}$ 	
is 	
an 	
abelian 	
sheaf 	
on 	
$ 	
\mathcal{C}_n$ 	
, 	

 	
then 	
we 	
proceed 	
in 	
exactly 	
the 	
same 	
ammner 	
as 	
above 	
, 	
except 	
that 	

 	
we 	
define 	
$ 	
\mathcal{H}$ 	
is 	
the 	
abelian 	
sheaf 	
on 	
$ 	
\mathcal{C}_{total}$ 	

 	
whose 	
degree 	
$ 	
m$ 	
part 	
is 	
the 	
sheaf 	

 	
$ 	
$ 	

 	
\bigoplus\nolimits_{\varphi 	
: 	
[ 	
n 	
] 	
\to 	
[ 	
m 	
] 	
} 	
f_\varphi^{-1}\mathcal{G 	
} 	

 	
$ 	
$ 	

 	
with 	
transition 	
maps 	
defined 	
exactly 	
as 	
above 	
. 	
The 	
bijection 	

 	
$ 	
$ 	

 	
\Mor_{\textit{Ab}(\mathcal{C}_n)}(\mathcal{G 	
} 	
, 	
\mathcal{F}_n 	
) 	
= 	

 	
\Mor_{\textit{Ab}(\mathcal{C}_{total})}(\mathcal{H 	
} 	
, 	
\mathcal{F 	
} 	
) 	

 	
$ 	
$ 	

 	
is 	
proved 	
exactly 	
as 	
above 	
. 	

 	
Thus 	
$ 	
\mathcal{H 	
} 	
= 	
g_{n!}\mathcal{G}$ 	
as 	
desired 	
. 	


 	
\medskip\noindent 	

 	
The 	
exactness 	
properties 	
of 	
$ 	
g^{Sh}_{n!}$ 	
and 	
$ 	
g_{n!}$ 	
follow 	

 	
from 	
formulas 	
given 	
for 	
these 	
functors 	
. 	

 	
\end{proof 	
} 	


 	
\begin{lemma 	
} 	

 	
\label{lemma 	
- 	
restriction 	
- 	
injective 	
- 	
to 	
- 	
component 	
- 	
site 	
} 	

 	
\begin{slogan 	
} 	

 	
An 	
injective 	
abelian 	
sheaf 	
on 	
a 	
simplicial 	
site 	
is 	
injective 	
on 	
each 	
component 	

 	
\end{slogan 	
} 	

 	
In 	
Situation 	
\ref{situation 	
- 	
simplicial 	
- 	
site}. 	

 	
If 	
$ 	
\mathcal{I}$ 	
is 	
injective 	
in 	
$ 	
\textit{Ab}(\mathcal{C}_{total})$ 	
, 	

 	
then 	
$ 	
\mathcal{I}_n$ 	
is 	
injective 	
in 	
$ 	
\textit{Ab}(\mathcal{C}_n)$. 	

 	
If 	
$ 	
\mathcal{I}^\bullet$ 	
is 	
a 	
K 	
- 	
injective 	
complex 	
in 	

 	
$ 	
\textit{Ab}(\mathcal{C}_{total})$ 	
, 	

 	
then 	
$ 	
\mathcal{I}_n^\bullet$ 	
is 	
K 	
- 	
injective 	
in 	
$ 	
\textit{Ab}(\mathcal{C}_n)$. 	

 	
\end{lemma 	
} 	


 	
\begin{proof 	
} 	

 	
The 	
first 	
statement 	
follows 	
from 	

 	
Homology 	
, 	
Lemma 	
\ref{homology 	
- 	
lemma 	
- 	
adjoint 	
- 	
preserve 	
- 	
injectives 	
} 	

 	
and 	

 	
Lemma 	
\ref{lemma 	
- 	
restriction 	
- 	
to 	
- 	
components 	
- 	
site}. 	

 	
The 	
second 	
statement 	
from 	

 	
Derived 	
Categories 	
, 	
Lemma 	
\ref{derived 	
- 	
lemma 	
- 	
adjoint 	
- 	
preserve 	
- 	
K 	
- 	
injectives 	
} 	

 	
and 	

 	
Lemma 	
\ref{lemma 	
- 	
restriction 	
- 	
to 	
- 	
components 	
- 	
site}. 	

 	
\end{proof 	
} 	








 	
\section{Augmentations 	
of 	
simplicial 	
sites 	
} 	

 	
\label{section 	
- 	
augmentation 	
- 	
simplicial 	
- 	
sites 	
} 	


 	
\noindent 	

 	
We 	
continue 	
in 	
the 	
fashion 	
described 	
in 	

 	
Section 	
\ref{section 	
- 	
simplicial 	
- 	
sites 	
} 	

 	
working 	
out 	
the 	
meaning 	
of 	
augmentations 	
in 	
cases 	
A 	
and 	
B 	

 	
treated 	
in 	
that 	
section 	
. 	


 	
\begin{remark 	
} 	

 	
\label{remark 	
- 	
augmentation 	
- 	
site 	
} 	

 	
In 	
Situation 	
\ref{situation 	
- 	
simplicial 	
- 	
site 	
} 	
an 	

 	
{ 	
\it 	
augmentation 	
$ 	
a_0 	
$ 	
towards 	
a 	
site 	
$ 	
\mathcal{D}$ 	
} 	
will 	
mean 	

 	
\begin{enumerate 	
} 	

 	
\item[(A 	
) 	
] 	
$ 	
a_0 	
: 	
\mathcal{C}_0 	
\to 	
\mathcal{D}$ 	
is 	
a 	
morphism 	
of 	
sites 	

 	
given 	
by 	
a 	
continuous 	
functor 	
$ 	
u_0 	
: 	
\mathcal{D 	
} 	
\to 	
\mathcal{C}_0 	
$ 	

 	
such 	
that 	
for 	
all 	
$ 	
\varphi 	
, 	
\psi 	
: 	
[ 	
0 	
] 	
\to 	
[ 	
n]$ 	
we 	
have 	

 	
$ 	
u_\varphi 	
\circ 	
u_0 	
= 	
u_\psi 	
\circ 	
u_0$. 	

 	
\item[(B 	
) 	
] 	
$ 	
a_0 	
: 	
\Sh(\mathcal{C}_0 	
) 	
\to 	
\Sh(\mathcal{D})$ 	
is 	
a 	
morphism 	

 	
of 	
topoi 	
given 	
by 	
a 	
cocontinuous 	
functor 	
$ 	
u_0 	
: 	
\mathcal{C}_0 	
\to 	
\mathcal{D}$ 	

 	
such 	
that 	
for 	
all 	
$ 	
\varphi 	
, 	
\psi 	
: 	
[ 	
0 	
] 	
\to 	
[ 	
n]$ 	
we 	
have 	

 	
$ 	
u_0 	
\circ 	
u_\varphi 	
= 	
u_0 	
\circ 	
u_\psi$. 	

 	
\end{enumerate 	
} 	

 	
\end{remark 	
} 	


 	
\begin{lemma 	
} 	

 	
\label{lemma 	
- 	
augmentation 	
- 	
site 	
} 	

 	
In 	
Situation 	
\ref{situation 	
- 	
simplicial 	
- 	
site 	
} 	
let 	
$ 	
a_0 	
$ 	
be 	
an 	

 	TYPE
augmentation 	
towards 	
a 	
site 	
$ 	
\mathcal{D}$ 	
as 	
in 	

 	
Remark 	
\ref{remark 	
- 	
augmentation 	
- 	
site}. 	
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 	
] 	
\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 	
] 	
\to 	
[ 	
n]$. 	

 	
Then 	
$ 	
u_n$ 	
is 	
cocontinuous 	
and 	
hence 	
defines 	
a 	
morphism 	
of 	
topoi 	

 	
$ 	
a_n 	
: 	
\Sh(\mathcal{C}_n 	
) 	
\to 	
\Sh(\mathcal{D)}$ 	
, 	
see 	

 	
Sites 	
, 	
Lemma 	
\ref{sites 	
- 	
lemma 	
- 	
composition 	
- 	
cocontinuous}. 	

 	
The 	
same 	
lemma 	
shows 	
that 	
for 	
all 	
$ 	
\varphi 	
: 	
[ 	
m 	
] 	
\to 	
[ 	
n]$ 	
we 	
have 	

 	
$ 	
a_m 	
\circ 	
f_\varphi 	
= 	
a_n$. 	


 	
\medskip\noindent 	

 	
Consider 	
the 	
functor 	
$ 	
a^{-1 	
} 	
: 	
\Sh(\mathcal{D 	
} 	
) 	
\to 	
\Sh(\mathcal{C}_{total})$ 	

 	
which 	
to 	
a 	
sheaf 	
of 	
sets 	
$ 	
\mathcal{G}$ 	
associates 	
the 	
sheaf 	

 	
$ 	
\mathcal{F 	
} 	
= 	
a^{-1}\mathcal{G}$ 	
whose 	
components 	
are 	
$ 	
a_n^{-1}\mathcal{G}$ 	

 	
and 	
whose 	
transition 	
maps 	
$ 	
\mathcal{F}(\varphi)$ 	
are 	
the 	
identifications 	

 	
$ 	
$ 	

 	
f_\varphi^{-1}\mathcal{F}_m 	
= 	

 	
f_\varphi^{-1 	
} 	
a_m^{-1}\mathcal{G 	
} 	
= 	

 	
a_n^{-1}\mathcal{G 	
} 	
= 	

 	
\mathcal{F}_n 	

 	
$ 	
$ 	

 	
for 	
$ 	
\varphi 	
: 	
[ 	
m 	
] 	
\to 	
[ 	
n]$ 	
, 	
see 	
the 	
description 	
of 	

 	
$ 	
\Sh(\mathcal{C}_{total})$ 	
in 	

 	
Lemma 	
\ref{lemma 	
- 	
describe 	
- 	
sheaves 	
- 	
simplicial 	
- 	
site 	
- 	
site}. 	

 	
Since 	
the 	
functors 	
$ 	
a_n^{-1}$ 	
are 	
exact 	
, 	
$ 	
a^{-1}$ 	
is 	
an 	
exact 	
functor 	
. 	

 	
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 	
. 	

 	
By 	
Simplicial 	
, 	
Lemma 	
\ref{simplicial 	
- 	
lemma 	
- 	
augmentation 	
- 	
howto 	
} 	

 	
this 	
is 	
the 	
same 	
thing 	
as 	
a 	
map 	
$ 	
\mathcal{G 	
} 	
\to 	
a_*\mathcal{F}$. 	

 	
Thus 	
$ 	
a^{-1}$ 	
and 	
$ 	
a_*$ 	
are 	
adjoint 	
functors 	
and 	
we 	
obtain 	

 	
our 	
morphism 	
of 	
topoi 	
$ 	
a$\footnote{In 	
case 	
B 	
the 	
morphism 	
$ 	
a$ 	

 	
corresponds 	
to 	
the 	
cocontinuous 	
functor 	

 	
$ 	
\mathcal{C}_{total 	
} 	
\to 	
\mathcal{D}$ 	
sending 	

 	
$ 	
U$ 	
in 	
$ 	
\mathcal{C}_n$ 	
to 	
$ 	
u_n(U)$.}. 	
The 	
equalities 	

 	
$ 	
a 	
\circ 	
g_n 	
= 	
f_n$ 	
follow 	
immediately 	
from 	
the 	
definitions 	
. 	

 	
\end{proof 	
} 	




 	
\section{Morphisms 	
of 	
simplicial 	
sites 	
} 	

 	
\label{section 	
- 	
morphism 	
- 	
simplicial 	
- 	
sites 	
} 	


 	
\noindent 	

 	
We 	
continue 	
in 	
the 	
fashion 	
described 	
in 	

 	
Section 	
\ref{section 	
- 	
simplicial 	
- 	
sites 	
} 	

 	
working 	
out 	
the 	
meaning 	
of 	
morphisms 	
of 	
simplicial 	
sites 	

 	
in 	
cases 	
A 	
and 	
B 	
treated 	
in 	
that 	
section 	
. 	


 	
\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 	
} 	

 	
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 	TYPE
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 	
} 	
\\ 	

 	
\Sh(\mathcal{C}_{total 	
} 	
) 	
\ar[r]^{h_{total 	
} 	
} 	
& 	

 	
\Sh(\mathcal{C}'_{total 	
} 	
) 	

 	
} 	

 	
$ 	
$ 	

 	
Moreover 	
, 	
we 	
have 	
$ 	
( 	
g'_n)^{-1 	
} 	
\circ 	
h_{total 	
, 	
* 	
} 	
= 	
h_{n 	
, 	
* 	
} 	
\circ 	
g_n^{-1}$. 	

 	
\end{lemma 	
} 	


 	
\begin{proof 	
} 	

 	
Case 	
A. 	
Say 	
$ 	
h_n$ 	
corresponds 	
to 	
the 	
continuous 	
functor 	

 	
$ 	
v_n 	
: 	
\mathcal{C}'_n 	
\to 	
\mathcal{C}_n$. 	
Then 	
we 	
can 	
define 	

 	
a 	
functor 	
$ 	
v_{total 	
} 	
: 	
\mathcal{C}'_{total 	
} 	
\to 	
\mathcal{C}_{total}$ 	

 	
by 	
using 	
$ 	
v_n$ 	
in 	
degree 	
$ 	
n$. 	
This 	
is 	
clearly 	
a 	
continuous 	
functor 	

 	
( 	
see 	
definition 	
of 	
coverings 	
in 	
Lemma 	
\ref{lemma 	
- 	
simplicial 	
- 	
site 	
- 	
site 	
} 	
) 	
. 	

 	
Let 	

 	
$ 	
h_{total}^{-1 	
} 	
= 	
v_{total 	
, 	
s 	
} 	
: 	

 	
\Sh(\mathcal{C}'_{total 	
} 	
) 	
\to 	
\Sh(\mathcal{C}_{total})$ 	
and 	

 	
$ 	
h_{total 	
, 	
* 	
} 	
= 	
v_{total}^s 	
= 	
v_{total}^p 	
: 	

 	
\Sh(\mathcal{C}_{total 	
} 	
) 	
\to 	
\Sh(\mathcal{C}'_{total})$ 	

 	
be 	
the 	
adjoint 	
pair 	
of 	
functors 	
constructed 	
and 	
studied 	
in 	

 	
Sites 	
, 	
Sections 	
\ref{sites 	
- 	
section 	
- 	
continuous 	
- 	
functors 	
} 	
and 	

 	
\ref{sites 	
- 	
section 	
- 	
morphism 	
- 	
sites}. 	

 	
To 	
see 	
that 	
$ 	
h_{total}$ 	
is 	
a 	
morphism 	
of 	
topoi 	

 	
we 	
still 	
have 	
to 	
verify 	
that 	
$ 	
h_{total}^{-1}$ 	
is 	
exact 	
. 	

 	
We 	
first 	
observe 	
that 	

 	
$ 	
( 	
g'_n)^{-1 	
} 	
\circ 	
h_{total 	
, 	
* 	
} 	
= 	
h_{n 	
, 	
* 	
} 	
\circ 	
g_n^{-1}$ 	
; 	

 	
this 	
is 	
immediate 	
by 	
computing 	
sections 	
over 	
an 	
object 	
$ 	
U$ 	

 	
of 	
$ 	
\mathcal{C}'_n$. 	
Thus 	
, 	
if 	
we 	
think 	
of 	
a 	
sheaf 	
$ 	
\mathcal{F}$ 	

 	
on 	
$ 	
\mathcal{C}_{total}$ 	
as 	
a 	
system 	
$ 	
( 	
\mathcal{F}_n 	
, 	
\mathcal{F}(\varphi))$ 	

 	
as 	
in 	
Lemma 	
\ref{lemma 	
- 	
describe 	
- 	
sheaves 	
- 	
simplicial 	
- 	
site 	
- 	
site 	
} 	
, 	
then 	

 	
$ 	
h_{total 	
, 	
* 	
} 	
\mathcal{F}$ 	
corresponds 	
to 	

 	
the 	
system 	
$ 	
( 	
h_{n 	
, 	
* 	
} 	
\mathcal{F}_n 	
, 	
h_{n 	
, 	
* 	
} 	
\mathcal{F}(\varphi))$. 	

 	
Clearly 	
, 	
the 	
functor 	

 	
$ 	
( 	
\mathcal{F}'_n 	
, 	
\mathcal{F}'(\varphi 	
) 	
) 	
\to 	

 	
( 	
h_n^{-1}\mathcal{F}'_n 	
, 	
h_n^{-1}\mathcal{F}'(\varphi))$ 	

 	
is 	
its 	
left 	
adjoint 	
. 	
By 	
uniqueness 	
of 	
adjoints 	
, 	
we 	
conclude 	
that 	

 	
$ 	
h_{total}^{-1}$ 	
is 	
given 	
by 	
this 	
rule 	
on 	
systems 	
. 	
In 	
particular 	
, 	

 	
$ 	
h_{total}^{-1}$ 	
is 	
exact 	
( 	
by 	
the 	
description 	
of 	
sheaves 	
on 	

 	
$ 	
\mathcal{C}_{total}$ 	
given 	
in 	
the 	
lemma 	
and 	
the 	
exactness 	
of 	

 	
the 	
functors 	
$ 	
h_n^{-1}$ 	
) 	
and 	
we 	
have 	
our 	
morphism 	
of 	
topoi 	
. 	

 	
Finally 	
, 	
we 	
obtain 	
$ 	
g_n^{-1 	
} 	
\circ 	
h_{total}^{-1 	
} 	
= 	

 	
h_n^{-1 	
} 	
\circ 	
( 	
g'_n)^{-1}$ 	
as 	
well 	
, 	
which 	
proves 	
that 	
the 	

 	
displayed 	
diagram 	
of 	
the 	
lemma 	
commutes 	
. 	


 	
\medskip\noindent 	

 	
Case 	
B. 	
Here 	
we 	
have 	
a 	
functor 	

 	
$ 	
v_{total 	
} 	
: 	
\mathcal{C}_{total 	
} 	
\to 	
\mathcal{C}'_{total}$ 	

 	
by 	
using 	
$ 	
v_n$ 	
in 	
degree 	
$ 	
n$. 	
This 	
is 	
clearly 	
a 	
cocontinuous 	
functor 	

 	
( 	
see 	
definition 	
of 	
coverings 	
in 	
Lemma 	
\ref{lemma 	
- 	
simplicial 	
- 	
cocontinuous 	
- 	
site 	
} 	
) 	
. 	

 	
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 	
\ref{sites 	
- 	
lemma 	
- 	
composition 	
- 	
cocontinuous}. 	

 	
Taking 	
left 	
adjoints 	
the 	
final 	
equality 	
of 	
the 	
lemma 	
becomes 	

 	
$ 	
$ 	

 	
h_{total}^{-1 	
} 	
\circ 	
( 	
g'_n)^{Sh 	
} 	
_ 	
! 	
= 	
g^{Sh}_{n 	
! 	
} 	
\circ 	
h_n^{-1 	
} 	

 	
$ 	
$ 	

 	
This 	
follows 	
immediately 	
from 	
the 	
explicit 	
description 	
of 	
the 	
functors 	

 	
$ 	
( 	
g'_n)^{Sh}_!$ 	
and 	
$ 	
g^{Sh}_{n!}$ 	
in 	

 	
Lemma 	
\ref{lemma 	
- 	
restriction 	
- 	
to 	
- 	
components 	
- 	
site 	
} 	
, 	

 	
the 	
fact 	
that 	
$ 	
h_n^{-1 	
} 	
\circ 	
( 	
f'_\varphi)^{-1 	
} 	
= 	

 	
f_\varphi^{-1 	
} 	
\circ 	
h_m^{-1}$ 	
for 	
$ 	
\varphi 	
: 	
[ 	
m 	
] 	
\to 	
[ 	
n]$ 	
, 	
and 	

 	
the 	
fact 	
that 	
we 	
already 	
know 	
$ 	
h_{total}^{-1}$ 	
commutes 	

 	
with 	
restrictions 	
to 	
the 	
degree 	
$ 	
n$ 	
parts 	
of 	
the 	
simplicial 	
sites 	
. 	

 	
\end{proof 	
} 	


 	
\begin{lemma 	
} 	

 	
\label{lemma 	
- 	
direct 	
- 	
image 	
- 	
morphism 	
- 	
simplicial 	
- 	
sites 	
} 	

 	
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 	TYPE
- 	
injective 	
complex 	
on 	
$ 	
\mathcal{C}_{total}$ 	

 	
representing 	
$ 	
K$. 	
Then 	
$ 	
g_n^{-1}K$ 	
is 	
represented 	
by 	

 	
$ 	
g_n^{-1}\mathcal{I}^\bullet 	
= 	
\mathcal{I}_n^\bullet$ 	

 	
which 	
is 	
K 	
- 	
injective 	
by 	

 	
Lemma 	
\ref{lemma 	
- 	
restriction 	
- 	
injective 	
- 	
to 	
- 	
component 	
- 	
site}. 	

 	
We 	
have 	
$ 	
( 	
g'_n)^{-1}h_{total 	
, 	
* 	
} 	
\mathcal{I}^\bullet 	
= 	

 	
h_{n 	
, 	
* 	
} 	
g_n^{-1}\mathcal{I}_n^\bullet$ 	
by 	

 	
Lemma 	
\ref{lemma 	
- 	
morphism 	
- 	
simplicial 	
- 	
sites 	
} 	

 	
which 	
gives 	
the 	
desired 	
equality 	
. 	

 	
\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 	TYPE

 	
towards 	
a 	
site 	
$ 	
\mathcal{D}$ 	
, 	
resp.\ 	
$ 	
\mathcal{D}'$ 	

 	
as 	
in 	
Remark 	
\ref{remark 	
- 	
augmentation 	
- 	
site}. 	

 	
Let 	
$ 	
h$ 	
be 	
a 	
morphism 	TYPE
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 	
} 	

 	
\item[(A 	
) 	
] 	
$ 	
h_{-1}$ 	
comes 	
from 	
a 	
morphism 	
of 	
sites 	

 	
$ 	
h_{-1 	
} 	
: 	
\mathcal{D 	
} 	
\to 	
\mathcal{D}'$ 	

 	
such 	
that 	
$ 	
a'_0 	
\circ 	
h_0 	
= 	
h_{-1 	
} 	
\circ 	
a_0 	
$ 	

 	
as 	
morphisms 	
of 	
sites 	
. 	

 	
\item[(B 	
) 	
] 	
$ 	
h_{-1}$ 	
comes 	
from 	
a 	
cocontinuous 	
functor 	

 	
$ 	
v_{-1 	
} 	
: 	
\mathcal{D 	
} 	
\to 	
\mathcal{D}'$ 	

 	
such 	
that 	
$ 	
u'_0 	
\circ 	
v_0 	
= 	
v_{-1 	
} 	
\circ 	
u_0 	
$ 	

 	
as 	
functors 	
. 	

 	
\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}. 	

 	
The 	
morphisms 	
$ 	
a$ 	
and 	
$ 	
a'$ 	
are 	
defined 	
in 	
Lemma 	
\ref{lemma 	
- 	
augmentation 	
- 	
site}. 	

 	
Thus 	
the 	
only 	
thing 	
is 	
to 	
prove 	
the 	
commutativity 	
of 	
the 	
diagram 	
. 	

 	
To 	
do 	
this 	
, 	
we 	
prove 	
that 	

 	
$ 	
a^{-1 	
} 	
\circ 	
h_{-1}^{-1 	
} 	
= 	
h_{total}^{-1 	
} 	
\circ 	
( 	
a')^{-1}$. 	

 	
By 	
the 	
commutative 	
diagrams 	
of 	

 	
Lemma 	
\ref{lemma 	
- 	
morphism 	
- 	
simplicial 	
- 	
sites 	
} 	

 	
and 	
the 	
description 	
of 	
$ 	
\Sh(\mathcal{C}_{total})$ 	

 	
and 	
$ 	
\Sh(\mathcal{C}'_{total})$ 	
in 	
terms 	
of 	
components 	

 	
in 	
Lemma 	
\ref{lemma 	
- 	
describe 	
- 	
sheaves 	
- 	
simplicial 	
- 	
site 	
- 	
site 	
} 	
, 	

 	
it 	
suffices 	
to 	
show 	
that 	

 	
$ 	
$ 	

 	
\xymatrix 	
{ 	

 	
\Sh(\mathcal{C}_n 	
) 	
\ar[d]_{a_n 	
} 	
\ar[r]_{h_n 	
} 	
& 	

 	
\Sh(\mathcal{C}'_n 	
) 	
\ar[d]^{a'_n 	
} 	
\\ 	

 	
\Sh(\mathcal{D 	
} 	
) 	
\ar[r]^{h_{-1 	
} 	
} 	
& 	

 	
\Sh(\mathcal{D 	
} 	
' 	
) 	

 	
} 	

 	
$ 	
$ 	

 	
commutes 	
for 	
all 	
$ 	
n$. 	
This 	
follows 	
from 	
the 	
case 	
for 	
$ 	
n 	
= 	
0 	
$ 	

 	
( 	
which 	
is 	
an 	
assumption 	
in 	

 	
Remark 	
\ref{remark 	
- 	
morphism 	
- 	
augmentation 	
- 	
simplicial 	
- 	
sites 	
} 	
) 	

 	
and 	
for 	
$ 	
n 	
> 	
0 	
$ 	
we 	
pick 	
$ 	
\varphi 	
: 	
[ 	
0 	
] 	
\to 	
[ 	
n]$ 	

 	
and 	
then 	
the 	
required 	
commutativity 	
follows 	
from 	
the 	
case 	
$ 	
n 	
= 	
0 	
$ 	

 	
and 	
the 	
relations 	
$ 	
a_n 	
= 	
a_0 	
\circ 	
f_\varphi$ 	

 	
and 	
$ 	
a'_n 	
= 	
a'_0 	
\circ 	
f'_\varphi$ 	

 	
as 	
well 	
as 	
the 	
commutation 	
relations 	

 	
$ 	
f'_\varphi 	
\circ 	
h_n 	
= 	
h_0 	
\circ 	
f_\varphi$. 	

 	
\end{proof 	
} 	





 	
\section{Ringed 	
simplicial 	
sites 	
} 	

 	
\label{section 	
- 	
simplicial 	
- 	
sites 	
- 	
modules 	
} 	


 	
\noindent 	

 	
Let 	
us 	
endow 	
our 	
simplicial 	
topos 	
with 	
a 	
sheaf 	
of 	
rings 	
. 	


 	
\begin{lemma 	
} 	

 	
\label{lemma 	
- 	
restriction 	
- 	
module 	
- 	
to 	
- 	
components 	
- 	
site 	
} 	

 	
In 	
Situation 	
\ref{situation 	
- 	
simplicial 	
- 	
site}. 	
Let 	
$ 	
\mathcal{O}$ 	

 	
be 	
a 	
sheaf 	TYPE
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 	
. 	

 	
The 	
functor 	

 	
$ 	
g_n^ 	
* 	
: 	
\textit{Mod}(\mathcal{O 	
} 	
) 	
\to 	
\textit{Mod}(\mathcal{O}_n)$ 	

 	
has 	
a 	
left 	
adjoint 	
$ 	
g_{n!}$. 	

 	
For 	
$ 	
\mathcal{G}$ 	
in 	
$ 	
\textit{Mod}(\mathcal{O}_n)$-modules 	
the 	

 	
restriction 	
of 	
$ 	
g_{n!}\mathcal{G}$ 	
to 	
$ 	
\mathcal{C}_m$ 	
is 	

 	
$ 	
$ 	

 	
\bigoplus\nolimits_{\varphi 	
: 	
[ 	
n 	
] 	
\to 	
[ 	
m 	
] 	
} 	
f_\varphi^*\mathcal{G 	
} 	

 	
$ 	
$ 	

 	
where 	
$ 	
f_\varphi 	
: 	
( 	
\Sh(\mathcal{C}_m 	
) 	
, 	
\mathcal{O}_m 	
) 	
\to 	

 	
( 	
\Sh(\mathcal{C}_n 	
) 	
, 	
\mathcal{O}_n)$ 	
is 	
the 	
morphism 	
of 	
ringed 	
topoi 	

 	
agreeing 	
with 	
the 	
previously 	
defined 	
$ 	
f_\varphi$ 	
on 	
topoi 	
and 	

 	
using 	
the 	
map 	

 	
$ 	
\mathcal{O}(\varphi 	
) 	
: 	
f_\varphi^{-1}\mathcal{O}_n 	
\to 	
\mathcal{O}_m$ 	

 	
on 	
sheaves 	
of 	
rings 	
. 	

 	
\end{lemma 	
} 	


 	
\begin{proof 	
} 	

 	
By 	
Lemma 	
\ref{lemma 	
- 	
restriction 	
- 	
to 	
- 	
components 	
- 	
site 	
} 	
we 	
have 	

 	
$ 	
g_n^{-1}\mathcal{O 	
} 	
= 	
\mathcal{O}_n$ 	
and 	
hence 	
we 	
obtain 	
our 	

 	
morphism 	
of 	
ringed 	
topoi 	
. 	
By 	
Modules 	
on 	
Sites 	
, 	
Lemma 	

 	
\ref{sites 	
- 	
modules 	
- 	
lemma 	
- 	
lower 	
- 	
shriek 	
- 	
modules 	
} 	

 	
we 	
obtain 	
the 	
adjoint 	
$ 	
g_{n!}$. 	
To 	
prove 	
the 	
formula 	
for 	
$ 	
g_{n!}$ 	

 	
we 	
first 	
define 	
a 	
sheaf 	
of 	
$ 	
\mathcal{O}$-modules 	
$ 	
\mathcal{H}$ 	

 	
on 	
$ 	
\mathcal{C}_{total}$ 	
with 	
degree 	
$ 	
m$ 	
component 	

 	
the 	
$ 	
\mathcal{O}_m$-module 	

 	
$ 	
$ 	

 	
\mathcal{H}_m 	
= 	

 	
\bigoplus\nolimits_{\varphi 	
: 	
[ 	
n 	
] 	
\to 	
[ 	
m 	
] 	
} 	
f_\varphi^*\mathcal{G 	
} 	

 	
$ 	
$ 	

 	
Given 	
a 	
map 	
$ 	
\psi 	
: 	
[ 	
m 	
] 	
\to 	
[ 	
m']$ 	
the 	
map 	

 	
$ 	
\mathcal{H}(\psi 	
) 	
: 	
f_\psi^{-1}\mathcal{H}_m 	
\to 	
\mathcal{H}_{m'}$ 	

 	
is 	
given 	
on 	
components 	
by 	

 	
$ 	
$ 	

 	
f_\psi^{-1 	
} 	
f_\varphi^*\mathcal{G 	
} 	
\to 	

 	
f_\psi^ 	
* 	
f_\varphi^*\mathcal{G 	
} 	
\to 	

 	
f_{\psi 	
\circ 	
\varphi}^*\mathcal{G 	
} 	

 	
$ 	
$ 	

 	
Since 	
this 	
map 	
$ 	
f_\psi^{-1}\mathcal{H}_m 	
\to 	
\mathcal{H}_{m'}$ 	
is 	

 	
$ 	
\mathcal{O}(\psi 	
) 	
: 	
f_\psi^{-1}\mathcal{O}_m 	
\to 	
\mathcal{O}_{m'}$-semi 	
- 	
linear 	
, 	

 	
this 	
indeed 	
does 	
define 	
an 	
$ 	
\mathcal{O}$-module 	

 	
( 	
use 	
Lemma 	
\ref{lemma 	
- 	
describe 	
- 	
sheaves 	
- 	
simplicial 	
- 	
site 	
- 	
site 	
} 	
) 	
. 	

 	
Then 	
one 	
proves 	
directly 	
that 	

 	
$ 	
$ 	

 	
\Mor_{\mathcal{O}_n}(\mathcal{G 	
} 	
, 	
\mathcal{F}_n 	
) 	
= 	

 	
\Mor_{\mathcal{O}}(\mathcal{H 	
} 	
, 	
\mathcal{F 	
} 	
) 	

 	
$ 	
$ 	

 	
proceeding 	
as 	
in 	
the 	
proof 	
of 	
Lemma 	
\ref{lemma 	
- 	
restriction 	
- 	
to 	
- 	
components 	
- 	
site}. 	

 	
Thus 	
$ 	
\mathcal{H 	
} 	
= 	
g_{n!}\mathcal{G}$ 	
as 	
desired 	
. 	

 	
\end{proof 	
} 	


 	
\begin{lemma 	
} 	

 	
\label{lemma 	
- 	
restriction 	
- 	
injective 	
- 	
to 	
- 	
component 	
- 	
limp 	
} 	

 	
In 	
Situation 	
\ref{situation 	
- 	
simplicial 	
- 	
site}. 	

 	
Let 	
$ 	
\mathcal{O}$ 	
be 	
a 	
sheaf 	TYPE
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 	
} 	

 	
With 	
assumptions 	
as 	
in 	

 	
Lemma 	
\ref{lemma 	
- 	
restriction 	
- 	
module 	
- 	
to 	
- 	
components 	
- 	
site 	
} 	
the 	
functor 	

 	
$ 	
g_{n 	
! 	
} 	
: 	
\textit{Mod}(\mathcal{O}_n 	
) 	
\to 	
\textit{Mod}(\mathcal{O})$ 	

 	
is 	
exact 	
if 	
the 	
maps 	
$ 	
f_\varphi^{-1}\mathcal{O}_n 	
\to 	
\mathcal{O}_m$ 	

 	
are 	
flat 	
for 	
all 	
$ 	
\varphi 	
: 	
[ 	
n 	
] 	
\to 	
[ 	
m]$. 	

 	
\end{lemma 	
} 	


 	
\begin{proof 	
} 	

 	
Recall 	
that 	
$ 	
g_{n!}\mathcal{G}$ 	
is 	
the 	
$ 	
\mathcal{O}$-module 	

 	
whose 	
degree 	
$ 	
m$ 	
part 	
is 	
the 	
$ 	
\mathcal{O}_m$-module 	

 	
$ 	
$ 	

 	
\bigoplus\nolimits_{\varphi 	
: 	
[ 	
n 	
] 	
\to 	
[ 	
m 	
] 	
} 	
f_\varphi^*\mathcal{G 	
} 	

 	
$ 	
$ 	

 	
Here 	
the 	
morphism 	
of 	
ringed 	
topoi 	

 	
$ 	
f_\varphi 	
: 	
( 	
\Sh(\mathcal{C}_m 	
) 	
, 	
\mathcal{O}_m 	
) 	
\to 	

 	
( 	
\Sh(\mathcal{C}_n 	
) 	
, 	
\mathcal{O}_n)$ 	
uses 	
the 	
map 	

 	
$ 	
f_\varphi^{-1}\mathcal{O}_n 	
\to 	
\mathcal{O}_m$ 	
of 	
the 	

 	
statement 	
of 	
the 	
lemma 	
. 	
If 	
these 	
maps 	
are 	
flat 	
, 	
then 	

 	
$ 	
f_\varphi^*$ 	
is 	
exact 	

 	
( 	
Modules 	
on 	
Sites 	
, 	
Lemma 	
\ref{sites 	
- 	
modules 	
- 	
lemma 	
- 	
flat 	
- 	
pullback 	
- 	
exact 	
} 	
) 	
. 	

 	
By 	
definition 	
of 	
the 	
site 	
$ 	
\mathcal{C}_{total}$ 	
we 	
see 	
that 	
these 	

 	
functors 	
have 	
the 	
desired 	
exactness 	
properties 	
and 	
we 	
conclude 	
. 	

 	
\end{proof 	
} 	


 	
\begin{lemma 	
} 	

 	
\label{lemma 	
- 	
restriction 	
- 	
injective 	
- 	
to 	
- 	
component 	
- 	
site 	
- 	
module 	
} 	

 	
In 	
Situation 	
\ref{situation 	
- 	
simplicial 	
- 	
site}. 	

 	
Let 	
$ 	
\mathcal{O}$ 	
be 	
a 	
sheaf 	TYPE
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 	
} 	

 	
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 	TYPE
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 	
. 	

 	
\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 	TYPE
of 	
rings 	
on 	
$ 	
\mathcal{C}_{total}$ 	
and 	
$ 	
\mathcal{C}'_{total}$. 	

 	
Let 	
$ 	
( 	
h 	
, 	
h^\sharp)$ 	
be 	
a 	
morphism 	TYPE
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}. 	

 	
Moreover 	
, 	
we 	
have 	

 	
$ 	
( 	
g'_n)^ 	
* 	
\circ 	
h_{total 	
, 	
* 	
} 	
= 	
h_{n 	
, 	
* 	
} 	
\circ 	
g_n^*$ 	

 	
as 	
functor 	
$ 	
\textit{Mod}(\mathcal{O 	
} 	
) 	
\to 	
\textit{Mod}(\mathcal{O}'_n)$. 	

 	
\end{lemma 	
} 	


 	
\begin{proof 	
} 	

 	
Follows 	
from 	

 	
Lemma 	
\ref{lemma 	
- 	
morphism 	
- 	
simplicial 	
- 	
sites 	
} 	
and 	

 	
\ref{lemma 	
- 	
restriction 	
- 	
module 	
- 	
to 	
- 	
components 	
- 	
site 	
} 	

 	
by 	
keeping 	
track 	
of 	
the 	
sheaves 	
of 	
rings 	
. 	

 	
A 	
small 	
point 	
is 	
that 	
in 	
order 	
to 	
define 	
$ 	
h_n$ 	
as 	
a 	
morphism 	

 	
of 	
ringed 	
topoi 	
we 	
set 	

 	
$ 	
h_n^\sharp 	
= 	
g_n^{-1}h^\sharp 	
: 	

 	
g_n^{-1}h_{total}^{-1}\mathcal{O 	
} 	
' 	
\to 	
g_n^{-1}\mathcal{O}$ 	

 	
which 	
makes 	
sense 	
because 	

 	
$ 	
g_n^{-1}h_{total}^{-1}\mathcal{O 	
} 	
' 	
= 	
h_n^{-1}(g'_n)^{-1}\mathcal{O 	
} 	
' 	
= 	

 	
h_n^{-1}\mathcal{O}'_n$ 	
and 	
$ 	
g_n^{-1}\mathcal{O 	
} 	
= 	
\mathcal{O}_n$. 	

 	
Note 	
that 	
$ 	
g_n^*\mathcal{F 	
} 	
= 	
g_n^{-1}\mathcal{F}$ 	

 	
for 	
a 	
sheaf 	
of 	
$ 	
\mathcal{O}$-modules 	
$ 	
\mathcal{F}$ 	

 	
and 	
similarly 	
for 	
$ 	
g'_n$ 	
and 	
this 	
helps 	
explain 	
why 	

 	
$ 	
( 	
g'_n)^ 	
* 	
\circ 	
h_{total 	
, 	
* 	
} 	
= 	
h_{n 	
, 	
* 	
} 	
\circ 	
g_n^*$ 	

 	
follows 	
from 	
the 	
corresponding 	
statement 	
of 	

 	
Lemma 	
\ref{lemma 	
- 	
morphism 	
- 	
simplicial 	
- 	
sites}. 	

 	
\end{proof 	
} 	


 	
\begin{lemma 	
} 	

 	
\label{lemma 	
- 	
direct 	
- 	
image 	
- 	
morphism 	
- 	
simplicial 	
- 	
sites 	
- 	
modules 	
} 	

 	
With 	
notation 	
and 	
hypotheses 	
as 	
in 	

 	
Lemma 	
\ref{lemma 	
- 	
morphism 	
- 	
simplicial 	
- 	
sites 	
- 	
modules}. 	

 	
For 	
$ 	
K 	
\in 	
D(\mathcal{O})$ 	
we 	
have 	

 	
$ 	
( 	
g'_n)^*Rh_{total 	
, 	
* 	
} 	
K 	
= 	
Rh_{n 	
, 	
* 	
} 	
g_n^*K$. 	

 	
\end{lemma 	
} 	


 	
\begin{proof 	
} 	

 	
Recall 	
that 	
$ 	
g_n^ 	
* 	
= 	
g_n^{-1}$ 	
because 	
$ 	
g_n^{-1}\mathcal{O 	
} 	
= 	
\mathcal{O}_n$ 	

 	
by 	
the 	
construction 	
in 	
Lemma 	
\ref{lemma 	
- 	
restriction 	
- 	
module 	
- 	
to 	
- 	
components 	
- 	
site}. 	

 	
In 	
particular 	
$ 	
g_n^*$ 	
is 	
exact 	
and 	
$ 	
Lg_n^*$ 	
is 	
given 	
by 	
applying 	
$ 	
g_n^*$ 	

 	
to 	
any 	
representative 	
complex 	
of 	
modules 	
. 	
Similarly 	
for 	
$ 	
g'_n$. 	

 	
There 	
is 	
a 	
canonical 	
base 	
change 	
map 	

 	
$ 	
( 	
g'_n)^*Rh_{total 	
, 	
* 	
} 	
K 	
\to 	
Rh_{n 	
, 	
* 	
} 	
g_n^*K$ 	
, 	
see 	

 	
Cohomology 	
on 	
Sites 	
, 	
Remark 	
\ref{sites 	
- 	
cohomology 	
- 	
remark 	
- 	
base 	
- 	
change}. 	

 	
By 	
Cohomology 	
on 	
Sites 	
, 	
Lemma 	

 	
\ref{sites 	
- 	
cohomology 	
- 	
lemma 	
- 	
modules 	
- 	
abelian 	
- 	
unbounded 	
} 	

 	
the 	
image 	
of 	
this 	
in 	
$ 	
D(\mathcal{C}'_n)$ 	
is 	
the 	
map 	

 	
$ 	
( 	
g'_n)^{-1}Rh_{total 	
, 	
* 	
} 	
K_{ab 	
} 	
\to 	
Rh_{n 	
, 	
* 	
} 	
g_n^{-1}K_{ab}$ 	

 	
where 	
$ 	
K_{ab}$ 	
is 	
the 	
image 	
of 	
$ 	
K$ 	
in 	
$ 	
D(\mathcal{C}_{total})$. 	

 	
This 	
we 	
proved 	
to 	
be 	
an 	
isomorphism 	TYPE
in 	

 	
Lemma 	
\ref{lemma 	
- 	
direct 	
- 	
image 	
- 	
morphism 	
- 	
simplicial 	
- 	
sites 	
} 	

 	
and 	
the 	
result 	
follows 	
. 	

 	
\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 	
} 	
\\ 	

 	
& 	
\Sh(\mathcal{C}_{total 	
} 	
) 	

 	
} 	

 	
$ 	
$ 	

 	
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 	
} 	

 	
\label{lemma 	
- 	
simplicial 	
- 	
resolution 	
- 	
Z 	
- 	
site 	
} 	

 	
In 	
Situation 	
\ref{situation 	
- 	
simplicial 	
- 	
site 	
} 	
and 	
with 	
notation 	
as 	
above 	

 	
there 	
is 	
a 	
complex 	

 	
$ 	
$ 	

 	
\ldots 	
\to 	
g_{2!}\mathbf{Z 	
} 	
\to 	
g_{1!}\mathbf{Z 	
} 	
\to 	
g_{0!}\mathbf{Z 	
} 	

 	
$ 	
$ 	

 	
of 	
abelian 	
sheaves 	
on 	
$ 	
\mathcal{C}_{total}$ 	
which 	
forms 	
a 	
resolution 	
of 	

 	
the 	
constant 	
sheaf 	
with 	
value 	
$ 	
\mathbf{Z}$ 	
on 	
$ 	
\mathcal{C}_{total}$. 	

 	
\end{lemma 	
} 	


 	
\begin{proof 	
} 	

 	
We 	
will 	
use 	
the 	
description 	
of 	
the 	
functors 	
$ 	
g_{n!}$ 	
in 	

 	
Lemma 	
\ref{lemma 	
- 	
restriction 	
- 	
to 	
- 	
components 	
- 	
site 	
} 	
without 	
further 	
mention 	
. 	

 	
As 	
maps 	
of 	
the 	
complex 	
we 	
take 	
$ 	
\sum 	
( 	
-1)^i 	
d^n_i$ 	
where 	

 	
$ 	
d^n_i 	
: 	
g_{n!}\mathbf{Z 	
} 	
\to 	
g_{n 	
- 	
1!}\mathbf{Z}$ 	
is 	
the 	

 	
adjoint 	
to 	
the 	
map 	
$ 	
\mathbf{Z 	
} 	
\to 	

 	
\bigoplus_{[n 	
- 	
1 	
] 	
\to 	
[ 	
n 	
] 	
} 	
\mathbf{Z 	
} 	
= 	
g_n^{-1}g_{n 	
- 	
1!}\mathbf{Z}$ 	

 	
corresponding 	
to 	
the 	
factor 	
labeled 	
with 	
$ 	
\delta^n_i 	
: 	
[ 	
n 	
- 	
1 	
] 	
\to 	
[ 	
n]$. 	

 	
Then 	
$ 	
g_m^{-1}$ 	
applied 	
to 	
the 	
complex 	
gives 	
the 	
complex 	

 	
$ 	
$ 	

 	
\ldots 	
\to 	

 	
\bigoplus\nolimits_{\alpha 	
\in 	
\Mor_\Delta([2 	
] 	
, 	
[ 	
m 	
] 	
) 	
] 	
} 	
\mathbf{Z 	
} 	
\to 	

 	
\bigoplus\nolimits_{\alpha 	
\in 	
\Mor_\Delta([1 	
] 	
, 	
[ 	
m 	
] 	
) 	
] 	
} 	
\mathbf{Z 	
} 	
\to 	

 	
\bigoplus\nolimits_{\alpha 	
\in 	
\Mor_\Delta([0 	
] 	
, 	
[ 	
m 	
] 	
) 	
] 	
} 	
\mathbf{Z 	
} 	

 	
$ 	
$ 	

 	
on 	
$ 	
\mathcal{C}_m$. 	

 	
In 	
other 	
words 	
, 	
this 	
is 	
the 	
complex 	
associated 	
to 	
the 	

 	
free 	
abelian 	
sheaf 	
on 	
the 	
simplicial 	
set 	
$ 	
\Delta[m]$ 	
, 	
see 	

 	
Simplicial 	
, 	
Example 	
\ref{simplicial 	
- 	
example 	
- 	
simplex 	
- 	
simplicial 	
- 	
set}. 	

 	
Since 	
$ 	
\Delta[m]$ 	
is 	
homotopy 	
equivalent 	
to 	
$ 	
\Delta[0]$ 	
, 	
see 	

 	
Simplicial 	
, 	
Example 	
\ref{simplicial 	
- 	
example 	
- 	
simplex 	
- 	
contractible 	
} 	
, 	

 	
and 	
since 	
`` 	
taking 	
free 	
abelian 	
sheaf 	
on 	
'' 	
is 	
a 	
functor 	
, 	

 	
we 	
see 	
that 	
the 	
complex 	
above 	
is 	
homotopy 	
equivalent 	
to 	

 	
the 	
free 	
abelian 	
sheaf 	
on 	
$ 	
\Delta[0]$ 	

 	
( 	
Simplicial 	
, 	
Remark 	
\ref{simplicial 	
- 	
remark 	
- 	
homotopy 	
- 	
better 	
} 	
and 	

 	
Lemma 	
\ref{simplicial 	
- 	
lemma 	
- 	
homotopy 	
- 	
equivalence 	
- 	
s 	
- 	
N 	
} 	
) 	
. 	

 	
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 	TYPE

 	
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 	
} 	

 	
Observe 	
that 	

 	
$ 	
\Hom(\mathbf{Z 	
} 	
, 	
\mathcal{F 	
} 	
) 	
= 	
\Gamma(\mathcal{C}_{total 	
} 	
, 	
\mathcal{F})$ 	

 	
and 	

 	
$ 	
\Hom(g_{n!}\mathbf{Z 	
} 	
, 	
\mathcal{F 	
} 	
) 	
= 	
\Gamma(\mathcal{C}_n 	
, 	
\mathcal{F}_n)$. 	

 	
Hence 	
this 	
lemma 	
is 	
an 	
immediate 	
consequence 	
of 	

 	
Lemma 	
\ref{lemma 	
- 	
simplicial 	
- 	
resolution 	
- 	
Z 	
- 	
site 	
} 	

 	
and 	
the 	
fact 	
that 	
$ 	
\Hom(- 	
, 	
\mathcal{F})$ 	
is 	
exact 	
if 	

 	
$ 	
\mathcal{F}$ 	
is 	
injective 	
. 	

 	
\end{proof 	
} 	


 	
\begin{lemma 	
} 	

 	
\label{lemma 	
- 	
simplicial 	
- 	
sheaf 	
- 	
cohomology 	
- 	
site 	
} 	

 	
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 	TYPE
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 	
} 	
) 	
. 	

 	
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})$. 	

 	
Then 	
$ 	
H^p(U 	
, 	
\mathcal{F 	
} 	
) 	
= 	
H^p(U 	
, 	
g_n^{-1}\mathcal{F})$ 	

 	
where 	
on 	
the 	
left 	
hand 	
side 	
$ 	
U$ 	
is 	
viewed 	
as 	
an 	
object 	
of 	
$ 	
\mathcal{C}_{total}$. 	

 	
\end{lemma 	
} 	


 	
\begin{proof 	
} 	

 	
Observe 	
that 	
`` 	
$ 	
U$ 	
viewed 	
as 	
object 	
of 	
$ 	
\mathcal{C}_{total}$ 	
'' 	

 	
is 	
explained 	
by 	
the 	
construction 	
of 	
$ 	
\mathcal{C}_{total}$ 	
in 	

 	
Lemma 	
\ref{lemma 	
- 	
simplicial 	
- 	
site 	
- 	
site 	
} 	
in 	
case 	
( 	
A 	
) 	
and 	

 	
Lemma 	
\ref{lemma 	
- 	
simplicial 	
- 	
cocontinuous 	
- 	
site 	
} 	
in 	
case 	
( 	
B 	
) 	
. 	

 	
The 	
equality 	
then 	
follows 	
from 	

 	
Lemma 	
\ref{lemma 	
- 	
restriction 	
- 	
injective 	
- 	
to 	
- 	
component 	
- 	
site 	
} 	

 	
and 	
the 	
definition 	
of 	
cohomology 	
. 	

 	
\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 	TYPE
towards 	
a 	
site 	
$ 	
\mathcal{D}$ 	
as 	
in 	

 	
Remark 	
\ref{remark 	
- 	
augmentation 	
- 	
site}. 	

 	
By 	
Lemma 	
\ref{lemma 	
- 	
augmentation 	
- 	
site 	
} 	
we 	
obtain 	
a 	
morphism 	
of 	
topoi 	

 	
$ 	
$ 	

 	
a 	
: 	
\Sh(\mathcal{C}_{total 	
} 	
) 	
\longrightarrow 	
\Sh(\mathcal{D 	
} 	
) 	

 	
$ 	
$ 	

 	
and 	
morphisms 	
of 	
topoi 	

 	
$ 	
g_n 	
: 	
\Sh(\mathcal{C}_n 	
) 	
\to 	
\Sh(\mathcal{C}_{total})$ 	

 	
as 	
in 	
Lemma 	
\ref{lemma 	
- 	
restriction 	
- 	
to 	
- 	
components 	
- 	
site}. 	

 	
The 	
compositions 	
$ 	
a 	
\circ 	
g_n$ 	
are 	
denoted 	

 	
$ 	
a_n 	
: 	
\Sh(\mathcal{C}_n 	
) 	
\to 	
\Sh(\mathcal{D})$. 	

 	
Furthermore 	
, 	
the 	
simplicial 	
structure 	
gives 	

 	
morphisms 	
of 	
topoi 	

 	
$ 	
f_\varphi 	
: 	
\Sh(\mathcal{C}_n 	
) 	
\to 	
\Sh(\mathcal{C}_m)$ 	

 	
such 	
that 	
$ 	
a_n 	
\circ 	
f_\varphi 	
= 	
a_m$ 	
for 	
all 	
$ 	
\varphi 	
: 	
[ 	
m 	
] 	
\to 	
[ 	
n]$. 	


 	
\begin{lemma 	
} 	

 	
\label{lemma 	
- 	
simplicial 	
- 	
resolution 	
- 	
augmentation 	
} 	

 	
In 	
Situation 	
\ref{situation 	
- 	
simplicial 	
- 	
site 	
} 	
let 	

 	
$ 	
a_0 	
$ 	
be 	
an 	
augmentation 	TYPE
towards 	
a 	
site 	
$ 	
\mathcal{D}$ 	

 	
as 	
in 	
Remark 	
\ref{remark 	
- 	
augmentation 	
- 	
site}. 	

 	
For 	
any 	
abelian 	
sheaf 	
$ 	
\mathcal{G}$ 	
on 	
$ 	
\mathcal{D}$ 	

 	
there 	
is 	
an 	
exact 	
complex 	

 	
$ 	
$ 	

 	
\ldots 	
\to 	

 	
g_{2!}(a_2^{-1}\mathcal{G 	
} 	
) 	
\to 	

 	
g_{1!}(a_1^{-1}\mathcal{G 	
} 	
) 	
\to 	

 	
g_{0!}(a_0^{-1}\mathcal{G 	
} 	
) 	
\to 	

 	
a^{-1}\mathcal{G 	
} 	
\to 	
0 	

 	
$ 	
$ 	

 	
of 	
abelian 	
sheaves 	
on 	
$ 	
\mathcal{C}_{total}$. 	

 	
\end{lemma 	
} 	


 	
\begin{proof 	
} 	

 	
We 	
encourage 	
the 	
reader 	
to 	
read 	
the 	
proof 	
of 	

 	
Lemma 	
\ref{lemma 	
- 	
simplicial 	
- 	
resolution 	
- 	
Z 	
- 	
site 	
} 	
first 	
. 	

 	
We 	
will 	
use 	
Lemma 	
\ref{lemma 	
- 	
augmentation 	
- 	
site 	
} 	
and 	

 	
the 	
description 	
of 	
the 	
functors 	
$ 	
g_{n!}$ 	
in 	

 	
Lemma 	
\ref{lemma 	
- 	
restriction 	
- 	
to 	
- 	
components 	
- 	
site 	
} 	
without 	
further 	
mention 	
. 	

 	
In 	
particular 	
$ 	
g_{n!}(a_n^{-1}\mathcal{G})$ 	
is 	
the 	

 	
sheaf 	
on 	
$ 	
\mathcal{C}_{total}$ 	
whose 	
restriction 	
to 	
$ 	
\mathcal{C}_m$ 	

 	
is 	
the 	
sheaf 	

 	
$ 	
$ 	

 	
\bigoplus\nolimits_{\varphi 	
: 	
[ 	
n 	
] 	
\to 	
[ 	
m 	
] 	
} 	
f_\varphi^{-1}a_n^{-1}\mathcal{G 	
} 	
= 	

 	
\bigoplus\nolimits_{\varphi 	
: 	
[ 	
n 	
] 	
\to 	
[ 	
m 	
] 	
} 	
a_m^{-1}\mathcal{G 	
} 	

 	
$ 	
$ 	

 	
As 	
maps 	
of 	
the 	
complex 	
we 	
take 	
$ 	
\sum 	
( 	
-1)^i 	
d^n_i$ 	
where 	

 	
$ 	
d^n_i 	
: 	
g_{n!}(a_n^{-1}\mathcal{G 	
} 	
) 	
\to 	
g_{n 	
- 	
1!}(a_{n 	
- 	
1}^{-1}\mathcal{G})$ 	

 	
is 	
the 	
adjoint 	
to 	
the 	
map 	

 	
$ 	
a_n^{-1}\mathcal{G 	
} 	
\to 	
\bigoplus_{[n 	
- 	
1 	
] 	
\to 	
[ 	
n 	
] 	
} 	
a_n^{-1}\mathcal{G 	
} 	
= 	

 	
g_n^{-1}g_{n 	
- 	
1!}(a_{n 	
- 	
1}^{-1}\mathcal{G})$ 	

 	
corresponding 	
to 	
the 	
factor 	
labeled 	
with 	
$ 	
\delta^n_i 	
: 	
[ 	
n 	
- 	
1 	
] 	
\to 	
[ 	
n]$. 	

 	
The 	
map 	
$ 	
g_{0!}(a_0^{-1}\mathcal{G 	
} 	
) 	
\to 	
a^{-1}\mathcal{G}$ 	
is 	
adjoint 	

 	
to 	
the 	
identity 	
map 	
of 	
$ 	
a_0^{-1}\mathcal{G}$. 	

 	
Then 	
$ 	
g_m^{-1}$ 	
applied 	
to 	
the 	
chain 	
complex 	
in 	
degrees 	

 	
$ 	
\ldots 	
, 	
2 	
, 	
1 	
, 	
0 	
$ 	
gives 	
the 	
complex 	

 	
$ 	
$ 	

 	
\ldots 	
\to 	

 	
\bigoplus\nolimits_{\alpha 	
\in 	
\Mor_\Delta([2 	
] 	
, 	
[ 	
m 	
] 	
) 	
] 	
} 	
a_m^{-1}\mathcal{G 	
} 	
\to 	

 	
\bigoplus\nolimits_{\alpha 	
\in 	
\Mor_\Delta([1 	
] 	
, 	
[ 	
m 	
] 	
) 	
] 	
} 	
a_m^{-1}\mathcal{G 	
} 	
\to 	

 	
\bigoplus\nolimits_{\alpha 	
\in 	
\Mor_\Delta([0 	
] 	
, 	
[ 	
m 	
] 	
) 	
] 	
} 	
a_m^{-1}\mathcal{G 	
} 	

 	
$ 	
$ 	

 	
on 	
$ 	
\mathcal{C}_m$. 	
This 	
is 	
equal 	
to 	
$ 	
a_m^{-1}\mathcal{G}$ 	

 	
tensored 	
over 	
the 	
constant 	
sheaf 	
$ 	
\mathbf{Z}$ 	
with 	
the 	
complex 	

 	
$ 	
$ 	

 	
\ldots 	
\to 	

 	
\bigoplus\nolimits_{\alpha 	
\in 	
\Mor_\Delta([2 	
] 	
, 	
[ 	
m 	
] 	
) 	
] 	
} 	
\mathbf{Z 	
} 	
\to 	

 	
\bigoplus\nolimits_{\alpha 	
\in 	
\Mor_\Delta([1 	
] 	
, 	
[ 	
m 	
] 	
) 	
] 	
} 	
\mathbf{Z 	
} 	
\to 	

 	
\bigoplus\nolimits_{\alpha 	
\in 	
\Mor_\Delta([0 	
] 	
, 	
[ 	
m 	
] 	
) 	
] 	
} 	
\mathbf{Z 	
} 	

 	
$ 	
$ 	

 	
discussed 	
in 	
the 	
proof 	
of 	
Lemma 	
\ref{lemma 	
- 	
simplicial 	
- 	
resolution 	
- 	
Z 	
- 	
site}. 	

 	
There 	
we 	
have 	
seen 	
that 	
this 	
complex 	
is 	
homotopy 	
equivalent 	
to 	

 	
$ 	
\mathbf{Z}$ 	
placed 	
in 	
degree 	
$ 	
0 	
$ 	
which 	
finishes 	
the 	
proof 	
. 	

 	
\end{proof 	
} 	


 	
\begin{lemma 	
} 	

 	
\label{lemma 	
- 	
augmentation 	
- 	
cech 	
- 	
complex 	
} 	

 	
In 	
Situation 	
\ref{situation 	
- 	
simplicial 	
- 	
site 	
} 	
let 	

 	
$ 	
a_0 	
$ 	
be 	
an 	
augmentation 	TYPE
towards 	
a 	
site 	
$ 	
\mathcal{D}$ 	

 	
as 	
in 	
Remark 	
\ref{remark 	
- 	
augmentation 	
- 	
site}. 	

 	
For 	
an 	
abelian 	
sheaf 	
$ 	
\mathcal{F}$ 	
on 	
$ 	
\mathcal{C}_{total}$ 	

 	
there 	
is 	
a 	
canonical 	
complex 	

 	
$ 	
$ 	

 	
0 	
\to 	
a_*\mathcal{F 	
} 	
\to 	
a_{0 	
, 	
* 	
} 	
\mathcal{F}_0 	
\to 	
a_{1 	
, 	
* 	
} 	
\mathcal{F}_1 	
\to 	

 	
a_{2 	
, 	
* 	
} 	
\mathcal{F}_2 	
\to 	
\ldots 	

 	
$ 	
$ 	

 	
on 	
$ 	
\mathcal{D}$ 	
which 	
is 	
exact 	
in 	
degrees 	
$ 	
-1 	
, 	
0 	
$ 	
and 	

 	
exact 	
everywhere 	
if 	
$ 	
\mathcal{F}$ 	
is 	
injective 	
. 	

 	
\end{lemma 	
} 	


 	
\begin{proof 	
} 	

 	
To 	
construct 	
the 	
complex 	
, 	
by 	
the 	
Yoneda 	
lemma 	
, 	
it 	
suffices 	
for 	
any 	

 	
abelian 	
sheaf 	
$ 	
\mathcal{G}$ 	
on 	
$ 	
\mathcal{D}$ 	
to 	
construct 	
a 	
complex 	

 	
$ 	
$ 	

 	
0 	
\to 	
\Hom(\mathcal{G 	
} 	
, 	
a_*\mathcal{F 	
} 	
) 	
\to 	

 	
\Hom(\mathcal{G 	
} 	
, 	
a_{0 	
, 	
* 	
} 	
\mathcal{F}_0 	
) 	
\to 	

 	
\Hom(\mathcal{G 	
} 	
, 	
a_{1 	
, 	
* 	
} 	
\mathcal{F}_1 	
) 	
\to 	
\ldots 	

 	
$ 	
$ 	

 	
functorially 	
in 	
$ 	
\mathcal{G}$. 	
To 	
do 	
this 	
apply 	
$ 	
\Hom(- 	
, 	
\mathcal{F})$ 	

 	
to 	
the 	
exact 	
complex 	
of 	
Lemma 	
\ref{lemma 	
- 	
simplicial 	
- 	
resolution 	
- 	
augmentation 	
} 	

 	
and 	
use 	
adjointness 	
of 	
pullback 	
and 	
pushforward 	
. 	

 	
The 	
exactness 	
properties 	
in 	
degrees 	
$ 	
-1 	
, 	
0 	
$ 	
follow 	
from 	

 	
the 	
construction 	
as 	
$ 	
\Hom(- 	
, 	
\mathcal{F})$ 	
is 	
left 	
exact 	
. 	

 	
If 	
$ 	
\mathcal{F}$ 	
is 	
an 	
injective 	
abelian 	
sheaf 	
, 	
then 	
the 	

 	
complex 	
is 	
exact 	
because 	
$ 	
\Hom(- 	
, 	
\mathcal{F})$ 	
is 	
exact 	
. 	

 	
\end{proof 	
} 	


 	
\begin{lemma 	
} 	

 	
\label{lemma 	
- 	
augmentation 	
- 	
spectral 	
- 	
sequence 	
} 	

 	
In 	
Situation 	
\ref{situation 	
- 	
simplicial 	
- 	
site 	
} 	
let 	

 	
$ 	
a_0 	
$ 	
be 	
an 	
augmentation 	TYPE
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 	TYPE
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 	
$ 	
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 	
} 	
) 	
. 	

 	
Hence 	
the 	
cohomology 	
of 	
the 	
columns 	
computes 	
$ 	
R^qa_{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 	
} 	






 	
\section{Cohomology 	
on 	
ringed 	
simplicial 	
sites 	
} 	

 	
\label{section 	
- 	
cohomology 	
- 	
simplicial 	
- 	
sites 	
- 	
modules 	
} 	


 	
\noindent 	

 	
This 	
section 	
is 	
the 	
analogue 	
of 	

 	
Section 	
\ref{section 	
- 	
cohomology 	
- 	
simplicial 	
- 	
sites 	
} 	

 	
for 	
sheaves 	
of 	
modules 	
. 	


 	
\medskip\noindent 	

 	
In 	
Situation 	
\ref{situation 	
- 	
simplicial 	
- 	
site 	
} 	
let 	
$ 	
\mathcal{O}$ 	

 	
be 	
a 	
sheaf 	TYPE
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 	
} 	

 	
\label{lemma 	
- 	
simplicial 	
- 	
resolution 	
- 	
ringed 	
} 	

 	
In 	
Situation 	
\ref{situation 	
- 	
simplicial 	
- 	
site 	
} 	
let 	
$ 	
\mathcal{O}$ 	

 	
be 	
a 	
sheaf 	TYPE
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]$. 	

 	
Then 	
$ 	
g_m^{-1}$ 	
applied 	
to 	
the 	
complex 	
gives 	
the 	
complex 	

 	
$ 	
$ 	

 	
\ldots 	
\to 	

 	
\bigoplus\nolimits_{\alpha 	
\in 	
\Mor_\Delta([2 	
] 	
, 	
[ 	
m 	
] 	
) 	
] 	
} 	
\mathcal{O}_m 	
\to 	

 	
\bigoplus\nolimits_{\alpha 	
\in 	
\Mor_\Delta([1 	
] 	
, 	
[ 	
m 	
] 	
) 	
] 	
} 	
\mathcal{O}_m 	
\to 	

 	
\bigoplus\nolimits_{\alpha 	
\in 	
\Mor_\Delta([0 	
] 	
, 	
[ 	
m 	
] 	
) 	
] 	
} 	
\mathcal{O}_m 	

 	
$ 	
$ 	

 	
on 	
$ 	
\mathcal{C}_m$. 	

 	
In 	
other 	
words 	
, 	
this 	
is 	
the 	
complex 	
associated 	
to 	
the 	

 	
free 	
$ 	
\mathcal{O}_m$-module 	
on 	
the 	
simplicial 	
set 	
$ 	
\Delta[m]$ 	
, 	
see 	

 	
Simplicial 	
, 	
Example 	
\ref{simplicial 	
- 	
example 	
- 	
simplex 	
- 	
simplicial 	
- 	
set}. 	

 	
Since 	
$ 	
\Delta[m]$ 	
is 	
homotopy 	
equivalent 	
to 	
$ 	
\Delta[0]$ 	
, 	
see 	

 	
Simplicial 	
, 	
Example 	
\ref{simplicial 	
- 	
example 	
- 	
simplex 	
- 	
contractible 	
} 	
, 	

 	
and 	
since 	
`` 	
taking 	
free 	
abelian 	
sheaf 	
on 	
'' 	
is 	
a 	
functor 	
, 	

 	
we 	
see 	
that 	
the 	
complex 	
above 	
is 	
homotopy 	
equivalent 	
to 	

 	
the 	
free 	
abelian 	
sheaf 	
on 	
$ 	
\Delta[0]$ 	

 	
( 	
Simplicial 	
, 	
Remark 	
\ref{simplicial 	
- 	
remark 	
- 	
homotopy 	
- 	
better 	
} 	
and 	

 	
Lemma 	
\ref{simplicial 	
- 	
lemma 	
- 	
homotopy 	
- 	
equivalence 	
- 	
s 	
- 	
N 	
} 	
) 	
. 	

 	
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 	TYPE
of 	
rings 	
. 	
Let 	
$ 	
\mathcal{F}$ 	
be 	
a 	

 	TYPE
sheaf 	
of 	
$ 	
\mathcal{O}$-modules 	
. 	
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 	
an 	
injective 	
$ 	
\mathcal{O}$-module 	
. 	

 	
\end{lemma 	
} 	


 	
\begin{proof 	
} 	

 	
Observe 	
that 	

 	
$ 	
\Hom(\mathcal{O 	
} 	
, 	
\mathcal{F 	
} 	
) 	
= 	
\Gamma(\mathcal{C}_{total 	
} 	
, 	
\mathcal{F})$ 	

 	
and 	

 	
$ 	
\Hom(g_{n!}\mathcal{O}_n 	
, 	
\mathcal{F 	
} 	
) 	
= 	
\Gamma(\mathcal{C}_n 	
, 	
\mathcal{F}_n)$. 	

 	
Hence 	
this 	
lemma 	
is 	
an 	
immediate 	
consequence 	
of 	

 	
Lemma 	
\ref{lemma 	
- 	
simplicial 	
- 	
resolution 	
- 	
ringed 	
} 	

 	
and 	
the 	
fact 	
that 	
$ 	
\Hom(- 	
, 	
\mathcal{F})$ 	
is 	
exact 	
if 	

 	
$ 	
\mathcal{F}$ 	
is 	
injective 	
. 	

 	
\end{proof 	
} 	


 	
\begin{lemma 	
} 	

 	
\label{lemma 	
- 	
simplicial 	
- 	
module 	
- 	
cohomology 	
- 	
site 	
} 	

 	
In 	
Situation 	
\ref{situation 	
- 	
simplicial 	
- 	
site 	
} 	
let 	
$ 	
\mathcal{O}$ 	

 	
be 	
a 	
sheaf 	TYPE
of 	
rings 	
. 	
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 	TYPE
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$. 	
Observe 	
that 	

 	
$ 	
\Gamma(\mathcal{D 	
} 	
, 	
- 	
) 	
= 	

 	
\Hom_{\mathcal{O}_\mathcal{D}}(\mathcal{O}_\mathcal{D 	
} 	
, 	
-)$ 	

 	
on 	
$ 	
\textit{Mod}(\mathcal{O}_\mathcal{D})$. 	
Hence 	
the 	
lemma 	

 	
says 	
rows 	
of 	
the 	
double 	
complex 	
are 	
exact 	

 	
in 	
positive 	
degrees 	
and 	
evaluate 	
to 	

 	
$ 	
\Gamma(\mathcal{C}_{total 	
} 	
, 	
\mathcal{I}^q)$ 	
in 	
degree 	
$ 	
0$. 	

 	
Thus 	
the 	
total 	
complex 	
associated 	
to 	
the 	
double 	
complex 	

 	
computes 	
$ 	
R\Gamma(\mathcal{C}_{total 	
} 	
, 	
K)$ 	
by 	

 	
Homology 	
, 	
Lemma 	
\ref{homology 	
- 	
lemma 	
- 	
double 	
- 	
complex 	
- 	
gives 	
- 	
resolution}. 	

 	
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 	

 	
are 	
limp 	
on 	
$ 	
\mathcal{C}_p$ 	

 	
( 	
Lemma 	
\ref{lemma 	
- 	
restriction 	
- 	
injective 	
- 	
to 	
- 	
component 	
- 	
limp 	
} 	
) 	
. 	

 	
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 	TYPE
of 	
rings 	
. 	
Let 	
$ 	
U 	
\in 	
\Ob(\mathcal{C}_n)$. 	
Let 	

 	
$ 	
\mathcal{F 	
} 	
\in 	
\textit{Mod}(\mathcal{O})$. 	

 	
Then 	
$ 	
H^p(U 	
, 	
\mathcal{F 	
} 	
) 	
= 	
H^p(U 	
, 	
g_n^*\mathcal{F})$ 	

 	
where 	
on 	
the 	
left 	
hand 	
side 	
$ 	
U$ 	
is 	
viewed 	
as 	
an 	
object 	
of 	

 	
$ 	
\mathcal{C}_{total}$. 	

 	
\end{lemma 	
} 	


 	
\begin{proof 	
} 	

 	
Observe 	
that 	
`` 	
$ 	
U$ 	
viewed 	
as 	
object 	
of 	
$ 	
\mathcal{C}_{total}$ 	
'' 	

 	
is 	
explained 	
by 	
the 	
construction 	
of 	
$ 	
\mathcal{C}_{total}$ 	
in 	

 	
Lemma 	
\ref{lemma 	
- 	
simplicial 	
- 	
site 	
- 	
site 	
} 	
in 	
case 	
( 	
A 	
) 	
and 	

 	
Lemma 	
\ref{lemma 	
- 	
simplicial 	
- 	
cocontinuous 	
- 	
site 	
} 	
in 	
case 	
( 	
B 	
) 	
. 	

 	
In 	
both 	
cases 	
the 	
functor 	
$ 	
\mathcal{C}_n 	
\to 	
\mathcal{C}$ 	

 	
is 	
continuous 	
and 	
cocontinuous 	
, 	
see 	

 	
Lemma 	
\ref{lemma 	
- 	
restriction 	
- 	
to 	
- 	
components 	
- 	
site 	
} 	
, 	
and 	

 	
$ 	
g_n^{-1}\mathcal{O 	
} 	
= 	
\mathcal{O}_n$ 	
by 	
definition 	
. 	

 	
Hence 	
the 	
result 	
is 	
a 	
special 	
case 	
of 	

 	
Cohomology 	
on 	
Sites 	
, 	
Lemma 	

 	
\ref{sites 	
- 	
cohomology 	
- 	
lemma 	
- 	
pullback 	
- 	
same 	
- 	
cohomology}. 	

 	
\end{proof 	
} 	







 	
\section{Cohomology 	
and 	
augmentations 	
of 	
ringed 	
simplicial 	
sites 	
} 	

 	
\label{section 	
- 	
cohomology 	
- 	
augmentation 	
- 	
ringed 	
- 	
simplicial 	
- 	
sites 	
} 	


 	
\noindent 	

 	
This 	
section 	
is 	
the 	
analogue 	
of 	

 	
Section 	
\ref{section 	
- 	
cohomology 	
- 	
augmentation 	
- 	
simplicial 	
- 	
sites 	
} 	

 	
for 	
sheaves 	
of 	
modules 	
. 	


 	
\medskip\noindent 	

 	
Consider 	
a 	
simplicial 	
site 	
$ 	
\mathcal{C}$ 	
as 	
in 	

 	
Situation 	
\ref{situation 	
- 	
simplicial 	
- 	
site}. 	

 	
Let 	
$ 	
a_0 	
$ 	
be 	
an 	
augmentation 	TYPE
towards 	
a 	
site 	
$ 	
\mathcal{D}$ 	
as 	
in 	

 	
Remark 	
\ref{remark 	
- 	
augmentation 	
- 	
site}. 	

 	
Let 	
$ 	
\mathcal{O}$ 	
be 	
a 	
sheaf 	TYPE
of 	
rings 	
on 	
$ 	
\mathcal{C}_{total}$. 	

 	
Let 	
$ 	
\mathcal{O}_\mathcal{D}$ 	
be 	
a 	
sheaf 	TYPE
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}. 	

 	
Consequently 	
, 	
we 	
obtain 	
a 	
morphism 	
of 	
ringed 	
topoi 	

 	
$ 	
$ 	

 	
a 	
: 	

 	
( 	
\Sh(\mathcal{C}_{total 	
} 	
) 	
, 	
\mathcal{O 	
} 	
) 	

 	
\longrightarrow 	

 	
( 	
\Sh(\mathcal{D 	
} 	
) 	
, 	
\mathcal{O}_\mathcal{D 	
} 	
) 	

 	
$ 	
$ 	

 	
We 	
will 	
think 	
of 	
$ 	
g_n 	
: 	
( 	
\Sh(\mathcal{C}_n 	
) 	
, 	
\mathcal{O}_n 	
) 	
\to 	

 	
( 	
\Sh(\mathcal{C}_{total 	
} 	
) 	
, 	
\mathcal{O})$ 	
as 	
a 	
morphism 	
of 	
ringed 	
topoi 	

 	
as 	
in 	

 	
Lemma 	
\ref{lemma 	
- 	
restriction 	
- 	
module 	
- 	
to 	
- 	
components 	
- 	
site 	
} 	
, 	
then 	

 	
taking 	
the 	
composition 	
$ 	
a_n 	
= 	
a 	
\circ 	
g_n$ 	

 	
( 	
Lemma 	
\ref{lemma 	
- 	
augmentation 	
- 	
site 	
} 	
) 	

 	
as 	
morphisms 	
of 	
ringed 	
topoi 	
we 	
obtain 	

 	
$ 	
$ 	

 	
a_n 	
: 	

 	
( 	
\Sh(\mathcal{C}_n 	
) 	
, 	
\mathcal{O}_n 	
) 	

 	
\longrightarrow 	

 	
( 	
\Sh(\mathcal{D 	
} 	
) 	
, 	
\mathcal{O}_\mathcal{D 	
} 	
) 	

 	
$ 	
$ 	

 	
Using 	
the 	
transition 	
maps 	
$ 	
f_\varphi^{-1}\mathcal{O}_m 	
\to 	
\mathcal{O}_n$ 	

 	
we 	
obtain 	
morphisms 	
of 	
ringed 	
topoi 	

 	
$ 	
$ 	

 	
f_\varphi 	
: 	
( 	
\Sh(\mathcal{C}_n 	
) 	
, 	
\mathcal{O}_n 	
) 	
\to 	

 	
( 	
\Sh(\mathcal{C}_m 	
) 	
, 	
\mathcal{O}_m 	
) 	

 	
$ 	
$ 	

 	
such 	
that 	
$ 	
a_n 	
\circ 	
f_\varphi 	
= 	
a_m$ 	
as 	
morphisms 	
of 	

 	
ringed 	
topoi 	
for 	
all 	
$ 	
\varphi 	
: 	
[ 	
m 	
] 	
\to 	
[ 	
n]$. 	


 	
\begin{lemma 	
} 	

 	
\label{lemma 	
- 	
flat 	
- 	
augmentation 	
- 	
modules 	
} 	

 	
With 	
notation 	
as 	
above 	
. 	
The 	
morphism 	

 	
$ 	
a 	
: 	
( 	
\Sh(\mathcal{C}_{total 	
} 	
) 	
, 	
\mathcal{O 	
} 	
) 	
\to 	

 	
( 	
\Sh(\mathcal{D 	
} 	
) 	
, 	
\mathcal{O}_\mathcal{D})$ 	

 	
is 	
flat 	
if 	
and 	
only 	
if 	

 	
$ 	
a_n 	
: 	
( 	
\Sh(\mathcal{C}_n 	
) 	
, 	
\mathcal{O}_n 	
) 	
\to 	

 	
( 	
\Sh(\mathcal{D 	
} 	
) 	
, 	
\mathcal{O}_\mathcal{D})$ 	

 	
is 	
flat 	
for 	
$ 	
n 	
\geq 	
0$. 	

 	
\end{lemma 	
} 	


 	
\begin{proof 	
} 	

 	
Since 	
$ 	
g_n 	
: 	
( 	
\Sh(\mathcal{C}_n 	
) 	
, 	
\mathcal{O}_n 	
) 	
\to 	

 	
( 	
\Sh(\mathcal{C}_{total 	
} 	
) 	
, 	
\mathcal{O})$ 	
is 	
flat 	
, 	
we 	
see 	

 	
that 	
if 	
$ 	
a$ 	
is 	
flat 	
, 	
then 	
$ 	
a_n 	
= 	
a 	
\circ 	
g_n$ 	
is 	
flat 	
as 	

 	
a 	
composition 	
. 	
Conversely 	
, 	
suppose 	
that 	
$ 	
a_n$ 	
is 	
flat 	
for 	
all 	
$ 	
n$. 	

 	
We 	
have 	
to 	
check 	
that 	
$ 	
\mathcal{O}$ 	
is 	
flat 	
as 	
a 	
sheaf 	
of 	

 	
$ 	
a^{-1}\mathcal{O}_\mathcal{D}$-modules 	
. 	
Let 	
$ 	
\mathcal{F 	
} 	
\to 	
\mathcal{G}$ 	

 	
be 	
an 	
injective 	TYPE
map 	
of 	
$ 	
a^{-1}\mathcal{O}_\mathcal{D}$-modules 	
. 	

 	
We 	
have 	
to 	
show 	
that 	

 	
$ 	
$ 	

 	
\mathcal{F 	
} 	
\otimes_{a^{-1}\mathcal{O}_\mathcal{D 	
} 	
} 	
\mathcal{O 	
} 	

 	
\to 	

 	
\mathcal{G 	
} 	
\otimes_{a^{-1}\mathcal{O}_\mathcal{D 	
} 	
} 	
\mathcal{O 	
} 	

 	
$ 	
$ 	

 	
is 	
injective 	
. 	
We 	
can 	
check 	
this 	
on 	
$ 	
\mathcal{C}_n$ 	
, 	
i.e. 	
, 	
after 	

 	
applying 	
$ 	
g_n^{-1}$. 	
Since 	
$ 	
g_n^ 	
* 	
= 	
g_n^{-1}$ 	
because 	

 	
$ 	
g_n^{-1}\mathcal{O 	
} 	
= 	
\mathcal{O}_n$ 	
we 	
obtain 	

 	
$ 	
$ 	

 	
g_n^{-1}\mathcal{F 	
} 	
\otimes_{g_n^{-1}a^{-1}\mathcal{O}_\mathcal{D 	
} 	
} 	

 	
\mathcal{O}_n 	

 	
\to 	

 	
g_n^{-1}\mathcal{G 	
} 	
\otimes_{g_n^{-1}a^{-1}\mathcal{O}_\mathcal{D 	
} 	
} 	

 	
\mathcal{O}_n 	

 	
$ 	
$ 	

 	
which 	
is 	
injective 	
because 	

 	
$ 	
g_n^{-1}a^{-1}\mathcal{O}_\mathcal{D 	
} 	
= 	
a_n^{-1}\mathcal{O}_\mathcal{D}$ 	

 	
and 	
we 	
assume 	
$ 	
a_n$ 	
was 	
flat 	
. 	

 	
\end{proof 	
} 	


 	
\begin{lemma 	
} 	

 	
\label{lemma 	
- 	
simplicial 	
- 	
resolution 	
- 	
augmentation 	
- 	
modules 	
} 	

 	
With 	
notation 	
as 	
above 	
. 	
For 	
a 	
$ 	
\mathcal{O}_\mathcal{D}$-module 	
$ 	
\mathcal{G}$ 	

 	
there 	
is 	
an 	
exact 	
complex 	

 	
$ 	
$ 	

 	
\ldots 	
\to 	

 	
g_{2!}(a_2^*\mathcal{G 	
} 	
) 	
\to 	

 	
g_{1!}(a_1^*\mathcal{G 	
} 	
) 	
\to 	

 	
g_{0!}(a_0^*\mathcal{G 	
} 	
) 	
\to 	

 	
a^*\mathcal{G 	
} 	
\to 	
0 	

 	
$ 	
$ 	

 	
of 	
sheaves 	
of 	
$ 	
\mathcal{O}$-modules 	
on 	
$ 	
\mathcal{C}_{total}$. 	

 	
Here 	
$ 	
g_{n!}$ 	
is 	
as 	
in 	
Lemma 	
\ref{lemma 	
- 	
restriction 	
- 	
module 	
- 	
to 	
- 	
components 	
- 	
site}. 	

 	
\end{lemma 	
} 	


 	
\begin{proof 	
} 	

 	
Observe 	
that 	
$ 	
a^*\mathcal{G}$ 	
is 	
the 	
$ 	
\mathcal{O}$-module 	
on 	

 	
$ 	
\mathcal{C}_{total}$ 	
whose 	
restriction 	
to 	
$ 	
\mathcal{C}_m$ 	

 	
is 	
the 	
$ 	
\mathcal{O}_m$-module 	
$ 	
a_m^*\mathcal{G}$. 	

 	
The 	
description 	
of 	
the 	
functors 	
$ 	
g_{n!}$ 	
on 	
modules 	

 	
in 	
Lemma 	
\ref{lemma 	
- 	
restriction 	
- 	
module 	
- 	
to 	
- 	
components 	
- 	
site 	
} 	

 	
shows 	
that 	
$ 	
g_{n!}(a_n^*\mathcal{G})$ 	
is 	
the 	

 	
$ 	
\mathcal{O}$-module 	
on 	
$ 	
\mathcal{C}_{total}$ 	

 	
whose 	
restriction 	
to 	
$ 	
\mathcal{C}_m$ 	
is 	
the 	
$ 	
\mathcal{O}_m$-module 	

 	
$ 	
$ 	

 	
\bigoplus\nolimits_{\varphi 	
: 	
[ 	
n 	
] 	
\to 	
[ 	
m 	
] 	
} 	
f_\varphi^*a_n^*\mathcal{G 	
} 	
= 	

 	
\bigoplus\nolimits_{\varphi 	
: 	
[ 	
n 	
] 	
\to 	
[ 	
m 	
] 	
} 	
a_m^*\mathcal{G 	
} 	

 	
$ 	
$ 	

 	
The 	
rest 	
of 	
the 	
proof 	
is 	
exactly 	
the 	
same 	
as 	
the 	
proof 	
of 	

 	
Lemma 	
\ref{lemma 	
- 	
simplicial 	
- 	
resolution 	
- 	
augmentation 	
} 	
, 	

 	
replacing 	
$ 	
a_m^{-1}\mathcal{G}$ 	
by 	
$ 	
a_m^*\mathcal{G}$. 	

 	
\end{proof 	
} 	


 	
\begin{lemma 	
} 	

 	
\label{lemma 	
- 	
augmentation 	
- 	
cech 	
- 	
complex 	
- 	
modules 	
} 	

 	
With 	
notation 	
as 	
above 	
. 	

 	
For 	
an 	
$ 	
\mathcal{O}$-module 	
$ 	
\mathcal{F}$ 	
on 	
$ 	
\mathcal{C}_{total}$ 	

 	
there 	
is 	
a 	
canonical 	
complex 	

 	
$ 	
$ 	

 	
0 	
\to 	
a_*\mathcal{F 	
} 	
\to 	
a_{0 	
, 	
* 	
} 	
\mathcal{F}_0 	
\to 	
a_{1 	
, 	
* 	
} 	
\mathcal{F}_1 	
\to 	

 	
a_{2 	
, 	
* 	
} 	
\mathcal{F}_2 	
\to 	
\ldots 	

 	
$ 	
$ 	

 	
of 	
$ 	
\mathcal{O}_\mathcal{D}$-modules 	
which 	
is 	
exact 	
in 	
degrees 	
$ 	
-1 	
, 	
0$. 	

 	
If 	
$ 	
\mathcal{F}$ 	
is 	
an 	
injective 	
$ 	
\mathcal{O}$-module 	
, 	
then 	
the 	
complex 	

 	
is 	
exact 	
in 	
all 	
degrees 	
and 	
remains 	
exact 	
on 	
applying 	
the 	
functor 	

 	
$ 	
\Hom_{\mathcal{O}_\mathcal{D}}(\mathcal{G 	
} 	
, 	
-)$ 	
for 	
any 	

 	
$ 	
\mathcal{O}_\mathcal{D}$-module 	
$ 	
\mathcal{G}$. 	

 	
\end{lemma 	
} 	


 	
\begin{proof 	
} 	

 	
To 	
construct 	
the 	
complex 	
, 	
by 	
the 	
Yoneda 	
lemma 	
, 	
it 	
suffices 	
for 	
any 	

 	
$ 	
\mathcal{O}_\mathcal{D}$-modules 	
$ 	
\mathcal{G}$ 	
on 	
$ 	
\mathcal{D}$ 	

 	
to 	
construct 	
a 	
complex 	

 	
$ 	
$ 	

 	
0 	
\to 	
\Hom_{\mathcal{O}_\mathcal{D}}(\mathcal{G 	
} 	
, 	
a_*\mathcal{F 	
} 	
) 	
\to 	

 	
\Hom_{\mathcal{O}_\mathcal{D}}(\mathcal{G 	
} 	
, 	
a_{0 	
, 	
* 	
} 	
\mathcal{F}_0 	
) 	
\to 	

 	
\Hom_{\mathcal{O}_\mathcal{D}}(\mathcal{G 	
} 	
, 	
a_{1 	
, 	
* 	
} 	
\mathcal{F}_1 	
) 	
\to 	
\ldots 	

 	
$ 	
$ 	

 	
functorially 	
in 	
$ 	
\mathcal{G}$. 	
To 	
do 	
this 	
apply 	

 	
$ 	
\Hom_\mathcal{O}(- 	
, 	
\mathcal{F})$ 	

 	
to 	
the 	
exact 	
complex 	
of 	

 	
Lemma 	
\ref{lemma 	
- 	
simplicial 	
- 	
resolution 	
- 	
augmentation 	
- 	
modules 	
} 	

 	
and 	
use 	
adjointness 	
of 	
pullback 	
and 	
pushforward 	
. 	

 	
The 	
exactness 	
properties 	
in 	
degrees 	
$ 	
-1 	
, 	
0 	
$ 	
follow 	
from 	

 	
the 	
construction 	
as 	
$ 	
\Hom_\mathcal{O}(- 	
, 	
\mathcal{F})$ 	
is 	
left 	
exact 	
. 	

 	
If 	
$ 	
\mathcal{F}$ 	
is 	
an 	
injective 	
$ 	
\mathcal{O}$-module 	
, 	
then 	
the 	

 	
complex 	
is 	
exact 	
because 	
$ 	
\Hom_\mathcal{O}(- 	
, 	
\mathcal{F})$ 	
is 	
exact 	
. 	

 	
\end{proof 	
} 	


 	
\begin{lemma 	
} 	

 	
\label{lemma 	
- 	
augmentation 	
- 	
spectral 	
- 	
sequence 	
- 	
modules 	
} 	

 	
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 	TYPE
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$. 	

 	
Thus 	
the 	
total 	
complex 	
associated 	
to 	
the 	
double 	
complex 	

 	
computes 	
$ 	
Ra_*K$ 	
by 	

 	
Homology 	
, 	
Lemma 	
\ref{homology 	
- 	
lemma 	
- 	
double 	
- 	
complex 	
- 	
gives 	
- 	
resolution}. 	

 	
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 	

 	
are 	
limp 	
on 	
$ 	
\mathcal{C}_p$ 	

 	
( 	
Lemma 	
\ref{lemma 	
- 	
restriction 	
- 	
injective 	
- 	
to 	
- 	
component 	
- 	
limp 	
} 	
) 	
. 	

 	
Hence 	
the 	
cohomology 	
of 	
the 	
columns 	
are 	
the 	
sheaves 	

 	
$ 	
R^qa_{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 	
} 	






 	
\section{Cartesian 	
sheaves 	
and 	
modules 	
} 	

 	
\label{section 	
- 	
cartesian 	
} 	


 	
\noindent 	

 	
Here 	
is 	
the 	
definition 	
. 	


 	
\begin{definition 	
} 	

 	
\label{definition 	
- 	
cartesian 	
- 	
sheaf 	
} 	

 	
In 	
Situation 	
\ref{situation 	
- 	
simplicial 	
- 	
site}. 	

 	
\begin{enumerate 	
} 	

 	
\item 	
A 	
sheaf 	
$ 	
\mathcal{F}$ 	
of 	
sets 	
or 	
of 	
abelian 	
groups 	
on 	

 	
$ 	
\mathcal{C}$ 	
is 	
{ 	
\it 	
cartesian 	
} 	
if 	
the 	
maps 	

 	
$ 	
\mathcal{F}(\varphi 	
) 	
: 	
f_\varphi^{-1}\mathcal{F}_m 	
\to 	
\mathcal{F}_n$ 	

 	
are 	
isomorphisms 	
for 	
all 	
$ 	
\varphi 	
: 	
[ 	
m 	
] 	
\to 	
[ 	
n]$. 	

 	
\item 	
If 	
$ 	
\mathcal{O}$ 	
is 	
a 	
sheaf 	
of 	
rings 	
on 	
$ 	
\mathcal{C}_{total}$ 	
, 	

 	
then 	
a 	
sheaf 	
$ 	
\mathcal{F}$ 	
of 	
$ 	
\mathcal{O}$-modules 	
is 	

 	
{ 	
\it 	
cartesian 	
} 	
if 	
  	
the 	
maps 	
$ 	
f_\varphi^*\mathcal{F}_m 	
\to 	
\mathcal{F}_n$ 	

 	
are 	
isomorphisms 	
for 	
all 	
$ 	
\varphi 	
: 	
[ 	
m 	
] 	
\to 	
[ 	
n]$. 	

 	
\item 	
An 	
object 	
$ 	
K$ 	
of 	
$ 	
D(\mathcal{C}_{total})$ 	
is 	
{ 	
\it 	
cartesian 	
} 	
if 	
the 	
maps 	

 	
$ 	
f_\varphi^{-1}K_m 	
\to 	
K_n$ 	

 	
are 	
isomorphisms 	
for 	
all 	
$ 	
\varphi 	
: 	
[ 	
m 	
] 	
\to 	
[ 	
n]$. 	

 	
\item 	
If 	
$ 	
\mathcal{O}$ 	
is 	
a 	
sheaf 	
of 	
rings 	
on 	
$ 	
\mathcal{C}_{total}$ 	
, 	
then 	

 	
an 	
object 	
$ 	
K$ 	
of 	
$ 	
D(\mathcal{O})$ 	
is 	
{ 	
\it 	
cartesian 	
} 	
if 	
the 	
maps 	

 	
$ 	
Lf_\varphi^*K_m 	
\to 	
K_n$ 	

 	
are 	
isomorphisms 	
for 	
all 	
$ 	
\varphi 	
: 	
[ 	
m 	
] 	
\to 	
[ 	
n]$. 	

 	
\end{enumerate 	
} 	

 	
\end{definition 	
} 	


 	
\noindent 	

 	
Of 	
course 	
there 	
is 	
a 	
general 	
notion 	
of 	
a 	
cartesian 	
section 	
of 	
a 	

 	
fibred 	
category 	
and 	
the 	
above 	
are 	
merely 	
examples 	
of 	
this 	
. 	

 	
The 	
property 	
on 	
pullbacks 	
needs 	
only 	
be 	
checked 	
for 	
the 	
degeneracies 	
. 	


 	
\begin{lemma 	
} 	

 	
\label{lemma 	
- 	
check 	
- 	
cartesian 	
- 	
module 	
} 	

 	
In 	
Situation 	
\ref{situation 	
- 	
simplicial 	
- 	
site}. 	

 	
\begin{enumerate 	
} 	

 	
\item 	
A 	
sheaf 	
$ 	
\mathcal{F}$ 	
of 	
sets 	
or 	
abelian 	
groups 	
is 	
cartesian 	

 	
if 	
and 	
only 	
if 	
the 	
maps 	

 	
$ 	
( 	
f_{\delta^n_j})^{-1}\mathcal{F}_{n 	
- 	
1 	
} 	
\to 	
\mathcal{F}_n$ 	

 	
are 	
isomorphisms 	
. 	

 	
\item 	
An 	
object 	
$ 	
K$ 	
of 	
$ 	
D(\mathcal{C}_{total})$ 	
is 	
cartesian 	

 	
if 	
and 	
only 	
if 	
the 	
maps 	

 	
$ 	
( 	
f_{\delta^n_j})^{-1}K_{n 	
- 	
1 	
} 	
\to 	
K_n$ 	

 	
are 	
isomorphisms 	
. 	

 	
\item 	
If 	
$ 	
\mathcal{O}$ 	
is 	
a 	
sheaf 	
of 	
rings 	
on 	
$ 	
\mathcal{C}_{total}$ 	

 	
a 	
sheaf 	
$ 	
\mathcal{F}$ 	
of 	
$ 	
\mathcal{O}$-modules 	
is 	
cartesian 	

 	
if 	
and 	
only 	
if 	
the 	
maps 	

 	
$ 	
( 	
f_{\delta^n_j})^*\mathcal{F}_{n 	
- 	
1 	
} 	
\to 	
\mathcal{F}_n$ 	

 	
are 	
isomorphisms 	
. 	

 	
\item 	
If 	
$ 	
\mathcal{O}$ 	
is 	
a 	
sheaf 	
of 	
rings 	
on 	
$ 	
\mathcal{C}_{total}$ 	

 	
an 	
object 	
$ 	
K$ 	
of 	
$ 	
D(\mathcal{O})$ 	
is 	
cartesian 	

 	
if 	
and 	
only 	
if 	
the 	
maps 	

 	
$ 	
L(f_{\delta^n_j})^*K_{n 	
- 	
1 	
} 	
\to 	
K_n$ 	

 	
are 	
isomorphisms 	
. 	

 	
\item 	
Add 	
more 	
here 	
. 	

 	
\end{enumerate 	
} 	

 	
\end{lemma 	
} 	


 	
\begin{proof 	
} 	

 	
In 	
each 	
case 	
the 	
key 	
is 	
that 	
the 	
pullback 	
functors 	

 	
compose 	
to 	
pullback 	
functor 	
; 	
for 	
part 	
( 	
4 	
) 	
see 	

 	
Cohomology 	
on 	
Sites 	
, 	
Lemma 	

 	
\ref{sites 	
- 	
cohomology 	
- 	
lemma 	
- 	
derived 	
- 	
pullback 	
- 	
composition}. 	

 	
We 	
show 	
how 	
the 	
argument 	
works 	
in 	
case 	
( 	
1 	
) 	
and 	
omit 	
the 	
proof 	

 	
in 	
the 	
other 	
cases 	
. 	

 	
The 	
category 	
$ 	
\Delta$ 	
is 	
generated 	
by 	
the 	
morphisms 	

 	
the 	
morphisms 	
$ 	
\delta^n_j$ 	
and 	
$ 	
\sigma^n_j$ 	
, 	
see 	

 	
Simplicial 	
, 	
Lemma 	
\ref{simplicial 	
- 	
lemma 	
- 	
face 	
- 	
degeneracy}. 	

 	
Hence 	
we 	
only 	
need 	
to 	
check 	
the 	
maps 	

 	
$ 	
( 	
f_{\delta^n_j})^{-1}\mathcal{F}_{n 	
- 	
1 	
} 	
\to 	
\mathcal{F}_n$ 	

 	
and 	
$ 	
( 	
f_{\sigma^n_j})^{-1}\mathcal{F}_{n 	
+ 	
1 	
} 	
\to 	
\mathcal{F}_n$ 	
are 	

 	
isomorphisms 	
, 	
see 	

 	
Simplicial 	
, 	
Lemma 	
\ref{simplicial 	
- 	
lemma 	
- 	
characterize 	
- 	
simplicial 	
- 	
object 	
} 	

 	
for 	
notation 	
. 	
Since 	
$ 	
\sigma^n_j 	
\circ 	
\delta_j^{n 	
+ 	
1 	
} 	
= 	
\text{id}_{[n]}$ 	

 	
the 	
composition 	

 	
$ 	
$ 	

 	
\mathcal{F}_n 	
= 	

 	
( 	
f_{\sigma^n_j})^{-1 	
} 	

 	
( 	
f_{\delta_j^{n 	
+ 	
1}})^{-1 	
} 	

 	
\mathcal{F}_n 	
\to 	

 	
( 	
f_{\sigma^n_j})^{-1 	
} 	

 	
\mathcal{F}_{n 	
+ 	
1 	
} 	
\to 	

 	
\mathcal{F}_n 	

 	
$ 	
$ 	

 	
is 	
the 	
identity 	
. 	
Thus 	
the 	
result 	
for 	
$ 	
\delta^{n 	
+ 	
1}_j$ 	
implies 	
the 	
result 	

 	
for 	
$ 	
\sigma^n_j$. 	

 	
\end{proof 	
} 	


 	
\begin{lemma 	
} 	

 	
\label{lemma 	
- 	
augmentation 	
- 	
cartesian 	
- 	
module 	
} 	

 	
In 	
Situation 	
\ref{situation 	
- 	
simplicial 	
- 	
site 	
} 	
let 	

 	
$ 	
a_0 	
$ 	
be 	
an 	
augmentation 	TYPE
towards 	
a 	
site 	
$ 	
\mathcal{D}$ 	
as 	
in 	

 	
Remark 	
\ref{remark 	
- 	
augmentation 	
- 	
site}. 	

 	
\begin{enumerate 	
} 	

 	
\item 	
The 	
pullback 	
$ 	
a^{-1}\mathcal{G}$ 	
of 	
a 	
sheaf 	
of 	
sets 	
or 	
abelian 	
groups 	

 	
on 	
$ 	
\mathcal{D}$ 	
is 	
cartesian 	
. 	

 	
\item 	
The 	
pullback 	
$ 	
a^{-1}K$ 	
of 	
an 	
object 	
$ 	
K$ 	
of 	
$ 	
D(\mathcal{D})$ 	

 	
is 	
cartesian 	
. 	

 	
\end{enumerate 	
} 	

 	
Let 	
$ 	
\mathcal{O}$ 	
be 	
a 	
sheaf 	TYPE
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 	
. 	

 	
\item[(4 	
) 	
] 	
The 	
derived 	
pullback 	
$ 	
La^*K$ 	
of 	
an 	
object 	

 	
$ 	
K$ 	
of 	
$ 	
D(\mathcal{O}_\mathcal{D})$ 	
is 	
cartesian 	
. 	

 	
\end{enumerate 	
} 	

 	
\end{lemma 	
} 	


 	
\begin{proof 	
} 	

 	
This 	
follows 	
immediately 	
from 	
the 	
identities 	

 	
$ 	
a_m 	
\circ 	
f_\varphi 	
= 	
a_n$ 	
for 	
all 	
$ 	
\varphi 	
: 	
[ 	
m 	
] 	
\to 	
[ 	
n]$. 	

 	
See 	
Lemma 	
\ref{lemma 	
- 	
augmentation 	
- 	
site 	
} 	
and 	
the 	
discussion 	
in 	

 	
Section 	
\ref{section 	
- 	
cohomology 	
- 	
augmentation 	
- 	
ringed 	
- 	
simplicial 	
- 	
sites}. 	

 	
\end{proof 	
} 	


 	
\begin{lemma 	
} 	

 	
\label{lemma 	
- 	
characterize 	
- 	
cartesian 	
} 	

 	
In 	
Situation 	
\ref{situation 	
- 	
simplicial 	
- 	
site}. 	

 	
The 	
category 	
of 	
cartesian 	
sheaves 	
of 	
sets 	
( 	
resp.\ 	
abelian 	
groups 	
) 	

 	
is 	
equivalent 	
to 	
the 	
category 	
of 	
pairs 	
$ 	
( 	
\mathcal{F 	
} 	
, 	
\alpha)$ 	

 	
where 	
$ 	
\mathcal{F}$ 	
is 	
a 	
a 	
sheaf 	
of 	
sets 	
( 	
resp.\ 	
abelian 	
groups 	
) 	

 	
on 	
$ 	
\mathcal{C}_0 	
$ 	
and 	

 	
$ 	
$ 	

 	
\alpha 	
: 	

 	
( 	
f_{\delta_1 	
^ 	
1})^{-1}\mathcal{F 	
} 	

 	
\longrightarrow 	
( 	
f_{\delta_0 	
^ 	
1})^{-1}\mathcal{F 	
} 	

 	
$ 	
$ 	

 	
is 	
an 	
isomorphism 	
of 	
sheaves 	
of 	
sets 	
( 	
resp.\ 	
abelian 	
groups 	
) 	

 	
on 	
$ 	
\mathcal{C}_1 	
$ 	
such 	
that 	

 	
$ 	
( 	
f_{\delta^2_1})^{-1}\alpha 	
= 	

 	
( 	
f_{\delta^2_0})^{-1}\alpha 	
\circ 	
( 	
f_{\delta^2_2})^{-1}\alpha$ 	

 	
as 	
maps 	
of 	
sheaves 	
on 	
$ 	
\mathcal{C}_2$. 	

 	
\end{lemma 	
} 	


 	
\begin{proof 	
} 	

 	
We 	
abbreviate 	

 	
$ 	
d^n_j 	
= 	
f_{\delta^n_j 	
} 	
: 	
\Sh(\mathcal{C}_n 	
) 	
\to 	
\Sh(\mathcal{C}_{n 	
- 	
1})$. 	

 	
The 	
condition 	
on 	
$ 	
\alpha$ 	
in 	
the 	
statement 	
of 	
the 	
lemma 	
makes 	
sense 	
because 	

 	
$ 	
$ 	

 	
d^1_1 	
\circ 	
d^2_2 	
= 	
d^1_1 	
\circ 	
d^2_1 	
, 	
\quad 	

 	
d^1_1 	
\circ 	
d^2_0 	
= 	
d^1_0 	
\circ 	
d^2_2 	
, 	
\quad 	

 	
d^1_0 	
\circ 	
d^2_0 	
= 	
d^1_0 	
\circ 	
d^2_1 	

 	
$ 	
$ 	

 	
as 	
morphisms 	
of 	
topoi 	
$ 	
\Sh(\mathcal{C}_2 	
) 	
\to 	
\Sh(\mathcal{C}_0)$ 	
, 	
see 	

 	
Simplicial 	
, 	
Remark 	
\ref{simplicial 	
- 	
remark 	
- 	
relations}. 	
Hence 	
we 	

 	
can 	
picture 	
these 	
maps 	
as 	
follows 	

 	
$ 	
$ 	

 	
\xymatrix 	
{ 	

 	
& 	
( 	
d^2_0)^{-1}(d^1_1)^{-1}\mathcal{F 	
} 	
\ar[r]_-{(d^2_0)^{-1}\alpha 	
} 	
& 	

 	
( 	
d^2_0)^{-1}(d^1_0)^{-1}\mathcal{F 	
} 	
\ar@{=}[rd 	
] 	
& 	
\\ 	

 	
( 	
d^2_2)^{-1}(d^1_0)^{-1}\mathcal{F 	
} 	
\ar@{=}[ru 	
] 	
& 	
& 	
& 	

 	
( 	
d^2_1)^{-1}(d^1_0)^{-1}\mathcal{F 	
} 	
\\ 	

 	
& 	
( 	
d^2_2)^{-1}(d^1_1)^{-1}\mathcal{F 	
} 	
\ar[lu]^{(d^2_2)^{-1}\alpha 	
} 	
\ar@{=}[r 	
] 	
& 	

 	
( 	
d^2_1)^{-1}(d^1_1)^{-1}\mathcal{F 	
} 	
\ar[ru]_{(d^2_1)^{-1}\alpha 	
} 	

 	
} 	

 	
$ 	
$ 	

 	
and 	
the 	
condition 	
signifies 	
the 	
diagram 	
is 	
commutative 	
. 	
It 	
is 	
clear 	
that 	

 	
given 	
a 	
cartesian 	
sheaf 	
$ 	
\mathcal{G}$ 	
of 	
sets 	
( 	
resp.\ 	
abelian 	
groups 	
) 	

 	
on 	
$ 	
\mathcal{C}_{total}$ 	

 	
we 	
can 	
set 	
$ 	
\mathcal{F 	
} 	
= 	
\mathcal{G}_0 	
$ 	
and 	
$ 	
\alpha$ 	
equal 	
to 	
the 	
composition 	

 	
$ 	
$ 	

 	
( 	
d_1 	
^ 	
1)^{-1}\mathcal{G}_0 	
\to 	
\mathcal{G}_1 	

 	
\leftarrow 	
( 	
d_1 	
^ 	
0)^{-1}\mathcal{G}_0 	

 	
$ 	
$ 	

 	
where 	
the 	
arrows 	
are 	
invertible 	
as 	
$ 	
\mathcal{G}$ 	
is 	
cartesian 	
. 	

 	
To 	
prove 	
this 	
functor 	

 	
is 	
an 	
equivalence 	
we 	
construct 	
a 	
quasi 	
- 	
inverse 	
. 	
The 	
construction 	
of 	

 	
the 	
quasi 	
- 	
inverse 	
is 	
analogous 	
to 	
the 	
construction 	
discussed 	
in 	

 	
Descent 	
, 	
Section 	
\ref{descent 	
- 	
section 	
- 	
descent 	
- 	
modules 	
} 	
from 	
which 	
we 	
borrow 	

 	
the 	
notation 	
$ 	
\tau^n_i 	
: 	
[ 	
0 	
] 	
\to 	
[ 	
n]$ 	
, 	
$ 	
0 	
\mapsto 	
i$ 	
and 	

 	
$ 	
\tau^n_{ij 	
} 	
: 	
[ 	
1 	
] 	
\to 	
[ 	
n]$ 	
, 	
$ 	
0 	
\mapsto 	
i$ 	
, 	
$ 	
1 	
\mapsto 	
j$. 	

 	
Namely 	
, 	
given 	
a 	
pair 	
$ 	
( 	
\mathcal{F 	
} 	
, 	
\alpha)$ 	

 	
as 	
in 	
the 	
lemma 	
we 	
set 	
$ 	
\mathcal{G}_n 	
= 	
( 	
f_{\tau^n_n})^{-1}\mathcal{F}$. 	

 	
Given 	
$ 	
\varphi 	
: 	
[ 	
n 	
] 	
\to 	
[ 	
m]$ 	
we 	
define 	

 	
$ 	
\mathcal{G}(\varphi 	
) 	
: 	
( 	
f_\varphi)^{-1}\mathcal{G}_n 	
\to 	
\mathcal{G}_m$ 	

 	
using 	

 	
$ 	
$ 	

 	
\xymatrix 	
{ 	

 	
( 	
f_\varphi)^{-1}\mathcal{G}_n 	
\ar@{=}[r 	
] 	
& 	

 	
( 	
f_\varphi)^{-1}(f_{\tau^n_n})^{-1}\mathcal{F 	
} 	
\ar@{=}[r 	
] 	
& 	

 	
( 	
f_{\tau^m_{\varphi(n)}})^{-1}\mathcal{F 	
} 	
\ar@{=}[r 	
] 	
& 	

 	
( 	
f_{\tau^m_{\varphi(n)m}})^{-1}(d^1_1)^{-1}\mathcal{F 	
} 	

 	
\ar[d]^{(f_{\tau^m_{\varphi(n)m}})^{-1}\alpha 	
} 	
\\ 	

 	
& 	

 	
\mathcal{G}_m 	
\ar@{=}[r 	
] 	
& 	

 	
( 	
f_{\tau^m_m})^{-1}\mathcal{F 	
} 	
\ar@{=}[r 	
] 	
& 	

 	
( 	
f_{\tau^m_{\varphi(n)m}})^{-1}(d^1_0)^{-1}\mathcal{F 	
} 	

 	
} 	

 	
$ 	
$ 	

 	
We 	
omit 	
the 	
verification 	
that 	
the 	
commutativity 	
of 	
the 	
displayed 	
diagram 	

 	
above 	
implies 	
the 	
maps 	
compose 	
correctly 	
and 	
hence 	
give 	
rise 	
to 	
a 	

 	
sheaf 	
on 	
$ 	
\mathcal{C}_{total}$ 	
, 	
see 	

 	
Lemma 	
\ref{lemma 	
- 	
describe 	
- 	
sheaves 	
- 	
simplicial 	
- 	
site 	
- 	
site}. 	

 	
We 	
also 	
omit 	
the 	
verification 	

 	
that 	
the 	
two 	
functors 	
are 	
quasi 	
- 	
inverse 	
to 	
each 	
other 	
. 	

 	
\end{proof 	
} 	


 	
\begin{lemma 	
} 	

 	
\label{lemma 	
- 	
characterize 	
- 	
cartesian 	
- 	
modules 	
} 	

 	
In 	
Situation 	
\ref{situation 	
- 	
simplicial 	
- 	
site 	
} 	

 	
let 	
$ 	
\mathcal{O}$ 	
be 	
a 	
sheaf 	TYPE
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 	
. 	

 	
\end{lemma 	
} 	


 	
\begin{proof 	
} 	

 	
The 	
proof 	
is 	
identical 	
to 	
the 	
proof 	
of 	

 	
Lemma 	
\ref{lemma 	
- 	
characterize 	
- 	
cartesian 	
} 	

 	
with 	
pullback 	
of 	
sheaves 	
of 	
abelian 	
groups 	
replaced 	

 	
by 	
pullback 	
of 	
modules 	
. 	

 	
\end{proof 	
} 	


 	
\begin{lemma 	
} 	

 	
\label{lemma 	
- 	
Serre 	
- 	
subcat 	
- 	
cartesian 	
- 	
modules 	
} 	

 	
In 	
Situation 	
\ref{situation 	
- 	
simplicial 	
- 	
site}. 	

 	
\begin{enumerate 	
} 	

 	
\item 	
The 	
full 	
subcategory 	
of 	
cartesian 	
abelian 	
sheaves 	
forms 	
a 	

 	
weak 	
Serre 	
subcategory 	
of 	
$ 	
\textit{Ab}(\mathcal{C}_{total})$. 	

 	
Colimits 	
of 	
systems 	
of 	
cartesian 	
abelian 	
sheaves 	
are 	
cartesian 	
. 	

 	
\item 	
Let 	
$ 	
\mathcal{O}$ 	
be 	
a 	
sheaf 	TYPE
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 	
. 	
The 	
full 	
subcategory 	
of 	
cartesian 	
$ 	
\mathcal{O}$-modules 	
forms 	
a 	

 	
weak 	
Serre 	
subcategory 	
of 	
$ 	
\textit{Mod}(\mathcal{O})$. 	

 	
Colimits 	
of 	
systems 	
of 	
cartesian 	
$ 	
\mathcal{O}$-modules 	
are 	
cartesian 	
. 	

 	
\end{enumerate 	
} 	

 	
\end{lemma 	
} 	


 	
\begin{proof 	
} 	

 	
To 	
see 	
we 	
obtain 	
a 	
weak 	
Serre 	
subcategory 	
in 	
( 	
1 	
) 	

 	
we 	
check 	
the 	
conditions 	
listed 	
in 	

 	
Homology 	
, 	
Lemma 	
\ref{homology 	
- 	
lemma 	
- 	
characterize 	
- 	
weak 	
- 	
serre 	
- 	
subcategory}. 	

 	
First 	
, 	
if 	
$ 	
\varphi 	
: 	
\mathcal{F 	
} 	
\to 	
\mathcal{G}$ 	
is 	
a 	
map 	

 	
between 	
cartesian 	
abelian 	
sheaves 	
, 	
then 	

 	
$ 	
\Ker(\varphi)$ 	
and 	
$ 	
\Coker(\varphi)$ 	
are 	
cartesian 	
too 	

 	
because 	
the 	
restriction 	
functors 	

 	
$ 	
\Sh(\mathcal{C}_{total 	
} 	
) 	
\to 	
\Sh(\mathcal{C}_n)$ 	

 	
and 	
the 	
functors 	
$ 	
f_\varphi^{-1}$ 	
are 	
exact 	
. 	

 	
Similarly 	
, 	
if 	

 	
$ 	
$ 	

 	
0 	
\to 	
\mathcal{F 	
} 	
\to 	
\mathcal{H 	
} 	
\to 	
\mathcal{G 	
} 	
\to 	
0 	

 	
$ 	
$ 	

 	
is 	
a 	
short 	
exact 	
sequence 	
of 	
abelian 	
sheaves 	
on 	
$ 	
\mathcal{C}_{total}$ 	

 	
with 	
$ 	
\mathcal{F}$ 	
and 	
$ 	
\mathcal{G}$ 	
cartesian 	
, 	
then 	
it 	
follows 	
that 	

 	
$ 	
\mathcal{H}$ 	
is 	
cartesian 	
from 	
the 	
5-lemma 	
. 	
To 	
see 	
the 	
property 	
of 	

 	
colimits 	
, 	
use 	
that 	
colimits 	
commute 	
with 	
pullback 	
as 	
pullback 	
is 	
a 	

 	
left 	
adjoint 	
. 	
In 	
the 	
case 	
of 	
modules 	

 	
we 	
argue 	
in 	
the 	
same 	
manner 	
, 	
using 	
the 	
exactness 	
of 	
flat 	
pullback 	

 	
( 	
Modules 	
on 	
Sites 	
, 	
Lemma 	
\ref{sites 	
- 	
modules 	
- 	
lemma 	
- 	
flat 	
- 	
pullback 	
- 	
exact 	
} 	
) 	

 	
and 	
the 	
fact 	
that 	
it 	
suffices 	
to 	
check 	
the 	
condition 	

 	
for 	
$ 	
f_{\delta^n_j}$ 	
, 	
see 	
Lemma 	
\ref{lemma 	
- 	
check 	
- 	
cartesian 	
- 	
module}. 	

 	
\end{proof 	
} 	


 	
\begin{remark}[Warning 	
] 	

 	
\label{remark 	
- 	
warning 	
- 	
cartesian 	
- 	
modules 	
} 	

 	
Lemma 	
\ref{lemma 	
- 	
Serre 	
- 	
subcat 	
- 	
cartesian 	
- 	
modules 	
} 	
notwithstanding 	
, 	
it 	

 	
can 	
happen 	
that 	
the 	
category 	
of 	
cartesian 	
$ 	
\mathcal{O}$-modules 	
is 	

 	
abelian 	
without 	
being 	
a 	
Serre 	
subcategory 	
of 	
$ 	
\textit{Mod}(\mathcal{O})$. 	

 	
Namely 	
, 	
suppose 	
that 	
we 	
only 	
know 	
that 	

 	
$ 	
f_{\delta_1 	
^ 	
1}$ 	
and 	
$ 	
f_{\delta_0 	
^ 	
1}$ 	
are 	
flat 	
. 	

 	
Then 	
it 	
follows 	
easily 	
from 	

 	
Lemma 	
\ref{lemma 	
- 	
characterize 	
- 	
cartesian 	
- 	
modules 	
} 	

 	
that 	
the 	
category 	
of 	
cartesian 	
$ 	
\mathcal{O}$-modules 	
is 	
abelian 	
. 	

 	
But 	
if 	
$ 	
f_{\delta_0 	
^ 	
2}$ 	
is 	
not 	
flat 	
( 	
for 	
example 	
) 	
, 	

 	
there 	
is 	
no 	
reason 	
for 	
the 	
inclusion 	
functor 	

 	
from 	
the 	
category 	
of 	
cartesian 	
$ 	
\mathcal{O}$-modules 	

 	
to 	
all 	
$ 	
\mathcal{O}$-modules 	
to 	
be 	
exact 	
. 	

 	
\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 	TYPE

 	
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 	
. 	

 	
Then 	
an 	
object 	
$ 	
K$ 	
of 	
$ 	
D(\mathcal{O})$ 	
is 	
cartesian 	
if 	
and 	
only 	

 	
if 	
$ 	
H^q(K)$ 	
is 	
a 	
cartesian 	
$ 	
\mathcal{O}$-module 	
for 	
all 	
$ 	
q$. 	

 	
\end{enumerate 	
} 	

 	
\end{lemma 	
} 	


 	
\begin{proof 	
} 	

 	
Part 	
( 	
1 	
) 	
is 	
true 	
because 	
the 	
pullback 	
functors 	
$ 	
( 	
f_\varphi)^{-1}$ 	

 	
are 	
exact 	
. 	
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 	TYPE
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 	
) 	
. 	
Since 	
$ 	
g_{n!}$ 	
is 	
exact 	
, 	
it 	
induces 	
a 	
functor 	

 	
on 	
derived 	
categories 	
adjoint 	
to 	
$ 	
g_n^{-1}$. 	

 	
The 	
map 	
is 	
the 	
adjoint 	
of 	
the 	
map 	

 	
$ 	
K_n 	
\to 	
( 	
g_n^{-1}g_{n!}\mathbf{Z 	
} 	
) 	
\otimes^\mathbf{L}_\mathbf{Z 	
} 	
K_n$ 	

 	
corresponding 	
to 	
$ 	
\mathbf{Z 	
} 	
\to 	
g_n^{-1}g_{n!}\mathbf{Z}$ 	

 	
which 	
in 	
turn 	
is 	
adjoint 	
to 	

 	
$ 	
\text{id 	
} 	
: 	
g_{n!}\mathbf{Z 	
} 	
\to 	
g_{n!}\mathbf{Z}$. 	

 	
Using 	
the 	
description 	
of 	
$ 	
g_{n!}$ 	

 	
given 	
in 	
Lemma 	
\ref{lemma 	
- 	
restriction 	
- 	
to 	
- 	
components 	
- 	
site 	
} 	

 	
we 	
see 	
that 	
the 	
restriction 	
to 	
$ 	
\mathcal{C}_m$ 	
of 	
this 	
map 	

 	
is 	

 	
$ 	
$ 	

 	
\bigoplus\nolimits_{\varphi 	
: 	
[ 	
n 	
] 	
\to 	
[ 	
m 	
] 	
} 	
f_\varphi^{-1}K_n 	

 	
\longrightarrow 	

 	
\bigoplus\nolimits_{\varphi 	
: 	
[ 	
n 	
] 	
\to 	
[ 	
m 	
] 	
} 	
K_m 	

 	
$ 	
$ 	

 	
Thus 	
the 	
statement 	
is 	
clear 	
. 	


 	
\medskip\noindent 	

 	
Proof 	
of 	
( 	
2 	
) 	
. 	
Since 	
$ 	
g_{n!}$ 	
is 	
exact 	

 	
( 	
Lemma 	
\ref{lemma 	
- 	
exactness 	
- 	
g 	
- 	
shriek 	
- 	
modules 	
} 	
) 	
, 	
it 	
induces 	
a 	
functor 	

 	
on 	
derived 	
categories 	
adjoint 	
to 	
$ 	
g_n^*$ 	
( 	
also 	
exact 	
) 	
. 	

 	
The 	
map 	
is 	
the 	
adjoint 	
of 	
the 	
map 	

 	
$ 	
K_n 	
\to 	
( 	
g_n^*g_{n!}\mathcal{O}_n 	
) 	
\otimes^\mathbf{L}_{\mathcal{O}_n 	
} 	
K_n$ 	

 	
corresponding 	
to 	
$ 	
\mathcal{O}_n 	
\to 	
g_n^*g_{n!}\mathcal{O}_n$ 	

 	
which 	
in 	
turn 	
is 	
adjoint 	
to 	

 	
$ 	
\text{id 	
} 	
: 	
g_{n!}\mathcal{O}_n 	
\to 	
g_{n!}\mathcal{O}_n$. 	

 	
Using 	
the 	
description 	
of 	
$ 	
g_{n!}$ 	

 	
given 	
in 	
Lemma 	
\ref{lemma 	
- 	
restriction 	
- 	
module 	
- 	
to 	
- 	
components 	
- 	
site 	
} 	

 	
we 	
see 	
that 	
the 	
restriction 	
to 	
$ 	
\mathcal{C}_m$ 	
of 	
this 	
map 	

 	
is 	

 	
$ 	
$ 	

 	
\bigoplus\nolimits_{\varphi 	
: 	
[ 	
n 	
] 	
\to 	
[ 	
m 	
] 	
} 	
f_\varphi^*K_n 	

 	
\longrightarrow 	

 	
\bigoplus\nolimits_{\varphi 	
: 	
[ 	
n 	
] 	
\to 	
[ 	
m 	
] 	
} 	

 	
f_\varphi^*\mathcal{O}_n 	
\otimes_{\mathcal{O}_m 	
} 	
K_m 	
= 	

 	
\bigoplus\nolimits_{\varphi 	
: 	
[ 	
n 	
] 	
\to 	
[ 	
m 	
] 	
} 	
K_m 	

 	
$ 	
$ 	

 	
Thus 	
the 	
statement 	
is 	
clear 	
. 	

 	
\end{proof 	
} 	


 	
\begin{lemma 	
} 	

 	
\label{lemma 	
- 	
quasi 	
- 	
coherent 	
- 	
sheaf 	
} 	

 	
In 	
Situation 	
\ref{situation 	
- 	
simplicial 	
- 	
site 	
} 	

 	
let 	
$ 	
\mathcal{O}$ 	
be 	
a 	
sheaf 	TYPE
of 	
rings 	
on 	
$ 	
\mathcal{C}_{total}$. 	

 	
Let 	
$ 	
\mathcal{F}$ 	
be 	
a 	
sheaf 	TYPE
of 	
$ 	
\mathcal{O}$-modules 	
. 	

 	
Then 	
$ 	
\mathcal{F}$ 	
is 	
quasi 	
- 	
coherent 	
in 	
the 	
sense 	
of 	

 	
Modules 	
on 	
Sites 	
, 	
Definition 	
\ref{sites 	
- 	
modules 	
- 	
definition 	
- 	
site 	
- 	
local 	
} 	

 	
if 	
and 	
only 	
if 	
$ 	
\mathcal{F}$ 	
is 	
cartesian 	

 	
and 	
$ 	
\mathcal{F}_n$ 	
is 	
a 	
quasi 	
- 	
coherent 	
$ 	
\mathcal{O}_n$-module 	
for 	
all 	
$ 	
n$. 	

 	
\end{lemma 	
} 	


 	
\begin{proof 	
} 	

 	
Assume 	
$ 	
\mathcal{F}$ 	
is 	
quasi 	
- 	
coherent 	
. 	
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 	TYPE
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}$. 	

 	
Next 	
, 	
let 	
$ 	
V$ 	
be 	
an 	
object 	TYPE
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]$. 	
The 	
fibre 	
products 	
$ 	
V_i 	
= 	
V 	
\times_U 	
U_i$ 	

 	
exist 	
and 	
we 	
get 	
an 	
induced 	
covering 	
$ 	
\{V_i 	
\to 	
V\}$ 	
in 	
$ 	
\mathcal{C}_m$. 	

 	
Restricting 	
the 	
presentation 	
above 	
to 	
the 	
sites 	

 	
$ 	
\mathcal{C}_n 	
/ 	
U_i$ 	
and 	
$ 	
\mathcal{C}_m 	
/ 	
V_i$ 	
we 	
obtain 	

 	
presentations 	

 	
$ 	
$ 	

 	
\bigoplus\nolimits_{j 	
\in 	
J_i 	
} 	
\mathcal{O}_{\mathcal{C}_m 	
/ 	
U_i 	
} 	
\to 	

 	
\bigoplus\nolimits_{k 	
\in 	
K_i 	
} 	
\mathcal{O}_{\mathcal{C}_m 	
/ 	
U_i 	
} 	
\to 	

 	
\mathcal{F}_n|_{\mathcal{C}_n 	
/ 	
U_i 	
} 	
\to 	
0 	

 	
$ 	
$ 	

 	
and 	

 	
$ 	
$ 	

 	
\bigoplus\nolimits_{j 	
\in 	
J_i 	
} 	
\mathcal{O}_{\mathcal{C}_m 	
/ 	
V_i 	
} 	
\to 	

 	
\bigoplus\nolimits_{k 	
\in 	
K_i 	
} 	
\mathcal{O}_{\mathcal{C}_m 	
/ 	
V_i 	
} 	
\to 	

 	
\mathcal{F}_m|_{\mathcal{C}_m 	
/ 	
V_i 	
} 	
\to 	
0 	

 	
$ 	
$ 	

 	
These 	
presentations 	
are 	
compatible 	
with 	
the 	
map 	

 	
$ 	
\mathcal{F}(\varphi 	
) 	
: 	
f_\varphi^*\mathcal{F}_n 	
\to 	
\mathcal{F}_m$ 	

 	
( 	
as 	
this 	
map 	
is 	
defined 	
using 	
the 	
restriction 	
maps 	
of 	
$ 	
\mathcal{F}$ 	

 	
along 	
morphisms 	
of 	
$ 	
\mathcal{C}_{total}$ 	
lying 	
over 	
$ 	
\varphi$ 	
) 	
. 	

 	
We 	
conclude 	
that 	
$ 	
\mathcal{F}(\varphi)|_{\mathcal{C}_m 	
/ 	
V_i}$ 	

 	
is 	
an 	
isomorphism 	
. 	
As 	
$ 	
\{V_i 	
\to 	
V\}$ 	
is 	
a 	
covering 	
we 	
conclude 	

 	
$ 	
\mathcal{F}(\varphi)|_{\mathcal{C}_m 	
/ 	
V}$ 	
is 	
an 	
isomorphism 	
. 	

 	
Since 	
$ 	
V$ 	
and 	
$ 	
U$ 	
were 	
arbitrary 	
this 	
proves 	
that 	
$ 	
\mathcal{F}$ 	
is 	
cartesian 	
. 	

 	
( 	
In 	
case 	
A 	
use 	
Sites 	
, 	
Lemma 	
\ref{sites 	
- 	
lemma 	
- 	
morphism 	
- 	
of 	
- 	
sites 	
- 	
covering}. 	
) 	


 	
\medskip\noindent 	

 	
Conversely 	
, 	
assume 	
$ 	
\mathcal{F}_n$ 	
is 	
quasi 	
- 	
coherent 	

 	
for 	
all 	
$ 	
n$ 	
and 	
that 	
$ 	
\mathcal{F}$ 	
is 	
cartesian 	
. 	

 	
Then 	
for 	
any 	
$ 	
n$ 	
and 	
object 	
$ 	
U$ 	
of 	
$ 	
\mathcal{C}_n$ 	
we 	

 	
can 	
choose 	
a 	
covering 	
$ 	
\{U_i 	
\to 	
U\}$ 	
of 	
$ 	
\mathcal{C}_n$ 	

 	
and 	
for 	
each 	
$ 	
i$ 	
a 	
presentation 	

 	
$ 	
$ 	

 	
\bigoplus\nolimits_{j 	
\in 	
J_i 	
} 	
\mathcal{O}_{\mathcal{C}_m 	
/ 	
U_i 	
} 	
\to 	

 	
\bigoplus\nolimits_{k 	
\in 	
K_i 	
} 	
\mathcal{O}_{\mathcal{C}_m 	
/ 	
U_i 	
} 	
\to 	

 	
\mathcal{F}_n|_{\mathcal{C}_n 	
/ 	
U_i 	
} 	
\to 	
0 	

 	
$ 	
$ 	

 	
Pulling 	
back 	
to 	
$ 	
\mathcal{C}_{total}/U_i$ 	
we 	
obtain 	
complexes 	

 	
$ 	
$ 	

 	
\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 	

 	
$ 	
$ 	

 	
of 	
modules 	
on 	
$ 	
\mathcal{C}_{total}/U_i$. 	
Then 	
the 	
property 	
that 	

 	
$ 	
\mathcal{F}$ 	
is 	
cartesian 	
implies 	
that 	
this 	
is 	
exact 	
. 	

 	
We 	
omit 	
the 	
details 	
. 	

 	
\end{proof 	
} 	







 	
\section{Formalities 	
on 	
cohomological 	
descent 	
} 	

 	
\label{section 	
- 	
formal 	
- 	
cohomological 	
- 	
descent 	
} 	


 	
\noindent 	

 	
In 	
this 	
section 	
we 	
discuss 	
only 	
to 	
what 	
extent 	
a 	
morphism 	

 	
of 	
ringed 	
topoi 	
determines 	
an 	
embedding 	
from 	
the 	
derived 	

 	
category 	
downstairs 	
to 	
the 	
derived 	
category 	
upstairs 	
. 	

 	
Here 	
is 	
a 	
typical 	
result 	
. 	


 	
\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 	
. 	

 	
Consider 	
the 	
full 	
subcategory 	
$ 	
D 	
' 	
\subset 	
D(\mathcal{O}_\mathcal{D})$ 	

 	
consisting 	
of 	
objects 	
$ 	
K$ 	
such 	
that 	

 	
$ 	
$ 	

 	
K 	
\longrightarrow 	
Rf_*Lf^*K 	

 	
$ 	
$ 	

 	
is 	
an 	
isomorphism 	
. 	
Then 	
$ 	
D'$ 	
is 	
a 	
saturated 	
triangulated 	
strictly 	
full 	

 	
subcategory 	
of 	
$ 	
D(\mathcal{O}_\mathcal{D})$ 	
and 	
the 	
functor 	

 	
$ 	
Lf^ 	
* 	
: 	
D 	
' 	
\to 	
D(\mathcal{O}_\mathcal{C})$ 	
is 	
fully 	
faithful 	
. 	

 	
\end{lemma 	
} 	


 	
\begin{proof 	
} 	

 	
See 	
Derived 	
Categories 	
, 	
Definition 	
\ref{derived 	
- 	
definition 	
- 	
saturated 	
} 	

 	
for 	
the 	
definition 	
of 	
saturated 	
in 	
this 	
setting 	
. 	
See 	

 	
Derived 	
Categories 	
, 	
Lemma 	
\ref{derived 	
- 	
lemma 	
- 	
triangulated 	
- 	
subcategory 	
} 	

 	
for 	
a 	
discussion 	
of 	
triangulated 	
subcategories 	
. 	

 	
The 	
canonical 	
map 	
of 	
the 	
lemma 	
is 	
the 	
unit 	
of 	
the 	
adjoint 	

 	
pair 	
of 	
functors 	
$ 	
( 	
Lf^ 	
* 	
, 	
Rf_*)$ 	
, 	
see 	

 	
Cohomology 	
on 	
Sites 	
, 	
Lemma 	
\ref{sites 	
- 	
cohomology 	
- 	
lemma 	
- 	
adjoint}. 	

 	
Having 	
said 	
this 	
the 	
proof 	
that 	
$ 	
D'$ 	
is 	
a 	
saturated 	
triangulated 	
subcategory 	

 	
is 	
omitted 	
; 	
it 	
follows 	
formally 	
from 	
the 	
fact 	
that 	

 	
$ 	
Lf^*$ 	
and 	
$ 	
Rf_*$ 	
are 	
exact 	
functors 	
of 	
triangulated 	
categories 	
. 	

 	
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 	
. 	

 	
Consider 	
the 	
full 	
subcategory 	
$ 	
D 	
' 	
\subset 	
D(\mathcal{O}_\mathcal{C})$ 	

 	
consisting 	
of 	
objects 	
$ 	
K$ 	
such 	
that 	

 	
$ 	
$ 	

 	
Lf^*Rf_*K 	
\longrightarrow 	
K 	

 	
$ 	
$ 	

 	
is 	
an 	
isomorphism 	
. 	
Then 	
$ 	
D'$ 	
is 	
a 	
saturated 	
triangulated 	
strictly 	
full 	

 	
subcategory 	
of 	
$ 	
D(\mathcal{O}_\mathcal{C})$ 	
and 	
the 	
functor 	

 	
$ 	
Rf 	
_ 	
* 	
: 	
D 	
' 	
\to 	
D(\mathcal{O}_\mathcal{D})$ 	
is 	
fully 	
faithful 	
. 	

 	
\end{lemma 	
} 	


 	
\begin{proof 	
} 	

 	
See 	
Derived 	
Categories 	
, 	
Definition 	
\ref{derived 	
- 	
definition 	
- 	
saturated 	
} 	

 	
for 	
the 	
definition 	
of 	
saturated 	
in 	
this 	
setting 	
. 	
See 	

 	
Derived 	
Categories 	
, 	
Lemma 	
\ref{derived 	
- 	
lemma 	
- 	
triangulated 	
- 	
subcategory 	
} 	

 	
for 	
a 	
discussion 	
of 	
triangulated 	
subcategories 	
. 	

 	
The 	
canonical 	
map 	
of 	
the 	
lemma 	
is 	
the 	
counit 	
of 	
the 	
adjoint 	

 	
pair 	
of 	
functors 	
$ 	
( 	
Lf^ 	
* 	
, 	
Rf_*)$ 	
, 	
see 	

 	
Cohomology 	
on 	
Sites 	
, 	
Lemma 	
\ref{sites 	
- 	
cohomology 	
- 	
lemma 	
- 	
adjoint}. 	

 	
Having 	
said 	
this 	
the 	
proof 	
that 	
$ 	
D'$ 	
is 	
a 	
saturated 	
triangulated 	
subcategory 	

 	
is 	
omitted 	
; 	
it 	
follows 	
formally 	
from 	
the 	
fact 	
that 	

 	
$ 	
Lf^*$ 	
and 	
$ 	
Rf_*$ 	
are 	
exact 	
functors 	
of 	
triangulated 	
categories 	
. 	

 	
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 	
. 	

 	
Let 	
$ 	
K$ 	
be 	
an 	
object 	
of 	
$ 	
D(\mathcal{O}_\mathcal{C})$. 	
Assume 	

 	
\begin{enumerate 	
} 	

 	
\item 	
$ 	
f$ 	
is 	
flat 	
, 	

 	
\item 	
$ 	
K$ 	
is 	
bounded 	
below 	
, 	

 	
\item 	
$ 	
f^*Rf_*H^q(K 	
) 	
\to 	
H^q(K)$ 	
is 	
an 	
isomorphism 	
. 	

 	
\end{enumerate 	
} 	

 	
Then 	
$ 	
f^*Rf_*K 	
\to 	
K$ 	
is 	
an 	
isomorphism 	
. 	

 	
\end{lemma 	
} 	


 	
\begin{proof 	
} 	

 	
Observe 	
that 	
$ 	
f^*Rf_*K 	
\to 	
K$ 	
is 	
an 	
isomorphism 	
if 	
and 	
only 	

 	
if 	
it 	
is 	
an 	
isomorphism 	
on 	
cohomology 	
sheaves 	
$ 	
H^j$. 	
Observe 	
that 	

 	
$ 	
H^j(f^*Rf_*K 	
) 	
= 	
f^*H^j(Rf_*K 	
) 	
= 	
f^*H^j(Rf_*\tau_{\leq 	
j}K 	
) 	
= 	

 	
H^j(f^*Rf_*\tau_{\leq 	
j}K)$. 	

 	
Hence 	
we 	
may 	
assume 	
that 	
$ 	
K$ 	
is 	
bounded 	
. 	
Then 	
property 	
( 	
3 	
) 	

 	
tells 	
us 	
the 	
cohomology 	
sheaves 	
are 	
in 	
the 	
triangulated 	

 	
subcategory 	
$ 	
D 	
' 	
\subset 	
D(\mathcal{O}_\mathcal{C})$ 	
of 	

 	
Lemma 	
\ref{lemma 	
- 	
upstairs}. 	
Hence 	
$ 	
K$ 	
is 	
in 	
it 	
too 	
. 	

 	
\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 	
. 	

 	
Let 	
$ 	
K$ 	
be 	
an 	
object 	
of 	
$ 	
D(\mathcal{O}_\mathcal{D})$. 	
Assume 	

 	
\begin{enumerate 	
} 	

 	
\item 	
$ 	
f$ 	
is 	
flat 	
, 	

 	
\item 	
$ 	
K$ 	
is 	
bounded 	
below 	
, 	

 	
\item 	
$ 	
H^q(K 	
) 	
\to 	
Rf_*f^*H^q(K)$ 	
is 	
an 	
isomorphism 	
. 	

 	
\end{enumerate 	
} 	

 	
Then 	
$ 	
K 	
\to 	
Rf_*f^*K$ 	
is 	
an 	
isomorphism 	
. 	

 	
\end{lemma 	
} 	


 	
\begin{proof 	
} 	

 	
Observe 	
that 	
$ 	
K 	
\to 	
Rf_*f^*K$ 	
is 	
an 	
isomorphism 	
if 	
and 	
only 	

 	
if 	
it 	
is 	
an 	
isomorphism 	
on 	
cohomology 	
sheaves 	
$ 	
H^j$. 	
Observe 	
that 	

 	
$ 	
H^j(Rf_*f^*K 	
) 	
= 	
H^j(Rf_*\tau_{\leq 	
j}f^*K 	
) 	
= 	
H^j(Rf_*f^*\tau_{\leq 	
j}K)$. 	

 	
Hence 	
we 	
may 	
assume 	
that 	
$ 	
K$ 	
is 	
bounded 	
. 	
Then 	
property 	
( 	
3 	
) 	

 	
tells 	
us 	
the 	
cohomology 	
sheaves 	
are 	
in 	
the 	
triangulated 	

 	
subcategory 	
$ 	
D 	
' 	
\subset 	
D(\mathcal{O}_\mathcal{D})$ 	
of 	

 	
Lemma 	
\ref{lemma 	
- 	
downstairs}. 	
Hence 	
$ 	
K$ 	
is 	
in 	
it 	
too 	
. 	

 	
\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 	
. 	

 	
Let 	
$ 	
\mathcal{A 	
} 	
\subset 	
\textit{Mod}(\mathcal{O})$ 	

 	
and 	
$ 	
\mathcal{A 	
} 	
' 	
\subset 	
\textit{Mod}(\mathcal{O}')$ 	

 	
be 	
weak 	
Serre 	
subcategories 	
. 	
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}')$. 	

 	
\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 	
} 	

 	
By 	
assumptions 	
( 	
2 	
) 	
and 	
( 	
3 	
) 	
and 	

 	
Lemmas 	
\ref{lemma 	
- 	
bounded 	
- 	
in 	
- 	
image 	
- 	
upstairs 	
} 	
and 	
\ref{lemma 	
- 	
downstairs 	
} 	

 	
we 	
see 	
that 	

 	
$ 	
f^ 	
* 	
: 	
D_{\mathcal{A}'}^+(\mathcal{O 	
} 	
' 	
) 	
\to 	
D_\mathcal{A}^+(\mathcal{O})$ 	

 	
is 	
fully 	
faithful 	
. 	

 	
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}')$. 	

 	
Thus 	
for 	
any 	
$ 	
K 	
\in 	
D^+_\mathcal{A}(\mathcal{O})$ 	
we 	
see 	
, 	

 	
using 	
the 	
spectral 	
sequence 	
$ 	
E_2^{p 	
, 	
q 	
} 	
= 	
R^pf_*H^q(K)$ 	

 	
converging 	
to 	
$ 	
R^{p 	
+ 	
q}f_*K$ 	
, 	

 	
that 	
$ 	
Rf_*K$ 	
is 	
in 	
$ 	
D^+_{\mathcal{A}'}(\mathcal{O}')$. 	

 	
Of 	
course 	
, 	
it 	
also 	
follows 	
from 	

 	
Lemmas 	
\ref{lemma 	
- 	
bounded 	
- 	
in 	
- 	
image 	
- 	
downstairs 	
} 	
and 	
\ref{lemma 	
- 	
upstairs 	
} 	

 	
that 	
$ 	
Rf 	
_ 	
* 	
: 	
D_\mathcal{A}^+(\mathcal{O 	
} 	
) 	
\to 	
D_{\mathcal{A}'}^+(\mathcal{O}')$ 	

 	
is 	
fully 	
faithful 	
. 	
Since 	
$ 	
f^*$ 	
and 	
$ 	
Rf_*$ 	
are 	
adjoint 	

 	
we 	
then 	
get 	
the 	
result 	
of 	
the 	
lemma 	
, 	
for 	
example 	
by 	

 	
Categories 	
, 	
Lemma 	
\ref{categories 	
- 	
lemma 	
- 	
adjoint 	
- 	
fully 	
- 	
faithful}. 	

 	
\end{proof 	
} 	


 	
\begin{lemma 	
} 	

 	
\label{lemma 	
- 	
equivalence 	
- 	
unbounded 	
- 	
one 	
} 	

 	
\begin{reference 	
} 	

 	
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 	
. 	

 	
Let 	
$ 	
\mathcal{A 	
} 	
\subset 	
\textit{Mod}(\mathcal{O})$ 	

 	
and 	
$ 	
\mathcal{A 	
} 	
' 	
\subset 	
\textit{Mod}(\mathcal{O}')$ 	

 	
be 	
weak 	
Serre 	
subcategories 	
. 	
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 	
$ 	
\mathcal{C 	
} 	
' 	
, 	
\mathcal{O 	
} 	
' 	
, 	
\mathcal{A}'$ 	
satisfy 	
the 	

 	
assumption 	
of 	

 	
Cohomology 	
on 	
Sites 	
, 	
Situation 	
\ref{sites 	
- 	
cohomology 	
- 	
situation 	
- 	
olsson 	
- 	
laszlo}. 	

 	
\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 	
} 	

 	
Since 	
$ 	
f^*$ 	
is 	
exact 	
, 	
it 	
is 	
clear 	
that 	
$ 	
f^*$ 	
defines 	
a 	
functor 	

 	
$ 	
f^ 	
* 	
: 	
D_{\mathcal{A}'}(\mathcal{O 	
} 	
' 	
) 	
\to 	
D_\mathcal{A}(\mathcal{O})$ 	

 	
as 	
in 	
the 	
statement 	
of 	
the 	
lemma 	
and 	
that 	
moreover 	
this 	

 	
functor 	
commutes 	
with 	
the 	
truncation 	
functors 	
$ 	
\tau_{\geq 	
-n}$. 	

 	
We 	
already 	
know 	
that 	
$ 	
f^*$ 	
and 	
$ 	
Rf_*$ 	
are 	
quasi 	
- 	
inverse 	

 	
equivalence 	
on 	
the 	
corresponding 	
bounded 	
below 	
categories 	
, 	

 	
see 	
Lemma 	
\ref{lemma 	
- 	
equivalence 	
- 	
bounded}. 	

 	
By 	
Cohomology 	
on 	
Sites 	
, 	
Lemma 	

 	
\ref{sites 	
- 	
cohomology 	
- 	
lemma 	
- 	
olsson 	
- 	
laszlo 	
- 	
map 	
- 	
version 	
- 	
one 	
} 	

 	
with 	
$ 	
N 	
= 	
0 	
$ 	
we 	
see 	
that 	
$ 	
Rf_*$ 	
indeed 	
defines 	
a 	
functor 	

 	
$ 	
Rf 	
_ 	
* 	
: 	
D_\mathcal{A}(\mathcal{O 	
} 	
) 	
\to 	
D_{\mathcal{A}'}(\mathcal{O}')$ 	

 	
and 	
that 	
moreover 	
this 	
functor 	
commutes 	
with 	

 	
the 	
truncation 	
functors 	
$ 	
\tau_{\geq 	
-n}$. 	

 	
Thus 	
for 	
$ 	
K$ 	
in 	
$ 	
D_\mathcal{A}(\mathcal{O})$ 	
the 	
map 	

 	
$ 	
f^*Rf_*K 	
\to 	
K$ 	
is 	
an 	
isomorphism 	
as 	
this 	
is 	
true 	

 	
on 	
trunctions 	
. 	

 	
Similarly 	
, 	
for 	
$ 	
K'$ 	
in 	
$ 	
D_{\mathcal{A}'}(\mathcal{O}')$ 	
the 	
map 	

 	
$ 	
K 	
' 	
\to 	
Rf_*f^*K'$ 	
is 	
an 	
isomorphism 	
as 	
this 	
is 	
true 	

 	
on 	
trunctions 	
. 	

 	
This 	
finishes 	
the 	
proof 	
. 	

 	
\end{proof 	
} 	


 	
\begin{lemma 	
} 	

 	
\label{lemma 	
- 	
equivalence 	
- 	
unbounded 	
- 	
two 	
} 	

 	
\begin{reference 	
} 	

 	
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 	
. 	

 	
Let 	
$ 	
\mathcal{A 	
} 	
\subset 	
\textit{Mod}(\mathcal{O})$ 	

 	
and 	
$ 	
\mathcal{A 	
} 	
' 	
\subset 	
\textit{Mod}(\mathcal{O}')$ 	

 	
be 	
weak 	
Serre 	
subcategories 	
. 	
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 	TYPE
. 	TYPE
Let 	

 	
$ 	
\text{Cov}(\mathcal{C 	
} 	
) 	
\supset 	
\text{Cov}'(\mathcal{C})$ 	

 	
be 	
two 	
ways 	
to 	
endow 	
$ 	
\mathcal{C}$ 	
with 	
the 	
structure 	
of 	
a 	
site 	
. 	

 	
Denote 	
$ 	
\tau$ 	
the 	
topology 	
corresponding 	
to 	
$ 	
\text{Cov}(\mathcal{C})$ 	

 	
and 	
$ 	
\tau'$ 	
the 	
topology 	
corresponding 	
to 	
$ 	
\text{Cov}'(\mathcal{C})$. 	

 	
Then 	
the 	
identity 	
functor 	
on 	
$ 	
\mathcal{C}$ 	
defines 	
a 	
morphism 	

 	
of 	
sites 	

 	
$ 	
$ 	

 	
\epsilon 	
: 	
\mathcal{C}_\tau 	
\longrightarrow 	
\mathcal{C}_{\tau 	
' 	
} 	

 	
$ 	
$ 	

 	
where 	
$ 	
\epsilon_*$ 	
is 	
the 	
identity 	
functor 	
on 	
underlying 	
presheaves 	
and 	

 	
where 	
$ 	
\epsilon^{-1}$ 	
is 	
the 	
$ 	
\tau$-sheafification 	
of 	
a 	
$ 	
\tau'$-sheaf 	

 	
( 	
hence 	
clearly 	
exact 	
) 	
. 	
Let 	
$ 	
\mathcal{O}$ 	
be 	
a 	
sheaf 	TYPE
of 	
rings 	
for 	
the 	

 	
$ 	
\tau$-topology 	
. 	
Then 	
$ 	
\mathcal{O}$ 	
is 	
also 	
a 	
sheaf 	
for 	
the 	
$ 	
\tau'$-topology 	

 	
and 	
$ 	
\epsilon$ 	
becomes 	
a 	
morphism 	
of 	
ringed 	
sites 	

 	
$ 	
$ 	

 	
\epsilon 	
: 	

 	
( 	
\mathcal{C}_\tau 	
, 	
\mathcal{O}_\tau 	
) 	

 	
\longrightarrow 	

 	
( 	
\mathcal{C}_{\tau 	
' 	
} 	
, 	
\mathcal{O}_{\tau 	
' 	
} 	
) 	

 	
$ 	
$ 	


 	
\begin{lemma 	
} 	

 	
\label{lemma 	
- 	
compare 	
- 	
topologies 	
- 	
derived 	
- 	
adequate 	
- 	
modules 	
} 	

 	
With 	
$ 	
\epsilon 	
: 	
( 	
\mathcal{C}_\tau 	
, 	
\mathcal{O}_\tau 	
) 	
\to 	

 	
( 	
\mathcal{C}_{\tau 	
' 	
} 	
, 	
\mathcal{O}_{\tau'})$ 	
as 	
above 	
. 	

 	
Let 	
$ 	
\mathcal{B 	
} 	
\subset 	
\Ob(\mathcal{C})$ 	
be 	
a 	
subset 	
. 	

 	
Let 	
$ 	
\mathcal{A 	
} 	
\subset 	
\textit{PMod}(\mathcal{O})$ 	

 	
be 	
a 	
full 	TYPE
subcategory 	
. 	
Assume 	

 	
\begin{enumerate 	
} 	

 	
\item 	
every 	
object 	
of 	
$ 	
\mathcal{A}$ 	
is 	
a 	
sheaf 	
for 	
the 	
$ 	
\tau$-topology 	
, 	

 	
\item 	
$ 	
\mathcal{A}$ 	
is 	
a 	
weak 	
Serre 	
subcategory 	
of 	

 	
$ 	
\textit{Mod}(\mathcal{O}_\tau)$ 	
, 	

 	
\item 	
every 	
object 	
of 	
$ 	
\mathcal{C}$ 	
has 	
a 	
$ 	
\tau'$-covering 	
whose 	

 	
members 	
are 	
elements 	
of 	
$ 	
\mathcal{B}$ 	
, 	
and 	

 	
\item 	
for 	
every 	
$ 	
U 	
\in 	
\mathcal{B}$ 	
we 	
have 	
$ 	
H^p_\tau(U 	
, 	
\mathcal{F 	
} 	
) 	
= 	
0 	
$ 	
, 	

 	
$ 	
p 	
> 	
0 	
$ 	
for 	
all 	
$ 	
\mathcal{F 	
} 	
\in 	
\mathcal{A}$. 	

 	
\end{enumerate 	
} 	

 	
Then 	
$ 	
\mathcal{A}$ 	
is 	
a 	
weak 	
Serre 	
subcategory 	
of 	

 	
$ 	
\textit{Mod}(\mathcal{O}_{\tau'})$ 	
and 	
there 	
is 	
an 	
equivalence 	

 	
of 	
triangulated 	
categories 	

 	
$ 	
D_\mathcal{A}(\mathcal{O}_\tau 	
) 	
= 	
D_\mathcal{A}(\mathcal{O}_{\tau'})$ 	

 	
given 	
by 	
$ 	
\epsilon^*$ 	
and 	
$ 	
R\epsilon_*$. 	

 	
\end{lemma 	
} 	


 	
\begin{proof 	
} 	

 	
Since 	
$ 	
\epsilon^{-1}\mathcal{O}_{\tau 	
' 	
} 	
= 	
\mathcal{O}_\tau$ 	

 	
we 	
see 	
that 	
$ 	
\epsilon$ 	
is 	
a 	
flat 	
morphism 	
of 	
ringed 	
sites 	

 	
and 	
that 	
in 	
fact 	
$ 	
\epsilon^{-1 	
} 	
= 	
\epsilon^*$ 	
on 	
sheaves 	

 	
of 	
modules 	
. 	
By 	
property 	
( 	
1 	
) 	
we 	
can 	
think 	
of 	
every 	
object 	
of 	

 	
$ 	
\mathcal{A}$ 	
as 	
a 	
sheaf 	
of 	
$ 	
\mathcal{O}_\tau$-modules 	

 	
and 	
as 	
a 	
sheaf 	
of 	
$ 	
\mathcal{O}_{\tau'}$-modules 	
. 	

 	
In 	
other 	
words 	
, 	
we 	
have 	
fully 	
faithful 	
inclusion 	
functors 	

 	
$ 	
$ 	

 	
\mathcal{A 	
} 	
\to 	
\textit{Mod}(\mathcal{O}_\tau 	
) 	
\to 	

 	
\textit{Mod}(\mathcal{O}_{\tau 	
' 	
} 	
) 	

 	
$ 	
$ 	

 	
To 	
avoid 	
confusion 	
we 	
will 	
denote 	

 	
$ 	
\mathcal{A 	
} 	
' 	
\subset 	
\textit{Mod}(\mathcal{O}_{\tau'})$ 	

 	
the 	
image 	
of 	
$ 	
\mathcal{A}$. 	
Then 	
it 	
is 	
clear 	
that 	

 	
$ 	
\epsilon 	
_ 	
* 	
: 	
\mathcal{A 	
} 	
\to 	
\mathcal{A}'$ 	
and 	

 	
$ 	
\epsilon^ 	
* 	
: 	
\mathcal{A 	
} 	
' 	
\to 	
\mathcal{A}$ 	
are 	

 	
quasi 	
- 	
inverse 	
equivalences 	
( 	
see 	
discussion 	
preceding 	

 	
the 	
lemma 	
and 	
use 	
that 	
objects 	
of 	
$ 	
\mathcal{A}'$ 	
are 	

 	
sheaves 	
in 	
the 	
$ 	
\tau$ 	
topology 	
) 	
. 	


 	
\medskip\noindent 	

 	
Conditions 	
( 	
3 	
) 	
and 	
( 	
4 	
) 	
imply 	
that 	
$ 	
R^p\epsilon_*\mathcal{F 	
} 	
= 	
0 	
$ 	

 	
for 	
$ 	
p 	
> 	
0 	
$ 	
and 	
$ 	
\mathcal{F 	
} 	
\in 	
\Ob(\mathcal{A})$. 	

 	
This 	
is 	
true 	
because 	
$ 	
R^p\epsilon_*$ 	
is 	
the 	
sheaf 	
associated 	

 	
to 	
the 	
presheave 	
$ 	
U 	
\mapsto 	
H^p_\tau(U 	
, 	
\mathcal{F})$ 	
, 	
see 	

 	
Cohomology 	
on 	
Sites 	
, 	
Lemma 	
\ref{sites 	
- 	
cohomology 	
- 	
lemma 	
- 	
higher 	
- 	
direct 	
- 	
images}. 	

 	
Thus 	
any 	
exact 	
complex 	
in 	
$ 	
\mathcal{A}$ 	
( 	
which 	
is 	
the 	
same 	
thing 	

 	
as 	
an 	
exact 	
complex 	
in 	
$ 	
\textit{Mod}(\mathcal{O}_\tau)$ 	

 	
whose 	
terms 	
are 	
in 	
$ 	
\mathcal{A}$ 	
, 	
see 	

 	
Homology 	
, 	
Lemma 	
\ref{homology 	
- 	
lemma 	
- 	
characterize 	
- 	
weak 	
- 	
serre 	
- 	
subcategory 	
} 	
) 	

 	
remains 	
exact 	
upon 	
applying 	
the 	
functor 	
$ 	
\epsilon_*$. 	


 	
\medskip\noindent 	

 	
Consider 	
an 	
exact 	
sequence 	

 	
$ 	
$ 	

 	
\mathcal{F}'_0 	
\to 	
\mathcal{F}'_1 	
\to 	

 	
\mathcal{F}'_2 	
\to 	
\mathcal{F}'_3 	
\to 	

 	
\mathcal{F}'_4 	

 	
$ 	
$ 	

 	
in 	
$ 	
\textit{Mod}(\mathcal{O}_{\tau'})$ 	
with 	

 	
$ 	
\mathcal{F}'_0 	
, 	
\mathcal{F}'_1 	
, 	
\mathcal{F}'_3 	
, 	
\mathcal{F}'_4 	
$ 	
in 	

 	
$ 	
\mathcal{A}'$. 	
Apply 	
the 	
exact 	
functor 	
$ 	
\epsilon^*$ 	
to 	
get 	

 	
an 	
exact 	
sequence 	

 	
$ 	
$ 	

 	
\epsilon^*\mathcal{F}'_0 	
\to 	
\epsilon^*\mathcal{F}'_1 	
\to 	

 	
\epsilon^*\mathcal{F}'_2 	
\to 	
\epsilon^*\mathcal{F}'_3 	
\to 	

 	
\epsilon^*\mathcal{F}'_4 	

 	
$ 	
$ 	

 	
in 	
$ 	
\textit{Mod}(\mathcal{O}_\tau)$. 	
Since 	
$ 	
\mathcal{A}$ 	
is 	

 	
a 	
weak 	
Serre 	
subcategory 	
and 	
since 	

 	
$ 	
\epsilon^*\mathcal{F}'_0 	
, 	
\epsilon^*\mathcal{F}'_1 	
, 	

 	
\epsilon^*\mathcal{F}'_3 	
, 	
\epsilon^*\mathcal{F}'_4 	
$ 	
are 	
in 	

 	
$ 	
\mathcal{A}$ 	
, 	
we 	
conclude 	
that 	
$ 	
\epsilon^*\mathcal{F}_2 	
$ 	

 	
is 	
in 	
$ 	
\mathcal{A}$ 	
by 	

 	
Homology 	
, 	
Definition 	
\ref{homology 	
- 	
definition 	
- 	
serre 	
- 	
subcategory}. 	

 	
Consider 	
the 	
map 	
of 	
sequences 	

 	
$ 	
$ 	

 	
\xymatrix 	
{ 	

 	
\mathcal{F}'_0 	
\ar[r 	
] 	
\ar[d 	
] 	
& 	

 	
\mathcal{F}'_1 	
\ar[r 	
] 	
\ar[d 	
] 	
& 	

 	
\mathcal{F}'_2 	
\ar[r 	
] 	
\ar[d 	
] 	
& 	

 	
\mathcal{F}'_3 	
\ar[r 	
] 	
\ar[d 	
] 	
& 	

 	
\mathcal{F}'_4 	
\ar[d 	
] 	
\\ 	

 	
\epsilon_*\epsilon^*\mathcal{F}'_0 	
\ar[r 	
] 	
& 	

 	
\epsilon_*\epsilon^*\mathcal{F}'_1 	
\ar[r 	
] 	
& 	

 	
\epsilon_*\epsilon^*\mathcal{F}'_2 	
\ar[r 	
] 	
& 	

 	
\epsilon_*\epsilon^*\mathcal{F}'_3 	
\ar[r 	
] 	
& 	

 	
\epsilon_*\epsilon^*\mathcal{F}'_4 	

 	
} 	

 	
$ 	
$ 	

 	
The 	
lower 	
row 	
is 	
exact 	
by 	
the 	
discussion 	
in 	
the 	
preceding 	

 	
paragraph 	
. 	
The 	
vertical 	
arrows 	
with 	
index 	
$ 	
0 	
$ 	
, 	
$ 	
1 	
$ 	
, 	
$ 	
3 	
$ 	
, 	
$ 	
4 	
$ 	

 	
are 	
isomorphisms 	
by 	
the 	
discussion 	
in 	
the 	
first 	
paragraph 	
. 	

 	
By 	
the 	
$ 	
5 	
$ 	
lemma 	
( 	
Homology 	
, 	
Lemma 	
\ref{homology 	
- 	
lemma 	
- 	
five 	
- 	
lemma 	
} 	
) 	

 	
we 	
find 	
that 	
$ 	
\mathcal{F}'_2 	
\cong 	
\epsilon_*\epsilon^*\mathcal{F}'_2 	
$ 	

 	
and 	
hence 	
$ 	
\mathcal{F}'_2 	
$ 	
is 	
in 	
$ 	
\mathcal{A}'$. 	

 	
In 	
this 	
way 	
we 	
see 	
that 	
$ 	
\mathcal{A}'$ 	
is 	
a 	
weak 	
Serre 	
subcategory 	

 	
of 	
$ 	
\textit{Mod}(\mathcal{O}_{\tau'})$ 	
, 	
see 	

 	
Homology 	
, 	
Definition 	
\ref{homology 	
- 	
definition 	
- 	
serre 	
- 	
subcategory}. 	


 	
\medskip\noindent 	

 	
At 	
this 	
point 	
it 	
makes 	
sense 	
to 	
talk 	
about 	
the 	

 	
derived 	
categories 	
$ 	
D_\mathcal{A}(\mathcal{O}_\tau)$ 	
and 	

 	
$ 	
D_{\mathcal{A}'}(\mathcal{O}_{\tau'})$ 	
, 	
see 	

 	
Derived 	
Categories 	
, 	
Section 	
\ref{derived 	
- 	
section 	
- 	
triangulated 	
- 	
sub}. 	

 	
To 	
finish 	
the 	
proof 	
we 	
show 	
that 	
conditions 	

 	
( 	
1 	
) 	
-- 	
( 	
5 	
) 	
of 	
Lemma 	
\ref{lemma 	
- 	
equivalence 	
- 	
unbounded 	
- 	
two 	
} 	
apply 	
. 	

 	
We 	
have 	
already 	
seen 	
( 	
1 	
) 	
, 	
( 	
2 	
) 	
, 	
( 	
3 	
) 	
above 	
. 	

 	
Note 	
that 	
since 	
every 	
object 	
has 	
a 	
$ 	
\tau'$-covering 	

 	
by 	
objects 	
of 	
$ 	
\mathcal{B}$ 	
, 	
a 	
fortiori 	
every 	
object 	
has 	

 	
a 	
$ 	
\tau$-covering 	
by 	
objects 	
of 	
$ 	
\mathcal{B}$. 	
Hence 	

 	
condition 	
( 	
4 	
) 	
of 	
Lemma 	
\ref{lemma 	
- 	
equivalence 	
- 	
unbounded 	
- 	
two 	
} 	
is 	
satisfied 	
. 	

 	
Similarly 	
, 	
condition 	
( 	
5 	
) 	
is 	
satisfied 	
as 	
well 	
. 	

 	
\end{proof 	
} 	






 	
\section{Simplicial 	
systems 	
of 	
the 	
derived 	
category 	
} 	

 	
\label{section 	
- 	
glueing 	
} 	


 	
\noindent 	

 	
In 	
this 	
section 	
we 	
are 	
going 	
to 	
prove 	
a 	
special 	
case 	
of 	

 	
\cite[Proposition 	
3.2.9]{BBD 	
} 	
in 	
the 	
setting 	
of 	
derived 	

 	
categories 	
of 	
abelian 	
sheaves 	
. 	
The 	
case 	
of 	
modules 	

 	
is 	
discussed 	
in 	
Section 	
\ref{section 	
- 	
glueing 	
- 	
modules}. 	


 	
\begin{definition 	
} 	

 	
\label{definition 	
- 	
cartesian 	
- 	
derived 	
} 	

 	
In 	
Situation 	
\ref{situation 	
- 	
simplicial 	
- 	
site}. 	
A 	

 	
{ 	
\it 	
simplicial 	
system 	
of 	
the 	
derived 	
category 	
} 	

 	
consists 	
of 	
the 	
following 	
data 	

 	
\begin{enumerate 	
} 	

 	
\item 	
for 	
every 	
$ 	
n$ 	
an 	
object 	
$ 	
K_n$ 	
of 	
$ 	
D(\mathcal{C}_n)$ 	
, 	

 	
\item 	
for 	
every 	
$ 	
\varphi 	
: 	
[ 	
m 	
] 	
\to 	
[ 	
n]$ 	
a 	
map 	

 	
$ 	
K_\varphi 	
: 	
f_\varphi^{-1}K_m 	
\to 	
K_n$ 	
in 	
$ 	
D(\mathcal{C}_n)$ 	

 	
\end{enumerate 	
} 	

 	
subject 	
to 	
the 	
condition 	
that 	

 	
$ 	
$ 	

 	
K_{\varphi 	
\circ 	
\psi 	
} 	
= 	
K_\varphi 	
\circ 	
f_\varphi^{-1}K_\psi 	
: 	

 	
f_{\varphi 	
\circ 	
\psi}^{-1}K_l 	
= 	
f_\varphi^{-1 	
} 	
f_\psi^{-1}K_l 	

 	
\longrightarrow 	

 	
K_n 	

 	
$ 	
$ 	

 	
for 	
any 	
morphisms 	
$ 	
\varphi 	
: 	
[ 	
m 	
] 	
\to 	
[ 	
n]$ 	
and 	
$ 	
\psi 	
: 	
[ 	
l 	
] 	
\to 	
[ 	
m]$ 	
of 	
$ 	
\Delta$. 	

 	
We 	
say 	
the 	
simplicial 	
system 	
is 	
{ 	
\it 	
cartesian 	
} 	
if 	
the 	
maps 	
$ 	
K_\varphi$ 	

 	
are 	
isomorphisms 	
for 	
all 	
$ 	
\varphi$. 	

 	
Given 	
two 	
simplicial 	
systems 	
of 	
the 	
derived 	
category 	

 	
there 	
is 	
an 	
obvious 	
notion 	
of 	
a 	

 	
{ 	
\it 	
morphism 	
of 	
simplicial 	
systems 	
of 	
the 	
derived 	
category}. 	

 	
\end{definition 	
} 	


 	
\noindent 	

 	
We 	
have 	
given 	
this 	
notion 	
a 	
ridiculously 	
long 	
name 	
intentionally 	
. 	

 	
The 	
goal 	
is 	
to 	
show 	
that 	
a 	
simplicial 	
system 	
of 	
the 	
derived 	
category 	

 	
comes 	
from 	
an 	
object 	
of 	
$ 	
D(\mathcal{C}_{total})$ 	
under 	
certain 	

 	
hypotheses 	
. 	


 	
\begin{lemma 	
} 	

 	
\label{lemma 	
- 	
cartesian 	
- 	
objects 	
- 	
derived 	
} 	

 	
In 	
Situation 	
\ref{situation 	
- 	
simplicial 	
- 	
site}. 	

 	
If 	
$ 	
K 	
\in 	
D(\mathcal{C}_{total})$ 	
is 	
an 	
object 	
, 	

 	
then 	
$ 	
( 	
K_n 	
, 	
K(\varphi))$ 	
is 	
a 	
simplicial 	
system 	
of 	
the 	
derived 	
category 	
. 	

 	
If 	
$ 	
K$ 	
is 	
cartesian 	
, 	
so 	
is 	
the 	
system 	
. 	

 	
\end{lemma 	
} 	


 	
\begin{proof 	
} 	

 	
This 	
is 	
obvious 	
. 	

 	
\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 	
\to 	
X_2 	
\to 	
X_1 	
\to 	
X_0 	
$ 	
, 	

 	
\item 	
$ 	
K 	
= 	
\text{hocolim 	
} 	
Y_n[n]$ 	
in 	
$ 	
D(\mathcal{C}_{total})$. 	

 	
\end{enumerate 	
} 	

 	
\end{lemma 	
} 	


 	
\begin{proof 	
} 	

 	
First 	
, 	
if 	
$ 	
K 	
= 	
\mathbf{Z}$ 	
, 	
then 	
this 	
is 	
the 	
construction 	
of 	

 	
Derived 	
Categories 	
, 	
Example 	
\ref{derived 	
- 	
example 	
- 	
key 	
- 	
postnikov 	
} 	

 	
applied 	
to 	
the 	
complex 	

 	
$ 	
$ 	

 	
\ldots 	
\to 	

 	
g_{2!}\mathbf{Z 	
} 	
\to 	

 	
g_{1!}\mathbf{Z 	
} 	
\to 	

 	
g_{0!}\mathbf{Z 	
} 	

 	
$ 	
$ 	

 	
in 	
$ 	
\textit{Ab}(\mathcal{C}_{total})$ 	
combined 	
with 	
the 	
fact 	
that 	

 	
this 	
complex 	
represents 	
$ 	
K 	
= 	
\mathbf{Z}$ 	
in 	
$ 	
D(\mathcal{C}_{total})$ 	

 	
by 	
Lemma 	
\ref{lemma 	
- 	
simplicial 	
- 	
resolution 	
- 	
Z 	
- 	
site}. 	

 	
The 	
general 	
case 	
follows 	
from 	
this 	
, 	
the 	
fact 	
that 	
the 	
exact 	
functor 	

 	
$ 	
- 	
\otimes^\mathbf{L}_\mathbf{Z 	
} 	
K$ 	
sends 	
Postnikov 	
systems 	
to 	

 	
Postnikov 	
systems 	
, 	
and 	

 	
that 	
$ 	
- 	
\otimes^\mathbf{L}_\mathbf{Z 	
} 	
K$ 	
commutes 	
with 	
homotopy 	
colimits 	
. 	

 	
\end{proof 	
} 	


 	
\begin{lemma 	
} 	

 	
\label{lemma 	
- 	
nullity 	
- 	
cartesian 	
- 	
objects 	
- 	
derived 	
} 	

 	
In 	
Situation 	
\ref{situation 	
- 	
simplicial 	
- 	
site}. 	

 	
If 	
$ 	
K 	
, 	
K 	
' 	
\in 	
D(\mathcal{C}_{total})$. 	

 	
Assume 	

 	
\begin{enumerate 	
} 	

 	
\item 	
$ 	
K$ 	
is 	
cartesian 	
, 	

 	
\item 	
$ 	
\Hom(K_i[i 	
] 	
, 	
K'_i 	
) 	
= 	
0 	
$ 	
for 	
$ 	
i 	
> 	
0 	
$ 	
, 	
and 	

 	
\item 	
$ 	
\Hom(K_i[i 	
+ 	
1 	
] 	
, 	
K'_i 	
) 	
= 	
0 	
$ 	
for 	
$ 	
i 	
\geq 	
0$. 	

 	
\end{enumerate 	
} 	

 	
Then 	
any 	
map 	
$ 	
K 	
\to 	
K'$ 	
which 	
induces 	
the 	
zero 	
map 	
$ 	
K_0 	
\to 	
K'_0 	
$ 	
is 	
zero 	
. 	

 	
\end{lemma 	
} 	


 	
\begin{proof 	
} 	

 	
Consider 	
the 	
objects 	
$ 	
X_n$ 	
and 	
the 	
Postnikov 	
system 	
$ 	
Y_n$ 	

 	
associated 	
to 	
$ 	
K$ 	
in 	
Lemma 	
\ref{lemma 	
- 	
abelian 	
- 	
postnikov}. 	

 	
As 	
$ 	
K 	
= 	
\text{hocolim 	
} 	
Y_n[n]$ 	
the 	
map 	
$ 	
K 	
\to 	
K'$ 	
induces 	

 	
a 	
compatible 	
family 	
of 	
morphisms 	
$ 	
Y_n[n 	
] 	
\to 	
K'$. 	

 	
By 	
( 	
1 	
) 	
and 	
Lemma 	
\ref{lemma 	
- 	
derived 	
- 	
cartesian 	
- 	
shriek 	
} 	
we 	
have 	

 	
$ 	
X_n 	
= 	
g_{n!}K_n$. 	
Since 	
$ 	
Y_0 	
= 	
X_0 	
$ 	
we 	
find 	
that 	

 	
$ 	
K_0 	
\to 	
K'_0 	
$ 	
being 	
zero 	
implies 	
$ 	
Y_0 	
\to 	
K'$ 	
is 	
zero 	
. 	

 	
Suppose 	
we 	
've 	
shown 	
that 	
the 	
map 	
$ 	
Y_n[n 	
] 	
\to 	
K'$ 	
is 	
zero 	

 	
for 	
some 	
$ 	
n 	
\geq 	
0$. 	
From 	
the 	
distinguished 	
triangle 	

 	
$ 	
$ 	

 	
Y_n[n 	
] 	
\to 	
Y_{n 	
+ 	
1}[n 	
+ 	
1 	
] 	
\to 	
X_{n 	
+ 	
1}[n 	
+ 	
1 	
] 	
\to 	
Y_n[n 	
+ 	
1 	
] 	

 	
$ 	
$ 	

 	
we 	
get 	
an 	
exact 	
sequence 	

 	
$ 	
$ 	

 	
\Hom(X_{n 	
+ 	
1}[n 	
+ 	
1 	
] 	
, 	
K 	
' 	
) 	
\to 	

 	
\Hom(Y_{n 	
+ 	
1}[n 	
+ 	
1 	
] 	
, 	
K 	
' 	
) 	
\to 	

 	
\Hom(Y_n[n 	
] 	
, 	
K 	
' 	
) 	

 	
$ 	
$ 	

 	
As 	
$ 	
X_{n 	
+ 	
1}[n 	
+ 	
1 	
] 	
= 	
g_{n 	
+ 	
1!}K_{n 	
+ 	
1}[n 	
+ 	
1]$ 	
the 	
first 	
group 	
is 	
equal 	
to 	

 	
$ 	
$ 	

 	
\Hom(K_{n 	
+ 	
1}[n 	
+ 	
1 	
] 	
, 	
K'_{n 	
+ 	
1 	
} 	
) 	

 	
$ 	
$ 	

 	
which 	
is 	
zero 	
by 	
assumption 	
( 	
2 	
) 	
. 	
By 	
induction 	
we 	
conclude 	
all 	
the 	
maps 	

 	
$ 	
Y_n[n 	
] 	
\to 	
K'$ 	
are 	
zero 	
. 	
Consider 	
the 	
defining 	
distinguished 	
triangle 	

 	
$ 	
$ 	

 	
\bigoplus 	
Y_n[n 	
] 	
\to 	

 	
\bigoplus 	
Y_n[n 	
] 	
\to 	

 	
K 	
\to 	

 	
( 	
\bigoplus 	
Y_n[n])[1 	
] 	

 	
$ 	
$ 	

 	
for 	
the 	
homotopy 	
colimit 	
. 	
Arguing 	
as 	
above 	
, 	
we 	
find 	
that 	
it 	
suffices 	

 	
to 	
show 	
that 	

 	
$ 	
$ 	

 	
\Hom((\bigoplus 	
Y_n[n])[1 	
] 	
, 	
K 	
' 	
) 	
= 	
\prod 	
\Hom(Y_n[n 	
+ 	
1 	
] 	
, 	
K 	
' 	
) 	

 	
$ 	
$ 	

 	
is 	
zero 	
for 	
all 	
$ 	
n 	
\geq 	
0$. 	
To 	
see 	
this 	
, 	
arguing 	
as 	
above 	
, 	

 	
it 	
suffices 	
to 	
show 	
that 	

 	
$ 	
$ 	

 	
\Hom(K_n[n 	
+ 	
1 	
] 	
, 	
K'_n 	
) 	
  	
= 	
0 	

 	
$ 	
$ 	

 	
for 	
all 	
$ 	
n 	
\geq 	
0 	
$ 	
which 	
follows 	
from 	
condition 	
( 	
3 	
) 	
. 	

 	
\end{proof 	
} 	


 	
\begin{lemma 	
} 	

 	
\label{lemma 	
- 	
hom 	
- 	
cartesian 	
- 	
objects 	
- 	
derived 	
} 	

 	
In 	
Situation 	
\ref{situation 	
- 	
simplicial 	
- 	
site}. 	

 	
If 	
$ 	
K 	
, 	
K 	
' 	
\in 	
D(\mathcal{C}_{total})$. 	

 	
Assume 	

 	
\begin{enumerate 	
} 	

 	
\item 	
$ 	
K$ 	
is 	
cartesian 	
, 	

 	
\item 	
$ 	
\Hom(K_i[i 	
- 	
1 	
] 	
, 	
K'_i 	
) 	
= 	
0 	
$ 	
for 	
$ 	
i 	
> 	
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 	
. 	

 	
Consider 	
the 	
objects 	
$ 	
X_n$ 	
and 	
Postnikov 	
system 	
$ 	
Y_n$ 	

 	
associated 	
to 	
$ 	
K$ 	
of 	
Lemma 	
\ref{lemma 	
- 	
abelian 	
- 	
postnikov}. 	

 	
By 	
( 	
1 	
) 	
and 	
Lemma 	
\ref{lemma 	
- 	
derived 	
- 	
cartesian 	
- 	
shriek 	
} 	
we 	
have 	

 	
$ 	
X_n 	
= 	
g_{n!}K_n$. 	
In 	
particular 	
, 	
the 	
map 	
$ 	
K_0 	
\to 	
K'_0 	
$ 	

 	
induces 	
a 	
morphism 	
$ 	
X_0 	
\to 	
K'$. 	
Since 	
$ 	
\{K_n 	
\to 	
K'_n\}$ 	

 	
is 	
a 	
morphism 	
of 	
systems 	
, 	
a 	
computation 	
( 	
omitted 	
) 	
shows 	
that 	

 	
the 	
composition 	

 	
$ 	
$ 	

 	
X_1 	
\to 	
X_0 	
\to 	
K 	
' 	

 	
$ 	
$ 	

 	
is 	
zero 	
. 	
As 	
$ 	
Y_0 	
= 	
X_0 	
$ 	
and 	
as 	
$ 	
Y_1 	
$ 	
fits 	
into 	
a 	
distinguished 	

 	
triangle 	

 	
$ 	
$ 	

 	
Y_1 	
\to 	
X_1 	
\to 	
Y_0 	
\to 	
Y_1[1 	
] 	

 	
$ 	
$ 	

 	
we 	
conclude 	
that 	
there 	
exists 	
a 	
morphism 	
$ 	
Y_1[1 	
] 	
\to 	
K'$ 	
whose 	

 	
composition 	
with 	
$ 	
X_0 	
= 	
Y_0 	
\to 	
Y_1[1]$ 	
is 	
the 	
morphism 	
$ 	
X_0 	
\to 	
K'$ 	

 	
given 	
above 	
. 	
Suppose 	
given 	
a 	
map 	
$ 	
Y_n[n 	
] 	
\to 	
K'$ 	
for 	
$ 	
n 	
\geq 	
1$. 	

 	
From 	
the 	
distinguished 	
triangle 	

 	
$ 	
$ 	

 	
X_{n 	
+ 	
1}[n 	
] 	
\to 	
Y_n[n 	
] 	
\to 	
Y_{n 	
+ 	
1}[n 	
+ 	
1 	
] 	
\to 	
X_{n 	
+ 	
1}[n 	
+ 	
1 	
] 	

 	
$ 	
$ 	

 	
we 	
get 	
an 	
exact 	
sequence 	

 	
$ 	
$ 	

 	
\Hom(Y_{n 	
+ 	
1}[n 	
+ 	
1 	
] 	
, 	
K 	
' 	
) 	
\to 	

 	
\Hom(Y_n[n 	
] 	
, 	
K 	
' 	
) 	
\to 	

 	
\Hom(X_{n 	
+ 	
1}[n 	
] 	
, 	
K 	
' 	
) 	

 	
$ 	
$ 	

 	
As 	
$ 	
X_{n 	
+ 	
1}[n 	
] 	
= 	
g_{n 	
+ 	
1!}K_{n 	
+ 	
1}[n]$ 	
the 	
last 	
group 	
is 	
equal 	
to 	

 	
$ 	
$ 	

 	
\Hom(K_{n 	
+ 	
1}[n 	
] 	
, 	
K'_{n 	
+ 	
1 	
} 	
) 	

 	
$ 	
$ 	

 	
which 	
is 	
zero 	
by 	
assumption 	
( 	
2 	
) 	
. 	
By 	
induction 	
we 	
get 	
a 	
system 	
of 	

 	
maps 	
$ 	
Y_n[n 	
] 	
\to 	
K'$ 	
compatible 	
with 	
transition 	
maps 	
and 	
reducing 	

 	
to 	
the 	
given 	
map 	
on 	
$ 	
Y_0$. 	
This 	
produces 	
a 	
map 	

 	
$ 	
$ 	

 	
\gamma 	
: 	

 	
K 	
= 	
\text{hocolim 	
} 	
Y_n[n 	
] 	

 	
\longrightarrow 	

 	
K 	
' 	

 	
$ 	
$ 	

 	
This 	
map 	
in 	
any 	
case 	
has 	
the 	
property 	
that 	
the 	
diagram 	

 	
$ 	
$ 	

 	
\xymatrix 	
{ 	

 	
X_0 	
\ar[rd 	
] 	
\ar[r 	
] 	
& 	

 	
K 	
\ar[d]^\gamma 	
\\ 	

 	
& 	
K 	
' 	

 	
} 	

 	
$ 	
$ 	

 	
is 	
commutative 	
. 	
Restricting 	
to 	

 	
$ 	
\mathcal{C}_0 	
$ 	
we 	
deduce 	
that 	
the 	
map 	
$ 	
\gamma_0 	
: 	
K_0 	
\to 	
K'_0 	
$ 	

 	
is 	
the 	
same 	
as 	
the 	
first 	
map 	
$ 	
K_0 	
\to 	
K'_0 	
$ 	
of 	
the 	
morphism 	

 	
of 	
simplicial 	
systems 	
. 	
Since 	
$ 	
K$ 	
is 	
cartesian 	
, 	
this 	
easily 	
gives 	
that 	

 	
$ 	
\{\gamma_n\}$ 	
is 	
the 	
map 	
of 	
simplicial 	
systems 	
we 	
started 	
out 	
with 	
. 	

 	
\end{proof 	
} 	


 	
\begin{lemma 	
} 	

 	
\label{lemma 	
- 	
cartesian 	
- 	
object 	
- 	
derived 	
- 	
from 	
- 	
simplicial 	
} 	

 	
In 	
Situation 	
\ref{situation 	
- 	
simplicial 	
- 	
site}. 	
Let 	

 	
$ 	
( 	
K_n 	
, 	
K_\varphi)$ 	
be 	
a 	
simplicial 	TYPE
system 	
of 	
the 	
derived 	
category 	
. 	

 	
Assume 	

 	
\begin{enumerate 	
} 	

 	
\item 	
$ 	
( 	
K_n 	
, 	
K_\varphi)$ 	
is 	
cartesian 	
, 	

 	
\item 	
$ 	
\Hom(K_i[t 	
] 	
, 	
K_i 	
) 	
= 	
0 	
$ 	
for 	
$ 	
i 	
\geq 	
0 	
$ 	
and 	
$ 	
t 	
> 	
0$. 	

 	
\end{enumerate 	
} 	

 	
Then 	
there 	
exists 	
a 	
cartesian 	
object 	
$ 	
K$ 	
of 	
$ 	
D(\mathcal{C}_{total})$ 	

 	
whose 	
associated 	
simplicial 	
system 	
is 	
isomorphic 	
to 	
$ 	
( 	
K_n 	
, 	
K_\varphi)$. 	

 	
\end{lemma 	
} 	


 	
\begin{proof 	
} 	

 	
Set 	
$ 	
X_n 	
= 	
g_{n!}K_n$ 	
in 	
$ 	
D(\mathcal{C}_{total})$. 	
For 	
each 	
$ 	
n 	
\geq 	
1 	
$ 	

 	
we 	
have 	

 	
$ 	
$ 	

 	
\Hom(X_n 	
, 	
X_{n 	
- 	
1 	
} 	
) 	
= 	

 	
\Hom(K_n 	
, 	
g_n^{-1}g_{n 	
- 	
1!}K_{n 	
- 	
1 	
} 	
) 	
= 	

 	
\bigoplus\nolimits_{\varphi 	
: 	
[ 	
n 	
- 	
1 	
] 	
\to 	
[ 	
n 	
] 	
} 	

 	
\Hom(K_n 	
, 	
f_\varphi^{-1}K_{n 	
- 	
1 	
} 	
) 	

 	
$ 	
$ 	

 	
Thus 	
we 	
get 	
a 	
map 	
$ 	
X_n 	
\to 	
X_{n 	
- 	
1}$ 	
corresponding 	
to 	
the 	

 	
alternating 	
sum 	
of 	
the 	
maps 	

 	
$ 	
K_\varphi^{-1 	
} 	
: 	
K_n 	
\to 	
f_\varphi^{-1}K_{n 	
- 	
1}$ 	

 	
where 	
$ 	
\varphi$ 	
runs 	
over 	
$ 	
\delta^n_0 	
, 	
\ldots 	
, 	
\delta^n_n$. 	

 	
We 	
can 	
do 	
this 	
because 	
$ 	
K_\varphi$ 	
is 	
invertible 	
by 	
assumption 	
( 	
1 	
) 	
. 	

 	
Please 	
observe 	
the 	
similarity 	
with 	
the 	
definition 	
of 	
the 	
maps 	

 	
in 	
the 	
proof 	
of 	
Lemma 	
\ref{lemma 	
- 	
simplicial 	
- 	
resolution 	
- 	
Z 	
- 	
site}. 	

 	
We 	
obtain 	
a 	
complex 	

 	
$ 	
$ 	

 	
\ldots 	
\to 	
X_2 	
\to 	
X_1 	
\to 	
X_0 	

 	
$ 	
$ 	

 	
in 	
$ 	
D(\mathcal{C}_{total})$. 	
We 	
omit 	
the 	
computation 	
which 	
shows 	

 	
that 	
the 	
compositions 	
are 	
zero 	
. 	
By 	

 	
Derived 	
Categories 	
, 	
Lemma 	
\ref{derived 	
- 	
lemma 	
- 	
existence 	
- 	
postnikov 	
- 	
system 	
} 	

 	
if 	
we 	
have 	

 	
$ 	
$ 	

 	
\Hom(X_i[i 	
- 	
j 	
- 	
2 	
] 	
, 	
X_j 	
) 	
= 	
0\text 	
{ 	
for 	
} 	
i 	
> 	
j 	
+ 	
2 	

 	
$ 	
$ 	

 	
then 	
we 	
can 	
extend 	
this 	
complex 	
to 	
a 	
Postnikov 	
system 	
. 	

 	
The 	
group 	
is 	
equal 	
to 	

 	
$ 	
$ 	

 	
\Hom(K_i[i 	
- 	
j 	
- 	
2 	
] 	
, 	
g_i^{-1}g_{j!}K_j 	
) 	

 	
$ 	
$ 	

 	
Again 	
using 	
that 	
$ 	
( 	
K_n 	
, 	
K_\varphi)$ 	
is 	
cartesian 	
we 	
see 	
that 	

 	
$ 	
g_i^{-1}g_{j!}K_j$ 	
is 	
isomorphic 	
to 	
a 	
finite 	
direct 	
sum 	
of 	
copies 	
of 	

 	
$ 	
K_i$. 	
Hence 	
the 	
group 	
vanishes 	
by 	
assumption 	
( 	
2 	
) 	
. 	

 	
Let 	
the 	
Postnikov 	
system 	
be 	
given 	
by 	
$ 	
Y_0 	
= 	
X_0 	
$ 	
and 	
distinguished 	

 	
sequences 	
$ 	
Y_n 	
\to 	
X_n 	
\to 	
Y_{n 	
- 	
1 	
} 	
\to 	
Y_n[1]$ 	
for 	
$ 	
n 	
\geq 	
1$. 	

 	
We 	
set 	

 	
$ 	
$ 	

 	
K 	
= 	
\text{hocolim 	
} 	
Y_n[n 	
] 	

 	
$ 	
$ 	

 	
To 	
finish 	
the 	
proof 	
we 	
have 	
to 	
show 	
that 	
$ 	
g_m^{-1}K$ 	
is 	
isomorphic 	

 	
to 	
$ 	
K_m$ 	
for 	
all 	
$ 	
m$ 	
compatible 	
with 	
the 	
maps 	
$ 	
K_\varphi$. 	
Observe 	
that 	

 	
$ 	
$ 	

 	
g_m^{-1 	
} 	
K 	
= 	
\text{hocolim 	
} 	
g_m^{-1}Y_n[n 	
] 	

 	
$ 	
$ 	

 	
and 	
that 	
$ 	
g_m^{-1}Y_n[n]$ 	
is 	
a 	
Postnikov 	
system 	
for 	
$ 	
g_m^{-1}X_n$. 	

 	
Consider 	
the 	
isomorphisms 	

 	
$ 	
$ 	

 	
g_m^{-1}X_n 	
= 	

 	
\bigoplus\nolimits_{\varphi 	
: 	
[ 	
n 	
] 	
\to 	
[ 	
m 	
] 	
} 	
f_\varphi^{-1}K_n 	

 	
\xrightarrow{\bigoplus 	
K_\varphi 	
} 	

 	
\bigoplus\nolimits_{\varphi 	
: 	
[ 	
n 	
] 	
\to 	
[ 	
m 	
] 	
} 	
K_m 	

 	
$ 	
$ 	

 	
These 	
maps 	
define 	
an 	
isomorphism 	
of 	
complexes 	

 	
$ 	
$ 	

 	
\xymatrix 	
{ 	

 	
\ldots 	
\ar[r 	
] 	
& 	

 	
g_m^{-1}X_2 	
\ar[r 	
] 	
\ar[d 	
] 	
& 	

 	
g_m^{-1}X_1 	
\ar[r 	
] 	
\ar[d 	
] 	
& 	

 	
g_m^{-1}X_0 	
\ar[d 	
] 	
\\ 	

 	
\ldots 	
\ar[r 	
] 	
& 	

 	
\bigoplus\nolimits_{\varphi 	
: 	
[ 	
2 	
] 	
\to 	
[ 	
m 	
] 	
} 	
K_m 	
\ar[r 	
] 	
& 	

 	
\bigoplus\nolimits_{\varphi 	
: 	
[ 	
1 	
] 	
\to 	
[ 	
m 	
] 	
} 	
K_m 	
\ar[r 	
] 	
& 	

 	
\bigoplus\nolimits_{\varphi 	
: 	
[ 	
0 	
] 	
\to 	
[ 	
m 	
] 	
} 	
K_m 	

 	
} 	

 	
$ 	
$ 	

 	
in 	
$ 	
D(\mathcal{C}_m)$ 	
where 	
the 	
arrows 	
in 	
the 	
bottom 	
row 	
are 	
as 	

 	
in 	
the 	
proof 	
of 	
Lemma 	
\ref{lemma 	
- 	
simplicial 	
- 	
resolution 	
- 	
Z 	
- 	
site}. 	

 	
The 	
squares 	
commute 	
by 	
our 	
choice 	
of 	
the 	
arrows 	
of 	
the 	
complex 	

 	
$ 	
\ldots 	
\to 	
X_2 	
\to 	
X_1 	
\to 	
X_0 	
$ 	
; 	
we 	
omit 	
the 	
computation 	
. 	

 	
The 	
bottom 	
row 	
complex 	
has 	
a 	
postnikov 	
tower 	
given 	
by 	

 	
$ 	
$ 	

 	
Y'_{m 	
, 	
n 	
} 	
= 	

 	
\left(\bigoplus\nolimits_{\varphi 	
: 	
[ 	
n 	
] 	
\to 	
[ 	
m 	
] 	
} 	
\mathbf{Z 	
} 	
\to 	

 	
\ldots 	
\to 	

 	
\bigoplus\nolimits_{\varphi 	
: 	
[ 	
0 	
] 	
\to 	
[ 	
m 	
] 	
} 	
\mathbf{Z}\right)[-n 	
] 	

 	
\otimes^\mathbf{L}_\mathbf{Z 	
} 	
K_m 	

 	
$ 	
$ 	

 	
and 	
$ 	
\text{hocolim 	
} 	
Y'_{m 	
, 	
n 	
} 	
= 	
K_m$ 	

 	
( 	
please 	
compare 	
with 	
the 	
proof 	
of 	
Lemma 	
\ref{lemma 	
- 	
abelian 	
- 	
postnikov 	
} 	

 	
and 	
Derived 	
Categories 	
, 	
Example 	
\ref{derived 	
- 	
example 	
- 	
key 	
- 	
postnikov 	
} 	
) 	
. 	

 	
Applying 	
the 	
second 	
part 	
of 	

 	
Derived 	
Categories 	
, 	
Lemma 	
\ref{derived 	
- 	
lemma 	
- 	
existence 	
- 	
postnikov 	
- 	
system 	
} 	

 	
the 	
vertical 	
maps 	
in 	
the 	
big 	
diagram 	
extend 	
to 	
an 	
isomorphism 	

 	
of 	
Postnikov 	
systems 	
provided 	
we 	
have 	

 	
$ 	
$ 	

 	
\Hom(g_m^{-1}X_i[i 	
- 	
j 	
- 	
1 	
] 	
, 	
\bigoplus\nolimits_{\varphi 	
: 	
[ 	
j 	
] 	
\to 	
[ 	
m 	
] 	
} 	
K_m 	
) 	

 	
= 	
0\text 	
{ 	
for 	
} 	
i 	
> 	
j 	
+ 	
1 	

 	
$ 	
$ 	

 	
The 	
is 	
true 	
if 	
$ 	
\Hom(K_m[i 	
- 	
j 	
- 	
1 	
] 	
, 	
K_m 	
) 	
= 	
0 	
$ 	
for 	
$ 	
i 	
> 	
j 	
+ 	
1 	
$ 	

 	
which 	
holds 	
by 	
assumption 	
( 	
2 	
) 	
. 	
Choose 	
an 	
isomorphism 	
given 	

 	
by 	
$ 	
\gamma_{m 	
, 	
n 	
} 	
: 	
g_m^{-1}Y_n 	
\to 	
Y'_{m 	
, 	
n}$ 	
of 	
Postnikov 	
systems 	

 	
in 	
$ 	
D(\mathcal{C}_m)$. 	
By 	
uniqueness 	
of 	
homotopy 	
colimits 	
, 	

 	
we 	
can 	
find 	
an 	
isomorphism 	

 	
$ 	
$ 	

 	
g_m^{-1 	
} 	
K 	
= 	
\text{hocolim 	
} 	
g_m^{-1}Y_n[n 	
] 	

 	
\xrightarrow{\gamma_m 	
} 	

 	
\text{hocolim 	
} 	
Y'_{m 	
, 	
n 	
} 	
= 	
K_m 	

 	
$ 	
$ 	

 	
compatible 	
with 	
$ 	
\gamma_{m 	
, 	
n}$. 	


 	
\medskip\noindent 	

 	
We 	
still 	
have 	
to 	
prove 	
that 	
the 	
maps 	
$ 	
\gamma_m$ 	
fit 	
into 	
commutative 	
diagrams 	

 	
$ 	
$ 	

 	
\xymatrix 	
{ 	

 	
f_\varphi^{-1}g_m^{-1}K 	
\ar[d]_{f_\varphi^{-1}\gamma_m 	
} 	
\ar[r]_{K(\varphi 	
) 	
} 	
& 	

 	
g_n^{-1}K 	
\ar[d]^{\gamma_n 	
} 	
\\ 	

 	
f_\varphi^{-1}K_m 	
\ar[r]^{K_\varphi 	
} 	
& 	

 	
K_n 	

 	
} 	

 	
$ 	
$ 	

 	
for 	
every 	
$ 	
\varphi 	
: 	
[ 	
m 	
] 	
\to 	
[ 	
n]$. 	
Consider 	
the 	
diagram 	

 	
$ 	
$ 	

 	
\xymatrix 	
{ 	

 	
f_\varphi^{-1}(\bigoplus_{\psi 	
: 	
[ 	
0 	
] 	
\to 	
[ 	
m 	
] 	
} 	
f_\psi^{-1}K_0 	
) 	

 	
\ar@{=}[r 	
] 	
\ar[d]_{f_\varphi^{-1}(\bigoplus 	
K_\psi 	
) 	
} 	
& 	

 	
f_\varphi^{-1}g_m^{-1}X_0 	
\ar[d 	
] 	
\ar[r]_{X_0(\varphi 	
) 	
} 	
& 	

 	
g_n^{-1}X_0 	
\ar[d 	
] 	
& 	

 	
\bigoplus_{\chi 	
: 	
[ 	
0 	
] 	
\to 	
[ 	
n 	
] 	
} 	
f_\chi^{-1}K_0 	

 	
\ar@{=}[l 	
] 	
\ar[d]^{\bigoplus 	
K_\chi 	
} 	
\\ 	

 	
f_\varphi^{-1}(\bigoplus_{\psi 	
: 	
[ 	
0 	
] 	
\to 	
[ 	
m 	
] 	
} 	
K_m 	
) 	
\ar@{=}[d 	
] 	
& 	

 	
f_\varphi^{-1}g_m^{-1}K 	
\ar[d]_{f_\varphi^{-1}\gamma_m 	
} 	
\ar[r]_{K(\varphi 	
) 	
} 	
& 	

 	
g_n^{-1}K 	
\ar[d]^{\gamma_n 	
} 	
& 	

 	
\bigoplus_{\chi 	
: 	
[ 	
0 	
] 	
\to 	
[ 	
n 	
] 	
} 	
K_n 	
\ar@{=}[d 	
] 	
\\ 	

 	
f_\varphi^{-1}Y'_{0 	
, 	
m 	
} 	
\ar[r 	
] 	
& 	

 	
f_\varphi^{-1}K_m 	
\ar[r]^{K_\varphi 	
} 	
& 	

 	
K_n 	
& 	

 	
Y'_{0 	
, 	
n 	
} 	
\ar[l 	
] 	

 	
} 	

 	
$ 	
$ 	

 	
The 	
top 	
middle 	
square 	
is 	
commutative 	
as 	
$ 	
X_0 	
\to 	
K$ 	
is 	
a 	
morphism 	

 	
of 	
simplicial 	
objects 	
. 	
The 	
left 	
, 	
resp.\ 	
the 	
right 	
rectangles 	
are 	

 	
commutative 	
as 	
$ 	
\gamma_m$ 	
, 	
resp.\ 	
$ 	
\gamma_n$ 	
is 	
compatible 	
with 	

 	
$ 	
\gamma_{0 	
, 	
m}$ 	
, 	
resp.\ 	
$ 	
\gamma_{0 	
, 	
n}$ 	
which 	
are 	
the 	
arrows 	

 	
$ 	
\bigoplus 	
K_\psi$ 	
and 	
$ 	
\bigoplus 	
K_\chi$ 	
in 	
the 	
diagram 	
. 	

 	
Going 	
around 	
the 	
outer 	
rectangle 	
of 	
the 	
diagram 	

 	
is 	
commutative 	
as 	
$ 	
( 	
K_n 	
, 	
K_\varphi)$ 	
is 	
a 	
simplical 	
system 	

 	
and 	
the 	
map 	
$ 	
X_0(\varphi)$ 	
is 	
given 	
by 	
the 	
obvious 	
identifications 	

 	
$ 	
f_\varphi^{-1}f_\psi^{-1}K_0 	
= 	
f_{\varphi 	
\circ 	
\psi}^{-1}K_0$. 	

 	
Note 	
that 	
the 	
arrow 	
$ 	
\bigoplus_\psi 	
K_m 	
\to 	
Y'_{0 	
, 	
m 	
} 	
\to 	
K_m$ 	

 	
induces 	
an 	
isomorphism 	
on 	
any 	
of 	
the 	
direct 	
summands 	

 	
( 	
because 	
of 	
our 	
explicit 	
construction 	
of 	
the 	
Postnikov 	

 	
systems 	
$ 	
Y'_{i 	
, 	
j}$ 	
above 	
) 	
. 	

 	
Hence 	
, 	
if 	
we 	
take 	
a 	
direct 	
summand 	
summand 	
of 	

 	
the 	
upper 	
left 	
and 	
corner 	
, 	
then 	
this 	
maps 	
isomorphically 	
to 	

 	
$ 	
f_\varphi^{-1}g_m^{-1}K$ 	
as 	
$ 	
\gamma_m$ 	
is 	
an 	
isomorphism 	
. 	

 	
Working 	
out 	
what 	
the 	
above 	
says 	
, 	

 	
but 	
looking 	
only 	
at 	
this 	
direct 	
summand 	
we 	
conclude 	
the 	
lower 	

 	
middle 	
square 	
commutes 	
as 	
we 	
well 	
. 	
This 	
concludes 	
the 	
proof 	
. 	

 	
\end{proof 	
} 	










 	
\section{Simplicial 	
systems 	
of 	
the 	
derived 	
category 	
: 	
modules 	
} 	

 	
\label{section 	
- 	
glueing 	
- 	
modules 	
} 	


 	
\noindent 	

 	
In 	
this 	
section 	
we 	
are 	
going 	
to 	
prove 	
a 	
special 	
case 	
of 	

 	
\cite[Proposition 	
3.2.9]{BBD 	
} 	
in 	
the 	
setting 	
of 	
derived 	

 	
categories 	
of 	
$ 	
\mathcal{O}$-modules 	
. 	
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 	TYPE
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 	
: 	

 	
Lf_{\varphi 	
\circ 	
\psi}^*K_l 	
= 	
Lf_\varphi^ 	
* 	
Lf_\psi^*K_l 	

 	
\longrightarrow 	

 	
K_n 	

 	
$ 	
$ 	

 	
for 	
any 	
morphisms 	
$ 	
\varphi 	
: 	
[ 	
m 	
] 	
\to 	
[ 	
n]$ 	
and 	
$ 	
\psi 	
: 	
[ 	
l 	
] 	
\to 	
[ 	
m]$ 	
of 	
$ 	
\Delta$. 	

 	
We 	
say 	
the 	
simplicial 	
system 	
is 	
{ 	
\it 	
cartesian 	
} 	
if 	
the 	
maps 	
$ 	
K_\varphi$ 	

 	
are 	
isomorphisms 	
for 	
all 	
$ 	
\varphi$. 	

 	
Given 	
two 	
simplicial 	
systems 	
of 	
the 	
derived 	
category 	

 	
there 	
is 	
an 	
obvious 	
notion 	
of 	
a 	

 	
{ 	
\it 	
morphism 	
of 	
simplicial 	
systems 	
of 	
the 	
derived 	
category 	
of 	
modules}. 	

 	
\end{definition 	
} 	


 	
\noindent 	

 	
We 	
have 	
given 	
this 	
notion 	
a 	
ridiculously 	
long 	
name 	
intentionally 	
. 	

 	
The 	
goal 	
is 	
to 	
show 	
that 	
a 	
simplicial 	
system 	
of 	
the 	
derived 	
category 	

 	
of 	
modules 	
comes 	
from 	
an 	
object 	
of 	
$ 	
D(\mathcal{O})$ 	
under 	
certain 	

 	
hypotheses 	
. 	


 	
\begin{lemma 	
} 	

 	
\label{lemma 	
- 	
cartesian 	
- 	
objects 	
- 	
derived 	
- 	
modules 	
} 	

 	
In 	
Situation 	
\ref{situation 	
- 	
simplicial 	
- 	
site 	
} 	
let 	
$ 	
\mathcal{O}$ 	
be 	
a 	

 	TYPE
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 	
. 	

 	
If 	
$ 	
K$ 	
is 	
cartesian 	
, 	
so 	
is 	
the 	
system 	
. 	

 	
\end{lemma 	
} 	


 	
\begin{proof 	
} 	

 	
This 	
is 	
immediate 	
from 	
the 	
definitions 	
. 	

 	
\end{proof 	
} 	


 	
\begin{lemma 	
} 	

 	
\label{lemma 	
- 	
modules 	
- 	
postnikov 	
} 	

 	
In 	
Situation 	
\ref{situation 	
- 	
simplicial 	
- 	
site 	
} 	
let 	
$ 	
\mathcal{O}$ 	

 	
be 	
a 	
sheaf 	TYPE
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 	
\to 	
X_2 	
\to 	
X_1 	
\to 	
X_0 	
$ 	
, 	

 	
\item 	
$ 	
K 	
= 	
\text{hocolim 	
} 	
Y_n[n]$ 	
in 	
$ 	
D(\mathcal{O})$. 	

 	
\end{enumerate 	
} 	

 	
\end{lemma 	
} 	


 	
\begin{proof 	
} 	

 	
First 	
, 	
if 	
$ 	
K 	
= 	
\mathcal{O}$ 	
, 	
then 	
this 	
is 	
the 	
construction 	
of 	

 	
Derived 	
Categories 	
, 	
Example 	
\ref{derived 	
- 	
example 	
- 	
key 	
- 	
postnikov 	
} 	

 	
applied 	
to 	
the 	
complex 	

 	
$ 	
$ 	

 	
\ldots 	
\to 	

 	
g_{2!}\mathcal{O}_2 	
\to 	

 	
g_{1!}\mathcal{O}_1 	
\to 	

 	
g_{0!}\mathcal{O}_0 	

 	
$ 	
$ 	

 	
in 	
$ 	
\textit{Ab}(\mathcal{C}_{total})$ 	
combined 	
with 	
the 	
fact 	
that 	

 	
this 	
complex 	
represents 	
$ 	
K 	
= 	
\mathcal{O}$ 	
in 	
$ 	
D(\mathcal{C}_{total})$ 	

 	
by 	
Lemma 	
\ref{lemma 	
- 	
simplicial 	
- 	
resolution 	
- 	
ringed}. 	

 	
The 	
general 	
case 	
follows 	
from 	
this 	
, 	
the 	
fact 	
that 	
the 	
exact 	
functor 	

 	
$ 	
- 	
\otimes^\mathbf{L}_\mathcal{O 	
} 	
K$ 	
sends 	
Postnikov 	
systems 	
to 	

 	
Postnikov 	
systems 	
, 	
and 	

 	
that 	
$ 	
- 	
\otimes^\mathbf{L}_\mathcal{O 	
} 	
K$ 	
commutes 	
with 	
homotopy 	
colimits 	
. 	

 	
\end{proof 	
} 	


 	
\begin{lemma 	
} 	

 	
\label{lemma 	
- 	
nullity 	
- 	
cartesian 	
- 	
modules 	
- 	
derived 	
} 	

 	
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 	
] 	
, 	
K'_i 	
) 	
= 	
0 	
$ 	
for 	
$ 	
i 	
> 	
0 	
$ 	
, 	
and 	

 	
\item 	
$ 	
\Hom(K_i[i 	
+ 	
1 	
] 	
, 	
K'_i 	
) 	
= 	
0 	
$ 	
for 	
$ 	
i 	
\geq 	
0$. 	

 	
\end{enumerate 	
} 	

 	
Then 	
any 	
map 	
$ 	
K 	
\to 	
K'$ 	
which 	
induces 	
the 	
zero 	
map 	
$ 	
K_0 	
\to 	
K'_0 	
$ 	
is 	
zero 	
. 	

 	
\end{lemma 	
} 	


 	
\begin{proof 	
} 	

 	
The 	
proof 	
is 	
exactly 	
the 	
same 	
as 	
the 	
proof 	
of 	

 	
Lemma 	
\ref{lemma 	
- 	
nullity 	
- 	
cartesian 	
- 	
objects 	
- 	
derived 	
} 	
except 	
using 	

 	
Lemma 	
\ref{lemma 	
- 	
modules 	
- 	
postnikov 	
} 	
instead 	
of 	

 	
Lemma 	
\ref{lemma 	
- 	
abelian 	
- 	
postnikov}. 	

 	
\end{proof 	
} 	


 	
\begin{lemma 	
} 	

 	
\label{lemma 	
- 	
hom 	
- 	
cartesian 	
- 	
modules 	
- 	
derived 	
} 	

 	
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 	
> 	
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{O})$. 	

 	
\end{lemma 	
} 	


 	
\begin{proof 	
} 	

 	
The 	
proof 	
is 	
exactly 	
the 	
same 	
as 	
the 	
proof 	
of 	

 	
Lemma 	
\ref{lemma 	
- 	
hom 	
- 	
cartesian 	
- 	
objects 	
- 	
derived 	
} 	
except 	
using 	

 	
Lemma 	
\ref{lemma 	
- 	
modules 	
- 	
postnikov 	
} 	
instead 	
of 	

 	
Lemma 	
\ref{lemma 	
- 	
abelian 	
- 	
postnikov}. 	

 	
\end{proof 	
} 	


 	
\begin{lemma 	
} 	

 	
\label{lemma 	
- 	
cartesian 	
- 	
module 	
- 	
derived 	
- 	
from 	
- 	
simplicial 	
} 	

 	
In 	
Situation 	
\ref{situation 	
- 	
simplicial 	
- 	
site 	
} 	
let 	
$ 	
\mathcal{O}$ 	
be 	

 	
a 	
sheaf 	
of 	
rings 	
on 	
$ 	
\mathcal{C}_{total}$. 	
Let 	

 	
$ 	
( 	
K_n 	
, 	
K_\varphi)$ 	
be 	
a 	
simplicial 	TYPE
system 	
of 	
the 	
derived 	
category 	

 	
of 	
modules 	
. 	
Assume 	

 	
\begin{enumerate 	
} 	

 	
\item 	
$ 	
f_\varphi^{-1}\mathcal{O}_n 	
\to 	
\mathcal{O}_m$ 	
is 	
flat 	
for 	

 	
$ 	
\varphi 	
: 	
[ 	
m 	
] 	
\to 	
[ 	
n]$ 	
, 	

 	
\item 	
$ 	
( 	
K_n 	
, 	
K_\varphi)$ 	
is 	
cartesian 	
, 	

 	
\item 	
$ 	
\Hom(K_i[t 	
] 	
, 	
K_i 	
) 	
= 	
0 	
$ 	
for 	
$ 	
i 	
\geq 	
0 	
$ 	
and 	
$ 	
t 	
> 	
0$. 	

 	
\end{enumerate 	
} 	

 	
Then 	
there 	
exists 	
a 	
cartesian 	
object 	
$ 	
K$ 	
of 	
$ 	
D(\mathcal{O})$ 	

 	
whose 	
associated 	
simplicial 	
system 	
is 	
isomorphic 	
to 	
$ 	
( 	
K_n 	
, 	
K_\varphi)$. 	

 	
\end{lemma 	
} 	


 	
\begin{proof 	
} 	

 	
The 	
proof 	
is 	
exactly 	
the 	
same 	
as 	
the 	
proof 	
of 	

 	
Lemma 	
\ref{lemma 	
- 	
cartesian 	
- 	
object 	
- 	
derived 	
- 	
from 	
- 	
simplicial 	
} 	

 	
with 	
the 	
following 	
changes 	

 	
\begin{enumerate 	
} 	

 	
\item 	
use 	
$ 	
g_n^ 	
* 	
= 	
Lg_n^*$ 	
everywhere 	
instead 	
of 	
$ 	
g_n^{-1}$ 	
, 	

 	
\item 	
use 	
$ 	
f_\varphi^ 	
* 	
= 	
Lf_\varphi^*$ 	
everywhere 	
instead 	
of 	
$ 	
f_\varphi^{-1}$ 	
, 	

 	
\item 	
refer 	
to 	
Lemma 	
\ref{lemma 	
- 	
simplicial 	
- 	
resolution 	
- 	
ringed 	
} 	

 	
instead 	
of 	
Lemma 	
\ref{lemma 	
- 	
simplicial 	
- 	
resolution 	
- 	
Z 	
- 	
site 	
} 	
, 	

 	
\item 	
in 	
the 	
construction 	
of 	
$ 	
Y'_{m 	
, 	
n}$ 	
use 	

 	
$ 	
\mathcal{O}_m$ 	
instead 	
of 	
$ 	
\mathbf{Z}$ 	
, 	

 	
\item 	
compare 	
with 	
the 	
proof 	
of 	
Lemma 	
\ref{lemma 	
- 	
modules 	
- 	
postnikov 	
} 	

 	
rather 	
than 	
the 	
proof 	
of 	
Lemma 	
\ref{lemma 	
- 	
abelian 	
- 	
postnikov}. 	

 	
\end{enumerate 	
} 	

 	
This 	
ends 	
the 	
proof 	
. 	

 	
\end{proof 	
} 	








 	
\section{The 	
site 	
associated 	
to 	
a 	
semi 	
- 	
representable 	
object 	
} 	

 	
\label{section 	
- 	
semi 	
- 	
representable 	
} 	


 	
\noindent 	

 	
Let 	
$ 	
\mathcal{C}$ 	
be 	
a 	
site 	TYPE
. 	TYPE
Recall 	
that 	
a 	
{ 	
\it 	
semi 	
- 	
representable 	
object 	
} 	

 	
of 	
$ 	
\mathcal{C}$ 	
is 	
simply 	
a 	
family 	
$ 	
\{U_i\}_{i 	
\in 	
I}$ 	

 	
of 	
objects 	
of 	
$ 	
\mathcal{C}$. 	
A 	

 	
{ 	
\it 	
morphism 	
$ 	
\{U_i\}_{i 	
\in 	
I 	
} 	
\to 	
\{V_j\}_{j 	
\in 	
J}$ 	
of 	

 	
semi 	
- 	
representable 	
objects 	
} 	
is 	
given 	
by 	
a 	
map 	
$ 	
\alpha 	
: 	
I 	
\to 	
J$ 	

 	
and 	
for 	
every 	
$ 	
i 	
\in 	
I$ 	
a 	
morphism 	
$ 	
f_i 	
: 	
U_i 	
\to 	
V_{\alpha(i)}$ 	

 	
of 	
$ 	
\mathcal{C}$. 	

 	
The 	
category 	
of 	
semi 	
- 	
representable 	
objects 	
of 	
$ 	
\mathcal{C}$ 	

 	
is 	
denoted 	
$ 	
\text{SR}(\mathcal{C})$. 	

 	
See 	
Hypercoverings 	
, 	
Definition 	
\ref{hypercovering 	
- 	
definition 	
- 	
SR 	
} 	

 	
and 	
the 	
enclosing 	
section 	
for 	
more 	
information 	
. 	


 	
\medskip\noindent 	

 	
For 	
a 	
semi 	
- 	
representable 	
object 	
$ 	
K 	
= 	
\{U_i\}_{i 	
\in 	
I}$ 	
of 	
$ 	
\mathcal{C}$ 	

 	
we 	
let 	

 	
$ 	
$ 	

 	
\mathcal{C}/K 	
= 	
\coprod\nolimits_{i 	
\in 	
I 	
} 	
\mathcal{C}/U_i 	

 	
$ 	
$ 	

 	
be 	
the 	
disjoint 	
union 	
of 	
the 	
localizations 	
of 	
$ 	
\mathcal{C}$ 	
at 	
$ 	
U_i$. 	

 	
There 	
is 	
a 	
natural 	
structure 	
of 	
a 	
site 	
on 	
this 	
category 	
, 	
with 	

 	
coverings 	
inherited 	
from 	
the 	
localizations 	
$ 	
\mathcal{C}/U_i$. 	

 	
The 	
site 	
$ 	
\mathcal{C}/K$ 	
is 	
called 	
the 	

 	
{ 	
\it 	
localization 	
of 	
$ 	
\mathcal{C}$ 	
at 	
$ 	
K$}. 	

 	
Observe 	
that 	
a 	
sheaf 	
on 	
$ 	
\mathcal{C}/K$ 	
is 	
the 	
same 	
thing 	
as 	

 	
a 	
family 	
of 	
sheaves 	
$ 	
\mathcal{F}_i$ 	
on 	
$ 	
\mathcal{C}/U_i$ 	
, 	
i.e. 	
, 	

 	
$ 	
$ 	

 	
\Sh(\mathcal{C}/K 	
) 	
= 	
\prod\nolimits_{i 	
\in 	
I 	
} 	
\Sh(\mathcal{C}/U_i 	
) 	

 	
$ 	
$ 	

 	
This 	
is 	
occasionally 	
usefull 	
to 	
understand 	
what 	
is 	
going 	
on 	
. 	


 	
\medskip\noindent 	

 	
Let 	
$ 	
\mathcal{C}$ 	
be 	
a 	
site 	TYPE
. 	TYPE
Let 	
$ 	
K 	
= 	
\{U_i\}_{i 	
\in 	
I}$ 	
be 	
an 	
object 	TYPE
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 	
\\ 	

 	
j^{-1 	
} 	
& 	
: 	
& 	

 	
\mathcal{G 	
} 	
& 	

 	
\longmapsto 	
& 	

 	
( 	
j_i^{-1}\mathcal{G})_{i 	
\in 	
I 	
} 	
\\ 	

 	
j 	
_ 	
* 	
& 	
: 	
& 	

 	
( 	
\mathcal{F}_i)_{i 	
\in 	
I 	
} 	
& 	

 	
\longmapsto 	
& 	

 	
\prod 	
j_{i 	
, 	
* 	
} 	
\mathcal{F}_i 	

 	
\end{matrix 	
} 	

 	
$ 	
$ 	

 	
as 	
the 	
reader 	
easily 	
verifies 	
. 	


 	
\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 	
. 	
More 	
precisely 	
, 	
suppose 	
$ 	
f 	
= 	
( 	
\alpha 	
, 	
f_i)$ 	

 	
where 	
$ 	
K 	
= 	
\{U_i\}_{i 	
\in 	
I}$ 	
, 	
$ 	
L 	
= 	
\{V_j\}_{j 	
\in 	
J}$ 	
, 	
$ 	
\alpha 	
: 	
I 	
\to 	
J$ 	
, 	

 	
and 	
$ 	
f_i 	
: 	
U_i 	
\to 	
V_{\alpha(i)}$. 	
Then 	
the 	
functor 	
$ 	
v$ 	
maps 	
the 	
component 	

 	
$ 	
\mathcal{C}/U_i$ 	
into 	
the 	
component 	
$ 	
\mathcal{C}/V_{\alpha(i)}$ 	

 	
via 	
the 	
construction 	
of 	
the 	
aforementioned 	
lemma 	
. 	
In 	
particular 	

 	
we 	
obtain 	
a 	
morphism 	

 	
$ 	
$ 	

 	
f 	
: 	
\Sh(\mathcal{C}/K 	
) 	
\to 	
\Sh(\mathcal{C}/L 	
) 	

 	
$ 	
$ 	

 	
of 	
topoi 	
. 	
In 	
terms 	
of 	
the 	
product 	
decompositions 	

 	
$ 	
\Sh(\mathcal{C}/K 	
) 	
= 	
\prod\nolimits_{i 	
\in 	
I 	
} 	
\Sh(\mathcal{C}/U_i)$ 	
and 	

 	
$ 	
\Sh(\mathcal{C}/L 	
) 	
= 	
\prod\nolimits_{j 	
\in 	
J 	
} 	
\Sh(\mathcal{C}/V_j)$ 	

 	
the 	
reader 	
verifies 	
that 	

 	
$ 	
$ 	

 	
\begin{matrix 	
} 	

 	
f 	
_ 	
! 	
& 	
: 	
& 	

 	
( 	
\mathcal{F}_i)_{i 	
\in 	
I 	
} 	
& 	

 	
\longmapsto 	
& 	

 	
( 	
\coprod\nolimits_{i 	
\in 	
I 	
, 	
\alpha(i 	
) 	
= 	
j 	
} 	
f_{i 	
, 	
! 	
} 	
\mathcal{F}_i)_{j 	
\in 	
J 	
} 	
\\ 	

 	
f^{-1 	
} 	
& 	
: 	
& 	

 	
( 	
\mathcal{G}_j)_{j 	
\in 	
J 	
} 	
& 	

 	
\longmapsto 	
& 	

 	
( 	
f_i^{-1}\mathcal{G}_{\alpha(i)})_{i 	
\in 	
I 	
} 	
\\ 	

 	
f 	
_ 	
* 	
& 	
: 	
& 	

 	
( 	
\mathcal{F}_i)_{i 	
\in 	
I 	
} 	
& 	

 	
\longmapsto 	
& 	

 	
( 	
\prod\nolimits_{i 	
\in 	
I 	
, 	
\alpha(i 	
) 	
= 	
j 	
} 	
f_{i 	
, 	
* 	
} 	
\mathcal{F}_i)_{j 	
\in 	
J 	
} 	

 	
\end{matrix 	
} 	

 	
$ 	
$ 	

 	
where 	
$ 	
f_i 	
: 	
\Sh(\mathcal{C}/U_i 	
) 	
\to 	
\Sh(\mathcal{C}/V_{\alpha(i)})$ 	

 	
is 	
the 	
morphism 	
associated 	
to 	
the 	
localization 	
functor 	

 	
$ 	
\mathcal{C}/U_i 	
\to 	
\mathcal{C}/V_{\alpha(i)}$ 	
corresponding 	
to 	

 	
$ 	
f_i 	
: 	
U_i 	
\to 	
V_{\alpha(i)}$. 	


 	
\begin{lemma 	
} 	

 	
\label{lemma 	
- 	
has 	
- 	
P 	
} 	

 	
Let 	
$ 	
\mathcal{C}$ 	
be 	
a 	
site 	TYPE
. 	TYPE

 	
\begin{enumerate 	
} 	

 	
\item 	
For 	
$ 	
K$ 	
in 	
$ 	
\text{SR}(\mathcal{C})$ 	
the 	
functor 	

 	
$ 	
j 	
: 	
\mathcal{C}/K 	
\to 	
\mathcal{C}$ 	
is 	
continuous 	
, 	

 	
cocontinuous 	
, 	
and 	
has 	
property 	
P 	
of 	

 	
Sites 	
, 	
Remark 	
\ref{sites 	
- 	
remark 	
- 	
cartesian 	
- 	
cocontinuous}. 	

 	
\item 	
For 	
$ 	
f 	
: 	
K 	
\to 	
L$ 	
in 	
$ 	
\text{SR}(\mathcal{C})$ 	

 	
the 	
functor 	
$ 	
v 	
: 	
\mathcal{C}/K 	
\to 	
\mathcal{C}/L$ 	
( 	
see 	
above 	
) 	

 	
is 	
continuous 	
, 	
cocontinuous 	
, 	
and 	
has 	
property 	
P 	
of 	

 	
Sites 	
, 	
Remark 	
\ref{sites 	
- 	
remark 	
- 	
cartesian 	
- 	
cocontinuous}. 	

 	
\end{enumerate 	
} 	

 	
\end{lemma 	
} 	


 	
\begin{proof 	
} 	

 	
Proof 	
of 	
( 	
2 	
) 	
. 	
In 	
the 	
notation 	
of 	
the 	
discussion 	
preceding 	
the 	
lemma 	
, 	

 	
the 	
localization 	
functors 	
$ 	
\mathcal{C}/U_i 	
\to 	
\mathcal{C}/V_{\alpha(i)}$ 	

 	
are 	
continuous 	
and 	
cocontinuous 	
by 	

 	
Sites 	
, 	
Section 	
\ref{sites 	
- 	
section 	
- 	
localize 	
} 	

 	
and 	
satisfy 	
$ 	
P$ 	
by 	

 	
Sites 	
, 	
Remark 	
\ref{sites 	
- 	
remark 	
- 	
localization 	
- 	
cartesian 	
- 	
cocontinuous}. 	

 	
It 	
is 	
formal 	
to 	
deduce 	
$ 	
v$ 	
is 	
continuous 	
and 	
cocontinuous 	
and 	
has 	
$ 	
P$. 	

 	
We 	
omit 	
the 	
details 	
. 	
We 	
also 	
omit 	
the 	
proof 	
of 	
( 	
1 	
) 	
. 	

 	
\end{proof 	
} 	


 	
\begin{lemma 	
} 	

 	
\label{lemma 	
- 	
push 	
- 	
pull 	
- 	
localization 	
} 	

 	
Let 	
$ 	
\mathcal{C}$ 	
be 	
a 	
site 	TYPE
and 	
$ 	
K$ 	
in 	
$ 	
\text{SR}(\mathcal{C})$. 	

 	
For 	
$ 	
\mathcal{F}$ 	
in 	
$ 	
\Sh(\mathcal{C})$ 	
we 	
have 	

 	
$ 	
$ 	

 	
j_*j^{-1}\mathcal{F 	
} 	
= 	
\SheafHom(F(K)^\ 	
# 	
, 	
\mathcal{F 	
} 	
) 	

 	
$ 	
$ 	

 	
where 	
$ 	
F$ 	
is 	
as 	
in 	

 	
Hypercoverings 	
, 	
Definition 	
\ref{hypercovering 	
- 	
definition 	
- 	
SR 	
- 	
F}. 	

 	
\end{lemma 	
} 	


 	
\begin{proof 	
} 	

 	
Say 	
$ 	
K 	
= 	
\{U_i\}_{i 	
\in 	
I}$. 	

 	
Using 	
the 	
description 	
of 	
the 	
functors 	
$ 	
j^{-1}$ 	
and 	
$ 	
j_*$ 	

 	
given 	
above 	
we 	
see 	
that 	

 	
$ 	
$ 	

 	
j_*j^{-1}\mathcal{F 	
} 	
= 	

 	
\prod\nolimits_{i 	
\in 	
I 	
} 	
j_{i 	
, 	
* 	
} 	
( 	
\mathcal{F}|_{\mathcal{C}/U_i 	
} 	
) 	
= 	

 	
\prod\nolimits_{i 	
\in 	
I 	
} 	
\SheafHom(h_{U_i}^\ 	
# 	
, 	
\mathcal{F 	
} 	
) 	

 	
$ 	
$ 	

 	
The 	
second 	
equality 	
by 	
Sites 	
, 	
Lemma 	
\ref{sites 	
- 	
lemma 	
- 	
hom 	
- 	
sheaf 	
- 	
hU}. 	

 	
Since 	
$ 	
F(K 	
) 	
= 	
\coprod 	
h_{U_i}$ 	
in 	
$ 	
\textit{PSh}(\mathcal{C}$ 	
, 	

 	
we 	
have 	
$ 	
F(K)^\ 	
# 	
= 	
\coprod 	
h_{U_i}^\#$ 	
in 	
$ 	
\Sh(\mathcal{C})$ 	

 	
and 	
since 	
$ 	
\SheafHom(- 	
, 	
\mathcal{F})$ 	
turns 	
coproducts 	
into 	

 	
products 	
( 	
immediate 	
from 	
the 	
construction 	
in 	

 	
Sites 	
, 	
Section 	
\ref{sites 	
- 	
section 	
- 	
glueing 	
- 	
sheaves 	
} 	
) 	
, 	
we 	
conclude 	
. 	

 	
\end{proof 	
} 	


 	
\begin{lemma 	
} 	

 	
\label{lemma 	
- 	
localize 	
- 	
compare 	
} 	

 	
Let 	
$ 	
\mathcal{C}$ 	
be 	
a 	
site 	TYPE
. 	TYPE

 	
\begin{enumerate 	
} 	

 	
\item 	
For 	
$ 	
K$ 	
in 	
$ 	
\text{SR}(\mathcal{C})$ 	
the 	
functor 	
$ 	
j_!$ 	

 	
gives 	
an 	
equivalence 	
$ 	
\Sh(\mathcal{C}/K 	
) 	
\to 	
\Sh(\mathcal{C})/F(K)^\#$ 	

 	
where 	
$ 	
F$ 	
is 	
as 	
in 	

 	
Hypercoverings 	
, 	
Definition 	
\ref{hypercovering 	
- 	
definition 	
- 	
SR 	
- 	
F}. 	

 	
\item 	
The 	
functor 	
$ 	
j^{-1 	
} 	
: 	
\Sh(\mathcal{C 	
} 	
) 	
\to 	
\Sh(\mathcal{C}/K)$ 	

 	
corresponds 	
via 	
the 	
identification 	
of 	
( 	
1 	
) 	
with 	

 	
$ 	
\mathcal{F 	
} 	
\mapsto 	
( 	
\mathcal{F 	
} 	
\times 	
F(K)^\ 	
# 	
\to 	
F(K)^\#)$. 	

 	
\item 	
For 	
$ 	
f 	
: 	
K 	
\to 	
L$ 	
in 	
$ 	
\text{SR}(\mathcal{C})$ 	
the 	
functor 	

 	
$ 	
f^{-1}$ 	
corresponds 	
via 	
the 	
identifications 	
of 	
( 	
1 	
) 	
to 	
the 	
functor 	

 	
$ 	
\Sh(\mathcal{C})/F(L)^\ 	
# 	
\to 	
\Sh(\mathcal{C})/F(K)^\#$ 	
, 	

 	
$ 	
( 	
\mathcal{G 	
} 	
\to 	
F(L)^\ 	
# 	
) 	
\mapsto 	

 	
( 	
\mathcal{G 	
} 	
\times_{F(L)^\ 	
# 	
} 	
F(K)^\ 	
# 	
\to 	
F(K)^\#)$. 	

 	
\end{enumerate 	
} 	

 	
\end{lemma 	
} 	


 	
\begin{proof 	
} 	

 	
Observe 	
that 	
if 	
$ 	
K 	
= 	
\{U_i\}_{i 	
\in 	
I}$ 	
then 	
the 	
category 	

 	
$ 	
\Sh(\mathcal{C}/K)$ 	
decomposes 	
as 	
the 	
product 	
of 	
the 	
categories 	

 	
$ 	
\Sh(\mathcal{C}/U_i)$. 	
Observe 	
that 	

 	
$ 	
F(K)^\ 	
# 	
= 	
\coprod_{i 	
\in 	
I 	
} 	
h_{U_i}^\#$ 	
( 	
coproduct 	
in 	
sheaves 	
) 	
. 	

 	
Hence 	
$ 	
\Sh(\mathcal{C})/F(K)^\#$ 	
is 	
the 	
product 	
of 	
the 	

 	
categories 	
$ 	
\Sh(\mathcal{C})/h_{U_i}^\#$. 	

 	
Thus 	
( 	
1 	
) 	
and 	
( 	
2 	
) 	
follow 	
from 	
the 	
corresponding 	

 	
statements 	
for 	
each 	
$ 	
i$ 	
, 	
see 	

 	
Sites 	
, 	
Lemmas 	
\ref{sites 	
- 	
lemma 	
- 	
essential 	
- 	
image 	
- 	
j 	
- 	
shriek 	
} 	
and 	

 	
\ref{sites 	
- 	
lemma 	
- 	
compute 	
- 	
j 	
- 	
shriek 	
- 	
restrict}. 	

 	
Similarly 	
, 	
if 	
$ 	
L 	
= 	
\{V_j\}_{j 	
\in 	
J}$ 	
and 	
$ 	
f$ 	
is 	
given 	

 	
by 	
$ 	
\alpha 	
: 	
I 	
\to 	
J$ 	
and 	
$ 	
f_i 	
: 	
U_i 	
\to 	
V_{\alpha(i)}$ 	
, 	

 	
then 	
we 	
can 	
apply 	

 	
Sites 	
, 	
Lemma 	
\ref{sites 	
- 	
lemma 	
- 	
relocalize 	
- 	
explicit 	
} 	

 	
to 	
each 	
of 	
the 	
re 	
- 	
localization 	
morphisms 	

 	
$ 	
\mathcal{C}/U_i 	
\to 	
\mathcal{C}/V_{\alpha(i)}$ 	

 	
to 	
get 	
( 	
3 	
) 	
. 	

 	
\end{proof 	
} 	


 	
\begin{lemma 	
} 	

 	
\label{lemma 	
- 	
localize 	
- 	
injective 	
} 	

 	
Let 	
$ 	
\mathcal{C}$ 	
be 	
a 	
site 	TYPE
. 	TYPE
For 	
$ 	
K$ 	
in 	
$ 	
\text{SR}(\mathcal{C})$ 	

 	
the 	
functor 	
$ 	
j^{-1}$ 	
sends 	
injective 	
abelian 	
sheaves 	
to 	
injective 	

 	
abelian 	
sheaves 	
. 	

 	
\end{lemma 	
} 	


 	
\begin{proof 	
} 	

 	
This 	
is 	
the 	
natural 	
generalization 	
of 	

 	
Cohomology 	
on 	
Sites 	
, 	
Lemma 	

 	
\ref{sites 	
- 	
cohomology 	
- 	
lemma 	
- 	
cohomology 	
- 	
of 	
- 	
open 	
} 	

 	
to 	
semi 	
- 	
representable 	
objects 	
. 	

 	
In 	
fact 	
, 	
it 	
follows 	
from 	
this 	
lemma 	

 	
by 	
the 	
product 	
decomposition 	
of 	
$ 	
\Sh(\mathcal{C}/K)$ 	

 	
and 	
the 	
description 	
of 	
the 	
functor 	
$ 	
j^{-1}$ 	
given 	
above 	
. 	

 	
\end{proof 	
} 	


 	
\begin{remark}[Variant 	
for 	
over 	
an 	
object 	
] 	

 	
\label{remark 	
- 	
semi 	
- 	
representable 	
- 	
over 	
- 	
object 	
} 	

 	
Let 	
$ 	
\mathcal{C}$ 	
be 	
a 	
site 	TYPE
. 	TYPE
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}. 	

 	
Thus 	
we 	
may 	
apply 	
the 	
above 	
discussion 	
to 	
the 	
site 	

 	
$ 	
\mathcal{C}/X$. 	
Briefly 	
, 	
the 	
constructions 	
above 	
give 	

 	
\begin{enumerate 	
} 	

 	
\item 	
a 	
site 	
$ 	
\mathcal{C}/K$ 	
for 	
$ 	
K$ 	
in 	
$ 	
\text{SR}(\mathcal{C 	
} 	
, 	
X)$ 	
, 	

 	
\item 	
a 	
decomposition 	

 	
$ 	
\Sh(\mathcal{C}/K 	
) 	
= 	
\prod 	
\Sh(\mathcal{C}/U_i)$ 	
if 	
$ 	
K 	
= 	
\{U_i 	
/ 	
X\}$ 	
, 	

 	
\item 	
a 	
localization 	
functor 	
$ 	
j 	
: 	
\mathcal{C}/K 	
\to 	
\mathcal{C}/X$ 	
, 	

 	
\item 	
a 	
morphism 	
$ 	
f 	
: 	
\Sh(\mathcal{C}/K 	
) 	
\to 	
\Sh(\mathcal{C}/L)$ 	

 	
for 	
$ 	
f 	
: 	
K 	
\to 	
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 	TYPE
. 	TYPE
Let 	
$ 	
\mathcal{O}_\mathcal{C}$ 	

 	
be 	
a 	
sheaf 	TYPE
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 	
. 	
For 	
example 	
, 	
the 	

 	
functor 	
$ 	
j^*$ 	
has 	
an 	
exact 	
left 	
adjoint 	

 	
$ 	
$ 	

 	
j 	
_ 	
! 	
: 	
\textit{Mod}(\mathcal{O}_K 	
) 	
\to 	
\textit{Mod}(\mathcal{O}_\mathcal{C 	
} 	
) 	
, 	

 	
$ 	
$ 	

 	
which 	
in 	
terms 	
of 	
the 	
product 	
decomposition 	
given 	
in 	
( 	
2 	
) 	
sends 	

 	
$ 	
( 	
\mathcal{F}_i)_{i 	
\in 	
I}$ 	
to 	
$ 	
\bigoplus 	
j_{i 	
, 	
! 	
} 	
\mathcal{F}_i$. 	

 	
Similarly 	
, 	
given 	
$ 	
f 	
: 	
K 	
\to 	
L$ 	
as 	
above 	
, 	
the 	
functor 	
$ 	
f^*$ 	
has 	

 	
an 	
exact 	
left 	
adjoint 	

 	
$ 	
f 	
_ 	
! 	
: 	
\textit{Mod}(\mathcal{O}_K 	
) 	
\to 	
\textit{Mod}(\mathcal{O}_L)$. 	

 	
Thus 	
the 	
functors 	
$ 	
j^*$ 	
and 	
$ 	
f^*$ 	
are 	
exact 	
, 	
i.e. 	
, 	

 	
$ 	
j$ 	
and 	
$ 	
f$ 	
are 	
flat 	
morphisms 	
of 	
ringed 	
topoi 	
( 	
also 	
follows 	

 	
from 	
the 	
equalities 	
$ 	
\mathcal{O}_K 	
= 	
j^{-1}\mathcal{O}_\mathcal{C}$ 	

 	
and 	
$ 	
\mathcal{O}_K 	
= 	
f^{-1}\mathcal{O}_L$ 	
) 	
. 	

 	
\end{remark 	
} 	


 	
\begin{remark}[Ringed 	
variant 	
over 	
an 	
object 	
] 	

 	
\label{remark 	
- 	
semi 	
- 	
representable 	
- 	
ringed 	
- 	
over 	
- 	
object 	
} 	

 	
Let 	
$ 	
\mathcal{C}$ 	
be 	
a 	
site 	TYPE
. 	TYPE
Let 	
$ 	
\mathcal{O}_\mathcal{C}$ 	

 	
be 	
a 	
sheaf 	TYPE
of 	
rings 	
on 	
$ 	
\mathcal{C}$. 	
Let 	
$ 	
X 	
\in 	
\Ob(\mathcal{C})$ 	

 	
and 	
denote 	
$ 	
\mathcal{O}_X 	
= 	
\mathcal{O}_\mathcal{C}|_{\mathcal{C}/U}$. 	

 	
Then 	
we 	
can 	
combine 	
the 	
constructions 	
given 	
in 	

 	
Remarks 	
\ref{remark 	
- 	
semi 	
- 	
representable 	
- 	
over 	
- 	
object 	
} 	

 	
and 	
\ref{remark 	
- 	
semi 	
- 	
representable 	
- 	
ringed 	
} 	
to 	
get 	

 	
\begin{enumerate 	
} 	

 	
\item 	
a 	
ringed 	
site 	
$ 	
( 	
\mathcal{C}/K 	
, 	
\mathcal{O}_K)$ 	

 	
for 	
$ 	
K$ 	
in 	
$ 	
\text{SR}(\mathcal{C 	
} 	
, 	
X)$ 	
, 	

 	
\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}/X 	
) 	
, 	
\mathcal{O}_X)$ 	

 	
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 	
} 	
, 	
X)$. 	

 	
\end{enumerate 	
} 	

 	
Of 	
course 	
all 	
of 	
the 	
results 	
mentioned 	
in 	

 	
Remark 	
\ref{remark 	
- 	
semi 	
- 	
representable 	
- 	
ringed 	
} 	

 	
hold 	
in 	
this 	
setting 	
as 	
well 	
. 	

 	
\end{remark 	
} 	






 	
\section{The 	
site 	
associate 	
to 	
a 	
simplicial 	
semi 	
- 	
representable 	
object 	
} 	

 	
\label{section 	
- 	
simplicial 	
- 	
semi 	
- 	
representable 	
} 	


 	
\noindent 	

 	
Let 	
$ 	
\mathcal{C}$ 	
be 	
a 	
site 	TYPE
. 	TYPE
Let 	
$ 	
K$ 	
be 	
a 	
simplicial 	TYPE
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]$. 	

 	
By 	
the 	
construction 	
in 	

 	
Section 	
\ref{section 	
- 	
semi 	
- 	
representable 	
} 	
we 	
obtain 	
a 	
simplicial 	
object 	

 	
$ 	
n 	
\mapsto 	
\mathcal{C}/K_n$ 	
in 	
the 	
category 	
whose 	
objects 	
are 	
sites 	
and 	

 	
whose 	
morphisms 	
are 	
cocontinuous 	
functors 	
. 	
In 	
other 	
words 	
, 	
we 	
get 	

 	
a 	
gadget 	
as 	
in 	
Case 	
B 	
of 	
Section 	
\ref{section 	
- 	
simplicial 	
- 	
sites}. 	

 	
The 	
functors 	
satisfy 	
property 	
P 	
by 	
Lemma 	
\ref{lemma 	
- 	
has 	
- 	
P}. 	

 	
Hence 	
we 	
may 	
apply 	

 	
Lemma 	
\ref{lemma 	
- 	
simplicial 	
- 	
cocontinuous 	
- 	
site 	
} 	

 	
to 	
obtain 	
a 	
site 	
$ 	
( 	
\mathcal{C}/K)_{total}$. 	


 	
\medskip\noindent 	

 	
We 	
can 	
describe 	
the 	
site 	
$ 	
( 	
\mathcal{C}/K)_{total}$ 	
explicitly 	
as 	
follows 	
. 	

 	
Say 	
$ 	
K_n 	
= 	
\{U_{n 	
, 	
i}\}_{i 	
\in 	
I_n}$. 	
For 	
$ 	
\varphi 	
: 	
[ 	
m 	
] 	
\to 	
[ 	
n]$ 	

 	
the 	
morphism 	
$ 	
K(\varphi 	
) 	
: 	
K_n 	
\to 	
K_m$ 	
is 	
given 	
by 	
a 	
map 	

 	
$ 	
\alpha(\varphi 	
) 	
: 	
I_n 	
\to 	
I_m$ 	
and 	
morphisms 	

 	
$ 	
f_{\varphi 	
, 	
i 	
} 	
: 	
U_{n 	
, 	
i 	
} 	
\to 	
U_{m 	
, 	
\alpha(\varphi)(i)}$ 	
for 	
$ 	
i 	
\in 	
I_n$. 	

 	
Then 	
we 	
have 	

 	
\begin{enumerate 	
} 	

 	
\item 	
an 	
object 	
of 	
$ 	
( 	
\mathcal{C}/K)_{total}$ 	

 	
corresponds 	
to 	
an 	
object 	
$ 	
( 	
U 	
/ 	
U_{n 	
, 	
i})$ 	

 	
of 	
$ 	
\mathcal{C}/U_{n 	
, 	
i}$ 	
for 	
some 	
$ 	
n$ 	
and 	
some 	
$ 	
i 	
\in 	
I_n$ 	
, 	

 	
\item 	
a 	
morphism 	
between 	
$ 	
U 	
/ 	
U_{n 	
, 	
i}$ 	
and 	
$ 	
V 	
/ 	
U_{m 	
, 	
j}$ 	

 	
is 	
a 	
pair 	
$ 	
( 	
\varphi 	
, 	
f)$ 	
where 	
$ 	
\varphi 	
: 	
[ 	
m 	
] 	
\to 	
[ 	
n]$ 	
, 	

 	
$ 	
j 	
= 	
\alpha(\varphi)(i)$ 	
, 	
and 	
$ 	
f 	
: 	
U 	
\to 	
V$ 	
is 	
a 	
morphism 	
of 	

 	
$ 	
\mathcal{C}$ 	
such 	
that 	

 	
$ 	
$ 	

 	
\vcenter 	
{ 	

 	
\xymatrix 	
{ 	

 	
U 	
\ar[r]_f 	
\ar[d 	
] 	
& 	
V 	
\ar[d 	
] 	
\\ 	

 	
U_{n 	
, 	
i 	
} 	
\ar[r]^-{f_{\varphi 	
, 	
i 	
} 	
} 	
& 	

 	
U_{m 	
, 	
j 	
} 	

 	
} 	

 	
} 	

 	
$ 	
$ 	

 	
is 	
commutative 	
, 	
and 	

 	
\item 	
coverings 	
of 	
the 	
object 	
$ 	
U 	
/ 	
U_{n 	
, 	
i}$ 	
are 	
constructed 	

 	
by 	
starting 	
with 	
a 	
covering 	
$ 	
\{f_j 	
: 	
U_j 	
\to 	
U\}$ 	
in 	
$ 	
\mathcal{C}$ 	

 	
and 	
letting 	
$ 	
\{(\text{id 	
} 	
, 	
f_j 	
) 	
: 	
U_j 	
/ 	
U_{n 	
, 	
i 	
} 	
\to 	
U 	
/ 	
U_{n 	
, 	
i}\}$ 	

 	
be 	
a 	
covering 	TYPE
in 	
$ 	
( 	
\mathcal{C}/K)_{total}$. 	

 	
\end{enumerate 	
} 	

 	
All 	
of 	
our 	
general 	
theory 	
developed 	
for 	
simplicial 	
sites 	
applies 	
to 	

 	
$ 	
( 	
\mathcal{C}/K)_{total}$. 	
Observe 	
that 	
the 	
obvious 	
forgetful 	
functor 	

 	
$ 	
$ 	

 	
j_{total 	
} 	
: 	
( 	
\mathcal{C}/K)_{total 	
} 	
\longrightarrow 	
\mathcal{C 	
} 	

 	
$ 	
$ 	

 	
is 	
continuous 	
and 	
cocontinuous 	
. 	
It 	
turns 	
out 	
that 	
the 	
associated 	

 	
morphism 	
of 	
topoi 	
comes 	
from 	
an 	
( 	
obvious 	
) 	
augmentation 	
. 	


 	
\begin{lemma 	
} 	

 	
\label{lemma 	
- 	
augmentation 	
- 	
simplicial 	
- 	
semi 	
- 	
representable 	
} 	

 	
Let 	
$ 	
\mathcal{C}$ 	
be 	
a 	
site 	TYPE
. 	TYPE
Let 	
$ 	
K$ 	
be 	
a 	
simplicial 	TYPE
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 	
. 	

 	
See 	
in 	
particular 	
the 	
footnote 	
in 	
the 	
proof 	
of 	

 	
Lemma 	
\ref{lemma 	
- 	
augmentation 	
- 	
site 	
} 	

 	
for 	
the 	
relationship 	
between 	
$ 	
a$ 	
and 	
$ 	
j_{total}$. 	

 	
\end{proof 	
} 	


 	
\begin{lemma 	
} 	

 	
\label{lemma 	
- 	
comparison 	
} 	

 	
With 	
assumption 	
and 	
notation 	
as 	
in 	

 	
Lemma 	
\ref{lemma 	
- 	
augmentation 	
- 	
simplicial 	
- 	
semi 	
- 	
representable 	
} 	

 	
we 	
have 	
the 	
following 	
properties 	
: 	

 	
\begin{enumerate 	
} 	

 	
\item 	
there 	
is 	
a 	
functor 	

 	
$ 	
a^{Sh 	
} 	
_ 	
! 	
: 	
\Sh((\mathcal{C}/K)_{total 	
} 	
) 	
\to 	
\Sh(\mathcal{C})$ 	

 	
left 	
adjoint 	
to 	
$ 	
a^{-1 	
} 	
: 	
\Sh(\mathcal{C 	
} 	
) 	
\to 	
\Sh((\mathcal{C}/K)_{total})$ 	
, 	

 	
\item 	
there 	
is 	
a 	
functor 	

 	
$ 	
a 	
_ 	
! 	
: 	
\textit{Ab}((\mathcal{C}/K)_{total 	
} 	
) 	
\to 	
\textit{Ab}(\mathcal{C})$ 	

 	
left 	
adjoint 	
to 	

 	
$ 	
a^{-1 	
} 	
: 	
\textit{Ab}(\mathcal{C 	
} 	
) 	
\to 	
\textit{Ab}((\mathcal{C}/K)_{total})$ 	
, 	

 	
\item 	
the 	
functor 	
$ 	
a^{-1}$ 	
associates 	
to 	

 	
$ 	
\mathcal{F}$ 	
in 	
$ 	
\Sh(\mathcal{C})$ 	
the 	
sheaf 	
on 	
$ 	
( 	
\mathcal{C}/K)_{total}$ 	

 	
wich 	
in 	
degree 	
$ 	
n$ 	
is 	
equal 	
to 	
$ 	
a_n^{-1}\mathcal{F}$ 	
, 	

 	
\item 	
the 	
functor 	
$ 	
a_*$ 	
associates 	
to 	
$ 	
\mathcal{G}$ 	
in 	

 	
$ 	
\textit{Ab}((\mathcal{C}/K)_{total})$ 	
the 	
equalizer 	
of 	
the 	
two 	
maps 	

 	
$ 	
j_{0 	
, 	
* 	
} 	
\mathcal{G}_0 	
\to 	
j_{1 	
, 	
* 	
} 	
\mathcal{G}_1 	
$ 	
, 	

 	
\end{enumerate 	
} 	

 	
\end{lemma 	
} 	


 	
\begin{proof 	
} 	

 	
Parts 	
( 	
3 	
) 	
and 	
( 	
4 	
) 	
hold 	
for 	
any 	
augmentation 	
of 	
a 	

 	
simplicial 	
site 	
, 	
see 	
Lemma 	
\ref{lemma 	
- 	
augmentation 	
- 	
site}. 	

 	
Parts 	
( 	
1 	
) 	
and 	
( 	
2 	
) 	
follow 	
as 	
$ 	
j_{total}$ 	
is 	
continuous 	
and 	
cocontinuous 	
. 	

 	
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 	TYPE
. 	TYPE
Let 	
$ 	
K$ 	
be 	
a 	
simplicial 	TYPE
object 	
of 	

 	
$ 	
\text{SR}(\mathcal{C})$. 	
Let 	
$ 	
U 	
/ 	
U_{n 	
, 	
i}$ 	
be 	
an 	
object 	TYPE
of 	

 	
$ 	
\mathcal{C}/K_n$. 	
Let 	

 	
$ 	
\mathcal{F 	
} 	
\in 	
\textit{Ab}((\mathcal{C}/K)_{total})$. 	

 	
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 	TYPE
. 	TYPE
Let 	
$ 	
X 	
\in 	
\Ob(\mathcal{C})$. 	

 	
Recall 	
that 	
we 	
have 	
a 	
category 	

 	
$ 	
\text{SR}(\mathcal{C 	
} 	
, 	
X 	
) 	
= 	
\text{SR}(\mathcal{C}/X)$ 	

 	
of 	
semi 	
- 	
representable 	
objects 	
over 	
$ 	
X$ 	
, 	

 	
see 	
Remark 	
\ref{remark 	
- 	
semi 	
- 	
representable 	
- 	
over 	
- 	
object}. 	

 	
We 	
may 	
apply 	
the 	
above 	
discussion 	
to 	
the 	
site 	

 	
$ 	
\mathcal{C}/X$. 	
Briefly 	
, 	
the 	
constructions 	
above 	
give 	

 	
\begin{enumerate 	
} 	

 	
\item 	
a 	
site 	
$ 	
( 	
\mathcal{C}/K)_{total}$ 	
for 	
a 	
simplicial 	
$ 	
K$ 	
object 	

 	
of 	
$ 	
\text{SR}(\mathcal{C 	
} 	
, 	
X)$ 	
, 	

 	
\item 	
a 	
localization 	
functor 	

 	
$ 	
j_{total 	
} 	
: 	
( 	
\mathcal{C}/K)_{total 	
} 	
\to 	
\mathcal{C}/X$ 	
, 	

 	
\item 	
localization 	
functors 	
$ 	
j_n 	
: 	
\mathcal{C}/K_n 	
\to 	
\mathcal{C}/X$ 	
, 	

 	
\item 	
a 	
morphism 	
of 	
topoi 	

 	
$ 	
a 	
: 	
\Sh((\mathcal{C}/K)_{total 	
} 	
) 	
\to 	
\Sh(\mathcal{C}/X)$ 	
, 	

 	
\item 	
morphisms 	
of 	
topoi 	

 	
$ 	
a_n 	
: 	
\Sh(\mathcal{C}/K_n 	
) 	
\to 	
\Sh(\mathcal{C}/X)$ 	
, 	

 	
\item 	
a 	
functor 	

 	
$ 	
a^{Sh 	
} 	
_ 	
! 	
: 	
\Sh((\mathcal{C}/K)_{total 	
} 	
) 	
\to 	
\Sh(\mathcal{C}/X)$ 	

 	
left 	
adjoint 	
to 	
$ 	
a^{-1}$ 	
, 	
and 	

 	
\item 	
a 	
functor 	

 	
$ 	
a 	
_ 	
! 	
: 	
\textit{Ab}((\mathcal{C}/K)_{total 	
} 	
) 	
\to 	
\textit{Ab}(\mathcal{C}/X)$ 	

 	
left 	
adjoint 	
to 	
$ 	
a^{-1}$. 	

 	
\end{enumerate 	
} 	

 	
All 	
of 	
the 	
results 	
of 	
this 	
section 	
hold 	
in 	
this 	
setting 	
. 	

 	
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 	TYPE
. 	TYPE
Let 	
$ 	
\mathcal{O}_\mathcal{C}$ 	
be 	
a 	
sheaf 	TYPE
of 	
rings 	
. 	

 	
Given 	
a 	
simplicial 	
semi 	
- 	
representable 	
object 	
$ 	
K$ 	
of 	
$ 	
\mathcal{C}$ 	

 	
we 	
set 	
$ 	
\mathcal{O 	
} 	
= 	
a^{-1}\mathcal{O}_\mathcal{C}$ 	
, 	
where 	
$ 	
a$ 	

 	
is 	
as 	
in 	
Lemmas 	
\ref{lemma 	
- 	
augmentation 	
- 	
simplicial 	
- 	
semi 	
- 	
representable 	
} 	
and 	

 	
\ref{lemma 	
- 	
comparison}. 	

 	
The 	
constructions 	
above 	
, 	
keeping 	
track 	
of 	
the 	
sheaves 	
of 	
rings 	

 	
as 	
in 	
Remark 	
\ref{remark 	
- 	
semi 	
- 	
representable 	
- 	
ringed 	
} 	
, 	
give 	

 	
\begin{enumerate 	
} 	

 	
\item 	
a 	
ringed 	
site 	
$ 	
( 	
( 	
\mathcal{C}/K)_{total 	
} 	
, 	
\mathcal{O})$ 	

 	
for 	
a 	
simplicial 	
$ 	
K$ 	
object 	
of 	
$ 	
\text{SR}(\mathcal{C})$ 	
, 	

 	
\item 	
a 	
morphism 	
of 	
ringed 	
topoi 	

 	
$ 	
a 	
: 	
( 	
\Sh((\mathcal{C}/K)_{total 	
} 	
) 	
, 	
\mathcal{O 	
} 	
) 	
\to 	

 	
( 	
\Sh(\mathcal{C 	
} 	
) 	
, 	
\mathcal{O}_\mathcal{C})$ 	
, 	

 	
\item 	
morphisms 	
of 	
ringed 	
topoi 	

 	
$ 	
a_n 	
: 	
( 	
\Sh(\mathcal{C}/K_n 	
) 	
, 	
\mathcal{O}_n 	
) 	
\to 	

 	
( 	
\Sh(\mathcal{C 	
} 	
) 	
, 	
\mathcal{O}_\mathcal{C})$ 	
, 	

 	
\item 	
a 	
functor 	

 	
$ 	
a 	
_ 	
! 	
: 	
\textit{Mod}(\mathcal{O 	
} 	
) 	
\to 	
\textit{Mod}(\mathcal{O}_\mathcal{C})$ 	

 	
left 	
adjoint 	
to 	
$ 	
a^*$. 	

 	
\end{enumerate 	
} 	

 	
The 	
functor 	
$ 	
a_!$ 	
exists 	
( 	
but 	
in 	
general 	
is 	
not 	
exact 	
) 	

 	
because 	
$ 	
a^{-1}\mathcal{O}_\mathcal{C 	
} 	
= 	
\mathcal{O}$ 	

 	
and 	
we 	
can 	
replace 	
the 	
use 	
of 	

 	
Modules 	
on 	
Sites 	
, 	
Lemma 	
\ref{sites 	
- 	
modules 	
- 	
lemma 	
- 	
g 	
- 	
shriek 	
- 	
adjoint 	
} 	

 	
in 	
the 	
proof 	
of 	
Lemma 	
\ref{lemma 	
- 	
comparison 	
} 	

 	
by 	
Modules 	
on 	
Sites 	
, 	
Lemma 	
\ref{sites 	
- 	
modules 	
- 	
lemma 	
- 	
lower 	
- 	
shriek 	
- 	
modules}. 	

 	
As 	
discussed 	
in 	
Remark 	
\ref{remark 	
- 	
semi 	
- 	
representable 	
- 	
ringed 	
} 	

 	
there 	
are 	
exact 	
functors 	

 	
$ 	
a_{n 	
! 	
} 	
: 	
\textit{Mod}(\mathcal{O}_n 	
) 	
\to 	

 	
\textit{Mod}(\mathcal{O}_\mathcal{C})$ 	

 	
left 	
adjoint 	
to 	
$ 	
a_n^*$. 	
Consequently 	
, 	
the 	
morphisms 	
$ 	
a$ 	
and 	
$ 	
a_n$ 	
are 	
flat 	
. 	

 	
Remark 	
\ref{remark 	
- 	
semi 	
- 	
representable 	
- 	
ringed 	
} 	

 	
implies 	
the 	
morphism 	
of 	
ringed 	
topoi 	

 	
$ 	
f_\varphi 	
: 	
( 	
\Sh(\mathcal{C}/K_n 	
) 	
, 	
\mathcal{O}_n 	
) 	
\to 	

 	
( 	
\Sh(\mathcal{C}/K_m 	
) 	
, 	
\mathcal{O}_m)$ 	

 	
for 	
$ 	
\varphi 	
: 	
[ 	
m 	
] 	
\to 	
[ 	
n]$ 	
is 	
flat 	
and 	
there 	
exists 	
an 	
exact 	
functor 	

 	
$ 	
f_{\varphi 	
! 	
} 	
: 	
\textit{Mod}(\mathcal{O}_n 	
) 	
\to 	
\textit{Mod}(\mathcal{O}_m)$ 	

 	
left 	
adjoint 	
to 	
$ 	
f_\varphi^*$. 	
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 	TYPE
. 	TYPE
Let 	
$ 	
\mathcal{O}_\mathcal{C}$ 	
be 	
a 	
sheaf 	TYPE
of 	
rings 	
. 	

 	
Let 	
$ 	
X 	
\in 	
\Ob(\mathcal{C})$ 	
and 	
denote 	

 	
$ 	
\mathcal{O}_X 	
= 	
\mathcal{O}_\mathcal{C}|_{\mathcal{C}/X}$. 	

 	
Then 	
we 	
can 	
combine 	
the 	
constructions 	
given 	
in 	

 	
Remarks 	
\ref{remark 	
- 	
augmentation 	
- 	
over 	
- 	
object 	
} 	
and 	

 	
\ref{remark 	
- 	
augmentation 	
- 	
ringed 	
} 	

 	
to 	
get 	

 	
\begin{enumerate 	
} 	

 	
\item 	
a 	
ringed 	
site 	
$ 	
( 	
( 	
\mathcal{C}/K)_{total 	
} 	
, 	
\mathcal{O})$ 	

 	
for 	
a 	
simplicial 	
$ 	
K$ 	
object 	
of 	
$ 	
\text{SR}(\mathcal{C 	
} 	
, 	
X)$ 	
, 	

 	
\item 	
a 	
morphism 	
of 	
ringed 	
topoi 	

 	
$ 	
a 	
: 	
( 	
\Sh((\mathcal{C}/K)_{total 	
} 	
) 	
, 	
\mathcal{O 	
} 	
) 	
\to 	

 	
( 	
\Sh(\mathcal{C}/X 	
) 	
, 	
\mathcal{O}_X)$ 	
, 	

 	
\item 	
morphisms 	
of 	
ringed 	
topoi 	

 	
$ 	
a_n 	
: 	
( 	
\Sh(\mathcal{C}/K_n 	
) 	
, 	
\mathcal{O}_n 	
) 	
\to 	

 	
( 	
\Sh(\mathcal{C}/X 	
) 	
, 	
\mathcal{O}_X)$ 	
, 	

 	
\item 	
a 	
functor 	

 	
$ 	
a 	
_ 	
! 	
: 	
\textit{Mod}(\mathcal{O 	
} 	
) 	
\to 	
\textit{Mod}(\mathcal{O}_X)$ 	

 	
left 	
adjoint 	
to 	
$ 	
a^*$. 	

 	
\end{enumerate 	
} 	

 	
Of 	
course 	
, 	
all 	
the 	
results 	
mentioned 	
in 	

 	
Remark 	
\ref{remark 	
- 	
augmentation 	
- 	
ringed 	
} 	

 	
hold 	
in 	
this 	
setting 	
as 	
well 	
. 	

 	
\end{remark 	
} 	







 	
\section{Cohomological 	
descent 	
for 	
hypercoverings 	
} 	

 	
\label{section 	
- 	
cohomological 	
- 	
descent 	
- 	
hypercoverings 	
} 	


 	
\noindent 	

 	
Let 	
$ 	
\mathcal{C}$ 	
be 	
a 	
site 	TYPE
. 	TYPE
In 	
this 	
section 	
we 	
assume 	
$ 	
\mathcal{C}$ 	

 	
has 	
equalizers 	
and 	
fibre 	
products 	
. 	
We 	
let 	
$ 	
K$ 	
be 	
a 	
hypercovering 	TYPE

 	
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 	TYPE
with 	
equalizers 	
and 	
fibre 	
products 	
. 	

 	
Let 	
$ 	
K$ 	
be 	
a 	
hypercovering 	
. 	
Then 	

 	
\begin{enumerate 	
} 	

 	
\item 	
$ 	
a^{-1 	
} 	
: 	
\Sh(\mathcal{C 	
} 	
) 	
\to 	
\Sh((\mathcal{C}/K)_{total})$ 	

 	
is 	
fully 	
faithful 	
with 	
essential 	
image 	
the 	
cartesian 	
sheaves 	
of 	
sets 	
, 	

 	
\item 	
$ 	
a^{-1 	
} 	
: 	
\textit{Ab}(\mathcal{C 	
} 	
) 	
\to 	

 	
\textit{Ab}((\mathcal{C}/K)_{total})$ 	

 	
is 	
fully 	
faithful 	
with 	
essential 	
image 	
the 	
cartesian 	
sheaves 	

 	
of 	
abelian 	
groups 	
. 	

 	
\end{enumerate 	
} 	

 	
In 	
both 	
cases 	
$ 	
a_*$ 	
provides 	
the 	
quasi 	
- 	
inverse 	
functor 	
. 	

 	
\end{lemma 	
} 	


 	
\begin{proof 	
} 	

 	
The 	
case 	
of 	
abelian 	
sheaves 	
follows 	
immediately 	
from 	
the 	
case 	

 	
of 	
sheaves 	
of 	
sets 	
as 	
the 	
functor 	
$ 	
a^{-1}$ 	
commutes 	
with 	
products 	
. 	

 	
In 	
the 	
rest 	
of 	
the 	
proof 	
we 	
work 	
with 	
sheaves 	
of 	
sets 	
. 	

 	
Observe 	
that 	
$ 	
a^{-1}\mathcal{F}$ 	
is 	
cartesian 	
for 	

 	
$ 	
\mathcal{F}$ 	
in 	
$ 	
\Sh(\mathcal{C})$ 	
by 	

 	
Lemma 	
\ref{lemma 	
- 	
augmentation 	
- 	
cartesian 	
- 	
module}. 	

 	
It 	
suffices 	
to 	
show 	
that 	
the 	
adjunction 	
map 	

 	
$ 	
\mathcal{F 	
} 	
\to 	
a_*a^{-1}\mathcal{F}$ 	

 	
is 	
an 	
isomorphism 	
$ 	
\mathcal{F}$ 	
in 	
$ 	
\Sh(\mathcal{C})$ 	

 	
and 	
that 	
for 	
a 	
cartesian 	
sheaf 	

 	
$ 	
\mathcal{G}$ 	
on 	
$ 	
( 	
\mathcal{C}/K)_{total}$ 	

 	
the 	
adjunction 	
map 	

 	
$ 	
a^{-1}a_*\mathcal{G 	
} 	
\to 	
\mathcal{G}$ 	
is 	
an 	
isomorphism 	
. 	


 	
\medskip\noindent 	

 	
Let 	
$ 	
\mathcal{F}$ 	
be 	
a 	
sheaf 	TYPE
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}. 	

 	
By 	
Lemma 	
\ref{lemma 	
- 	
push 	
- 	
pull 	
- 	
localization 	
} 	

 	
$ 	
$ 	

 	
a_{0 	
, 	
* 	
} 	
a_0^{-1}\mathcal{F 	
} 	
= 	
\SheafHom(F(K_0)^\ 	
# 	
, 	
\mathcal{F 	
} 	
) 	

 	
\quad\text{and}\quad 	

 	
a_{1 	
, 	
* 	
} 	
a_1^{-1}\mathcal{F 	
} 	
= 	
\SheafHom(F(K_1)^\ 	
# 	
, 	
\mathcal{F 	
} 	
) 	

 	
$ 	
$ 	

 	
On 	
the 	
other 	
hand 	
, 	
we 	
know 	
that 	

 	
$ 	
$ 	

 	
\xymatrix 	
{ 	

 	
F(K_1)^\ 	
# 	
\ar@<1ex>[r 	
] 	
\ar@<-1ex>[r 	
] 	
& 	

 	
F(K_0)^\ 	
# 	
\ar[r 	
] 	
& 	
\text{final 	
object 	
} 	
* 	
\text 	
{ 	
of 	
} 	
\Sh(\mathcal{C 	
} 	
) 	

 	
} 	

 	
$ 	
$ 	

 	
is 	
a 	
coequalizer 	
diagram 	
in 	
sheaves 	
of 	
sets 	
by 	
definition 	
of 	

 	
a 	
hypercovering 	
. 	
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 	TYPE
sheaf 	
on 	
$ 	
( 	
\mathcal{C}/K)_{total}$. 	

 	
We 	
will 	
show 	
that 	
$ 	
\mathcal{G 	
} 	
= 	
a^{-1}\mathcal{F}$ 	
for 	
some 	
sheaf 	

 	
$ 	
\mathcal{F}$ 	
on 	
$ 	
\mathcal{C}$. 	
This 	
will 	
finish 	
the 	
proof 	
because 	

 	
then 	
$ 	
a^{-1}a_*\mathcal{G 	
} 	
= 	
a^{-1}a_*a^{-1}\mathcal{F 	
} 	
= 	

 	
a^{-1}\mathcal{F 	
} 	
= 	
\mathcal{G}$ 	
by 	
the 	
result 	
of 	
the 	
previous 	
paragraph 	
. 	

 	
Set 	
$ 	
\mathcal{K}_n 	
= 	
F(K_n)^\#$ 	
for 	
$ 	
n 	
\geq 	
0$. 	
Then 	
we 	
have 	
maps 	
of 	
sheaves 	

 	
$ 	
$ 	

 	
\xymatrix 	
{ 	

 	
\mathcal{K}_2 	

 	
\ar@<1ex>[r 	
] 	

 	
\ar@<0ex>[r 	
] 	

 	
\ar@<-1ex>[r 	
] 	

 	
& 	

 	
\mathcal{K}_1 	

 	
\ar@<0.5ex>[r 	
] 	

 	
\ar@<-0.5ex>[r 	
] 	

 	
& 	

 	
\mathcal{K}_0 	

 	
} 	

 	
$ 	
$ 	

 	
coming 	
from 	
the 	
fact 	
that 	
$ 	
K$ 	
is 	
a 	
simplicial 	
semi 	
- 	
representable 	
object 	
. 	

 	
The 	
fact 	
that 	
$ 	
K$ 	
is 	
a 	
hypercovering 	
means 	
that 	

 	
$ 	
$ 	

 	
\mathcal{K}_1 	
\to 	
\mathcal{K}_0 	
\times 	
\mathcal{K}_0 	

 	
\quad\text{and}\quad 	

 	
\mathcal{K}_2 	
\to 	

 	
\left(\text{cosq}_1 	
( 	

 	
\xymatrix 	
{ 	

 	
\mathcal{K}_1 	

 	
\ar@<0.5ex>[r 	
] 	

 	
\ar@<-0.5ex>[r 	
] 	

 	
& 	

 	
\mathcal{K}_0 	
\ar[l 	
] 	

 	
} 	
) 	
\right)_2 	

 	
$ 	
$ 	

 	
are 	
surjective 	
maps 	
of 	
sheaves 	
. 	
Using 	
the 	
description 	
of 	
cartesian 	
sheaves 	
on 	

 	
$ 	
( 	
\mathcal{C}/K)_{total}$ 	
given 	
in 	
Lemma 	
\ref{lemma 	
- 	
characterize 	
- 	
cartesian 	
} 	

 	
and 	
using 	
the 	
description 	
of 	
$ 	
\Sh(\mathcal{C}/K_n)$ 	
in 	

 	
Lemma 	
\ref{lemma 	
- 	
localize 	
- 	
compare 	
} 	

 	
we 	
find 	
that 	
our 	
problem 	
can 	
be 	
entirely 	
formulated\footnote{Even 	
though 	
it 	

 	
does 	
not 	
matter 	
what 	
the 	
precise 	
formulation 	
is 	
, 	
we 	
spell 	
it 	
out 	
: 	

 	
the 	
problem 	
is 	
to 	
show 	
that 	
given 	
an 	
object 	

 	
$ 	
\mathcal{G}_0/\mathcal{K}_0 	
$ 	
of 	
$ 	
\Sh(\mathcal{C})/\mathcal{K}_0 	
$ 	

 	
and 	
an 	
isomorphism 	

 	
$ 	
$ 	

 	
\alpha 	
: 	

 	
\mathcal{G}_0 	
\times_{\mathcal{K}_0 	
, 	
\mathcal{K}(\delta^1_1 	
) 	
} 	
\mathcal{K}_1 	
\to 	

 	
\mathcal{G}_0 	
\times_{\mathcal{K}_0 	
, 	
\mathcal{K}(\delta^1_0 	
) 	
} 	
\mathcal{K}_1 	

 	
$ 	
$ 	

 	
over 	
$ 	
\mathcal{K}_1 	
$ 	
satisfying 	
a 	
cocycle 	
condtion 	
in 	

 	
$ 	
\Sh(\mathcal{C})/\mathcal{K}_2 	
$ 	
, 	
there 	
exists 	

 	
$ 	
\mathcal{F}$ 	
in 	
$ 	
\Sh(\mathcal{C})$ 	
and 	
an 	
isomorphism 	

 	
$ 	
\mathcal{F 	
} 	
\times 	
\mathcal{K}_0 	
\to 	
\mathcal{G}_0 	
$ 	
over 	
$ 	
\mathcal{K}_0 	
$ 	

 	
compatible 	
with 	
$ 	
\alpha$. 	
} 	
in 	
terms 	
of 	

 	
\begin{enumerate 	
} 	

 	
\item 	
the 	
topos 	
$ 	
\Sh(\mathcal{C})$ 	
, 	
and 	

 	
\item 	
the 	
simplicial 	
object 	
$ 	
\mathcal{K}$ 	
in 	
$ 	
\Sh(\mathcal{C})$ 	

 	
whose 	
terms 	
are 	
$ 	
\mathcal{K}_n$. 	

 	
\end{enumerate 	
} 	

 	
Thus 	
, 	
after 	
replacing 	
$ 	
\mathcal{C}$ 	
by 	
a 	
different 	
site 	
$ 	
\mathcal{C}'$ 	

 	
as 	
in 	
Sites 	
, 	
Lemma 	
\ref{sites 	
- 	
lemma 	
- 	
topos 	
- 	
good 	
- 	
site 	
} 	
, 	
we 	
may 	
assume 	

 	
$ 	
\mathcal{C}$ 	
has 	
all 	
finite 	
limits 	
, 	

 	
the 	
topology 	
on 	
$ 	
\mathcal{C}$ 	
is 	
subcanonical 	
, 	

 	
a 	
family 	
$ 	
\{V_j 	
\to 	
V\}$ 	
of 	
morphisms 	
of 	
$ 	
\mathcal{C}$ 	

 	
is 	
a 	
covering 	
if 	
and 	
only 	
if 	
$ 	
\coprod 	
h_{V_j 	
} 	
\to 	
V$ 	
is 	
surjective 	
, 	
and 	

 	
there 	
exists 	
a 	
simplicial 	
object 	
$ 	
U$ 	
of 	
$ 	
\mathcal{C}$ 	

 	
such 	
that 	
$ 	
\mathcal{K}_n 	
= 	
h_{U_n}$ 	
as 	
simplicial 	
sheaves 	
. 	

 	
Working 	
backwards 	
through 	
the 	
equivalences 	
we 	
may 	
assume 	

 	
$ 	
K_n 	
= 	
\{U_n\}$ 	
for 	
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 	
U)_2\}$ 	
is 	
a 	
covering 	
. 	

 	
Let 	
us 	
use 	
$ 	
d^n_i 	
: 	
U_n 	
\to 	
U_{n 	
- 	
1}$ 	
and 	

 	
$ 	
s^n_j 	
: 	
U_n 	
\to 	
U_{n 	
+ 	
1}$ 	
the 	
morphisms 	
corresponding 	

 	
to 	
$ 	
\delta^n_i$ 	
and 	
$ 	
\sigma^n_j$ 	
as 	
in 	

 	
Simplicial 	
, 	
Definition 	
\ref{simplicial 	
- 	
definition 	
- 	
face 	
- 	
degeneracy}. 	

 	
By 	
abuse 	
of 	
notation 	
, 	
given 	
a 	
morphism 	

 	
$ 	
c 	
: 	
V 	
\to 	
W$ 	
of 	
$ 	
\mathcal{C}$ 	
we 	
denote 	
the 	
morphism 	
of 	
topoi 	

 	
$ 	
c 	
: 	
\Sh(\mathcal{C}/V 	
) 	
\to 	
\Sh(\mathcal{C}/W)$ 	
by 	
the 	
same 	
letter 	
. 	

 	
Now 	
$ 	
\mathcal{G}$ 	
is 	
given 	
by 	
a 	
sheaf 	
$ 	
\mathcal{G}_0 	
$ 	

 	
on 	
$ 	
\mathcal{C}/U_0 	
$ 	
and 	
an 	
isomorphism 	

 	
$ 	
\alpha 	
: 	
( 	
d^1_1)^{-1}\mathcal{G}_0 	
\to 	
( 	
d^1_0)^{-1}\mathcal{G}_0 	
$ 	

 	
satisfying 	
the 	
cocycle 	
condition 	
on 	
$ 	
\mathcal{C}/U_2 	
$ 	

 	
formulated 	
in 	
Lemma 	
\ref{lemma 	
- 	
characterize 	
- 	
cartesian}. 	

 	
Since 	
$ 	
\{U_2 	
\to 	
( 	
\text{cosq}_1 	
\text{sk}_1 	
U)_2\}$ 	

 	
is 	
a 	
covering 	
, 	
the 	
corresponding 	
pullback 	
functor 	

 	
on 	
sheaves 	
is 	
faithful 	
( 	
small 	
detail 	
omitted 	
) 	
. 	

 	
Hence 	
we 	
may 	
replace 	
$ 	
U$ 	
by 	
$ 	
\text{cosk}_1 	
\text{sk}_1 	
U$ 	
, 	
because 	

 	
this 	
replaces 	
$ 	
U_2 	
$ 	
by 	
$ 	
( 	
\text{cosq}_1 	
\text{sk}_1 	
U)_2 	
$ 	
and 	
leaves 	

 	
$ 	
U_1 	
$ 	
and 	
$ 	
U_0 	
$ 	
unchanged 	
. 	
Then 	

 	
$ 	
$ 	

 	
( 	
d^2_0 	
, 	
d^2_1 	
, 	
d^2_2 	
) 	
: 	
U_2 	
\to 	
U_1 	
\times 	
U_1 	
\times 	
U_1 	

 	
$ 	
$ 	

 	
is 	
a 	
monomorphism 	
whose 	
its 	
image 	
on 	
$ 	
T$-valued 	
points 	
is 	

 	
described 	
in 	
Simplicial 	
, 	
Lemma 	
\ref{simplicial 	
- 	
lemma 	
- 	
work 	
- 	
out}. 	

 	
In 	
particular 	
, 	
there 	
is 	
a 	
morphism 	
$ 	
c$ 	
fitting 	
into 	
a 	
commutative 	
diagram 	

 	
$ 	
$ 	

 	
\xymatrix 	
{ 	

 	
U_1 	
\times_{(d^1_1 	
, 	
d^1_0 	
) 	
, 	
U_0 	
\times 	
U_0 	
, 	
( 	
d^1_1 	
, 	
d^1_0 	
) 	
} 	
U_1 	

 	
\ar[d 	
] 	
\ar[rr]_c 	
& 	
& 	
U_2 	
\ar[d 	
] 	
\\ 	

 	
U_1 	
\times 	
U_1 	

 	
\ar[rr]^{(\text{pr}_1 	
, 	
\text{pr}_2 	
, 	
s^0_0 	
\circ 	
d^1_1 	
\circ 	
\text{pr}_1 	
) 	
} 	
& 	
& 	

 	
U_1 	
\times 	
U_1 	
\times 	
U_1 	

 	
} 	

 	
$ 	
$ 	

 	
as 	
going 	
around 	
the 	
other 	
way 	
defines 	
a 	
point 	
of 	
$ 	
U_2$. 	

 	
Pulling 	
back 	
the 	
cocycle 	
condition 	
for 	
$ 	
\alpha$ 	
on 	
$ 	
U_2 	
$ 	

 	
translates 	
into 	
the 	
condition 	
that 	
the 	
pullbacks 	
of 	
$ 	
\alpha$ 	

 	
via 	
the 	
projections 	
to 	

 	
$ 	
U_1 	
\times_{(d^1_1 	
, 	
d^1_0 	
) 	
, 	
U_0 	
\times 	
U_0 	
, 	
( 	
d^1_1 	
, 	
d^1_0 	
) 	
} 	
U_1 	
$ 	

 	
are 	
the 	
same 	
as 	
the 	
pullback 	
of 	
$ 	
\alpha$ 	
via 	

 	
$ 	
s^0_0 	
\circ 	
d^1_1 	
\circ 	
\text{pr}_1 	
$ 	
is 	
the 	
identity 	
map 	

 	
( 	
namely 	
, 	
the 	
pullback 	
of 	
$ 	
\alpha$ 	
by 	
$ 	
s^0_0 	
$ 	
is 	
the 	
identity 	
) 	
. 	

 	
By 	
Sites 	
, 	
Lemma 	
\ref{sites 	
- 	
lemma 	
- 	
glue 	
- 	
maps 	
} 	

 	
this 	
means 	
that 	
$ 	
\alpha$ 	
comes 	
from 	
an 	
isomorphism 	

 	
$ 	
$ 	

 	
\alpha 	
' 	
: 	
\text{pr}_1^{-1}\mathcal{G}_0 	
\to 	
\text{pr}_2^{-1}\mathcal{G}_0 	

 	
$ 	
$ 	

 	
of 	
sheaves 	
on 	
$ 	
\mathcal{C}/U_0 	
\times 	
U_0$. 	

 	
Then 	
finally 	
, 	
the 	
morphism 	
$ 	
U_2 	
\to 	
U_0 	
\times 	
U_0 	
\times 	
U_0 	
$ 	

 	
is 	
surjective 	
on 	
associated 	
sheaves 	
as 	
is 	
easily 	
seen 	
using 	
the 	

 	
surjectivity 	
of 	
$ 	
U_1 	
\to 	
U_0 	
\times 	
U_0 	
$ 	

 	
and 	
the 	
description 	
of 	
$ 	
U_2 	
$ 	
given 	
above 	
. 	
Therefore 	
$ 	
\alpha'$ 	

 	
satisfies 	
the 	
cocycle 	
condition 	
on 	
$ 	
U_0 	
\times 	
U_0 	
\times 	
U_0$. 	

 	
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 	TYPE
with 	
equalizers 	
and 	
fibre 	
products 	
. 	

 	
Let 	
$ 	
K$ 	
be 	
a 	
hypercovering 	
. 	
The 	
{ 	
\v 	
C}ech 	
complex 	

 	
of 	
Lemma 	
\ref{lemma 	
- 	
augmentation 	
- 	
cech 	
- 	
complex 	
} 	
associated 	
to 	

 	
$ 	
a^{-1}\mathcal{F}$ 	

 	
$ 	
$ 	

 	
a_{0 	
, 	
* 	
} 	
a_0^{-1}\mathcal{F 	
} 	
\to 	
a_{1 	
, 	
* 	
} 	
a_1^{-1}\mathcal{F 	
} 	
\to 	

 	
a_{2 	
, 	
* 	
} 	
a_2^{-1}\mathcal{F 	
} 	
\to 	
\ldots 	

 	
$ 	
$ 	

 	
is 	
equal 	
to 	
the 	
complex 	
$ 	
\SheafHom(s(\mathbf{Z}_{F(K)}^\ 	
# 	
) 	
, 	
\mathcal{F})$. 	

 	
Here 	
$ 	
s(\mathbf{Z}_{F(K)}^\#)$ 	
is 	
as 	
in 	

 	
Hypercoverings 	
, 	
Definition 	
\ref{hypercovering 	
- 	
definition 	
- 	
homology}. 	

 	
\end{lemma 	
} 	


 	
\begin{proof 	
} 	

 	
By 	
Lemma 	
\ref{lemma 	
- 	
push 	
- 	
pull 	
- 	
localization 	
} 	
we 	
have 	

 	
$ 	
$ 	

 	
a_{n 	
, 	
* 	
} 	
a_n^{-1}\mathcal{F 	
} 	
= 	
\SheafHom'(F(K_n)^\ 	
# 	
, 	
\mathcal{F 	
} 	
) 	

 	
$ 	
$ 	

 	
where 	
$ 	
\SheafHom'$ 	
is 	
as 	
in 	
Sites 	
, 	
Section 	
\ref{sites 	
- 	
section 	
- 	
glueing 	
- 	
sheaves}. 	

 	
The 	
boundary 	
maps 	
in 	
the 	
complex 	
of 	

 	
Lemma 	
\ref{lemma 	
- 	
augmentation 	
- 	
cech 	
- 	
complex 	
} 	

 	
come 	
from 	
the 	
simplicial 	
structure 	
. 	

 	
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 	TYPE
with 	
equalizers 	
and 	
fibre 	
products 	
. 	

 	
Let 	
$ 	
K$ 	
be 	
a 	
hypercovering 	
. 	
For 	

 	
$ 	
E 	
\in 	
D^+(\mathcal{C})$ 	
the 	
map 	

 	
$ 	
$ 	

 	
E 	
\longrightarrow 	
Ra_*a^{-1}E 	

 	
$ 	
$ 	

 	
is 	
an 	
isomorphism 	
. 	

 	
\end{lemma 	
} 	


 	
\begin{proof 	
} 	

 	
First 	
, 	
let 	
$ 	
\mathcal{I}$ 	
be 	
an 	
injective 	TYPE
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}. 	

 	
Thus 	
the 	
complex 	

 	
$ 	
$ 	

 	
a_{0 	
, 	
* 	
} 	
a_0^{-1}\mathcal{I 	
} 	
\to 	

 	
a_{1 	
, 	
* 	
} 	
a_1^{-1}\mathcal{I 	
} 	
\to 	

 	
a_{2 	
, 	
* 	
} 	
a_2^{-1}\mathcal{I 	
} 	
\to\ldots 	

 	
$ 	
$ 	

 	
computes 	
$ 	
Ra_*a^{-1}\mathcal{I}$. 	
By 	

 	
Lemma 	
\ref{lemma 	
- 	
hypercovering 	
- 	
cech 	
- 	
complex 	
} 	

 	
this 	
is 	
equal 	
to 	
the 	
complex 	

 	
$ 	
\SheafHom(s(\mathbf{Z}_{F(K)}^\ 	
# 	
) 	
, 	
\mathcal{I})$. 	

 	
Because 	
$ 	
K$ 	
is 	
a 	
hypercovering 	
, 	
we 	
see 	
that 	

 	
$ 	
s(\mathbf{Z}_{F(K)}^\#)$ 	
is 	
exact 	
in 	
degrees 	
$ 	
> 	
0 	
$ 	
by 	

 	
Hypercoverings 	
, 	
Lemma 	
\ref{hypercovering 	
- 	
lemma 	
- 	
acyclic 	
- 	
hypercover 	
- 	
sheaves 	
} 	

 	
applied 	
to 	
the 	
simplicial 	
presheaf 	
$ 	
F(K)$. 	

 	
Since 	
$ 	
\mathcal{I}$ 	
is 	
injective 	
, 	
the 	
functor 	
$ 	
\SheafHom(- 	
, 	
\mathcal{I})$ 	

 	
is 	
exact 	
and 	
we 	
conclude 	
that 	

 	
$ 	
\SheafHom(s(\mathbf{Z}_{F(K)}^\ 	
# 	
) 	
, 	
\mathcal{I})$ 	

 	
is 	
exact 	
in 	
positive 	
degrees 	
. 	
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 	
. 	

 	
Choose 	
a 	
bounded 	
below 	
complex 	
$ 	
\mathcal{I}^\bullet$ 	
of 	
injectives 	

 	
representing 	
$ 	
E$. 	
By 	
the 	
result 	
of 	
the 	
first 	
paragraph 	
and 	

 	
Leray 	
's 	
acyclicity 	
lemma 	

 	
( 	
Derived 	
Categories 	
, 	
Lemma 	
\ref{derived 	
- 	
lemma 	
- 	
leray 	
- 	
acyclicity 	
} 	
) 	

 	
$ 	
Ra_*a^{-1}\mathcal{I}^\bullet$ 	

 	
is 	
computed 	
by 	
the 	
complex 	

 	
$ 	
a_*a^{-1}\mathcal{I}^\bullet 	
= 	
\mathcal{I}^\bullet$ 	

 	
and 	
we 	
conclude 	
the 	
lemma 	
is 	
true 	
. 	

 	
\end{proof 	
} 	


 	
\begin{lemma 	
} 	

 	
\label{lemma 	
- 	
compare 	
- 	
cohomology 	
- 	
hypercovering 	
} 	

 	
Let 	
$ 	
\mathcal{C}$ 	
be 	
a 	
site 	TYPE
with 	
equalizers 	
and 	
fibre 	
products 	
. 	

 	
Let 	
$ 	
K$ 	
be 	
a 	
hypercovering 	
. 	

 	
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 	TYPE
with 	
equalizers 	
and 	
fibre 	
products 	
. 	

 	
Let 	
$ 	
K$ 	
be 	
a 	
hypercovering 	
. 	

 	
Let 	
$ 	
\mathcal{A 	
} 	
\subset 	
\textit{Ab}((\mathcal{C}/K)_{total})$ 	

 	
denote 	
the 	
weak 	
Serre 	
subcategory 	
of 	
cartesian 	
abelian 	
sheaves 	
. 	

 	
Then 	
the 	
functor 	
$ 	
a^{-1}$ 	
defines 	
an 	
equivalence 	

 	
$ 	
$ 	

 	
D^+(\mathcal{C 	
} 	
) 	
\longrightarrow 	
D_\mathcal{A}^+((\mathcal{C}/K)_{total 	
} 	
) 	

 	
$ 	
$ 	

 	
with 	
quasi 	
- 	
inverse 	
$ 	
Ra_*$. 	

 	
\end{lemma 	
} 	


 	
\begin{proof 	
} 	

 	
Observe 	
that 	
$ 	
\mathcal{A}$ 	
is 	
a 	
weak 	
Serre 	
subcategory 	
by 	

 	
Lemma 	
\ref{lemma 	
- 	
Serre 	
- 	
subcat 	
- 	
cartesian 	
- 	
modules}. 	

 	
The 	
equivalence 	
is 	
a 	

 	
formal 	
consequence 	
of 	
the 	
results 	
obtained 	
so 	
far 	
. 	
Use 	

 	
Lemmas 	
\ref{lemma 	
- 	
equivalence 	
- 	
bounded 	
} 	
, 	

 	
\ref{lemma 	
- 	
hypercovering 	
- 	
descent 	
- 	
sheaves 	
} 	
, 	
and 	

 	
\ref{lemma 	
- 	
hypercovering 	
- 	
descent 	
- 	
bounded 	
- 	
abelian}. 	

 	
\end{proof 	
} 	


 	
\noindent 	

 	
We 	
urge 	
the 	
reader 	
to 	
skip 	
the 	
following 	
remark 	
. 	


 	
\begin{remark 	
} 	

 	
\label{remark 	
- 	
compare 	
- 	
cohomology 	
- 	
hypercovering 	
- 	
presheaf 	
} 	

 	
Let 	
$ 	
\mathcal{C}$ 	
be 	
a 	
site 	TYPE
. 	TYPE
Let 	
$ 	
\mathcal{G}$ 	
be 	
a 	
presheaf 	TYPE
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 	
. 	

 	
To 	
see 	
this 	
, 	

 	
define 	
the 	
localization 	
$ 	
\mathcal{C}/\mathcal{G}$ 	

 	
of 	
$ 	
\mathcal{C}$ 	
at 	
$ 	
\mathcal{G}$ 	
exactly 	
as 	
in 	

 	
Sites 	
, 	
Lemma 	
\ref{sites 	
- 	
lemma 	
- 	
localize 	
- 	
topos 	
- 	
site 	
} 	

 	
( 	
which 	
is 	
stated 	
only 	
for 	
sheaves 	
; 	
the 	
topos 	

 	
$ 	
\Sh(\mathcal{C}/\mathcal{G})$ 	
is 	
equal 	
to 	
the 	
localization 	

 	
of 	
the 	
topos 	
$ 	
\Sh(\mathcal{C})$ 	
at 	
the 	
sheaf 	
$ 	
\mathcal{G}^\#$ 	
) 	
. 	

 	
Then 	
the 	
reader 	
easily 	
shows 	
that 	
the 	
site 	

 	
$ 	
\mathcal{C}/\mathcal{G}$ 	
has 	
fibre 	
products 	
and 	
equalizers 	

 	
and 	
that 	
a 	
hypercovering 	
of 	
$ 	
\mathcal{G}$ 	
in 	
$ 	
\mathcal{C}$ 	

 	
is 	
the 	
same 	
thing 	
as 	
a 	
hypercovering 	
for 	
the 	
site 	
$ 	
\mathcal{C}/\mathcal{G}$. 	

 	
Hence 	
replacing 	
the 	
site 	
$ 	
\mathcal{C}$ 	
by 	
$ 	
\mathcal{C}/\mathcal{G}$ 	

 	
in 	
the 	
lemmas 	
on 	
hypercoverings 	
above 	
we 	
obtain 	
proofs 	
of 	
the 	

 	
corresponding 	
results 	
for 	
hypercoverings 	
of 	
$ 	
\mathcal{G}$. 	

 	
Example 	
: 	
for 	
a 	
hypercovering 	
$ 	
K$ 	
of 	
$ 	
\mathcal{G}$ 	
we 	
have 	

 	
$ 	
$ 	

 	
R\Gamma(\mathcal{C}/\mathcal{G 	
} 	
, 	
E 	
) 	
= 	

 	
R\Gamma((\mathcal{C}/K)_{total 	
} 	
, 	
a^{-1}E 	
) 	

 	
$ 	
$ 	

 	
for 	
$ 	
E 	
\in 	
D^+(\mathcal{C}/\mathcal{G})$ 	
where 	

 	
$ 	
a 	
: 	
\Sh((\mathcal{C}/K)_{total 	
} 	
) 	
\to 	
\Sh(\mathcal{C}/\mathcal{G})$ 	

 	
is 	
the 	
canonical 	
augmentation 	
. 	
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 	TYPE
. 	TYPE
Let 	
$ 	
\mathcal{O}_\mathcal{C}$ 	

 	
be 	
a 	
sheaf 	TYPE
of 	
rings 	
. 	
Assume 	
$ 	
\mathcal{C}$ 	

 	
has 	
equalizers 	
and 	
fibre 	
products 	
and 	
let 	
$ 	
K$ 	
be 	
a 	
hypercovering 	TYPE

 	
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 	TYPE
with 	
equalizers 	
and 	
fibre 	
products 	
. 	

 	
Let 	
$ 	
\mathcal{O}_\mathcal{C}$ 	
be 	
a 	
sheaf 	TYPE
of 	
rings 	
. 	

 	
Let 	
$ 	
K$ 	
be 	
a 	
hypercovering 	TYPE
. 	
With 	
notation 	
as 	
above 	

 	
$ 	
$ 	

 	
a^ 	
* 	
: 	
\textit{Mod}(\mathcal{O}_\mathcal{C 	
} 	
) 	
\to 	
\textit{Mod}(\mathcal{O 	
} 	
) 	

 	
$ 	
$ 	

 	
is 	
fully 	
faithful 	
with 	
essential 	
image 	
the 	
cartesian 	
$ 	
\mathcal{O}$-modules 	
. 	

 	
The 	
functor 	
$ 	
a_*$ 	
provides 	
the 	
quasi 	
- 	
inverse 	
. 	

 	
\end{lemma 	
} 	


 	
\begin{proof 	
} 	

 	
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 	TYPE
with 	
equalizers 	
and 	
fibre 	
products 	
. 	

 	
Let 	
$ 	
\mathcal{O}_\mathcal{C}$ 	
be 	
a 	
sheaf 	TYPE
of 	
rings 	
. 	

 	
Let 	
$ 	
K$ 	
be 	
a 	
hypercovering 	TYPE
. 	
For 	

 	
$ 	
E 	
\in 	
D^+(\mathcal{O}_\mathcal{C})$ 	
the 	
map 	

 	
$ 	
$ 	

 	
E 	
\longrightarrow 	
Ra_*La^*E 	

 	
$ 	
$ 	

 	
is 	
an 	
isomorphism 	
. 	

 	
\end{lemma 	
} 	


 	
\begin{proof 	
} 	

 	
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 	TYPE
with 	
equalizers 	
and 	
fibre 	
products 	
. 	

 	
Let 	
$ 	
\mathcal{O}_\mathcal{C}$ 	
be 	
a 	
sheaf 	TYPE
of 	
rings 	
. 	

 	
Let 	
$ 	
K$ 	
be 	
a 	
hypercovering 	TYPE
. 	

 	
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 	TYPE
with 	
equalizers 	
and 	
fibre 	
products 	
. 	

 	
Let 	
$ 	
\mathcal{O}_\mathcal{C}$ 	
be 	
a 	
sheaf 	TYPE
of 	
rings 	
. 	

 	
Let 	
$ 	
K$ 	
be 	
a 	
hypercovering 	TYPE
. 	

 	
Let 	
$ 	
\mathcal{A 	
} 	
\subset 	
\textit{Mod}(\mathcal{O})$ 	

 	
denote 	
the 	
weak 	
Serre 	
subcategory 	
of 	
cartesian 	
$ 	
\mathcal{O}$-modules 	
. 	

 	
Then 	
the 	
functor 	
$ 	
La^*$ 	
defines 	
an 	
equivalence 	

 	
$ 	
$ 	

 	
D^+(\mathcal{O}_\mathcal{C 	
} 	
) 	
\longrightarrow 	
D_\mathcal{A}^+(\mathcal{O 	
} 	
) 	

 	
$ 	
$ 	

 	
with 	
quasi 	
- 	
inverse 	
$ 	
Ra_*$. 	

 	
\end{lemma 	
} 	


 	
\begin{proof 	
} 	

 	
Observe 	
that 	
$ 	
\mathcal{A}$ 	
is 	
a 	
weak 	
Serre 	
subcategory 	
by 	

 	
Lemma 	
\ref{lemma 	
- 	
Serre 	
- 	
subcat 	
- 	
cartesian 	
- 	
modules 	
} 	

 	
( 	
the 	
required 	
hypotheses 	
hold 	
by 	
the 	
discussion 	
in 	

 	
Remark 	
\ref{remark 	
- 	
augmentation 	
- 	
ringed 	
} 	
) 	
. 	

 	
The 	
equivalence 	
is 	
a 	

 	
formal 	
consequence 	
of 	
the 	
results 	
obtained 	
so 	
far 	
. 	
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 	TYPE
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 	TYPE
map 	

 	
of 	
sheaves 	
. 	
} 	
by 	
Hypercoverings 	
, 	
Lemma 	

 	
\ref{hypercovering 	
- 	
lemma 	
- 	
hypercovering 	
- 	
F}. 	

 	
This 	
means 	
that 	
every 	
single 	
result 	
proved 	
for 	
hypercoverings 	

 	
in 	
Section 	
\ref{section 	
- 	
cohomological 	
- 	
descent 	
- 	
hypercoverings 	
} 	

 	
has 	
an 	
immediate 	
analogue 	
in 	
the 	
situation 	
in 	
this 	
section 	
. 	


 	
\begin{lemma 	
} 	

 	
\label{lemma 	
- 	
hypercovering 	
- 	
X 	
- 	
descent 	
- 	
sheaves 	
} 	

 	
Let 	
$ 	
\mathcal{C}$ 	
be 	
a 	
site 	TYPE
with 	
fibre 	
products 	
and 	
$ 	
X 	
\in 	
\Ob(\mathcal{C})$. 	

 	
Let 	
$ 	
K$ 	
be 	
a 	
hypercovering 	TYPE
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 	
. 	

 	
\end{enumerate 	
} 	

 	
In 	
both 	
cases 	
$ 	
a_*$ 	
provides 	
the 	
quasi 	
- 	
inverse 	
functor 	
. 	

 	
\end{lemma 	
} 	


 	
\begin{proof 	
} 	

 	
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 	TYPE
with 	
fibre 	
product 	
and 	
$ 	
X 	
\in 	
\Ob(\mathcal{C})$. 	

 	
Let 	
$ 	
K$ 	
be 	
a 	
hypercovering 	TYPE
of 	
$ 	
X$. 	
For 	

 	
$ 	
E 	
\in 	
D^+(\mathcal{C}/X)$ 	
the 	
map 	

 	
$ 	
$ 	

 	
E 	
\longrightarrow 	
Ra_*a^{-1}E 	

 	
$ 	
$ 	

 	
is 	
an 	
isomorphism 	
. 	

 	
\end{lemma 	
} 	


 	
\begin{proof 	
} 	

 	
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 	TYPE
with 	
fibre 	
products 	
and 	
$ 	
X 	
\in 	
\Ob(\mathcal{C})$. 	

 	
Let 	
$ 	
K$ 	
be 	
a 	
hypercovering 	TYPE
of 	
$ 	
X$. 	

 	
Then 	
we 	
have 	
a 	
canonical 	
isomorphism 	

 	
$ 	
$ 	

 	
R\Gamma(X 	
, 	
E 	
) 	
= 	
R\Gamma((\mathcal{C}/K)_{total 	
} 	
, 	
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 	TYPE
with 	
fibre 	
products 	
and 	
$ 	
X 	
\in 	
\Ob(\mathcal{C})$. 	

 	
Let 	
$ 	
K$ 	
be 	
a 	
hypercovering 	TYPE
of 	
$ 	
X$. 	

 	
Let 	
$ 	
\mathcal{A 	
} 	
\subset 	
\textit{Ab}((\mathcal{C}/K)_{total})$ 	

 	
denote 	
the 	
weak 	
Serre 	
subcategory 	
of 	
cartesian 	
abelian 	
sheaves 	
. 	

 	
Then 	
the 	
functor 	
$ 	
a^{-1}$ 	
defines 	
an 	
equivalence 	

 	
$ 	
$ 	

 	
D^+(\mathcal{C}/X 	
) 	
\longrightarrow 	
D_\mathcal{A}^+((\mathcal{C}/K)_{total 	
} 	
) 	

 	
$ 	
$ 	

 	
with 	
quasi 	
- 	
inverse 	
$ 	
Ra_*$. 	

 	
\end{lemma 	
} 	


 	
\begin{proof 	
} 	

 	
Via 	
Remarks 	
\ref{remark 	
- 	
semi 	
- 	
representable 	
- 	
over 	
- 	
object 	
} 	
and 	

 	
\ref{remark 	
- 	
augmentation 	
- 	
over 	
- 	
object 	
} 	

 	
this 	
follows 	
from 	
Lemma 	
\ref{lemma 	
- 	
hypercovering 	
- 	
equivalence 	
- 	
bounded}. 	

 	
\end{proof 	
} 	









 	
\section{Cohomological 	
descent 	
for 	
hypercoverings 	
of 	
an 	
object 	
: 	
modules 	
} 	

 	
\label{section 	
- 	
cohomological 	
- 	
descent 	
- 	
hypercoverings 	
- 	
X 	
- 	
modules 	
} 	


 	
\noindent 	

 	
In 	
this 	
section 	
we 	
assume 	
$ 	
\mathcal{C}$ 	
has 	
fibre 	
products 	

 	
and 	
$ 	
X 	
\in 	
\Ob(\mathcal{C})$. 	
We 	
let 	
$ 	
K$ 	
be 	
a 	
hypercovering 	TYPE
of 	
$ 	
X$ 	

 	
as 	
defined 	
in 	

 	
Hypercoverings 	
, 	
Definition 	
\ref{hypercovering 	
- 	
definition 	
- 	
hypercovering}. 	

 	
Let 	
$ 	
\mathcal{O}_\mathcal{C}$ 	
be 	
a 	
sheaf 	TYPE
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 	TYPE
with 	
fibre 	
products 	
and 	
$ 	
X 	
\in 	
\Ob(\mathcal{C})$. 	

 	
Let 	
$ 	
\mathcal{O}_\mathcal{C}$ 	
be 	
a 	
sheaf 	TYPE
of 	
rings 	
. 	

 	
Let 	
$ 	
K$ 	
be 	
a 	
hypercovering 	TYPE
of 	
$ 	
X$. 	
With 	
notation 	
as 	
above 	

 	
$ 	
$ 	

 	
a^ 	
* 	
: 	
\textit{Mod}(\mathcal{O}_X 	
) 	
\to 	
\textit{Mod}(\mathcal{O 	
} 	
) 	

 	
$ 	
$ 	

 	
is 	
fully 	
faithful 	
with 	
essential 	
image 	
the 	
cartesian 	
$ 	
\mathcal{O}$-modules 	
. 	

 	
The 	
functor 	
$ 	
a_*$ 	
provides 	
the 	
quasi 	
- 	
inverse 	
. 	

 	
\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 	
- 	
hypercovering 	
- 	
descent 	
- 	
modules}. 	

 	
\end{proof 	
} 	


 	
\begin{lemma 	
} 	

 	
\label{lemma 	
- 	
hypercovering 	
- 	
X 	
- 	
descent 	
- 	
bounded 	
- 	
modules 	
} 	

 	
Let 	
$ 	
\mathcal{C}$ 	
be 	
a 	
site 	TYPE
with 	
fibre 	
products 	
and 	
$ 	
X 	
\in 	
\Ob(\mathcal{C})$. 	

 	
Let 	
$ 	
\mathcal{O}_\mathcal{C}$ 	
be 	
a 	
sheaf 	TYPE
of 	
rings 	
. 	

 	
Let 	
$ 	
K$ 	
be 	
a 	
hypercovering 	TYPE
of 	
$ 	
X$. 	
For 	

 	
$ 	
E 	
\in 	
D^+(\mathcal{O}_X)$ 	
the 	
map 	

 	
$ 	
$ 	

 	
E 	
\longrightarrow 	
Ra_*La^*E 	

 	
$ 	
$ 	

 	
is 	
an 	
isomorphism 	
. 	

 	
\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 	
- 	
hypercovering 	
- 	
descent 	
- 	
bounded 	
- 	
modules}. 	

 	
\end{proof 	
} 	


 	
\begin{lemma 	
} 	

 	
\label{lemma 	
- 	
compare 	
- 	
cohomology 	
- 	
hypercovering 	
- 	
X 	
- 	
modules 	
} 	

 	
Let 	
$ 	
\mathcal{C}$ 	
be 	
a 	
site 	TYPE
with 	
fibre 	
products 	
and 	
$ 	
X 	
\in 	
\Ob(\mathcal{C})$. 	

 	
Let 	
$ 	
\mathcal{O}_\mathcal{C}$ 	
be 	
a 	
sheaf 	TYPE
of 	
rings 	
. 	

 	
Let 	
$ 	
K$ 	
be 	
a 	
hypercovering 	TYPE
of 	
$ 	
X$. 	

 	
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 	TYPE
with 	
fibre 	
products 	
and 	
$ 	
X 	
\in 	
\Ob(\mathcal{C})$. 	

 	
Let 	
$ 	
\mathcal{O}_\mathcal{C}$ 	
be 	
a 	
sheaf 	TYPE
of 	
rings 	
. 	

 	
Let 	
$ 	
K$ 	
be 	
a 	
hypercovering 	TYPE
of 	
$ 	
X$. 	

 	
Let 	
$ 	
\mathcal{A 	
} 	
\subset 	
\textit{Mod}(\mathcal{O})$ 	

 	
denote 	
the 	
weak 	
Serre 	
subcategory 	
of 	
cartesian 	
$ 	
\mathcal{O}$-modules 	
. 	

 	
Then 	
the 	
functor 	
$ 	
La^*$ 	
defines 	
an 	
equivalence 	

 	
$ 	
$ 	

 	
D^+(\mathcal{O}_X 	
) 	
\longrightarrow 	
D_\mathcal{A}^+(\mathcal{O 	
} 	
) 	

 	
$ 	
$ 	

 	
with 	
quasi 	
- 	
inverse 	
$ 	
Ra_*$. 	

 	
\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 	
- 	
hypercovering 	
- 	
equivalence 	
- 	
bounded 	
- 	
modules}. 	

 	
\end{proof 	
} 	











 	
\section{Hypercovering 	
by 	
a 	
simplicial 	
object 	
of 	
the 	
site 	
} 	

 	
\label{section 	
- 	
hypercovering 	
} 	


 	
\noindent 	

 	
Let 	
$ 	
\mathcal{C}$ 	
be 	
a 	
site 	TYPE
with 	
fibre 	
products 	
and 	

 	
let 	
$ 	
X 	
\in 	
\Ob(\mathcal{C})$. 	

 	
In 	
this 	
section 	
we 	
elucidate 	
the 	
results 	
of 	

 	
Section 	
\ref{section 	
- 	
cohomological 	
- 	
descent 	
- 	
hypercoverings 	
- 	
X 	
} 	

 	
in 	
the 	
case 	
that 	
our 	
hypercovering 	
is 	
given 	
by 	

 	
a 	
simplicial 	
object 	
of 	
the 	
site 	
. 	

 	
Let 	
$ 	
U$ 	
be 	
a 	
simplicial 	TYPE
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 	
. 	
The 	
morphism 	
of 	
topoi 	
$ 	
a$ 	
is 	
given 	
by 	
the 	
cocontinuous 	

 	
functor 	
$ 	
V 	
/ 	
U_n 	
\mapsto 	
V 	
/ 	
X$. 	
That 	
's 	
all 	
folks 	
! 	


 	
\medskip\noindent 	

 	
Let 	
us 	
say 	
that 	
the 	
augmentation 	
$ 	
a 	
: 	
U 	
\to 	
X$ 	
is 	
a 	

 	
{ 	
\it 	
hypercovering 	
of 	
$ 	
X$ 	
in 	
$ 	
\mathcal{C}$ 	
} 	

 	
if 	
the 	
following 	
hold 	

 	
\begin{enumerate 	
} 	

 	
\item 	
$ 	
\{U_0 	
\to 	
X\}$ 	
is 	
a 	
covering 	
of 	
$ 	
\mathcal{C}$ 	
, 	

 	
\item 	
$ 	
\{U_1 	
\to 	
U_0 	
\times_X 	
U_0\}$ 	
is 	
a 	
covering 	
of 	
$ 	
\mathcal{C}$ 	
, 	

 	
\item 	
$ 	
\{U_{n 	
+ 	
1 	
} 	
\to 	
( 	
\text{cosk}_n\text{sk}_n 	
U)_{n 	
+ 	
1}\}$ 	

 	
is 	
a 	
covering 	
of 	
$ 	
\mathcal{C}$ 	
for 	
$ 	
n 	
\geq 	
1$. 	

 	
\end{enumerate 	
} 	

 	
The 	
category 	
$ 	
\mathcal{C}/X$ 	
has 	
all 	
connected 	
finite 	
limits 	
, 	
hence 	
the 	

 	
coskeleta 	
used 	
in 	
the 	
formulation 	
above 	
exist 	
. 	
Of 	
course 	
, 	
we 	
see 	

 	
that 	
$ 	
U$ 	
is 	
a 	
hypercovering 	
of 	
$ 	
X$ 	
in 	
$ 	
\mathcal{C}$ 	
if 	
and 	
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 	TYPE
with 	
fibre 	
product 	
and 	
$ 	
X 	
\in 	
\Ob(\mathcal{C})$. 	

 	
Let 	
$ 	
a 	
: 	
U 	
\to 	
X$ 	
be 	
a 	
hypercovering 	TYPE
of 	
$ 	
X$ 	
in 	
$ 	
\mathcal{C}$ 	
as 	
defined 	
above 	
. 	

 	
Then 	

 	
\begin{enumerate 	
} 	

 	
\item 	
$ 	
a^{-1 	
} 	
: 	
\Sh(\mathcal{C}/X 	
) 	
\to 	
\Sh((\mathcal{C}/U)_{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}/U)_{total})$ 	

 	
is 	
fully 	
faithful 	
with 	
essential 	
image 	
the 	
cartesian 	
sheaves 	

 	
of 	
abelian 	
groups 	
. 	

 	
\end{enumerate 	
} 	

 	
In 	
both 	
cases 	
$ 	
a_*$ 	
provides 	
the 	
quasi 	
- 	
inverse 	
functor 	
. 	

 	
\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 	TYPE
with 	
fibre 	
product 	
and 	
$ 	
X 	
\in 	
\Ob(\mathcal{C})$. 	

 	
Let 	
$ 	
a 	
: 	
U 	
\to 	
X$ 	
be 	
a 	
hypercovering 	TYPE
of 	
$ 	
X$ 	
in 	
$ 	
\mathcal{C}$ 	
as 	
defined 	
above 	
. 	

 	
For 	
$ 	
E 	
\in 	
D^+(\mathcal{C}/X)$ 	
the 	
map 	

 	
$ 	
$ 	

 	
E 	
\longrightarrow 	
Ra_*a^{-1}E 	

 	
$ 	
$ 	

 	
is 	
an 	
isomorphism 	
. 	

 	
\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 	TYPE
with 	
fibre 	
products 	
and 	
$ 	
X 	
\in 	
\Ob(\mathcal{C})$. 	

 	
Let 	
$ 	
a 	
: 	
U 	
\to 	
X$ 	
be 	
a 	
hypercovering 	TYPE
of 	
$ 	
X$ 	
in 	
$ 	
\mathcal{C}$ 	
as 	
defined 	
above 	
. 	

 	
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 	TYPE
with 	
fibre 	
product 	
and 	
$ 	
X 	
\in 	
\Ob(\mathcal{C})$. 	

 	
Let 	
$ 	
a 	
: 	
U 	
\to 	
X$ 	
be 	
a 	
hypercovering 	TYPE
of 	
$ 	
X$ 	
in 	
$ 	
\mathcal{C}$ 	
as 	
defined 	
above 	
. 	

 	
Let 	
$ 	
\mathcal{A 	
} 	
\subset 	
\textit{Ab}((\mathcal{C}/U)_{total})$ 	

 	
denote 	
the 	
weak 	
Serre 	
subcategory 	
of 	
cartesian 	
abelian 	
sheaves 	
. 	

 	
Then 	
the 	
functor 	
$ 	
a^{-1}$ 	
defines 	
an 	
equivalence 	

 	
$ 	
$ 	

 	
D^+(\mathcal{C}/X 	
) 	
\longrightarrow 	
D_\mathcal{A}^+((\mathcal{C}/U)_{total 	
} 	
) 	

 	
$ 	
$ 	

 	
with 	
quasi 	
- 	
inverse 	
$ 	
Ra_*$. 	

 	
\end{lemma 	
} 	


 	
\begin{proof 	
} 	

 	
This 	
is 	
a 	
special 	
case 	
of 	

 	
Lemma 	
\ref{lemma 	
- 	
hypercovering 	
- 	
X 	
- 	
equivalence 	
- 	
bounded 	
} 	

 	
\end{proof 	
} 	


 	
\begin{lemma 	
} 	

 	
\label{lemma 	
- 	
sr 	
- 	
when 	
- 	
fibre 	
- 	
products 	
} 	

 	
Let 	
$ 	
U$ 	
be 	
a 	
simplicial 	TYPE
object 	
of 	
a 	
site 	
$ 	
\mathcal{C}$ 	

 	
with 	
fibre 	
products 	
. 	

 	
\begin{enumerate 	
} 	

 	
\item 	
$ 	
\mathcal{C}/U$ 	
has 	
the 	
structure 	
of 	
a 	
simplicial 	
object 	

 	
in 	
the 	
category 	
whose 	
objects 	
are 	
sites 	
and 	

 	
whose 	
morphisms 	
are 	
morphisms 	
of 	
sites 	
, 	

 	
\item 	
the 	
construction 	
of 	
Lemma 	
\ref{lemma 	
- 	
simplicial 	
- 	
site 	
- 	
site 	
} 	

 	
applied 	
to 	
the 	
structure 	
in 	
( 	
1 	
) 	

 	
reproduces 	
the 	
site 	
$ 	
( 	
\mathcal{C}/U)_{total}$ 	
above 	
, 	

 	
\item 	
if 	
$ 	
a 	
: 	
U 	
\to 	
X$ 	
is 	
an 	
augmentation 	
, 	
then 	

 	
$ 	
a_0 	
: 	
\mathcal{C}/U_0 	
\to 	
\mathcal{C}/X$ 	
is 	
an 	
augmentation 	

 	
as 	
in 	
Remark 	
\ref{remark 	
- 	
augmentation 	
- 	
site 	
} 	
part 	
( 	
A 	
) 	
and 	
gives 	
the 	

 	
same 	
morphism 	
of 	
topoi 	

 	
$ 	
a 	
: 	
\Sh((\mathcal{C}/U)_{total 	
} 	
) 	
\to 	
\Sh(\mathcal{C}/X)$ 	

 	
as 	
the 	
one 	
above 	
. 	

 	
\end{enumerate 	
} 	

 	
\end{lemma 	
} 	


 	
\begin{proof 	
} 	

 	
Given 	
a 	
morphism 	
of 	
objects 	
$ 	
V 	
\to 	
W$ 	
of 	
$ 	
\mathcal{C}$ 	
the 	
localization 	

 	
morphism 	
$ 	
j 	
: 	
\mathcal{C}/V 	
\to 	
\mathcal{C}/W$ 	
is 	
a 	
left 	
adjoint 	
to 	

 	
the 	
base 	
change 	
functor 	
$ 	
\mathcal{C}/W 	
\to 	
\mathcal{C}/V$. 	

 	
The 	
base 	
change 	
functor 	
is 	
continuous 	
and 	
induces 	
the 	
same 	
morphism 	
of 	

 	
topoi 	
as 	
$ 	
j$. 	
See 	

 	
Sites 	
, 	
Lemma 	
\ref{sites 	
- 	
lemma 	
- 	
relocalize 	
- 	
given 	
- 	
fibre 	
- 	
products}. 	

 	
This 	
proves 	
( 	
1 	
) 	
. 	


 	
\medskip\noindent 	

 	
Part 	
( 	
2 	
) 	
holds 	
because 	
a 	
morphism 	
$ 	
V 	
/ 	
U_n 	
\to 	
W 	
/ 	
U_m$ 	

 	
of 	
the 	
category 	
constructed 	

 	
in 	
Lemma 	
\ref{lemma 	
- 	
simplicial 	
- 	
site 	
- 	
site 	
} 	

 	
is 	
a 	
morphism 	
$ 	
V 	
\to 	
W 	
\times_{U_m 	
, 	
f_\varphi 	
} 	
U_n$ 	
over 	
$ 	
U_n$ 	

 	
which 	
is 	
the 	
same 	
thing 	
as 	
a 	
morphism 	
$ 	
f 	
: 	
V 	
\to 	
W$ 	

 	
over 	
the 	
morphism 	
$ 	
f_\varphi 	
: 	
U_n 	
\to 	
U_m$ 	
, 	
i.e. 	
, 	

 	
the 	
same 	
thing 	
as 	
a 	
morphism 	
in 	
the 	
category 	
$ 	
( 	
\mathcal{C}/U)_{total}$ 	

 	
defined 	
above 	
. 	
Equality 	
of 	
sets 	
of 	
coverings 	
is 	

 	
immediate 	
from 	
the 	
definition 	
. 	


 	
\medskip\noindent 	

 	
We 	
omit 	
the 	
proof 	
of 	
( 	
3 	
) 	
. 	

 	
\end{proof 	
} 	








 	
\section{Hypercovering 	
by 	
a 	
simplicial 	
object 	
of 	
the 	
site 	
: 	
modules 	
} 	

 	
\label{section 	
- 	
hypercovering 	
- 	
modules 	
} 	


 	
\noindent 	

 	
Let 	
$ 	
\mathcal{C}$ 	
be 	
a 	
site 	TYPE
with 	
fibre 	
products 	
and 	
$ 	
X 	
\in 	
\Ob(\mathcal{C})$. 	

 	
Let 	
$ 	
\mathcal{O}_\mathcal{C}$ 	
be 	
a 	
sheaf 	TYPE
of 	
rings 	
on 	
$ 	
\mathcal{C}$. 	

 	
Let 	
$ 	
U 	
\to 	
X$ 	
be 	
a 	
hypercovering 	TYPE
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}. 	

 	
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 	TYPE
with 	
fibre 	
products 	
and 	
$ 	
X 	
\in 	
\Ob(\mathcal{C})$. 	

 	
Let 	
$ 	
\mathcal{O}_\mathcal{C}$ 	
be 	
a 	
sheaf 	TYPE
of 	
rings 	
. 	

 	
Let 	
$ 	
U$ 	
be 	
a 	
hypercovering 	TYPE
of 	
$ 	
X$ 	
in 	
$ 	
\mathcal{C}$. 	
With 	
notation 	
as 	
above 	

 	
$ 	
$ 	

 	
a^ 	
* 	
: 	
\textit{Mod}(\mathcal{O}_X 	
) 	
\to 	
\textit{Mod}(\mathcal{O 	
} 	
) 	

 	
$ 	
$ 	

 	
is 	
fully 	
faithful 	
with 	
essential 	
image 	
the 	
cartesian 	
$ 	
\mathcal{O}$-modules 	
. 	

 	
The 	
functor 	
$ 	
a_*$ 	
provides 	
the 	
quasi 	
- 	
inverse 	
. 	

 	
\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 	TYPE
with 	
fibre 	
products 	
and 	
$ 	
X 	
\in 	
\Ob(\mathcal{C})$. 	

 	
Let 	
$ 	
\mathcal{O}_\mathcal{C}$ 	
be 	
a 	
sheaf 	TYPE
of 	
rings 	
. 	

 	
Let 	
$ 	
U$ 	
be 	
a 	
hypercovering 	TYPE
of 	
$ 	
X$ 	
in 	
$ 	
\mathcal{C}$. 	
For 	

 	
$ 	
E 	
\in 	
D^+(\mathcal{O}_X)$ 	
the 	
map 	

 	
$ 	
$ 	

 	
E 	
\longrightarrow 	
Ra_*La^*E 	

 	
$ 	
$ 	

 	
is 	
an 	
isomorphism 	
. 	

 	
\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 	TYPE
with 	
fibre 	
products 	
and 	
$ 	
X 	
\in 	
\Ob(\mathcal{C})$. 	

 	
Let 	
$ 	
\mathcal{O}_\mathcal{C}$ 	
be 	
a 	
sheaf 	TYPE
of 	
rings 	
. 	

 	
Let 	
$ 	
U$ 	
be 	
a 	
hypercovering 	TYPE
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 	
} 	

 	
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 	TYPE
with 	
fibre 	
products 	
and 	
$ 	
X 	
\in 	
\Ob(\mathcal{C})$. 	

 	
Let 	
$ 	
\mathcal{O}_\mathcal{C}$ 	
be 	
a 	
sheaf 	TYPE
of 	
rings 	
. 	

 	
Let 	
$ 	
U$ 	
be 	
a 	
hypercovering 	TYPE
of 	
$ 	
X$ 	
in 	
$ 	
\mathcal{C}$. 	

 	
Let 	
$ 	
\mathcal{A 	
} 	
\subset 	
\textit{Mod}(\mathcal{O})$ 	

 	
denote 	
the 	
weak 	
Serre 	
subcategory 	
of 	
cartesian 	
$ 	
\mathcal{O}$-modules 	
. 	

 	
Then 	
the 	
functor 	
$ 	
La^*$ 	
defines 	
an 	
equivalence 	

 	
$ 	
$ 	

 	
D^+(\mathcal{O}_X 	
) 	
\longrightarrow 	
D_\mathcal{A}^+(\mathcal{O 	
} 	
) 	

 	
$ 	
$ 	

 	
with 	
quasi 	
- 	
inverse 	
$ 	
Ra_*$. 	

 	
\end{lemma 	
} 	


 	
\begin{proof 	
} 	

 	
This 	
is 	
a 	
special 	
case 	
of 	

 	
Lemma 	
\ref{lemma 	
- 	
hypercovering 	
- 	
X 	
- 	
equivalence 	
- 	
bounded 	
- 	
modules}. 	

 	
\end{proof 	
} 	








 	
\section{Unbounded 	
cohomological 	
descent 	
for 	
hypercoverings 	
} 	

 	
\label{section 	
- 	
unbounded 	
- 	
cohomological 	
- 	
descent 	
} 	


 	
\noindent 	

 	
In 	
this 	
section 	
we 	
discuss 	
unbounded 	
cohomological 	
descent 	
. 	

 	
The 	
results 	
themselves 	
will 	
be 	
immediate 	
consequences 	
of 	

 	
our 	
results 	
on 	
bounded 	
cohomological 	
descent 	
in 	
the 	
previous 	

 	
sections 	
and 	
Lemmas 	
\ref{lemma 	
- 	
equivalence 	
- 	
unbounded 	
- 	
one 	
} 	
and/or 	

 	
\ref{lemma 	
- 	
equivalence 	
- 	
unbounded 	
- 	
two 	
} 	
; 	
the 	
real 	
work 	
lies 	

 	
in 	
setting 	
up 	
notation 	
and 	
choosing 	
appropriate 	
assumptions 	
. 	

 	
Our 	
discussion 	
is 	
motivated 	
by 	
the 	
discussion 	
in 	
\cite{six 	
- 	
I 	
} 	

 	
although 	
the 	
details 	
are 	
a 	
good 	
bit 	
different 	
. 	


 	
\medskip\noindent 	

 	
Let 	
$ 	
( 	
\mathcal{C 	
} 	
, 	
\mathcal{O}_\mathcal{C})$ 	
be 	
a 	
ringed 	TYPE
site 	
. 	

 	
Assume 	
given 	
for 	
every 	
object 	
$ 	
U$ 	
of 	
$ 	
\mathcal{C}$ 	

 	
a 	
weak 	
Serre 	
subcategory 	
$ 	
\mathcal{A}_U 	
\subset 	
\textit{Mod}(\mathcal{O}_U)$ 	

 	
satisfying 	
the 	
following 	
properties 	

 	
\begin{enumerate 	
} 	

 	
\item 	

 	
\label{item 	
- 	
restriction 	
} 	

 	
given 	
a 	
morphism 	
$ 	
U 	
\to 	
V$ 	
of 	
$ 	
\mathcal{C}$ 	
the 	
restriction 	

 	
functor 	
$ 	
\textit{Mod}(\mathcal{O}_V 	
) 	
\to 	
\textit{Mod}(\mathcal{O}_U)$ 	

 	
sends 	
$ 	
\mathcal{A}_V$ 	
into 	
$ 	
\mathcal{A}_U$ 	
, 	

 	
\item 	

 	
\label{item 	
- 	
local 	
} 	

 	
given 	
a 	
covering 	
$ 	
\{U_i 	
\to 	
U\}_{i 	
\in 	
I}$ 	
of 	
$ 	
\mathcal{C}$ 	

 	
an 	
object 	
$ 	
\mathcal{F}$ 	
of 	
$ 	
\textit{Mod}(\mathcal{O}_U)$ 	

 	
is 	
in 	
$ 	
\mathcal{A}_U$ 	
if 	
and 	
only 	
if 	
the 	
restriction 	
of 	

 	
$ 	
\mathcal{F}$ 	
to 	
$ 	
\mathcal{C}/U_i$ 	
is 	
in 	
$ 	
\mathcal{A}_{U_i}$ 	

 	
for 	
all 	
$ 	
i 	
\in 	
I$. 	

 	
\item 	

 	
\label{item 	
- 	
bounded 	
- 	
dimension 	
} 	

 	
there 	
exists 	
a 	
subset 	
$ 	
\mathcal{B 	
} 	
\subset 	
\Ob(\mathcal{C})$ 	

 	
such 	
that 	

 	
\begin{enumerate 	
} 	

 	
\item 	
every 	
object 	
of 	
$ 	
\mathcal{C}$ 	
has 	
a 	
covering 	
whose 	

 	
members 	
are 	
in 	
$ 	
\mathcal{B}$ 	
, 	
and 	

 	
\item 	
for 	
every 	
$ 	
V 	
\in 	
\mathcal{B}$ 	
there 	
exists 	
an 	
integer 	
$ 	
d_V$ 	

 	
and 	
a 	
cofinal 	
system 	
$ 	
\text{Cov}_V$ 	
of 	
coverings 	
of 	
$ 	
V$ 	
such 	

 	
that 	

 	
$ 	
$ 	

 	
H^p(V_i 	
, 	
\mathcal{F 	
} 	
) 	
= 	
0 	
\text 	
{ 	
for 	
} 	

 	
\{V_i 	
\to 	
V\ 	
} 	
\in 	
\text{Cov}_V,\ 	
p 	
> 	
d_V 	
, 	
\text 	
{ 	
and 	
} 	

 	
\mathcal{F 	
} 	
\in 	
\Ob(\mathcal{A}_V 	
) 	

 	
$ 	
$ 	

 	
\end{enumerate 	
} 	

 	
\end{enumerate 	
} 	

 	
Note 	
that 	
we 	
require 	
this 	
to 	
be 	
true 	
for 	
$ 	
\mathcal{F}$ 	
in 	

 	
$ 	
\mathcal{A}_V$ 	
and 	
not 	
just 	
for 	
`` 	
global 	
'' 	
objects 	

 	
( 	
and 	
thus 	
it 	
is 	
stronger 	
than 	
the 	
condition 	
imposed 	
in 	

 	
Cohomology 	
on 	
Sites 	
, 	
Situation 	
\ref{sites 	
- 	
cohomology 	
- 	
situation 	
- 	
olsson 	
- 	
laszlo 	
} 	
) 	
. 	

 	
In 	
this 	
situation 	
, 	
there 	
is 	
a 	
weak 	
Serre 	
subcategory 	

 	
$ 	
\mathcal{A 	
} 	
\subset 	
\textit{Mod}(\mathcal{O}_\mathcal{C})$ 	

 	
consisting 	
of 	
objects 	
whose 	
restriction 	
to 	
$ 	
\mathcal{C}/U$ 	

 	
is 	
in 	
$ 	
\mathcal{A}_U$ 	
for 	
all 	
$ 	
U 	
\in 	
\Ob(\mathcal{C})$. 	

 	
Moreover 	
, 	
there 	
are 	
derived 	
categories 	

 	
$ 	
D_\mathcal{A}(\mathcal{O}_\mathcal{C})$ 	
and 	

 	
$ 	
D_{\mathcal{A}_U}(\mathcal{O}_U)$ 	
and 	
the 	
restriction 	

 	
functors 	
send 	
these 	
into 	
each 	
other 	
. 	


 	
\begin{example 	
} 	

 	
\label{example 	
- 	
quasi 	
- 	
coherent 	
- 	
spaces 	
- 	
etale 	
} 	

 	
Let 	
$ 	
S$ 	
be 	
a 	
scheme 	TYPE
and 	
let 	
$ 	
X$ 	
be 	
an 	
algebraic 	TYPE
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 	
. 	

 	
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 	
. 	

 	
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 	
. 	

 	
\end{enumerate 	
} 	

 	
Let 	
$ 	
\Lambda$ 	
be 	
a 	
finite 	TYPE
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 	
. 	

 	
Let 	
$ 	
\mathcal{O}_\mathcal{C 	
} 	
= 	
\underline{\Lambda}$ 	
be 	
the 	

 	
constant 	
sheaf 	
. 	
Set 	
$ 	
\mathcal{A}_U 	
= 	
\textit{Mod}(\mathcal{O}_U)$ 	
, 	

 	
in 	
other 	
words 	
, 	
we 	
consider 	
all 	
\'etale 	
sheaves 	
of 	
$ 	
\Lambda$-modules 	
. 	

 	
Let 	
$ 	
\mathcal{B 	
} 	
\subset 	
\Ob(\mathcal{C})$ 	

 	
be 	
the 	
set 	
of 	
quasi 	
- 	
compact 	
objects 	
. 	
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 	
. 	
To 	
see 	
that 	
condition 	
( 	
3 	
) 	

 	
holds 	
with 	
this 	
choice 	
, 	
use 	

 	
\cite[Expos\'e 	
VIII 	
- 	
A 	
, 	
Corollary 	
1.2 	
and 	
Lemma 	
2.2]{Traveaux 	
} 	

 	
plus 	
elementary 	
arguments 	
on 	
cohomological 	
dimensions 	
of 	
fields 	
. 	

 	
We 	
add 	
$ 	
1 	
$ 	
to 	
the 	
formula 	
because 	
our 	
list 	
contains 	
cases 	
where 	
we 	
allow 	
$ 	
S$ 	

 	
to 	
have 	
finite 	
residue 	
field 	
. 	

 	
We 	
will 	
come 	
back 	
to 	
this 	
example 	
later 	
( 	
insert 	
future 	
reference 	
) 	
. 	

 	
\end{example 	
} 	


 	
\noindent 	

 	
Let 	
$ 	
( 	
\mathcal{C 	
} 	
, 	
\mathcal{O}_\mathcal{C})$ 	
be 	
a 	
ringed 	TYPE
site 	
. 	

 	
Assume 	
given 	
weak 	
Serre 	
subcategories 	

 	
$ 	
\mathcal{A}_U 	
\subset 	
\textit{Mod}(\mathcal{O}_U)$ 	

 	
satisfying 	
condition 	
( 	
\ref{item 	
- 	
restriction 	
} 	
) 	
. 	

 	
Then 	

 	
\begin{enumerate 	
} 	

 	
\item 	
given 	
a 	
semi 	
- 	
representable 	
object 	
$ 	
K 	
= 	
\{U_i\}_{i 	
\in 	
I}$ 	

 	
we 	
get 	
a 	
weak 	
Serre 	
subcategory 	

 	
$ 	
\mathcal{A}_K 	
\subset 	
\textit{Mod}(\mathcal{O}_K)$ 	

 	
by 	
taking 	
$ 	
\prod 	
\mathcal{A}_{U_i 	
} 	
\subset 	

 	
\prod 	
\textit{Mod}(\mathcal{O}_{U_i 	
} 	
) 	
= 	
\textit{Mod}(\mathcal{O}_K)$ 	
, 	
and 	

 	
\item 	
given 	
a 	
morphism 	
of 	
semi 	
- 	
representable 	
objects 	

 	
$ 	
f 	
: 	
K 	
\to 	
L$ 	
the 	
pullback 	
map 	

 	
$ 	
f^ 	
* 	
: 	
\textit{Mod}(\mathcal{O}_L 	
) 	
\to 	
\textit{Mod}(\mathcal{O}_L)$ 	

 	
sends 	
$ 	
\mathcal{A}_L$ 	
into 	
$ 	
\mathcal{A}_K$. 	

 	
\end{enumerate 	
} 	

 	
See 	
Remark 	
\ref{remark 	
- 	
semi 	
- 	
representable 	
- 	
ringed 	
} 	
for 	
notation 	
and 	

 	
explanation 	
. 	
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 	TYPE
site 	
. 	

 	
Assume 	
given 	
weak 	
Serre 	
subcategories 	

 	
$ 	
\mathcal{A}_U 	
\subset 	
\textit{Mod}(\mathcal{O}_U)$ 	

 	
satisfying 	
conditions 	
( 	
\ref{item 	
- 	
restriction 	
} 	
) 	
, 	

 	
( 	
\ref{item 	
- 	
local 	
} 	
) 	
, 	
and 	
( 	
\ref{item 	
- 	
bounded 	
- 	
dimension 	
} 	
) 	
above 	
. 	

 	
Assume 	
$ 	
\mathcal{C}$ 	
has 	
equalizers 	
and 	
fibre 	
products 	
and 	

 	
let 	
$ 	
K$ 	
be 	
a 	
hypercovering 	
. 	

 	
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 	
) 	
. 	

 	
Then 	
the 	
functor 	
$ 	
La^*$ 	
defines 	
an 	
equivalence 	

 	
$ 	
$ 	

 	
D_\mathcal{A}(\mathcal{O}_\mathcal{C 	
} 	
) 	

 	
\longrightarrow 	

 	
D_{\mathcal{A}_{total}}(\mathcal{O 	
} 	
) 	

 	
$ 	
$ 	

 	
with 	
quasi 	
- 	
inverse 	
$ 	
Ra_*$. 	

 	
\end{lemma 	
} 	


 	
\begin{proof 	
} 	

 	
The 	
cartesian 	
$ 	
\mathcal{O}$-modules 	
form 	
a 	
weak 	
Serre 	
subcategory 	
by 	

 	
Lemma 	
\ref{lemma 	
- 	
Serre 	
- 	
subcat 	
- 	
cartesian 	
- 	
modules 	
} 	

 	
( 	
the 	
required 	
hypotheses 	
hold 	
by 	
the 	
discussion 	
in 	

 	
Remark 	
\ref{remark 	
- 	
augmentation 	
- 	
ringed 	
} 	
) 	
. 	

 	
Since 	
the 	
restriction 	
functor 	

 	
$ 	
g_n^ 	
* 	
: 	
\textit{Mod}(\mathcal{O 	
} 	
) 	
\to 	
\textit{Mod}(\mathcal{O}_n)$ 	

 	
are 	
exact 	
, 	
it 	
follows 	
that 	
$ 	
\mathcal{A}_{total}$ 	
is 	
a 	
weak 	
Serre 	

 	
subcategory 	
. 	


 	
\medskip\noindent 	

 	
Let 	
us 	
show 	
that 	
$ 	
a^ 	
* 	
: 	
\mathcal{A 	
} 	
\to 	
\mathcal{A}_{total}$ 	

 	
is 	
an 	
equivalence 	
of 	
categories 	
with 	
inverse 	
given 	
by 	
$ 	
La_*$. 	

 	
We 	
already 	
know 	
that 	
$ 	
La_*a^*\mathcal{F 	
} 	
= 	
\mathcal{F}$ 	
by 	
the 	

 	
bounded 	
version 	

 	
( 	
Lemma 	
\ref{lemma 	
- 	
hypercovering 	
- 	
equivalence 	
- 	
bounded 	
- 	
modules 	
} 	
) 	
. 	

 	
It 	
is 	
clear 	
that 	
$ 	
a^*\mathcal{F}$ 	
is 	
in 	
$ 	
\mathcal{A}_{total}$ 	

 	
for 	
$ 	
\mathcal{F}$ 	
in 	
$ 	
\mathcal{A}$. 	
Conversely 	
, 	
assume 	
that 	

 	
$ 	
\mathcal{G 	
} 	
\in 	
\mathcal{A}_{total}$. 	
Because 	
$ 	
\mathcal{G}$ 	

 	
is 	
cartesian 	
we 	
see 	
that 	
$ 	
\mathcal{G 	
} 	
= 	
a^*\mathcal{F}$ 	

 	
for 	
some 	
$ 	
\mathcal{O}_\mathcal{C}$-module 	
$ 	
\mathcal{F}$ 	
by 	

 	
Lemma 	
\ref{lemma 	
- 	
hypercovering 	
- 	
descent 	
- 	
modules}. 	

 	
We 	
want 	
to 	
show 	
that 	
$ 	
\mathcal{F}$ 	
is 	
in 	
$ 	
\mathcal{A}$. 	

 	
Take 	
$ 	
U 	
\in 	
\Ob(\mathcal{C})$. 	
We 	
have 	
to 	
show 	
that 	
the 	

 	
restriction 	
of 	
$ 	
\mathcal{F}$ 	
to 	
$ 	
\mathcal{C}/U$ 	
is 	
in 	
$ 	
\mathcal{A}_U$. 	

 	
As 	
usual 	
, 	
write 	
$ 	
K_0 	
= 	
\{U_{0 	
, 	
i}\}_{i 	
\in 	
I_0}$. 	

 	
Since 	
$ 	
K$ 	
is 	
a 	
hypercovering 	
, 	
the 	
map 	
$ 	
\coprod_{i 	
\in 	
I_0 	
} 	
h_{U_{0 	
, 	
i 	
} 	
} 	
\to 	
* 	
$ 	

 	
becomes 	
surjective 	
after 	
sheafification 	
. 	
This 	
implies 	
there 	
is 	

 	
a 	
covering 	
$ 	
\{U_j 	
\to 	
U\}_{j 	
\in 	
J}$ 	
and 	
a 	
map 	
$ 	
\tau 	
: 	
J 	
\to 	
I_0 	
$ 	

 	
and 	
for 	
each 	
$ 	
j 	
\in 	
J$ 	
a 	
morphism 	
$ 	
\varphi_j 	
: 	
U_j 	
\to 	
U_{0 	
, 	
\tau(j)}$. 	

 	
Since 	
$ 	
\mathcal{G}_0 	
= 	
a_0^*\mathcal{F}$ 	
we 	
find 	

 	
that 	
the 	
restriction 	
of 	
$ 	
\mathcal{F}$ 	
to 	
$ 	
\mathcal{C}/U_j$ 	

 	
is 	
equal 	
to 	
the 	
restriction 	
of 	
the 	
$ 	
\tau(j)$th 	
component 	
of 	

 	
$ 	
\mathcal{G}_0 	
$ 	
to 	
$ 	
\mathcal{C}/U_j$ 	
via 	
the 	
morphism 	

 	
$ 	
\varphi_j 	
: 	
U_j 	
\to 	
U_{0 	
, 	
\tau(i)}$. 	
Hence 	
by 	

 	
( 	
\ref{item 	
- 	
restriction 	
} 	
) 	
we 	
find 	
that 	
$ 	
\mathcal{F}|_{\mathcal{C}/U_j}$ 	

 	
is 	
in 	
$ 	
\mathcal{A}_{U_j}$ 	
and 	
in 	
turn 	
by 	

 	
( 	
\ref{item 	
- 	
local 	
} 	
) 	
we 	
find 	
that 	
$ 	
\mathcal{F}|_{\mathcal{C}/U}$ 	

 	
is 	
in 	
$ 	
\mathcal{A}_U$. 	


 	
\medskip\noindent 	

 	
In 	
particular 	
the 	
statement 	
of 	
the 	
lemma 	
makes 	
sense 	
. 	

 	
The 	
lemma 	
now 	
follows 	
from 	

 	
Lemma 	
\ref{lemma 	
- 	
equivalence 	
- 	
unbounded 	
- 	
one}. 	

 	
Assumption 	
( 	
1 	
) 	
is 	
clear 	
( 	
see 	
Remark 	
\ref{remark 	
- 	
augmentation 	
- 	
ringed 	
} 	
) 	
. 	

 	
Assumptions 	
( 	
2 	
) 	
and 	
( 	
3 	
) 	
we 	
proved 	
in 	
the 	
preceding 	
paragraph 	
. 	

 	
Assumption 	
( 	
4 	
) 	
is 	
immediate 	
from 	
( 	
\ref{item 	
- 	
bounded 	
- 	
dimension 	
} 	
) 	
. 	

 	
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 	
} 	
) 	
. 	
Here 	
we 	
use 	
the 	
description 	
of 	

 	
the 	
site 	
$ 	
( 	
\mathcal{C}/K)_{total}$ 	
given 	
in 	

 	
Section 	
\ref{section 	
- 	
simplicial 	
- 	
semi 	
- 	
representable}. 	

 	
Moreover 	
, 	
we 	
set 	
$ 	
\text{Cov}_{U 	
/ 	
U_{n 	
, 	
i}}$ 	
equal 	
to 	
$ 	
\text{Cov}_U$ 	

 	
and 	
$ 	
d_{U 	
/ 	
U_{n 	
, 	
i}}$ 	
equal 	
$ 	
d_U$ 	
where 	
$ 	
\text{Cov}_U$ 	
and 	
$ 	
d_U$ 	

 	
are 	
given 	
to 	
us 	
by 	
( 	
\ref{item 	
- 	
bounded 	
- 	
dimension 	
} 	
) 	
. 	

 	
Then 	
we 	
claim 	
that 	
condition 	
( 	
5 	
) 	
holds 	
with 	
these 	
choices 	
. 	

 	
This 	
follows 	
immediately 	
from 	

 	
Lemma 	
\ref{lemma 	
- 	
sanity 	
- 	
check 	
- 	
simplicial 	
- 	
semi 	
- 	
representable 	
} 	

 	
and 	
the 	
fact 	
that 	
$ 	
\mathcal{F 	
} 	
\in 	
\mathcal{A}_{total}$ 	

 	
implies 	
$ 	
\mathcal{F}_n 	
\in 	
\mathcal{A}_{K_n}$ 	
and 	
hence 	

 	
$ 	
\mathcal{F}_{n 	
, 	
i 	
} 	
\in 	
\mathcal{A}_{U_{n 	
, 	
i}}$. 	

 	
( 	
The 	
reader 	
who 	
worries 	
about 	
the 	
difference 	
between 	

 	
cohomology 	
of 	
abelian 	
sheaves 	
versus 	
cohomology 	

 	
of 	
sheaves 	
of 	
modules 	
may 	
consult 	
Cohomology 	
on 	
Sites 	
, 	
Lemma 	

 	
\ref{sites 	
- 	
cohomology 	
- 	
lemma 	
- 	
cohomology 	
- 	
modules 	
- 	
abelian 	
- 	
agree}. 	
) 	

 	
\end{proof 	
} 	











 	
\section{Glueing 	
complexes 	
} 	

 	
\label{section 	
- 	
glueing 	
- 	
complexes 	
} 	


 	
\noindent 	

 	
This 	
section 	
is 	
the 	
continuation 	
of 	

 	
Cohomology 	
, 	
Section 	
\ref{cohomology 	
- 	
section 	
- 	
glueing 	
- 	
complexes}. 	

 	
The 	
goal 	
is 	
to 	
prove 	
a 	
slight 	
generalizaton 	
of 	
\cite[Theorem 	
3.2.4]{BBD}. 	

 	
Our 	
method 	
will 	
be 	
a 	
tiny 	TYPE
bit 	
different 	
in 	
that 	
we 	
use 	

 	
the 	
material 	
from 	
Sections 	
\ref{section 	
- 	
glueing 	
} 	
and 	

 	
\ref{section 	
- 	
glueing 	
- 	
modules}. 	
We 	
will 	
also 	
reprove 	
the 	

 	
unbounded 	
version 	
as 	
it 	
is 	
proved 	
in 	
\cite{six 	
- 	
I}. 	


 	
\medskip\noindent 	

 	
Here 	
is 	
the 	
situation 	
we 	
are 	
interested 	
in 	
. 	


 	
\begin{situation 	
} 	

 	
\label{situation 	
- 	
locally 	
- 	
given 	
} 	

 	
Let 	
$ 	
( 	
\mathcal{C 	
} 	
, 	
\mathcal{O}_\mathcal{C})$ 	
be 	
a 	
ringed 	TYPE
site 	
. 	
We 	
are 	
given 	

 	
\begin{enumerate 	
} 	

 	
\item 	
a 	
category 	
$ 	
\mathcal{B}$ 	
and 	
a 	
functor 	

 	
$ 	
u 	
: 	
\mathcal{B 	
} 	
\to 	
\mathcal{C}$ 	
, 	

 	
\item 	
an 	
object 	
$ 	
E_U$ 	
in 	
$ 	
D(\mathcal{O}_{u(U)})$ 	
for 	
$ 	
U 	
\in 	
\Ob(\mathcal{B})$ 	
, 	

 	
\item 	
an 	
isomorphism 	
$ 	
\rho_a 	
: 	
E_U|_{\mathcal{C}/u(V 	
) 	
} 	
\to 	
E_V$ 	
in 	

 	
$ 	
D(\mathcal{O}_{u(V)})$ 	
for 	
$ 	
a 	
: 	
V 	
\to 	
U$ 	
in 	
$ 	
\mathcal{B}$ 	

 	
\end{enumerate 	
} 	

 	
such 	
that 	
whenever 	
we 	
have 	
composable 	
arrows 	

 	
$ 	
b 	
: 	
W 	
\to 	
V$ 	
and 	
$ 	
a 	
: 	
V 	
\to 	
U$ 	
of 	
$ 	
\mathcal{B}$ 	
, 	
then 	

 	
$ 	
\rho_{a 	
\circ 	
b 	
} 	
= 	
\rho_b 	
\circ 	
\rho_a|_{\mathcal{C}/u(W)}$. 	

 	
\end{situation 	
} 	


 	
\noindent 	

 	
We 	
wo 	
n't 	
be 	
able 	
to 	
prove 	
anything 	
about 	
this 	
without 	
making 	
more 	

 	
assumptions 	
. 	
An 	
interesting 	
case 	
is 	
where 	
$ 	
\mathcal{B}$ 	
is 	
a 	
full 	

 	
subcategory 	
such 	
that 	
every 	
object 	
of 	
$ 	
\mathcal{C}$ 	
has 	
a 	
covering 	

 	
whose 	
members 	
are 	
objects 	
of 	
$ 	
\mathcal{B}$ 	
( 	
this 	
is 	
the 	
case 	
considered 	

 	
in 	
\cite{BBD 	
} 	
) 	
. 	
For 	
us 	
it 	
is 	
important 	
to 	
allow 	
cases 	
where 	
this 	
is 	
not 	

 	
the 	
case 	
; 	
the 	
main 	
alternative 	
case 	
is 	
where 	
we 	
have 	
a 	
morphism 	

 	
of 	
sites 	
$ 	
f 	
: 	
\mathcal{C 	
} 	
\to 	
\mathcal{D}$ 	
and 	
$ 	
\mathcal{B}$ 	

 	
is 	
a 	
full 	
subcategory 	
of 	
$ 	
\mathcal{D}$ 	
such 	
that 	
every 	
object 	
of 	

 	
$ 	
\mathcal{D}$ 	
has 	
a 	
covering 	
whose 	
members 	
are 	
objects 	
of 	
$ 	
\mathcal{B}$. 	


 	
\medskip\noindent 	

 	
In 	
Situation 	
\ref{situation 	
- 	
locally 	
- 	
given 	
} 	
a 	
{ 	
\it 	
solution 	
} 	

 	
will 	
be 	
a 	
pair 	TYPE
$ 	
( 	
E 	
, 	
\rho_U)$ 	
where 	
$ 	
E$ 	
is 	
an 	
object 	
of 	

 	
$ 	
D(\mathcal{O}_\mathcal{C})$ 	

 	
and 	
$ 	
\rho_U 	
: 	
E|_{\mathcal{C}/u(U 	
) 	
} 	
\to 	
E_U$ 	

 	
for 	
$ 	
U 	
\in 	
\Ob(\mathcal{B})$ 	

 	
are 	
isomorphisms 	
such 	
that 	

 	
we 	
have 	
$ 	
\rho_a 	
\circ 	
\rho_U|_{\mathcal{C}/u(V 	
) 	
} 	
= 	
\rho_V$ 	

 	
for 	
$ 	
a 	
: 	
V 	
\to 	
U$ 	
in 	
$ 	
\mathcal{B}$. 	


 	
\begin{lemma 	
} 	

 	
\label{lemma 	
- 	
prepare 	
- 	
bbd 	
- 	
glueing 	
} 	

 	
In 	
Situation 	
\ref{situation 	
- 	
locally 	
- 	
given}. 	

 	
Assume 	
negative 	
self 	
- 	
exts 	
of 	
$ 	
E_U$ 	
in 	
$ 	
D(\mathcal{O}_{u(U)})$ 	
are 	
zero 	
. 	

 	
Let 	
$ 	
L$ 	
be 	
a 	
simplicial 	TYPE
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 	
) 	
. 	

 	
Moreover 	
, 	
$ 	
E$ 	
is 	
unique 	
up 	
to 	
unique 	
isomorphism 	
. 	

 	
\end{lemma 	
} 	


 	
\begin{proof 	
} 	

 	
Recall 	
that 	

 	
$ 	
\Sh(\mathcal{C}/K_n 	
) 	
= 	
\prod_{i 	
\in 	
I_n 	
} 	
\Sh(\mathcal{C}/u(U_{n 	
, 	
i}))$ 	

 	
and 	
similarly 	
for 	
the 	
categories 	
of 	
modules 	
. 	
This 	
product 	
decomposition 	

 	
is 	
also 	
inherited 	
by 	
the 	
derived 	
categories 	
of 	
sheaves 	
of 	
modules 	
. 	

 	
Moreover 	
, 	
this 	
product 	
decomposition 	
is 	
compatible 	
with 	

 	
the 	
morphisms 	
in 	
the 	
simplicial 	
semi 	
- 	
representable 	
object 	
$ 	
K$. 	

 	
See 	
Section 	
\ref{section 	
- 	
semi 	
- 	
representable}. 	

 	
Hence 	
we 	
can 	
set 	
$ 	
E_n 	
= 	
\prod_{i 	
\in 	
I_n 	
} 	
E_{U_{n 	
, 	
i}}$ 	

 	
( 	
`` 	
formal 	
'' 	
product 	
) 	
in 	
$ 	
D(\mathcal{O}_n)$. 	

 	
Taking 	
( 	
formal 	
) 	
products 	
of 	
the 	
maps 	
$ 	
\rho_a$ 	
of 	

 	
Situation 	
\ref{situation 	
- 	
locally 	
- 	
given 	
} 	

 	
we 	
obtain 	
isomorphisms 	
$ 	
E_\varphi 	
: 	
f_\varphi^*E_n 	
\to 	
E_m$. 	

 	
The 	
assumption 	
about 	
compostions 	
of 	
the 	
maps 	
$ 	
\rho_a$ 	

 	
immediately 	
implies 	
that 	
$ 	
( 	
E_n 	
, 	
E_\varphi)$ 	

 	
defines 	
a 	
simplicial 	
system 	
of 	
the 	
derived 	
category 	
of 	
modules 	

 	
as 	
in 	
Definition 	
\ref{definition 	
- 	
cartesian 	
- 	
derived 	
- 	
modules}. 	

 	
The 	
vanishing 	
of 	
negative 	
exts 	
assumed 	
in 	
the 	
lemma 	
implies 	
that 	

 	
$ 	
\Hom(E_n[t 	
] 	
, 	
E_n 	
) 	
= 	
0 	
$ 	
for 	
$ 	
n 	
\geq 	
0 	
$ 	
and 	
$ 	
t 	
> 	
0$. 	

 	
Thus 	
by 	

 	
Lemma 	
\ref{lemma 	
- 	
cartesian 	
- 	
module 	
- 	
derived 	
- 	
from 	
- 	
simplicial 	
} 	

 	
we 	
obtain 	
$ 	
E$. 	

 	
Uniqueness 	
up 	
to 	
unique 	
isomorphism 	
follows 	
from 	

 	
Lemmas 	
\ref{lemma 	
- 	
nullity 	
- 	
cartesian 	
- 	
modules 	
- 	
derived 	
} 	
and 	

 	
\ref{lemma 	
- 	
hom 	
- 	
cartesian 	
- 	
modules 	
- 	
derived}. 	

 	
\end{proof 	
} 	


 	
\begin{lemma}[BBD 	
glueing 	
lemma 	
] 	

 	
\label{lemma 	
- 	
bbd 	
- 	
glueing 	
} 	

 	
In 	
Situation 	
\ref{situation 	
- 	
locally 	
- 	
given}. 	
Assume 	

 	
\begin{enumerate 	
} 	

 	
\item 	
$ 	
\mathcal{C}$ 	
has 	
equalizers 	
and 	
fibre 	
products 	
, 	

 	
\item 	
there 	
is 	
a 	
morphism 	
of 	
sites 	
$ 	
f 	
: 	
\mathcal{C 	
} 	
\to 	
\mathcal{D}$ 	

 	
given 	
by 	
a 	
continuous 	
functor 	
$ 	
u 	
: 	
\mathcal{D 	
} 	
\to 	
\mathcal{C}$ 	

 	
such 	
that 	

 	
\begin{enumerate 	
} 	

 	
\item 	
$ 	
\mathcal{D}$ 	
has 	
equalizers 	
and 	
fibre 	
products 	
and 	
$ 	
u$ 	

 	
commutes 	
with 	
them 	
, 	

 	
\item 	
$ 	
\mathcal{B}$ 	
is 	
a 	
full 	
subcategory 	
of 	
$ 	
\mathcal{D}$ 	

 	
and 	
$ 	
u 	
: 	
\mathcal{B 	
} 	
\to 	
\mathcal{C}$ 	
is 	
the 	
restriction 	
of 	
$ 	
u$ 	
, 	

 	
\item 	
every 	
object 	
of 	
$ 	
\mathcal{D}$ 	
has 	
a 	
covering 	
whose 	
members 	

 	
are 	
objects 	
of 	
$ 	
\mathcal{B}$ 	
, 	

 	
\end{enumerate 	
} 	

 	
\item 	
all 	
negative 	
self 	
- 	
exts 	
of 	
$ 	
E_U$ 	
in 	
$ 	
D(\mathcal{O}_{u(U)})$ 	
are 	
zero 	
, 	
and 	

 	
\item 	
there 	
exists 	
a 	
$ 	
t 	
\in 	
\mathbf{Z}$ 	
such 	
that 	
$ 	
H^i(E_U 	
) 	
= 	
0 	
$ 	
for 	
$ 	
i 	
< 	
t$ 	

 	
and 	
$ 	
U 	
\in 	
\Ob(\mathcal{B})$. 	

 	
\end{enumerate 	
} 	

 	
Then 	
there 	
exists 	
a 	
solution 	
unique 	
up 	
to 	
unique 	
isomorphism 	
. 	

 	
\end{lemma 	
} 	


 	
\begin{proof 	
} 	

 	
By 	
Hypercoverings 	
, 	
Lemma 	
\ref{hypercovering 	
- 	
lemma 	
- 	
hypercovering 	
- 	
site 	
} 	

 	
there 	
exists 	
a 	
hypercovering 	
$ 	
L$ 	
for 	
the 	
site 	
$ 	
\mathcal{D}$ 	
such 	
that 	

 	
$ 	
L_n 	
= 	
\{U_{n 	
, 	
i}\}_{i 	
\in 	
I_n}$ 	
with 	
$ 	
U_{i 	
, 	
n 	
} 	
\in 	
\Ob(\mathcal{B})$. 	

 	
Set 	
$ 	
K 	
= 	
u(L)$. 	
Apply 	
Lemma 	
\ref{lemma 	
- 	
prepare 	
- 	
bbd 	
- 	
glueing 	
} 	

 	
to 	
get 	
a 	
cartesian 	
object 	
$ 	
E$ 	
of 	
$ 	
D(\mathcal{O})$ 	
on 	
the 	
site 	

 	
$ 	
( 	
\mathcal{C}/K)_{total}$ 	
restricting 	
to 	
$ 	
E_{U_{n 	
, 	
i}}$ 	
on 	

 	
$ 	
\mathcal{C}/u(U_{n 	
, 	
i})$ 	
compatibly 	
. 	

 	
The 	
assumption 	
on 	
$ 	
t$ 	
implies 	
that 	
$ 	
E 	
\in 	
D^+(\mathcal{O})$. 	

 	
By 	
Hypercoverings 	
, 	
Lemma 	
\ref{hypercovering 	
- 	
lemma 	
- 	
hypercovering 	
- 	
morphism 	
- 	
sites 	
} 	

 	
we 	
see 	
that 	
$ 	
K$ 	
is 	
a 	
hypercovering 	
too 	
. 	

 	
By 	
Lemma 	
\ref{lemma 	
- 	
hypercovering 	
- 	
equivalence 	
- 	
bounded 	
- 	
modules 	
} 	

 	
we 	
find 	
that 	
$ 	
E 	
= 	
a^*F$ 	
for 	
some 	
$ 	
F$ 	
in 	
$ 	
D^+(\mathcal{O}_\mathcal{C})$. 	


 	
\medskip\noindent 	

 	
To 	
prove 	
that 	
$ 	
F$ 	
is 	
a 	
solution 	
we 	
will 	
use 	
the 	
construction 	
of 	

 	
$ 	
L_0 	
$ 	
and 	
$ 	
L_1 	
$ 	
given 	
in 	
the 	
proof 	
of 	

 	
Hypercoverings 	
, 	
Lemma 	
\ref{hypercovering 	
- 	
lemma 	
- 	
hypercovering 	
- 	
site}. 	

 	
( 	
This 	
is 	
a 	
bit 	
inelegant 	
but 	
there 	
does 	
not 	
seem 	
to 	
be 	
a 	
completely 	TYPE

 	
straightforward 	
way 	
around 	
it 	
. 	
) 	


 	
\medskip\noindent 	

 	
Namely 	
, 	
we 	
have 	
$ 	
I_0 	
= 	
\Ob(\mathcal{B})$ 	
and 	
so 	

 	
$ 	
L_0 	
= 	
\{U\}_{U 	
\in 	
\Ob(\mathcal{B})}$. 	

 	
Hence 	
the 	
isomorphism 	
$ 	
a^*F 	
\to 	
E$ 	
restricted 	
to 	
the 	
components 	

 	
$ 	
\mathcal{C}/u(U)$ 	
of 	
$ 	
\mathcal{C}/K_0 	
$ 	
defines 	
isomorphisms 	

 	
$ 	
\rho_U 	
: 	
F|_{\mathcal{C}/u(U 	
) 	
} 	
\to 	
E_U$ 	
for 	
$ 	
U 	
\in 	
\Ob(\mathcal{B})$ 	

 	
by 	
our 	
choice 	
of 	
$ 	
E$. 	


 	
\medskip\noindent 	

 	
To 	
prove 	
that 	
$ 	
\rho_U$ 	
satisfy 	
the 	
requirement 	
of 	
compatibility 	

 	
with 	
the 	
maps 	
$ 	
\rho_a$ 	
of 	
Situation 	
\ref{situation 	
- 	
locally 	
- 	
given 	
} 	

 	
we 	
use 	
that 	
$ 	
I_1 	
$ 	
contains 	
the 	
set 	

 	
$ 	
$ 	

 	
\Omega 	
= 	

 	
\{(U 	
, 	
V 	
, 	
W 	
, 	
a 	
, 	
b 	
) 	
\mid 	
U 	
, 	
V 	
, 	
W 	
\in 	
\mathcal{B 	
} 	
, 	
a 	
: 	
U 	
\to 	
V 	
, 	
b 	
: 	
U 	
\to 	
W\ 	
} 	

 	
$ 	
$ 	

 	
and 	
that 	
for 	
$ 	
i 	
= 	
( 	
U 	
, 	
V 	
, 	
W 	
, 	
a 	
, 	
b)$ 	
in 	
$ 	
\Omega$ 	
we 	
have 	

 	
$ 	
U_{1 	
, 	
i 	
} 	
= 	
U$. 	
Moreover 	
, 	
the 	
component 	
maps 	
$ 	
f_{\delta^1_0 	
, 	
i}$ 	
and 	

 	
$ 	
f_{\delta^1_1 	
, 	
i}$ 	
of 	
the 	
two 	
morphisms 	
$ 	
K_1 	
\to 	
K_0 	
$ 	
are 	
the 	
morphisms 	

 	
$ 	
$ 	

 	
a 	
: 	
U 	
\to 	
V 	
\quad\text{and}\quad 	
b 	
: 	
U 	
\to 	
V 	

 	
$ 	
$ 	

 	
Hence 	
the 	
compatibility 	
mentioned 	
in 	

 	
Lemma 	
\ref{lemma 	
- 	
prepare 	
- 	
bbd 	
- 	
glueing 	
} 	
gives 	
that 	

 	
$ 	
$ 	

 	
\rho_a 	
\circ 	
\rho_V|_{\mathcal{C}/u(U 	
) 	
} 	
= 	
\rho_U 	

 	
\quad\text{and}\quad 	

 	
\rho_b 	
\circ 	
\rho_W|_{\mathcal{C}/u(U 	
) 	
} 	
= 	
\rho_U 	

 	
$ 	
$ 	

 	
Taking 	
$ 	
i 	
= 	
( 	
U 	
, 	
V 	
, 	
U 	
, 	
a 	
, 	
\text{id}_U 	
) 	
\in 	
\Omega$ 	
for 	
example 	
, 	
we 	
find 	

 	
that 	
we 	
have 	
the 	
desired 	
compatibility 	
. 	
The 	
uniqueness 	
of 	
$ 	
F$ 	
follows 	

 	
from 	
the 	
uniqueness 	
of 	
$ 	
E$ 	
in 	
the 	
previous 	
lemma 	
( 	
small 	
detail 	
omitted 	
) 	
. 	

 	
\end{proof 	
} 	


 	
\begin{lemma}[Unbounded 	
BBD 	
glueing 	
lemma 	
] 	

 	
\label{lemma 	
- 	
bbd 	
- 	
unbounded 	
- 	
glueing 	
} 	

 	
In 	
Situation 	
\ref{situation 	
- 	
locally 	
- 	
given}. 	
Assume 	

 	
\begin{enumerate 	
} 	

 	
\item 	
$ 	
\mathcal{C}$ 	
has 	
equalizers 	
and 	
fibre 	
products 	
, 	

 	
\item 	
there 	
is 	
a 	
morphism 	
of 	
sites 	
$ 	
f 	
: 	
\mathcal{C 	
} 	
\to 	
\mathcal{D}$ 	

 	
given 	
by 	
a 	
continuous 	
functor 	
$ 	
u 	
: 	
\mathcal{D 	
} 	
\to 	
\mathcal{C}$ 	

 	
such 	
that 	

 	
\begin{enumerate 	
} 	

 	
\item 	
$ 	
\mathcal{D}$ 	
has 	
equalizers 	
and 	
fibre 	
products 	
and 	
$ 	
u$ 	

 	
commutes 	
with 	
them 	
, 	

 	
\item 	
$ 	
\mathcal{B}$ 	
is 	
a 	
full 	
subcategory 	
of 	
$ 	
\mathcal{D}$ 	

 	
and 	
$ 	
u 	
: 	
\mathcal{B 	
} 	
\to 	
\mathcal{C}$ 	
is 	
the 	
restriction 	
of 	
$ 	
u$ 	
, 	

 	
\item 	
every 	
object 	
of 	
$ 	
\mathcal{D}$ 	
has 	
a 	
covering 	
whose 	
members 	

 	
are 	
objects 	
of 	
$ 	
\mathcal{B}$ 	
, 	

 	
\end{enumerate 	
} 	

 	
\item 	
all 	
negative 	
self 	
- 	
exts 	
of 	
$ 	
E_U$ 	
in 	
$ 	
D(\mathcal{O}_{u(U)})$ 	
are 	
zero 	
, 	
and 	

 	
\item 	
there 	
exist 	
weak 	
Serre 	
subcategories 	

 	
$ 	
\mathcal{A}_U 	
\subset 	
\textit{Mod}(\mathcal{O}_U)$ 	
for 	
all 	

 	
$ 	
U 	
\in 	
\Ob(\mathcal{C})$ 	
satisfying 	
conditions 	
( 	
\ref{item 	
- 	
restriction 	
} 	
) 	
, 	

 	
( 	
\ref{item 	
- 	
local 	
} 	
) 	
, 	
and 	
( 	
\ref{item 	
- 	
bounded 	
- 	
dimension 	
} 	
) 	
, 	

 	
\item 	
$ 	
E_U 	
\in 	
D_{\mathcal{A}_U}(\mathcal{O}_U)$. 	

 	
\end{enumerate 	
} 	

 	
Then 	
there 	
exists 	
a 	
solution 	
unique 	
up 	
to 	
unique 	
isomorphism 	
. 	

 	
\end{lemma 	
} 	


 	
\begin{proof 	
} 	

 	
The 	
proof 	
is 	
{ 	
\bf 	
exactly 	
} 	
the 	
same 	
as 	
the 	
proof 	
of 	

 	
Lemma 	
\ref{lemma 	
- 	
bbd 	
- 	
glueing}. 	
The 	
only 	
change 	
is 	
that 	

 	
$ 	
E$ 	
is 	
an 	
object 	
of 	
$ 	
D_{\mathcal{A}_{total}}(\mathcal{O})$ 	

 	
and 	
hence 	
we 	
use 	
Lemma 	
\ref{lemma 	
- 	
hypercovering 	
- 	
equivalence 	
- 	
modules 	
} 	

 	
to 	
obtain 	
$ 	
F$ 	
with 	
$ 	
E 	
= 	
a^*F$ 	

 	
instead 	
of 	
Lemma 	
\ref{lemma 	
- 	
hypercovering 	
- 	
equivalence 	
- 	
bounded 	
- 	
modules}. 	

 	
\end{proof 	
} 	






 	
\section{Proper 	
hypercoverings 	
in 	
topology 	
} 	

 	
\label{section 	
- 	
proper 	
- 	
hypercovering 	
} 	


 	
\noindent 	

 	
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 	TYPE

 	
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 	
. 	

 	
$ 	
$ 	

 	
\fbox{Principle 	
: 	
Proper 	
hypercoverings 	
can 	
be 	
used 	
to 	
compute 	
cohomology 	
. 	
} 	

 	
$ 	
$ 	

 	
A 	
key 	
idea 	
behind 	
the 	
proof 	
of 	
the 	
principle 	
is 	
to 	
find 	
a 	
topology 	

 	
on 	
$ 	
\textit{LC}$ 	
which 	
is 	
stronger 	
than 	
the 	
usual 	
one 	
such 	
that 	

 	
( 	
a 	
) 	
a 	
surjective 	
proper 	
map 	
defines 	
a 	
covering 	
, 	
and 	

 	
( 	
b 	
) 	
cohomology 	
of 	
usual 	
sheaves 	
with 	
respect 	
to 	
this 	
stronger 	

 	
topology 	
agrees 	
with 	
the 	
usual 	
cohomology 	
. 	

 	
Properties 	
( 	
a 	
) 	
and 	
( 	
b 	
) 	
hold 	
for 	
the 	
qc 	
topology 	
, 	
see 	

 	
Cohomology 	
on 	
Sites 	
, 	
Section 	
\ref{sites 	
- 	
cohomology 	
- 	
section 	
- 	
cohomology 	
- 	
LC}. 	

 	
Once 	
we 	
have 	
( 	
a 	
) 	
and 	
( 	
b 	
) 	
we 	
deduce 	
the 	
principle 	
via 	

 	
the 	
earlier 	
work 	
done 	
in 	
this 	
chapter 	
. 	


 	
\begin{lemma 	
} 	

 	
\label{lemma 	
- 	
compare 	
- 	
simplicial 	
- 	
objects 	
} 	

 	
Let 	
$ 	
U$ 	
be 	
a 	
simplicial 	TYPE
object 	
of 	
$ 	
\textit{LC}$ 	
and 	
let 	
$ 	
a 	
: 	
U 	
\to 	
X$ 	

 	
be 	
an 	
augmentation 	TYPE
. 	
There 	
is 	
a 	
commutative 	
diagram 	

 	
$ 	
$ 	

 	
\xymatrix 	
{ 	

 	
\Sh((\textit{LC}_{qc}/U)_{total 	
} 	
) 	
\ar[r]_-h 	
\ar[d]_{a_{qc 	
} 	
} 	
& 	

 	
\Sh(U_{Zar 	
} 	
) 	
\ar[d]^a 	
\\ 	

 	
\Sh(\textit{LC}_{qc}/X 	
) 	
\ar[r]^-{h_{-1 	
} 	
} 	
& 	

 	
\Sh(X 	
) 	

 	
} 	

 	
$ 	
$ 	

 	
where 	
the 	
left 	
vertical 	
arrow 	
is 	
defined 	
in 	

 	
Section 	
\ref{section 	
- 	
hypercovering 	
} 	

 	
and 	
the 	
right 	
vertical 	
arrow 	
is 	
defined 	
in 	

 	
Lemma 	
\ref{lemma 	
- 	
augmentation}. 	

 	
\end{lemma 	
} 	


 	
\begin{proof 	
} 	

 	
Write 	
$ 	
\Sh(X 	
) 	
= 	
\Sh(X_{Zar})$. 	
Observe 	
that 	
both 	

 	
$ 	
( 	
\textit{LC}_{qc}/U)_{total}$ 	
and 	
$ 	
U_{Zar}$ 	
fall 	

 	
into 	
case 	
A 	
of 	
Situation 	
\ref{situation 	
- 	
simplicial 	
- 	
site}. 	

 	
This 	
is 	
immediate 	
from 	
the 	
construction 	
of 	

 	
$ 	
U_{Zar}$ 	
in 	
Section 	
\ref{section 	
- 	
simplicial 	
- 	
top 	
} 	

 	
and 	
it 	
follows 	
from 	
Lemma 	
\ref{lemma 	
- 	
sr 	
- 	
when 	
- 	
fibre 	
- 	
products 	
} 	

 	
for 	
$ 	
( 	
\textit{LC}_{qc}/U)_{total}$. 	

 	
Next 	
, 	
consider 	
the 	
functors 	

 	
$ 	
U_{n 	
, 	
Zar 	
} 	
\to 	
\textit{LC}_{qc}/U_n$ 	
, 	
$ 	
U 	
\mapsto 	
U 	
/ 	
U_n$ 	

 	
and 	

 	
$ 	
X_{Zar 	
} 	
\to 	
\textit{LC}_{qc}/X$ 	
, 	
$ 	
U 	
\mapsto 	
U 	
/ 	
X$. 	

 	
We 	
have 	
seen 	
that 	
these 	
define 	
morphisms 	
of 	
sites 	

 	
in 	
Cohomology 	
on 	
Sites 	
, 	
Section 	
\ref{sites 	
- 	
cohomology 	
- 	
section 	
- 	
cohomology 	
- 	
LC}. 	

 	
Thus 	
we 	
obtain 	
a 	
morphism 	
of 	
simplicial 	
sites 	
compatible 	
with 	

 	
augmentations 	
as 	
in 	
Remark 	
\ref{remark 	
- 	
morphism 	
- 	
augmentation 	
- 	
simplicial 	
- 	
sites 	
} 	

 	
and 	
we 	
may 	
apply 	

 	
Lemma 	
\ref{lemma 	
- 	
morphism 	
- 	
augmentation 	
- 	
simplicial 	
- 	
sites 	
} 	
to 	
conclude 	
. 	

 	
\end{proof 	
} 	


 	
\begin{lemma 	
} 	

 	
\label{lemma 	
- 	
descent 	
- 	
sheaves 	
- 	
for 	
- 	
proper 	
- 	
hypercovering 	
} 	

 	
Let 	
$ 	
U$ 	
be 	
a 	
simplicial 	TYPE
object 	
of 	
$ 	
\textit{LC}$ 	
and 	
let 	
$ 	
a 	
: 	
U 	
\to 	
X$ 	

 	
be 	
an 	
augmentation 	TYPE
. 	
If 	
$ 	
a 	
: 	
U 	
\to 	
X$ 	
gives 	
a 	
proper 	
hypercovering 	
of 	
$ 	
X$ 	
, 	

 	
then 	

 	
$ 	
$ 	

 	
a^{-1 	
} 	
: 	
\Sh(X 	
) 	
\to 	
\Sh(U_{Zar 	
} 	
) 	

 	
\quad\text{and}\quad 	

 	
a^{-1 	
} 	
: 	
\textit{Ab}(X 	
) 	
\to 	
\textit{Ab}(U_{Zar 	
} 	
) 	

 	
$ 	
$ 	

 	
are 	
fully 	
faithful 	
with 	
essential 	
image 	
the 	
cartesian 	
sheaves 	
and 	

 	
quasi 	
- 	
inverse 	
given 	
by 	
$ 	
a_*$. 	
Here 	
$ 	
a 	
: 	
\Sh(U_{Zar 	
} 	
) 	
\to 	
\Sh(X)$ 	
is 	
as 	
in 	

 	
Lemma 	
\ref{lemma 	
- 	
augmentation}. 	

 	
\end{lemma 	
} 	


 	
\begin{proof 	
} 	

 	
We 	
will 	
prove 	
the 	
statement 	
for 	
sheaves 	
of 	
sets 	
. 	
It 	
will 	
be 	
an 	

 	TYPE
almost 	
formal 	
consequence 	
of 	
results 	
already 	
established 	
. 	

 	
Consider 	
the 	
diagram 	
of 	
Lemma 	
\ref{lemma 	
- 	
compare 	
- 	
simplicial 	
- 	
objects}. 	

 	
By 	
Cohomology 	
on 	
Sites 	
, 	
Lemma 	
\ref{sites 	
- 	
cohomology 	
- 	
lemma 	
- 	
describe 	
- 	
pullback 	
- 	
pi 	
} 	

 	
the 	
functor 	
$ 	
( 	
h_{-1})^{-1}$ 	
is 	
fully 	
faithful 	
with 	
quasi 	
- 	
inverse 	
$ 	
h_{-1 	
, 	
* 	
} 	
$ 	
. 	

 	
The 	
same 	
holds 	
true 	
for 	
the 	
components 	
$ 	
h_n$ 	
of 	
$ 	
h$. 	

 	
By 	
the 	
description 	
of 	
the 	
functors 	
$ 	
h^{-1}$ 	
and 	
$ 	
h_*$ 	
of 	

 	
Lemma 	
\ref{lemma 	
- 	
morphism 	
- 	
simplicial 	
- 	
sites 	
} 	

 	
we 	
conclude 	
that 	
$ 	
h^{-1}$ 	
is 	
fully 	
faithful 	
with 	
quasi 	
- 	
inverse 	
$ 	
h_*$. 	

 	
Observe 	
that 	
$ 	
U$ 	
is 	
a 	
hypercovering 	
of 	
$ 	
X$ 	
in 	
$ 	
\textit{LC}_{qc}$ 	

 	
( 	
as 	
defined 	
in 	
Section 	
\ref{section 	
- 	
hypercovering 	
} 	
) 	
by 	

 	
Cohomology 	
on 	
Sites 	
, 	
Lemma 	

 	
\ref{sites 	
- 	
cohomology 	
- 	
lemma 	
- 	
proper 	
- 	
surjective 	
- 	
is 	
- 	
qc 	
- 	
covering}. 	

 	
By 	
Lemma 	
\ref{lemma 	
- 	
hypercovering 	
- 	
X 	
- 	
simple 	
- 	
descent 	
- 	
sheaves 	
} 	

 	
we 	
see 	
that 	
$ 	
a_{qc}^{-1}$ 	
is 	
fully 	
faithful 	
with 	
quasi 	
- 	
inverse 	
$ 	
a_{qc 	
, 	
* 	
} 	
$ 	

 	
and 	
with 	
essential 	
image 	
the 	
cartesian 	
sheaves 	
on 	

 	
$ 	
( 	
\textit{LC}_{qc}/U)_{total}$. 	

 	
A 	
formal 	
argument 	
( 	
chasing 	
around 	
the 	
diagram 	
) 	
now 	
shows 	
that 	

 	
$ 	
a^{-1}$ 	
is 	
fully 	
faithful 	
. 	


 	
\medskip\noindent 	

 	
Finally 	
, 	
suppose 	
that 	
$ 	
\mathcal{G}$ 	
is 	
a 	
cartesian 	
sheaf 	
on 	
$ 	
U_{Zar}$. 	

 	
Then 	
$ 	
h^{-1}\mathcal{G}$ 	
is 	
a 	
cartesian 	
sheaf 	
on 	
$ 	
\textit{LC}_{qc}/U$. 	

 	
Hence 	
$ 	
h^{-1}\mathcal{G 	
} 	
= 	
a_{qc}^{-1}\mathcal{H}$ 	
for 	
some 	
sheaf 	

 	
$ 	
\mathcal{H}$ 	
on 	
$ 	
\textit{LC}_{qc}/X$. 	

 	
We 	
compute 	

 	
\begin{align 	
* 	
} 	

 	
( 	
h_{-1})^{-1}(a_*\mathcal{G 	
} 	
) 	

 	
& 	
= 	

 	
( 	
h_{-1})^{-1 	
} 	

 	
\text{Eq 	
} 	
( 	

 	
\xymatrix 	
{ 	

 	
a_{0 	
, 	
* 	
} 	
\mathcal{G}_0 	

 	
\ar@<1ex>[r 	
] 	
\ar@<-1ex>[r 	
] 	
& 	

 	
a_{1 	
, 	
* 	
} 	
\mathcal{G}_1 	

 	
} 	

 	
) 	
\\ 	

 	
& 	
= 	

 	
\text{Eq 	
} 	
( 	

 	
\xymatrix 	
{ 	

 	
( 	
h_{-1})^{-1}a_{0 	
, 	
* 	
} 	
\mathcal{G}_0 	

 	
\ar@<1ex>[r 	
] 	
\ar@<-1ex>[r 	
] 	
& 	

 	
( 	
h_{-1})^{-1}a_{1 	
, 	
* 	
} 	
\mathcal{G}_1 	

 	
} 	

 	
) 	
\\ 	

 	
& 	
= 	

 	
\text{Eq 	
} 	
( 	

 	
\xymatrix 	
{ 	

 	
a_{qc 	
, 	
0 	
, 	
* 	
} 	
h_0^{-1}\mathcal{G}_0 	

 	
\ar@<1ex>[r 	
] 	
\ar@<-1ex>[r 	
] 	
& 	

 	
a_{qc 	
, 	
1 	
, 	
* 	
} 	
h_1^{-1}\mathcal{G}_1 	

 	
} 	

 	
) 	
\\ 	

 	
& 	
= 	

 	
\text{Eq 	
} 	
( 	

 	
\xymatrix 	
{ 	

 	
a_{qc 	
, 	
0 	
, 	
* 	
} 	
a_{qc 	
, 	
0}^{-1}\mathcal{H 	
} 	

 	
\ar@<1ex>[r 	
] 	
\ar@<-1ex>[r 	
] 	
& 	

 	
a_{qc 	
, 	
1 	
, 	
* 	
} 	
a_{qc 	
, 	
1}^{-1}\mathcal{H 	
} 	

 	
} 	

 	
) 	
\\ 	

 	
& 	
= 	

 	
a_{qc 	
, 	
* 	
} 	
a_{qc}^{-1}\mathcal{H 	
} 	
\\ 	

 	
& 	
= 	

 	
\mathcal{H 	
} 	

 	
\end{align 	
* 	
} 	

 	
Here 	
the 	
first 	
equality 	
follows 	
from 	
Lemma 	
\ref{lemma 	
- 	
augmentation 	
} 	
, 	

 	
the 	
second 	
equality 	
follows 	
as 	
$ 	
( 	
h_{-1})^{-1}$ 	
is 	
an 	
exact 	
functor 	
, 	

 	
the 	
third 	
equality 	
follows 	
from 	

 	
Cohomology 	
on 	
Sites 	
, 	
Lemma 	
\ref{sites 	
- 	
cohomology 	
- 	
lemma 	
- 	
push 	
- 	
pull 	
- 	
LC 	
} 	

 	
( 	
here 	
we 	
use 	
that 	
$ 	
a_0 	
: 	
U_0 	
\to 	
X$ 	
and 	
$ 	
a_1 	
: 	
U_1 	
\to 	
X$ 	
are 	
proper 	
) 	
, 	

 	
the 	
fourth 	
follows 	
from 	
$ 	
a_{qc}^{-1}\mathcal{H 	
} 	
= 	
h^{-1}\mathcal{G}$ 	
, 	

 	
the 	
fifth 	
from 	
Lemma 	
\ref{lemma 	
- 	
augmentation 	
- 	
site 	
} 	
, 	
and 	
the 	

 	
sixth 	
we 	
've 	
seen 	
above 	
. 	
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 	TYPE
object 	
of 	
$ 	
\textit{LC}$ 	
and 	
let 	
$ 	
a 	
: 	
U 	
\to 	
X$ 	

 	
be 	
an 	
augmentation 	TYPE
. 	
If 	
$ 	
a 	
: 	
U 	
\to 	
X$ 	
gives 	
a 	
proper 	
hypercovering 	
of 	
$ 	
X$ 	
, 	

 	
then 	
for 	
$ 	
K 	
\in 	
D^+(X)$ 	

 	
$ 	
$ 	

 	
K 	
\to 	
Ra_*(a^{-1}K 	
) 	

 	
$ 	
$ 	

 	
is 	
an 	
isomorphism 	
where 	
$ 	
a 	
: 	
\Sh(U_{Zar 	
} 	
) 	
\to 	
\Sh(X)$ 	
is 	
as 	
in 	

 	
Lemma 	
\ref{lemma 	
- 	
augmentation}. 	

 	
\end{lemma 	
} 	


 	
\begin{proof 	
} 	

 	
Consider 	
the 	
diagram 	
of 	
Lemma 	
\ref{lemma 	
- 	
compare 	
- 	
simplicial 	
- 	
objects}. 	

 	
Observe 	
that 	
$ 	
Rh_{n 	
, 	
* 	
} 	
h_n^{-1}$ 	
is 	
the 	
identity 	
functor 	

 	
on 	
$ 	
D^+(U_n)$ 	
by 	
Cohomology 	
on 	
Sites 	
, 	
Lemma 	

 	
\ref{sites 	
- 	
cohomology 	
- 	
lemma 	
- 	
cohomological 	
- 	
descent 	
- 	
LC}. 	

 	
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 	TYPE
object 	
of 	
$ 	
\textit{LC}$ 	
and 	
let 	
$ 	
a 	
: 	
U 	
\to 	
X$ 	

 	
be 	
an 	
augmentation 	TYPE
. 	
If 	
$ 	
U$ 	
is 	
a 	
proper 	
hypercovering 	
of 	
$ 	
X$ 	
, 	
then 	

 	
$ 	
$ 	

 	
R\Gamma(X 	
, 	
K 	
) 	
= 	
R\Gamma(U_{Zar 	
} 	
, 	
a^{-1}K 	
) 	

 	
$ 	
$ 	

 	
for 	
$ 	
K 	
\in 	
D^+(X)$ 	
where 	
$ 	
a 	
: 	
\Sh(U_{Zar 	
} 	
) 	
\to 	
\Sh(X)$ 	

 	
is 	
as 	
in 	
Lemma 	
\ref{lemma 	
- 	
augmentation}. 	

 	
\end{lemma 	
} 	


 	
\begin{proof 	
} 	

 	
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 	TYPE
object 	
of 	
$ 	
\textit{LC}$ 	
and 	
let 	
$ 	
a 	
: 	
U 	
\to 	
X$ 	

 	
be 	
an 	
augmentation 	TYPE
. 	

 	
Let 	
$ 	
\mathcal{A 	
} 	
\subset 	
\textit{Ab}(U_{Zar})$ 	

 	
denote 	
the 	
weak 	
Serre 	
subcategory 	
of 	
cartesian 	
abelian 	
sheaves 	
. 	

 	
If 	
$ 	
U$ 	
is 	
a 	
proper 	
hypercovering 	
of 	
$ 	
X$ 	
, 	
then 	

 	
the 	
functor 	
$ 	
a^{-1}$ 	
defines 	
an 	
equivalence 	

 	
$ 	
$ 	

 	
D^+(X 	
) 	
\longrightarrow 	
D_\mathcal{A}^+(U_{Zar 	
} 	
) 	

 	
$ 	
$ 	

 	
with 	
quasi 	
- 	
inverse 	
$ 	
Ra_*$ 	
where 	
$ 	
a 	
: 	
\Sh(U_{Zar 	
} 	
) 	
\to 	
\Sh(X)$ 	

 	
is 	
as 	
in 	
Lemma 	
\ref{lemma 	
- 	
augmentation}. 	

 	
\end{lemma 	
} 	


 	
\begin{proof 	
} 	

 	
Observe 	
that 	
$ 	
\mathcal{A}$ 	
is 	
a 	
weak 	
Serre 	
subcategory 	
by 	

 	
Lemma 	
\ref{lemma 	
- 	
Serre 	
- 	
subcat 	
- 	
cartesian 	
- 	
modules}. 	

 	
The 	
equivalence 	
is 	
a 	

 	
formal 	
consequence 	
of 	
the 	
results 	
obtained 	
so 	
far 	
. 	
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 	TYPE
object 	
of 	
$ 	
\textit{LC}$ 	
and 	
let 	

 	
$ 	
a 	
: 	
U 	
\to 	
X$ 	
be 	
an 	
augmentation 	TYPE
. 	
Let 	
$ 	
\mathcal{F}$ 	
be 	
an 	
abelian 	TYPE
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 	
] 	

 	
} 	

 	
$ 	
$ 	

 	
Here 	
there 	
are 	
two 	
morphisms 	
$ 	
d^1_0 	
, 	
d^1_1 	
: 	
X_1 	
\to 	
X_0 	
$ 	

 	
and 	
a 	
single 	
morphism 	
$ 	
s^0_0 	
: 	
X_0 	
\to 	
X_1 	
$ 	
, 	
etc 	
. 	

 	
These 	
morphisms 	
satisfy 	
some 	
required 	
relations 	
such 	
as 	

 	
$ 	
d^1_0 	
\circ 	
s^0_0 	
= 	
\text{id}_{X_0 	
} 	
= 	
d^1_1 	
\circ 	
s^0_0 	
$ 	
, 	
see 	

 	
Simplicial 	
, 	
Lemma 	
\ref{simplicial 	
- 	
lemma 	
- 	
characterize 	
- 	
simplicial 	
- 	
object}. 	

 	
It 	
is 	
useful 	
to 	
think 	
of 	
$ 	
d^n_i 	
: 	
X_n 	
\to 	
X_{n 	
- 	
1}$ 	

 	
as 	
the 	
`` 	
projection 	
forgetting 	
the 	
$ 	
i$th 	
coordinate 	
'' 	
and 	

 	
to 	
think 	
of 	
$ 	
s^n_j 	
: 	
X_n 	
\to 	
X_{n 	
+ 	
1}$ 	
as 	
the 	
`` 	
diagonal 	
map 	
repeating 	

 	
the 	
$ 	
j$th 	
coordinate 	
'' 	
. 	


 	
\medskip\noindent 	

 	
A 	
{ 	
\it 	
morphism 	
of 	
simplicial 	
schemes 	
} 	
$ 	
h 	
: 	
X 	
\to 	
Y$ 	
is 	
the 	
same 	

 	
thing 	
as 	
a 	
morphism 	
of 	
simplicial 	
objects 	
in 	
the 	
category 	
of 	
schemes 	
, 	

 	
see 	
Simplicial 	
, 	
Definition 	
\ref{simplicial 	
- 	
definition 	
- 	
simplicial 	
- 	
object}. 	

 	
Thus 	
$ 	
h$ 	
consists 	
of 	
morphisms 	
of 	
schemes 	
$ 	
h_n 	
: 	
X_n 	
\to 	
Y_n$ 	

 	
such 	
that 	
$ 	
h_{n 	
- 	
1 	
} 	
\circ 	
d^n_j 	
= 	
d^n_j 	
\circ 	
h_n$ 	
and 	

 	
$ 	
h_{n 	
+ 	
1 	
} 	
\circ 	
s^n_j 	
= 	
s^n_j 	
\circ 	
h_n$ 	
whenever 	
this 	
makes 	
sense 	
. 	


 	
\medskip\noindent 	

 	
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 	TYPE
scheme 	
. 	
The 	
construction 	
of 	

 	
Section 	
\ref{section 	
- 	
simplicial 	
- 	
top 	
} 	
applied 	
to 	
the 	
underlying 	

 	
simplicial 	
topological 	
space 	
gives 	
a 	
site 	
$ 	
X_{Zar}$. 	

 	
On 	
the 	
other 	
hand 	
, 	
for 	
every 	
$ 	
n$ 	
we 	
have 	
the 	
small 	
Zariski 	
site 	

 	
$ 	
X_{n 	
, 	
Zar}$ 	
( 	
Topologies 	
, 	
Definition 	

 	
\ref{topologies 	
- 	
definition 	
- 	
big 	
- 	
small 	
- 	
Zariski 	
} 	
) 	

 	
and 	
for 	
every 	
morphism 	
$ 	
\varphi 	
: 	
[ 	
m 	
] 	
\to 	
[ 	
n]$ 	

 	
we 	
have 	
a 	
morphism 	
of 	
sites 	

 	
$ 	
f_\varphi 	
= 	
X(\varphi)_{small 	
} 	
: 	
X_{n 	
, 	
Zar 	
} 	
\to 	
X_{m 	
, 	
Zar}$ 	
, 	

 	
associated 	
to 	
the 	
morphism 	
of 	
schemes 	

 	
$ 	
X(\varphi 	
) 	
: 	
X_n 	
\to 	
X_m$ 	
( 	
Topologies 	
, 	
Lemma 	

 	
\ref{topologies 	
- 	
lemma 	
- 	
morphism 	
- 	
big 	
- 	
small 	
} 	
) 	
. 	

 	
This 	
gives 	
a 	
simplicial 	
object 	
$ 	
\mathcal{C}$ 	
in 	
the 	
category 	
of 	
sites 	
. 	

 	
In 	
Lemma 	
\ref{lemma 	
- 	
simplicial 	
- 	
site 	
- 	
site 	
} 	
we 	
constructed 	
an 	
associated 	

 	
site 	
$ 	
\mathcal{C}_{total}$. 	
Assigning 	
to 	
an 	
open 	
immersion 	
its 	
image 	

 	
defines 	
an 	
equivalence 	
$ 	
\mathcal{C}_{total 	
} 	
\to 	
X_{Zar}$ 	
which 	

 	
identifies 	
sheaves 	
, 	
i.e. 	
, 	
$ 	
\Sh(\mathcal{C}_{total 	
} 	
) 	
= 	
\Sh(X_{Zar})$. 	

 	
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 	
$ 	
\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 	TYPE
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}. 	

 	
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 	
. 	
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 	TYPE
scheme 	
with 	
structure 	
sheaf 	
$ 	
\mathcal{O}$. 	

 	
Let 	
$ 	
S$ 	
be 	
a 	
scheme 	TYPE
and 	
let 	
$ 	
a_0 	
: 	
X_0 	
\to 	
S$ 	
be 	
an 	
augmentation 	TYPE
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 	
) 	
. 	
This 	
puts 	
us 	
in 	
the 	
situation 	

 	
discussed 	
in 	

 	
Section 	
\ref{section 	
- 	
cohomology 	
- 	
augmentation 	
- 	
ringed 	
- 	
simplicial 	
- 	
sites}. 	

 	
Therefore 	
we 	
obtain 	
a 	
morphism 	
of 	
ringed 	
topoi 	

 	
$ 	
$ 	

 	
a 	
: 	
( 	
\Sh(X_{Zar 	
} 	
) 	
, 	
\mathcal{O 	
} 	
) 	
\longrightarrow 	
( 	
\Sh(S 	
) 	
, 	
\mathcal{O}_S 	
) 	

 	
$ 	
$ 	


 	
\medskip\noindent 	

 	
A 	
final 	
observation 	
is 	
the 	
following 	
. 	
Suppose 	
we 	
are 	
given 	
a 	
morphism 	

 	
$ 	
h 	
: 	
X 	
\to 	
Y$ 	
of 	
simplicial 	
schemes 	
$ 	
X$ 	
and 	
$ 	
Y$ 	
with 	
structure 	
sheaves 	

 	
$ 	
\mathcal{O}_X$ 	
, 	
$ 	
\mathcal{O}_Y$ 	
, 	
augmentations 	

 	
$ 	
a_0 	
: 	
X_0 	
\to 	
X_{-1}$ 	
, 	
$ 	
b_0 	
: 	
Y_0 	
\to 	
Y_{-1}$ 	
and 	
a 	
morphism 	

 	
$ 	
h_{-1 	
} 	
: 	
X_{-1 	
} 	
\to 	
Y_{-1}$ 	
such 	
that 	

 	
$ 	
$ 	

 	
\xymatrix 	
{ 	

 	
X_0 	
\ar[r]_{h_0 	
} 	
\ar[d]_{a_0 	
} 	
& 	
Y_0 	
\ar[d]^{b_0 	
} 	
\\ 	

 	
X_{-1 	
} 	
\ar[r]^{h_{-1 	
} 	
} 	
& 	
Y_{-1 	
} 	

 	
} 	

 	
$ 	
$ 	

 	
commutes 	
. 	
Then 	
from 	
the 	
constructions 	
elucidated 	
above 	

 	
we 	
obtain 	
a 	
commutative 	
diagram 	
of 	
morphisms 	
of 	
ringed 	
topoi 	
as 	
follows 	

 	
$ 	
$ 	

 	
\xymatrix 	
{ 	

 	
( 	
\Sh(X_{Zar 	
} 	
) 	
, 	
\mathcal{O}_X 	
) 	
\ar[r]_{h_{Zar 	
} 	
} 	
\ar[d]_a 	
& 	

 	
( 	
\Sh(Y_{Zar 	
} 	
) 	
, 	
\mathcal{O}_Y 	
) 	
\ar[d]^b 	
\\ 	

 	
( 	
\Sh(X_{-1 	
} 	
) 	
, 	
\mathcal{O}_{X_{-1 	
} 	
} 	
) 	
\ar[r]^{h_{-1 	
} 	
} 	
& 	

 	
( 	
\Sh(Y_{-1 	
} 	
) 	
, 	
\mathcal{O}_{Y_{-1 	
} 	
} 	
) 	

 	
} 	

 	
$ 	
$ 	









 	
\section{Descent 	
in 	
terms 	
of 	
simplicial 	
schemes 	
} 	

 	
\label{section 	
- 	
simplicial 	
- 	
descent 	
} 	


 	
\noindent 	

 	
Cartesian 	
morphisms 	
are 	
defined 	
as 	
follows 	
. 	


 	
\begin{definition 	
} 	

 	
\label{definition 	
- 	
cartesian 	
- 	
morphism 	
} 	

 	
Let 	
$ 	
a 	
: 	
Y 	
\to 	
X$ 	
be 	
a 	
morphism 	
of 	
simplicial 	
schemes 	
. 	

 	
We 	
say 	
$ 	
a$ 	
is 	
{ 	
\it 	
cartesian 	
} 	
, 	
or 	
that 	
{ 	
\it 	
$ 	
Y$ 	
is 	
cartesian 	
over 	
$ 	
X$ 	
} 	
, 	

 	
if 	
for 	
every 	
morphism 	
$ 	
\varphi 	
: 	
[ 	
n 	
] 	
\to 	
[ 	
m]$ 	
of 	
$ 	
\Delta$ 	
the 	
corresponding 	

 	
diagram 	

 	
$ 	
$ 	

 	
\xymatrix 	
{ 	

 	
Y_m 	
\ar[r]_a 	
\ar[d]_{Y(\varphi 	
) 	
} 	
& 	
X_m 	
\ar[d]^{X(\varphi)}\\ 	

 	
Y_n 	
\ar[r]^{a 	
} 	
& 	
X_n 	

 	
} 	

 	
$ 	
$ 	

 	
is 	
a 	
fibre 	
square 	
in 	
the 	
category 	
of 	
schemes 	
. 	

 	
\end{definition 	
} 	


 	
\noindent 	

 	
Cartesian 	
morphisms 	
are 	
related 	
to 	
descent 	
data 	
. 	
First 	
we 	
prove 	
a 	
general 	

 	
lemma 	
describing 	
the 	
category 	
of 	
cartesian 	
simplicial 	
schemes 	
over 	
a 	

 	
fixed 	
simplicial 	
scheme 	
. 	
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 	TYPE
scheme 	
. 	
The 	
category 	
of 	
simplicial 	
schemes 	
cartesian 	

 	
over 	
$ 	
X$ 	
is 	
equivalent 	
to 	
the 	
category 	
of 	
pairs 	
$ 	
( 	
V 	
, 	
\varphi)$ 	

 	
where 	
$ 	
V$ 	
is 	
a 	
scheme 	
over 	
$ 	
X_0 	
$ 	
and 	

 	
$ 	
$ 	

 	
\varphi 	
: 	

 	
V 	
\times_{X_0 	
, 	
d^1_1 	
} 	
X_1 	

 	
\longrightarrow 	

 	
X_1 	
\times_{d^1_0 	
, 	
X_0 	
} 	
V 	

 	
$ 	
$ 	

 	
is 	
an 	
isomorphism 	
over 	
$ 	
X_1 	
$ 	
such 	
that 	

 	
$ 	
( 	
s_0 	
^ 	
0)^*\varphi 	
= 	
\text{id}_V$ 	
and 	
such 	
that 	

 	
$ 	
$ 	

 	
( 	
d^2_1)^*\varphi 	
= 	
( 	
d^2_0)^*\varphi 	
\circ 	
( 	
d^2_2)^*\varphi 	

 	
$ 	
$ 	

 	
as 	
morphisms 	
of 	
schemes 	
over 	
$ 	
X_2$. 	

 	
\end{lemma 	
} 	


 	
\begin{proof 	
} 	

 	
The 	
statement 	
of 	
the 	
displayed 	
equality 	
makes 	
sense 	
because 	

 	
$ 	
d^1_1 	
\circ 	
d^2_2 	
= 	
d^1_1 	
\circ 	
d^2_1 	
$ 	
, 	

 	
$ 	
d^1_1 	
\circ 	
d^2_0 	
= 	
d^1_0 	
\circ 	
d^2_2 	
$ 	
, 	
and 	

 	
$ 	
d^1_0 	
\circ 	
d^2_0 	
= 	
d^1_0 	
\circ 	
d^2_1 	
$ 	
as 	
morphisms 	
$ 	
X_2 	
\to 	
X_0 	
$ 	
, 	
see 	

 	
Simplicial 	
, 	
Remark 	
\ref{simplicial 	
- 	
remark 	
- 	
relations 	
} 	
hence 	
we 	

 	
can 	
picture 	
these 	
maps 	
as 	
follows 	

 	
$ 	
$ 	

 	
\xymatrix 	
{ 	

 	
& 	

 	
X_2 	
\times_{d^1_1 	
\circ 	
d^2_0 	
, 	
X_0 	
} 	
V 	

 	
\ar[r]_-{(d^2_0)^*\varphi 	
} 	
& 	

 	
X_2 	
\times_{d^1_0 	
\circ 	
d^2_0 	
, 	
X_0 	
} 	
V 	

 	
\ar@{=}[rd 	
] 	
& 	
\\ 	

 	
X_2 	
\times_{d^1_0 	
\circ 	
d^2_2 	
, 	
X_0 	
} 	
V 	

 	
\ar@{=}[ru 	
] 	
& 	
& 	
& 	

 	
X_2 	
\times_{d^1_0 	
\circ 	
d^2_1 	
, 	
X_0 	
} 	
V 	
\\ 	

 	
& 	

 	
X_2 	
\times_{d^1_1 	
\circ 	
d^2_2 	
, 	
X_0 	
} 	
V 	

 	
\ar[lu]^{(d^2_2)^*\varphi 	
} 	
\ar@{=}[r 	
] 	
& 	

 	
X_2 	
\times_{d^1_1 	
\circ 	
d^2_1 	
, 	
X_0 	
} 	
V 	

 	
\ar[ru]_{(d^2_1)^*\varphi 	
} 	

 	
} 	

 	
$ 	
$ 	

 	
and 	
the 	
condition 	
signifies 	
the 	
diagram 	
is 	
commutative 	
. 	
It 	
is 	
clear 	
that 	

 	
given 	
a 	
simplicial 	
scheme 	
$ 	
Y$ 	
cartesian 	
over 	
$ 	
X$ 	
we 	
can 	

 	
set 	
$ 	
V 	
= 	
Y_0 	
$ 	
and 	
$ 	
\varphi$ 	
equal 	
to 	
the 	
composition 	

 	
$ 	
$ 	

 	
V 	
\times_{X_0 	
, 	
d^1_1 	
} 	
X_1 	
= 	

 	
Y_0 	
\times_{X_0 	
, 	
d^1_1 	
} 	
X_1 	
= 	
Y_1 	
= 	

 	
X_1 	
\times_{X_0 	
, 	
d^1_0 	
} 	
Y_0 	
= 	

 	
X_1 	
\times_{X_0 	
, 	
d^1_0 	
} 	
V 	

 	
$ 	
$ 	

 	
of 	
identifications 	
given 	
by 	
the 	
cartesian 	
structure 	
. 	
To 	
prove 	
this 	
functor 	

 	
is 	
an 	
equivalence 	
we 	
construct 	
a 	
quasi 	
- 	
inverse 	
. 	
The 	
construction 	
of 	

 	
the 	
quasi 	
- 	
inverse 	
is 	
analogous 	
to 	
the 	
construction 	
discussed 	
in 	

 	
Descent 	
, 	
Section 	
\ref{descent 	
- 	
section 	
- 	
descent 	
- 	
modules 	
} 	
from 	
which 	
we 	
borrow 	

 	
the 	
notation 	
$ 	
\tau^n_i 	
: 	
[ 	
0 	
] 	
\to 	
[ 	
n]$ 	
, 	
$ 	
0 	
\mapsto 	
i$ 	
and 	

 	
$ 	
\tau^n_{ij 	
} 	
: 	
[ 	
1 	
] 	
\to 	
[ 	
n]$ 	
, 	
$ 	
0 	
\mapsto 	
i$ 	
, 	
$ 	
1 	
\mapsto 	
j$. 	

 	
Namely 	
, 	
given 	
a 	
pair 	
$ 	
( 	
V 	
, 	
\varphi)$ 	

 	
as 	
in 	
the 	
lemma 	
we 	
set 	
$ 	
Y_n 	
= 	
X_n 	
\times_{X(\tau^n_n 	
) 	
, 	
X_0 	
} 	
V$. 	

 	
Then 	
given 	
$ 	
\beta 	
: 	
[ 	
n 	
] 	
\to 	
[ 	
m]$ 	
we 	
define 	

 	
$ 	
V(\beta 	
) 	
: 	
Y_m 	
\to 	
Y_n$ 	
as 	
the 	
pullback 	
by 	
$ 	
X(\tau^m_{\beta(n)m})$ 	

 	
of 	
the 	
map 	
$ 	
\varphi$ 	
postcomposed 	
by 	
the 	
projection 	

 	
$ 	
X_m 	
\times_{X(\beta 	
) 	
, 	
X_n 	
} 	
Y_n 	
\to 	
Y_n$. 	
This 	
makes 	
sense 	
because 	

 	
$ 	
$ 	

 	
X_m 	
\times_{X(\tau^m_{\beta(n)m 	
} 	
) 	
, 	
X_1 	
} 	
X_1 	
\times_{d^1_1 	
, 	
X_0 	
} 	
V 	

 	
= 	

 	
X_m 	
\times_{X(\tau^m_m 	
) 	
, 	
X_0 	
} 	
V 	
= 	
Y_m 	

 	
$ 	
$ 	

 	
and 	

 	
$ 	
$ 	

 	
X_m 	
\times_{X(\tau^m_{\beta(n)m 	
} 	
) 	
, 	
X_1 	
} 	
X_1 	
\times_{d^1_0 	
, 	
X_0 	
} 	
V 	
= 	

 	
X_m 	
\times_{X(\tau^m_{\beta(n 	
) 	
} 	
) 	
, 	
X_0 	
} 	
V 	
= 	

 	
X_m 	
\times_{X(\beta 	
) 	
, 	
X_n 	
} 	
Y_n 	
. 	

 	
$ 	
$ 	

 	
We 	
omit 	
the 	
verification 	
that 	
the 	
commutativity 	

 	
of 	
the 	
displayed 	
diagram 	

 	
above 	
implies 	
the 	
maps 	
compose 	
correctly 	
. 	
We 	
also 	
omit 	
the 	
verification 	

 	
that 	
the 	
two 	
functors 	
are 	
quasi 	
- 	
inverse 	
to 	
each 	
other 	
. 	

 	
\end{proof 	
} 	


 	
\begin{definition 	
} 	

 	
\label{definition 	
- 	
fibre 	
- 	
products 	
- 	
simplicial 	
- 	
scheme 	
} 	

 	
Let 	
$ 	
f 	
: 	
X 	
\to 	
S$ 	
be 	
a 	
morphism 	
of 	
schemes 	
. 	
The 	
{ 	
\it 	
simplicial 	
scheme 	

 	
associated 	
to 	
$ 	
f$ 	
} 	
, 	
denoted 	
$ 	
( 	
X 	
/ 	
S)_\bullet$ 	
, 	
is 	
the 	
functor 	

 	
$ 	
\Delta^{opp 	
} 	
\to 	
\Sch$ 	
, 	
$ 	
[ 	
n 	
] 	
\mapsto 	
X 	
\times_S 	
\ldots 	
\times_S 	
X$ 	

 	
described 	
in 	

 	
Simplicial 	
, 	
Example 	
\ref{simplicial 	
- 	
example 	
- 	
fibre 	
- 	
products 	
- 	
simplicial 	
- 	
object}. 	

 	
\end{definition 	
} 	


 	
\noindent 	

 	
Thus 	
$ 	
( 	
X 	
/ 	
S)_n$ 	
is 	
the 	
$ 	
( 	
n 	
+ 	
1)$-fold 	
fibre 	
product 	
of 	
$ 	
X$ 	
over 	
$ 	
S$. 	

 	
The 	
morphism 	
$ 	
d^1_0 	
: 	
X 	
\times_S 	
X 	
\to 	
X$ 	
is 	
the 	
map 	

 	
$ 	
( 	
x_0 	
, 	
x_1 	
) 	
\mapsto 	
x_1 	
$ 	
and 	
the 	
morphism 	
$ 	
d^1_1 	
$ 	
is 	
the 	
other 	

 	
projection 	
. 	
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 	
. 	

 	
Let 	
$ 	
\pi 	
: 	
Y 	
\to 	
( 	
X 	
/ 	
S)_\bullet$ 	
be 	
a 	
cartesian 	TYPE
morphism 	

 	
of 	
simplicial 	
schemes 	
. 	

 	
Set 	
$ 	
V 	
= 	
Y_0 	
$ 	
considered 	
as 	
a 	
scheme 	
over 	
$ 	
X$. 	

 	
The 	
morphisms 	
$ 	
d^1_0 	
, 	
d^1_1 	
: 	
Y_1 	
\to 	
Y_0 	
$ 	
and 	
the 	
morphism 	

 	
$ 	
\pi_1 	
: 	
Y_1 	
\to 	
X 	
\times_S 	
X$ 	
induce 	
isomorphisms 	

 	
$ 	
$ 	

 	
\xymatrix 	
{ 	

 	
V 	
\times_S 	
X 	
& 	
& 	

 	
Y_1 	
\ar[ll]_-{(d^1_1 	
, 	
\text{pr}_1 	
\circ 	
\pi_1 	
) 	
} 	

 	
\ar[rr]^-{(\text{pr}_0 	
\circ 	
\pi_1 	
, 	
d^1_0 	
) 	
} 	
& 	
& 	

 	
X 	
\times_S 	
V. 	

 	
} 	

 	
$ 	
$ 	

 	
Denote 	
$ 	
\varphi 	
: 	
V 	
\times_S 	
X 	
\to 	
X 	
\times_S 	
V$ 	
the 	

 	
resulting 	
isomorphism 	
. 	

 	
Then 	
the 	
pair 	
$ 	
( 	
V 	
, 	
\varphi)$ 	
is 	
a 	
descent 	
datum 	
relative 	

 	
to 	
$ 	
X 	
\to 	
S$. 	

 	
\end{lemma 	
} 	


 	
\begin{proof 	
} 	

 	
This 	
is 	
a 	
special 	
case 	
of 	
( 	
part 	
of 	
) 	

 	
Lemma 	
\ref{lemma 	
- 	
characterize 	
- 	
cartesian 	
- 	
schemes 	
} 	

 	
as 	
the 	
displayed 	
equation 	
of 	
that 	
lemma 	
is 	

 	
equivalent 	
to 	
the 	
cocycle 	
condition 	
of 	

 	
Descent 	
, 	
Definition 	
\ref{descent 	
- 	
definition 	
- 	
descent 	
- 	
datum}. 	

 	
\end{proof 	
} 	


 	
\begin{lemma 	
} 	

 	
\label{lemma 	
- 	
cartesian 	
- 	
equivalent 	
- 	
descent 	
- 	
datum 	
} 	

 	
Let 	
$ 	
f 	
: 	
X 	
\to 	
S$ 	
be 	
a 	
morphism 	
of 	
schemes 	
. 	
The 	
construction 	

 	
$ 	
$ 	

 	
\begin{matrix 	
} 	

 	
\text{category 	
of 	
cartesian 	
} 	
\\ 	

 	
\text{schemes 	
over 	
} 	
( 	
X 	
/ 	
S)_\bullet 	

 	
\end{matrix 	
} 	

 	
\longrightarrow 	

 	
\begin{matrix 	
} 	

 	
\text 	
{ 	
category 	
of 	
descent 	
data 	
} 	
\\ 	

 	
\text 	
{ 	
relative 	
to 	
} 	
X 	
/ 	
S 	

 	
\end{matrix 	
} 	

 	
$ 	
$ 	

 	
of 	
Lemma 	
\ref{lemma 	
- 	
cartesian 	
- 	
over 	
} 	

 	
is 	
an 	
equivalence 	
of 	
categories 	
. 	

 	
\end{lemma 	
} 	


 	
\begin{proof 	
} 	

 	
The 	
functor 	
from 	
left 	
to 	
right 	
is 	
given 	
in 	

 	
Lemma 	
\ref{lemma 	
- 	
cartesian 	
- 	
over}. 	

 	
Hence 	
this 	
is 	
a 	
special 	
case 	
of 	

 	
Lemma 	
\ref{lemma 	
- 	
characterize 	
- 	
cartesian 	
- 	
schemes}. 	

 	
\end{proof 	
} 	


 	
\noindent 	

 	
We 	
may 	
reinterpret 	
the 	
pullback 	
of 	

 	
Descent 	
, 	
Lemma 	
\ref{descent 	
- 	
lemma 	
- 	
pullback 	
} 	
as 	
follows 	
. 	

 	
Suppose 	
given 	
a 	
morphism 	
of 	
simplicial 	
schemes 	
$ 	
f 	
: 	
X 	
' 	
\to 	
X$ 	
and 	
a 	

 	
cartesian 	
morphism 	
of 	
simplicial 	
schemes 	
$ 	
Y 	
\to 	
X$. 	
Then 	

 	
the 	
fibre 	
product 	
( 	
viewed 	
as 	
a 	
`` 	
pullback 	
'' 	
) 	

 	
$ 	
$ 	

 	
f^*Y 	
= 	
Y 	
\times_X 	
X 	
' 	

 	
$ 	
$ 	

 	
of 	
simplicial 	
schemes 	
is 	
a 	
simplicial 	
scheme 	
cartesian 	
over 	
$ 	
X'$. 	

 	
Suppose 	
given 	
a 	
commutative 	
diagram 	
of 	
morphisms 	
of 	
schemes 	

 	
$ 	
$ 	

 	
\xymatrix 	
{ 	

 	
X 	
' 	
\ar[r]_f 	
\ar[d 	
] 	
& 	
X 	
\ar[d 	
] 	
\\ 	

 	
S 	
' 	
\ar[r 	
] 	
& 	
S. 	

 	
} 	

 	
$ 	
$ 	

 	
This 	
gives 	
rise 	
to 	
a 	
morphism 	
of 	
simplicial 	
schemes 	

 	
$ 	
$ 	

 	
f_\bullet 	
: 	
( 	
X'/S')_\bullet 	
\longrightarrow 	
( 	
X 	
/ 	
S)_\bullet 	
. 	

 	
$ 	
$ 	

 	
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 	
. 	
Given 	
a 	

 	
quasi 	
- 	
coherent 	
module 	
$ 	
\mathcal{F}$ 	
on 	
$ 	
U_{Zar}$ 	
the 	
pullback 	

 	
$ 	
f^*\mathcal{F}$ 	
is 	
a 	
quasi 	
- 	
coherent 	
module 	
on 	
$ 	
V_{Zar}$. 	

 	
\end{lemma 	
} 	


 	
\begin{proof 	
} 	

 	
Recall 	
that 	
$ 	
\mathcal{F}$ 	
is 	
cartesian 	
with 	
$ 	
\mathcal{F}_n$ 	

 	
quasi 	
- 	
coherent 	
, 	
see 	
Lemma 	
\ref{lemma 	
- 	
quasi 	
- 	
coherent 	
- 	
sheaf}. 	

 	
By 	
Lemma 	
\ref{lemma 	
- 	
describe 	
- 	
functoriality 	
} 	
we 	
see 	
that 	

 	
$ 	
( 	
f^*\mathcal{F})_n 	
= 	
f_n^*\mathcal{F}_n$ 	
( 	
some 	
details 	
omitted 	
) 	
. 	

 	
Hence 	
$ 	
( 	
f^*\mathcal{F})_n$ 	
is 	
quasi 	
- 	
coherent 	
. 	

 	
The 	
same 	
fact 	
and 	
the 	
cartesian 	
property 	
for 	
$ 	
\mathcal{F}$ 	

 	
imply 	
the 	
cartesian 	
property 	
for 	
$ 	
f^*\mathcal{F}$. 	

 	
Thus 	
$ 	
\mathcal{F}$ 	
is 	
quasi 	
- 	
coherent 	
by 	

 	
Lemma 	
\ref{lemma 	
- 	
quasi 	
- 	
coherent 	
- 	
sheaf 	
} 	
again 	
. 	

 	
\end{proof 	
} 	


 	
\begin{lemma 	
} 	

 	
\label{lemma 	
- 	
pushforward 	
- 	
cartesian 	
- 	
module 	
} 	

 	
Let 	
$ 	
f 	
: 	
V 	
\to 	
U$ 	
be 	
a 	
cartesian 	TYPE
morphism 	
of 	
simplicial 	
schemes 	
. 	

 	
Assume 	
the 	
morphisms 	
$ 	
d^n_j 	
: 	
U_n 	
\to 	
U_{n 	
- 	
1}$ 	
are 	

 	
flat 	
and 	
the 	
morphisms 	
$ 	
V_n 	
\to 	
U_n$ 	
are 	
quasi 	
- 	
compact 	
and 	
quasi 	
- 	
separated 	
. 	

 	
For 	
a 	
quasi 	
- 	
coherent 	
module 	
$ 	
\mathcal{G}$ 	
on 	
$ 	
V_{Zar}$ 	

 	
the 	
pushforward 	
$ 	
f_*\mathcal{G}$ 	
is 	
a 	
quasi 	
- 	
coherent 	
module 	
on 	
$ 	
U_{Zar}$. 	

 	
\end{lemma 	
} 	


 	
\begin{proof 	
} 	

 	
If 	
$ 	
\mathcal{F 	
} 	
= 	
f 	
_ 	
* 	
\mathcal{G}$ 	
, 	
then 	

 	
$ 	
\mathcal{F}_n 	
= 	
f_{n 	
, 	
* 	
} 	
\mathcal{G}_n$ 	
by 	

 	
Lemma 	
\ref{lemma 	
- 	
describe 	
- 	
functoriality}. 	

 	
The 	
maps 	
$ 	
\mathcal{F}(\varphi)$ 	
are 	
defined 	
using 	
the 	
base 	
change 	
maps 	
, 	
see 	

 	
Cohomology 	
, 	
Section 	
\ref{cohomology 	
- 	
section 	
- 	
base 	
- 	
change 	
- 	
map}. 	

 	
The 	
sheaves 	
$ 	
\mathcal{F}_n$ 	
are 	
quasi 	
- 	
coherent 	
by 	

 	
Schemes 	
, 	
Lemma 	
\ref{schemes 	
- 	
lemma 	
- 	
push 	
- 	
forward 	
- 	
quasi 	
- 	
coherent 	
} 	

 	
and 	
the 	
fact 	
that 	
$ 	
\mathcal{G}_n$ 	
is 	
quasi 	
- 	
coherent 	

 	
by 	
Lemma 	
\ref{lemma 	
- 	
quasi 	
- 	
coherent 	
- 	
sheaf}. 	

 	
The 	
base 	
change 	
maps 	
along 	
the 	
degeneracies 	

 	
$ 	
d^n_j$ 	
are 	
isomorphisms 	
by 	
Cohomology 	
of 	
Schemes 	
, 	
Lemma 	

 	
\ref{coherent 	
- 	
lemma 	
- 	
flat 	
- 	
base 	
- 	
change 	
- 	
cohomology 	
} 	

 	
and 	
the 	
fact 	
that 	
$ 	
\mathcal{G}$ 	
is 	
cartesian 	

 	
by 	
Lemma 	
\ref{lemma 	
- 	
quasi 	
- 	
coherent 	
- 	
sheaf}. 	

 	
Hence 	
$ 	
\mathcal{F}$ 	
is 	
cartesian 	
by 	

 	
Lemma 	
\ref{lemma 	
- 	
check 	
- 	
cartesian 	
- 	
module}. 	

 	
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 	TYPE
morphism 	
of 	

 	
simplicial 	
schemes 	
. 	
Assume 	
the 	
morphisms 	
$ 	
d^n_j 	
: 	
U_n 	
\to 	
U_{n 	
- 	
1}$ 	
are 	

 	
flat 	
and 	
the 	
morphisms 	
$ 	
V_n 	
\to 	
U_n$ 	
are 	
quasi 	
- 	
compact 	
and 	
quasi 	
- 	
separated 	
. 	

 	
Then 	
$ 	
f^*$ 	
and 	
$ 	
f_*$ 	
form 	
an 	
adjoint 	
pair 	
of 	
functors 	

 	
between 	
the 	
categories 	
of 	
quasi 	
- 	
coherent 	
modules 	
on 	
$ 	
U_{Zar}$ 	
and 	
$ 	
V_{Zar}$. 	

 	
\end{lemma 	
} 	


 	
\begin{proof 	
} 	

 	
We 	
have 	
seen 	
in 	
Lemmas 	
\ref{lemma 	
- 	
pullback 	
- 	
cartesian 	
- 	
module 	
} 	
and 	

 	
\ref{lemma 	
- 	
pushforward 	
- 	
cartesian 	
- 	
module 	
} 	

 	
that 	
the 	
statement 	
makes 	
sense 	
. 	
The 	
adjointness 	
property 	
follows 	

 	
immediately 	
from 	
the 	
fact 	
that 	
each 	
$ 	
f_n^*$ 	
is 	
adjoint 	
to 	
$ 	
f_{n 	
, 	
* 	
} 	
$ 	
. 	

 	
\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}.}. 	

 	
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 	TYPE
of 	
$ 	
f$. 	
Let 	
$ 	
( 	
\mathcal{F 	
} 	
, 	
\alpha)$ 	

 	
be 	
a 	
pair 	TYPE
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$. 	
Hence 	
we 	

 	
see 	
that 	
$ 	
( 	
\sigma 	
\circ 	
f 	
, 	
1)^*\alpha$ 	
defines 	
an 	
isomorphism 	

 	
$ 	
$ 	

 	
f^*\mathcal{G 	
} 	
= 	
( 	
\sigma 	
\circ 	
f 	
, 	
1)^*\text{pr}_0^*\mathcal{F 	
} 	

 	
\longrightarrow 	

 	
( 	
\sigma 	
\circ 	
f 	
, 	
1)^*\text{pr}_1^*\mathcal{F 	
} 	
= 	
\mathcal{F 	
} 	

 	
$ 	
$ 	

 	
We 	
omit 	
the 	
verification 	
that 	
this 	
isomorphism 	
is 	
compatible 	

 	
with 	
$ 	
\alpha$ 	
and 	
the 	
canonical 	
isomorphism 	

 	
$ 	
\text{pr}_0^*f^*\mathcal{G 	
} 	
\to 	
\text{pr}_1^*f^*\mathcal{G}$. 	

 	
\end{proof 	
} 	







 	
\section{Groupoids 	
and 	
simplicial 	
schemes 	
} 	

 	
\label{section 	
- 	
groupoids 	
- 	
simplicial 	
} 	


 	
\noindent 	

 	
Given 	
a 	
groupoid 	
in 	
schemes 	
we 	
can 	
build 	
a 	
simplicial 	
scheme 	
. 	

 	
It 	
will 	
turn 	
out 	
that 	
the 	
category 	
of 	
quasi 	
- 	
coherent 	
sheaves 	
on 	
a 	

 	
groupoid 	
is 	
equivalent 	
to 	
the 	
category 	
of 	
cartesian 	
quasi 	
- 	
coherent 	

 	
sheaves 	
on 	
the 	
associated 	
simplicial 	
scheme 	
. 	


 	
\begin{lemma 	
} 	

 	
\label{lemma 	
- 	
groupoid 	
- 	
simplicial 	
} 	

 	
Let 	
$ 	
( 	
U 	
, 	
R 	
, 	
s 	
, 	
t 	
, 	
c 	
, 	
e 	
, 	
i)$ 	
be 	
a 	
groupoid 	TYPE
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 	
. 	
The 	
map 	
$ 	
d^n_j 	
: 	
X_n 	
\to 	
X_{n 	
- 	
1}$ 	
is 	
given 	
on 	

 	
functors 	
of 	
points 	
by 	

 	
$ 	
$ 	

 	
( 	
r_1 	
, 	
\ldots 	
, 	
r_n 	
) 	
\longmapsto 	
( 	
r_1 	
, 	
\ldots 	
, 	
c(r_j 	
, 	
r_{j 	
+ 	
1 	
} 	
) 	
, 	
\ldots 	
, 	
r_n 	
) 	

 	
$ 	
$ 	

 	
for 	
$ 	
1 	
\leq 	
j 	
\leq 	
n 	
- 	
1 	
$ 	
whereas 	

 	
$ 	
d^n_0(r_1 	
, 	
\ldots 	
, 	
r_n 	
) 	
= 	
( 	
r_2 	
, 	
\ldots 	
, 	
r_n)$ 	
and 	

 	
$ 	
d^n_n(r_1 	
, 	
\ldots 	
, 	
r_n 	
) 	
= 	
( 	
r_1 	
, 	
\ldots 	
, 	
r_{n 	
- 	
1})$. 	

 	
\end{lemma 	
} 	


 	
\begin{proof 	
} 	

 	
We 	
only 	
have 	
to 	
verify 	
that 	
the 	
rules 	
prescribed 	
in 	
( 	
1 	
) 	
, 	
( 	
2 	
) 	
, 	
( 	
3 	
) 	
, 	
( 	
4 	
) 	
, 	
( 	
5 	
) 	

 	
define 	
a 	
$ 	
2$-truncated 	
simplicial 	
scheme 	
$ 	
U'$ 	
over 	
$ 	
S$ 	
, 	
since 	
then 	
( 	
6 	
) 	

 	
allows 	
us 	
to 	
set 	
$ 	
X 	
= 	
\text{cosk}_2 	
U'$ 	
, 	
see 	

 	
Simplicial 	
, 	
Lemma 	
\ref{simplicial 	
- 	
lemma 	
- 	
existence 	
- 	
cosk}. 	

 	
Using 	
the 	
functor 	
of 	
points 	
approach 	
, 	
all 	
we 	
have 	
to 	
verify 	
is 	
that 	

 	
if 	
$ 	
( 	
\text{Ob 	
} 	
, 	
\text{Arrows 	
} 	
, 	
s 	
, 	
t 	
, 	
c 	
, 	
e 	
, 	
i)$ 	
is 	
a 	
groupoid 	
, 	
then 	

 	
$ 	
$ 	

 	
\xymatrix 	
{ 	

 	
\text{Arrows 	
} 	
\times_{s 	
, 	
\text{Ob 	
} 	
, 	
t 	
} 	
\text{Arrows 	
} 	

 	
\ar@<8ex>[d]^{\text{pr}_0 	
} 	

 	
\ar@<0ex>[d]_c 	

 	
\ar@<-8ex>[d]_{\text{pr}_1 	
} 	

 	
\\ 	

 	
\text{Arrows 	
} 	

 	
\ar@<4ex>[d]^t 	

 	
\ar@<-4ex>[d]_s 	

 	
\ar@<4ex>[u]^{1 	
, 	
e 	
} 	

 	
\ar@<-4ex>[u]_{e 	
, 	
1 	
} 	

 	
\\ 	

 	
\text{Ob 	
} 	

 	
\ar@<0ex>[u]_e 	

 	
} 	

 	
$ 	
$ 	

 	
is 	
a 	
$ 	
2$-truncated 	
simplicial 	
set 	
. 	
We 	
omit 	
the 	
details 	
. 	


 	
\medskip\noindent 	

 	
Finally 	
, 	
the 	
description 	
of 	
$ 	
X_n$ 	
for 	
$ 	
n 	
> 	
2 	
$ 	
follows 	
by 	
induction 	
from 	

 	
the 	
description 	
of 	
$ 	
X_0 	
$ 	
, 	
$ 	
X_1 	
$ 	
, 	
$ 	
X_2 	
$ 	
, 	
and 	

 	
Simplicial 	
, 	
Remark 	
\ref{simplicial 	
- 	
remark 	
- 	
inductive 	
- 	
coskeleton 	
} 	
and 	

 	
Lemma 	
\ref{simplicial 	
- 	
lemma 	
- 	
work 	
- 	
out}. 	
Alternately 	
, 	
one 	
shows 	
that 	

 	
$ 	
\text{cosk}_2 	
$ 	
applied 	
to 	
the 	
$ 	
2$-truncated 	
simplicial 	
set 	
displayed 	
above 	

 	
gives 	
a 	
simplicial 	
set 	
whose 	
$ 	
n$th 	
term 	
equals 	

 	
$ 	
\text{Arrows 	
} 	
\times_{s 	
, 	
\text{Ob 	
} 	
, 	
t 	
} 	
\ldots 	
\times_{s 	
, 	
\text{Ob 	
} 	
, 	
t 	
} 	

 	
\text{Arrows}$ 	
with 	
$ 	
n$ 	
factors 	
and 	
degeneracy 	
maps 	
as 	
given 	
in 	
the 	
lemma 	
. 	

 	
Some 	
details 	
omitted 	
. 	

 	
\end{proof 	
} 	


 	
\begin{lemma 	
} 	

 	
\label{lemma 	
- 	
quasi 	
- 	
coherent 	
- 	
groupoid 	
- 	
simplicial 	
} 	

 	
Let 	
$ 	
S$ 	
be 	
a 	
scheme 	TYPE
. 	TYPE
Let 	
$ 	
( 	
U 	
, 	
R 	
, 	
s 	
, 	
t 	
, 	
c)$ 	
be 	
a 	
groupoid 	TYPE
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 	TYPE
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 	
. 	
In 	
exactly 	
the 	
same 	
manner 	

 	
one 	
shows 	
all 	
the 	
other 	
squares 	
are 	
cartesian 	
too 	
. 	
Hence 	

 	
the 	
morphism 	
is 	
cartesian 	
. 	

 	
\end{proof 	
} 	





 	
\section{Descent 	
data 	
give 	
equivalence 	
relations 	
} 	

 	
\label{section 	
- 	
equivalence 	
- 	
relation 	
} 	


 	
\noindent 	

 	
In 	
Section 	
\ref{section 	
- 	
simplicial 	
- 	
descent 	
} 	
we 	
saw 	
how 	
descent 	
data 	
relative 	
to 	

 	
$ 	
X 	
\to 	
S$ 	
can 	
be 	
formulated 	
in 	
terms 	
of 	
cartesian 	
simplicial 	

 	
schemes 	
over 	
$ 	
( 	
X 	
/ 	
S)_\bullet$. 	
Here 	
we 	
link 	
this 	
to 	
equivalence 	

 	
relations 	
as 	
follows 	
. 	


 	
\begin{lemma 	
} 	

 	
\label{lemma 	
- 	
equivalence 	
- 	
relation 	
} 	

 	
Let 	
$ 	
f 	
: 	
X 	
\to 	
S$ 	
be 	
a 	
morphism 	
of 	
schemes 	
. 	

 	
Let 	
$ 	
\pi 	
: 	
Y 	
\to 	
( 	
X 	
/ 	
S)_\bullet$ 	
be 	
a 	
cartesian 	TYPE
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 	
. 	
Namely 	
the 	

 	
composition 	
$ 	
Y_1 	
\to 	
Y_0 	
\times_S 	
Y_0 	
\to 	
Y_0 	
\times_S 	
X$ 	

 	
is 	
an 	
isomorphism 	
as 	
$ 	
\pi$ 	
is 	
cartesian 	
. 	


 	
\medskip\noindent 	

 	
Consider 	
the 	
morphism 	

 	
$ 	
$ 	

 	
( 	
d^2_2 	
, 	
d^2_0 	
) 	
: 	
Y_2 	
\to 	
Y_1 	
\times_{d^1_0 	
, 	
Y_0 	
, 	
d^1_1 	
} 	
Y_1 	
. 	

 	
$ 	
$ 	

 	
This 	
works 	
because 	
$ 	
d_0 	
\circ 	
d_2 	
= 	
d_1 	
\circ 	
d_0 	
$ 	
, 	

 	
see 	
Simplicial 	
, 	
Remark 	
\ref{simplicial 	
- 	
remark 	
- 	
relations}. 	

 	
Also 	
, 	
it 	
is 	
a 	
morphism 	
over 	
$ 	
( 	
X 	
/ 	
S)_2$. 	
It 	
is 	
an 	
isomorphism 	

 	
because 	
$ 	
Y 	
\to 	
( 	
X 	
/ 	
S)_\bullet$ 	
is 	
cartesian 	
. 	
Note 	
for 	
example 	
that 	
the 	

 	
right 	
hand 	
side 	
is 	
isomorphic 	
to 	

 	
$ 	
Y_0 	
\times_{\pi_0 	
, 	
X 	
, 	
\text{pr}_1 	
} 	
( 	
X 	
\times_S 	
X 	
\times_S 	
X 	
) 	
= 	

 	
X 	
\times_S 	
Y_0 	
\times_S 	
X$ 	

 	
because 	
$ 	
\pi$ 	
is 	
cartesian 	
. 	
Details 	
omitted 	
. 	


 	
\medskip\noindent 	

 	
As 	
in 	
Groupoids 	
, 	
Definition 	
\ref{groupoids 	
- 	
definition 	
- 	
equivalence 	
- 	
relation 	
} 	

 	
we 	
denote 	
$ 	
t 	
= 	
\text{pr}_0 	
\circ 	
j 	
= 	
d^1_1 	
$ 	
and 	

 	
$ 	
s 	
= 	
\text{pr}_1 	
\circ 	
j 	
= 	
d^1_0$. 	

 	
The 	
isomorphism 	
above 	
, 	
combined 	
with 	
the 	
morphism 	

 	
$ 	
d^2_1 	
: 	
Y_2 	
\to 	
Y_1 	
$ 	
give 	
us 	
a 	
composition 	
morphism 	

 	
$ 	
$ 	

 	
c 	
: 	
Y_1 	
\times_{s 	
, 	
Y_0 	
, 	
t 	
} 	
Y_1 	
\longrightarrow 	
Y_1 	

 	
$ 	
$ 	

 	
over 	
$ 	
Y_0 	
\times_S 	
Y_0$. 	
This 	
immediately 	
implies 	

 	
that 	
for 	
any 	
scheme 	
$ 	
T 	
/ 	
S$ 	
the 	
relation 	

 	
$ 	
Y_1(T 	
) 	
\subset 	
Y_0(T 	
) 	
\times 	
Y_0(T)$ 	
is 	
transitive 	
. 	


 	
\medskip\noindent 	

 	
Reflexivity 	
follows 	
from 	
the 	
fact 	
that 	
the 	

 	
restriction 	
of 	
the 	
morphism 	
$ 	
j$ 	
to 	
the 	
diagonal 	

 	
$ 	
\Delta 	
: 	
X 	
\to 	
X 	
\times_S 	
X$ 	
is 	
an 	
isomorphism 	

 	
( 	
again 	
use 	
the 	
cartesian 	
property 	
of 	
$ 	
\pi$ 	
) 	
. 	


 	
\medskip\noindent 	

 	
To 	
see 	
symmetry 	
we 	
consider 	
the 	
morphism 	

 	
$ 	
$ 	

 	
( 	
d^2_2 	
, 	
d^2_1 	
) 	
: 	
Y_2 	
\to 	
Y_1 	
\times_{d^1_1 	
, 	
Y_0 	
, 	
d^1_1 	
} 	
Y_1 	
. 	

 	
$ 	
$ 	

 	
This 	
works 	
because 	
$ 	
d_1 	
\circ 	
d_2 	
= 	
d_1 	
\circ 	
d_1 	
$ 	
, 	

 	
see 	
Simplicial 	
, 	
Remark 	
\ref{simplicial 	
- 	
remark 	
- 	
relations}. 	

 	
It 	
is 	
an 	
isomorphism 	

 	
because 	
$ 	
Y 	
\to 	
( 	
X 	
/ 	
S)_\bullet$ 	
is 	
cartesian 	
. 	

 	
Note 	
for 	
example 	
that 	
the 	

 	
right 	
hand 	
side 	
is 	
isomorphic 	
to 	

 	
$ 	
Y_0 	
\times_{\pi_0 	
, 	
X 	
, 	
\text{pr}_0 	
} 	
( 	
X 	
\times_S 	
X 	
\times_S 	
X 	
) 	
= 	

 	
Y_0 	
\times_S 	
X 	
\times_S 	
X$ 	

 	
because 	
$ 	
\pi$ 	
is 	
cartesian 	
. 	
Details 	
omitted 	
. 	


 	
\medskip\noindent 	

 	
Let 	
$ 	
T 	
/ 	
S$ 	
be 	
a 	
scheme 	TYPE
. 	TYPE
Let 	
$ 	
a 	
\sim 	
b$ 	
for 	
$ 	
a 	
, 	
b 	
\in 	
Y_0(T)$ 	

 	
be 	
synonymous 	
with 	
$ 	
( 	
a 	
, 	
b 	
) 	
\in 	
Y_1(T)$. 	

 	
The 	
isomorphism 	
$ 	
( 	
d^2_2 	
, 	
d^2_1)$ 	
above 	

 	
implies 	
that 	
if 	
$ 	
a 	
\sim 	
b$ 	
and 	
$ 	
a 	
\sim 	
c$ 	
, 	
then 	
$ 	
b 	
\sim 	
c$. 	

 	
Combined 	
with 	
reflexivity 	
this 	
shows 	
that 	
$ 	
\sim$ 	
is 	

 	
an 	
equivalence 	
relation 	
. 	

 	
\end{proof 	
} 	








 	
\section{An 	
example 	
case 	
} 	

 	
\label{section 	
- 	
example 	
} 	


 	
\noindent 	

 	
In 	
this 	
section 	
we 	
show 	
that 	
disjoint 	
unions 	
of 	
spectra 	

 	
of 	
Artinian 	
rings 	
can 	
be 	
descended 	
along 	
a 	
quasi 	
- 	
compact 	

 	
surjective 	
flat 	
morphism 	
of 	
schemes 	
. 	


 	
\begin{lemma 	
} 	

 	
\label{lemma 	
- 	
equivalence 	
- 	
classes 	
- 	
points 	
} 	

 	
Let 	
$ 	
X 	
\to 	
S$ 	
be 	
a 	
morphism 	
of 	
schemes 	
. 	
Suppose 	
$ 	
Y 	
\to 	
( 	
X 	
/ 	
S)_\bullet$ 	

 	
is 	
a 	
cartesian 	
morphism 	
of 	
simplicial 	
schemes 	
. 	
For 	
$ 	
y 	
\in 	
Y_0 	
$ 	
a 	
point 	
define 	

 	
$ 	
$ 	

 	
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'\ 	
} 	

 	
$ 	
$ 	

 	
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 	
. 	

 	
Suppose 	
$ 	
Y 	
\to 	
( 	
X 	
/ 	
S)_\bullet$ 	
is 	
a 	
cartesian 	
morphism 	
of 	
simplicial 	
schemes 	
. 	

 	
Let 	
$ 	
y 	
\in 	
Y_0 	
$ 	
be 	
a 	
point 	TYPE
. 	TYPE
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 	
\\ 	

 	
y 	
\ar[r 	
] 	
& 	

 	
Y_0 	
& 	

 	
} 	

 	
$ 	
$ 	

 	
Note 	
that 	
$ 	
F_y$ 	
is 	
quasi 	
- 	
compact 	
. 	
This 	
proves 	
the 	
lemma 	
. 	

 	
\end{proof 	
} 	


 	
\begin{lemma 	
} 	

 	
\label{lemma 	
- 	
descent 	
- 	
disjoint 	
- 	
union 	
- 	
Artinian 	
- 	
along 	
- 	
fields 	
} 	

 	
Let 	
$ 	
X 	
\to 	
S$ 	
be 	
a 	
quasi 	TYPE
- 	
compact 	
flat 	
surjective 	
morphism 	
. 	

 	
Let 	
$ 	
( 	
V 	
, 	
\varphi)$ 	
be 	
a 	
descent 	TYPE
datum 	
relative 	

 	
to 	
$ 	
X 	
\to 	
S$. 	
If 	
$ 	
V$ 	
is 	
a 	
disjoint 	
union 	
of 	

 	
spectra 	
of 	
Artinian 	
rings 	
, 	
then 	
$ 	
( 	
V 	
, 	
\varphi)$ 	
is 	
effective 	
. 	

 	
\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 	
. 	
Moreover 	
, 	
let 	
$ 	
v_i 	
\in 	
V$ 	
be 	
the 	
unique 	
closed 	
point 	
of 	

 	
$ 	
\Spec(A_i)$ 	
for 	
all 	
$ 	
i 	
\in 	
I$. 	
Write 	
$ 	
i 	
\sim 	
j$ 	
if 	
and 	
only 	
if 	

 	
$ 	
v_i 	
\in 	
T_{v_j}$ 	
with 	
notation 	
as 	
in 	

 	
Lemma 	
\ref{lemma 	
- 	
equivalence 	
- 	
classes 	
- 	
points 	
} 	
above 	
. 	

 	
By 	
Lemmas 	
\ref{lemma 	
- 	
equivalence 	
- 	
classes 	
- 	
points 	
} 	
and 	
\ref{lemma 	
- 	
quasi 	
- 	
compact 	
} 	

 	
this 	
is 	
an 	
equivalence 	
relation 	
with 	
finite 	
equivalence 	

 	
classes 	
. 	
Let 	
$ 	
\overline{I 	
} 	
= 	
I/\sim$. 	
Then 	
we 	
can 	
write 	

 	
$ 	
V 	
= 	
\coprod_{\overline{i 	
} 	
\in 	
\overline{I 	
} 	
} 	
V_{\overline{i}}$ 	

 	
with 	

 	
$ 	
V_{\overline{i 	
} 	
} 	
= 	
\coprod_{i 	
\in 	
\overline{i 	
} 	
} 	
\Spec(A_i)$. 	

 	
By 	
construction 	
we 	
see 	
that 	

 	
$ 	
\varphi 	
: 	
V 	
\times_S 	
X 	
\to 	
X 	
\times_S 	
V$ 	
maps 	

 	
the 	
open 	
and 	
closed 	
subspaces 	
$ 	
V_{\overline{i 	
} 	
} 	
\times_S 	
X$ 	

 	
into 	
the 	
open 	
and 	
closed 	
subspaces 	
$ 	
X 	
\times_S 	
V_{\overline{i}}$. 	

 	
In 	
other 	
words 	
, 	
we 	
get 	
descent 	
data 	

 	
$ 	
( 	
V_{\overline{i 	
} 	
} 	
, 	
\varphi_{\overline{i}})$ 	
, 	
and 	

 	
$ 	
( 	
V 	
, 	
\varphi)$ 	
is 	
the 	
coproduct 	
of 	
them 	
in 	
the 	
category 	
of 	

 	
descent 	
data 	
. 	

 	
Since 	
each 	
of 	
the 	
$ 	
V_{\overline{i}}$ 	
is 	
a 	
finite 	
union 	
of 	

 	
spectra 	
of 	
Artinian 	
local 	
rings 	
the 	
morphism 	
$ 	
V_{\overline{i 	
} 	
} 	
\to 	
X$ 	

 	
is 	
affine 	
, 	
see 	
Morphisms 	
, 	
Lemma 	
\ref{morphisms 	
- 	
lemma 	
- 	
Artinian 	
- 	
affine}. 	

 	
Since 	
$ 	
\{X 	
\to 	
S\}$ 	
is 	
an 	
fpqc 	
covering 	
we 	
see 	
that 	
all 	

 	
the 	
descent 	
data 	
$ 	
( 	
V_{\overline{i 	
} 	
} 	
, 	
\varphi_{\overline{i}})$ 	
are 	
effective 	

 	
by 	
Descent 	
, 	
Lemma 	
\ref{descent 	
- 	
lemma 	
- 	
affine}. 	

 	
\end{proof 	
} 	


 	
\noindent 	

 	
To 	
be 	
sure 	
, 	
the 	
lemma 	
above 	
has 	
very 	
limited 	
applicability 	
! 	










 	
\section{Simplicial 	
algebraic 	
spaces 	
} 	

 	
\label{section 	
- 	
simplicial 	
- 	
algebraic 	
- 	
spaces 	
} 	


 	
\noindent 	

 	
Let 	
$ 	
S$ 	
be 	
a 	
scheme 	TYPE
. 	TYPE
A 	
{ 	
\it 	
simplicial 	
algebraic 	
space 	
} 	

 	
is 	
a 	
simplicial 	
object 	
in 	
the 	
category 	
of 	
algebraic 	
spaces 	
over 	
$ 	
S$ 	
, 	

 	
see 	
Simplicial 	
, 	
Definition 	
\ref{simplicial 	
- 	
definition 	
- 	
simplicial 	
- 	
object}. 	

 	
Recall 	
that 	
a 	
simplicial 	
algebraic 	
space 	
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 	
] 	

 	
} 	

 	
$ 	
$ 	

 	
Here 	
there 	
are 	
two 	
morphisms 	
$ 	
d^1_0 	
, 	
d^1_1 	
: 	
X_1 	
\to 	
X_0 	
$ 	

 	
and 	
a 	
single 	
morphism 	
$ 	
s^0_0 	
: 	
X_0 	
\to 	
X_1 	
$ 	
, 	
etc 	
. 	

 	
These 	
morphisms 	
satisfy 	
some 	
required 	
relations 	
such 	
as 	

 	
$ 	
d^1_0 	
\circ 	
s^0_0 	
= 	
\text{id}_{X_0 	
} 	
= 	
d^1_1 	
\circ 	
s^0_0 	
$ 	
, 	
see 	

 	
Simplicial 	
, 	
Lemma 	
\ref{simplicial 	
- 	
lemma 	
- 	
characterize 	
- 	
simplicial 	
- 	
object}. 	

 	
It 	
is 	
useful 	
to 	
think 	
of 	
$ 	
d^n_i 	
: 	
X_n 	
\to 	
X_{n 	
- 	
1}$ 	

 	
as 	
the 	
`` 	
projection 	
forgetting 	
the 	
$ 	
i$th 	
coordinate 	
'' 	
and 	

 	
to 	
think 	
of 	
$ 	
s^n_j 	
: 	
X_n 	
\to 	
X_{n 	
+ 	
1}$ 	
as 	
the 	
`` 	
diagonal 	
map 	
repeating 	

 	
the 	
$ 	
j$th 	
coordinate 	
'' 	
. 	


 	
\medskip\noindent 	

 	
A 	
{ 	
\it 	
morphism 	
of 	
simplicial 	
algebraic 	
spaces 	
} 	
$ 	
h 	
: 	
X 	
\to 	
Y$ 	
is 	
the 	
same 	

 	
thing 	
as 	
a 	
morphism 	
of 	
simplicial 	
objects 	
in 	

 	
the 	
category 	
of 	
algebraic 	
spaces 	
over 	
$ 	
S$ 	
, 	

 	
see 	
Simplicial 	
, 	
Definition 	
\ref{simplicial 	
- 	
definition 	
- 	
simplicial 	
- 	
object}. 	

 	
Thus 	
$ 	
h$ 	
consists 	
of 	
morphisms 	
of 	
algebraic 	
spaces 	
$ 	
h_n 	
: 	
X_n 	
\to 	
Y_n$ 	

 	
such 	
that 	
$ 	
h_{n 	
- 	
1 	
} 	
\circ 	
d^n_j 	
= 	
d^n_j 	
\circ 	
h_n$ 	
and 	

 	
$ 	
h_{n 	
+ 	
1 	
} 	
\circ 	
s^n_j 	
= 	
s^n_j 	
\circ 	
h_n$ 	
whenever 	
this 	
makes 	
sense 	
. 	


 	
\medskip\noindent 	

 	
An 	
{ 	
\it 	
augmentation 	
} 	
$ 	
a 	
: 	
X 	
\to 	
X_{-1}$ 	

 	
of 	
a 	
simplicial 	
algebraic 	
space 	
$ 	
X$ 	
is 	
given 	
by 	
a 	
morphism 	

 	
of 	
algebraic 	
spaces 	
$ 	
a_0 	
: 	
X_0 	
\to 	
X_{-1}$ 	

 	
such 	
that 	
$ 	
a_0 	
\circ 	
d^1_0 	
= 	
a_0 	
\circ 	
d^1_1$. 	

 	
See 	
Simplicial 	
, 	
Section 	
\ref{simplicial 	
- 	
section 	
- 	
augmentation}. 	

 	
In 	
this 	
situation 	
we 	
always 	
indicate 	
$ 	
a_n 	
: 	
X_n 	
\to 	
X_{-1}$ 	
the 	
induced 	

 	
morphisms 	
for 	
$ 	
n 	
\geq 	
0$. 	


 	
\medskip\noindent 	

 	
Let 	
$ 	
X$ 	
be 	
a 	
simplicial 	TYPE
algebraic 	
space 	
. 	
For 	
every 	
$ 	
n$ 	
we 	
have 	
the 	

 	
site 	
$ 	
X_{n 	
, 	
spaces 	
, 	
\etale}$ 	
( 	
Properties 	
of 	
Spaces 	
, 	
Definition 	

 	
\ref{spaces 	
- 	
properties 	
- 	
definition 	
- 	
spaces 	
- 	
etale 	
- 	
site 	
} 	
) 	

 	
and 	
for 	
every 	
morphism 	
$ 	
\varphi 	
: 	
[ 	
m 	
] 	
\to 	
[ 	
n]$ 	
we 	
have 	
a 	
morphism 	
of 	
sites 	

 	
$ 	
$ 	

 	
f_\varphi 	
= 	
X(\varphi)_{spaces 	
, 	
\etale 	
} 	
: 	

 	
X_{n 	
, 	
spaces 	
, 	
\etale 	
} 	
\to 	
X_{m 	
, 	
spaces 	
, 	
\etale 	
} 	
, 	

 	
$ 	
$ 	

 	
associated 	
to 	
the 	
morphism 	
of 	
algebraic 	
spaces 	

 	
$ 	
X(\varphi 	
) 	
: 	
X_n 	
\to 	
X_m$ 	
( 	
Properties 	
of 	
Spaces 	
, 	
Lemma 	

 	
\ref{spaces 	
- 	
properties 	
- 	
lemma 	
- 	
functoriality 	
- 	
etale 	
- 	
site 	
} 	
) 	
. 	

 	
This 	
gives 	
a 	
simplicial 	
object 	
in 	
the 	
category 	
of 	
sites 	
. 	

 	
In 	
Lemma 	
\ref{lemma 	
- 	
simplicial 	
- 	
site 	
- 	
site 	
} 	
we 	
constructed 	
an 	
associated 	

 	
site 	
which 	
we 	
denote 	
$ 	
X_{spaces 	
, 	
\etale}$. 	

 	
An 	
object 	
of 	
the 	
site 	
$ 	
X_{spaces 	
, 	
\etale}$ 	
is 	
a 	

 	
an 	
algebraic 	
space 	
$ 	
U$ 	
\'etale 	
over 	
$ 	
X_n$ 	
for 	
some 	
$ 	
n$ 	

 	
and 	
a 	
morphism 	
$ 	
( 	
\varphi 	
, 	
f 	
) 	
: 	
U 	
/ 	
X_n 	
\to 	
V 	
/ 	
X_m$ 	
is 	
given 	

 	
by 	
a 	
morphism 	
$ 	
\varphi 	
: 	
[ 	
m 	
] 	
\to 	
[ 	
n]$ 	
in 	
$ 	
\Delta$ 	
and 	
a 	
morphism 	

 	
$ 	
f 	
: 	
U 	
\to 	
V$ 	
of 	
algebraic 	
spaces 	
such 	
that 	
the 	
diagram 	

 	
$ 	
$ 	

 	
\xymatrix 	
{ 	

 	
U 	
\ar[r]_f 	
\ar[d 	
] 	
& 	
V 	
\ar[d 	
] 	
\\ 	

 	
X_n 	
\ar[r]^{f_\varphi 	
} 	
& 	
X_m 	

 	
} 	

 	
$ 	
$ 	

 	
is 	
commutative 	
. 	
Consider 	
the 	
full 	
subcategories 	

 	
$ 	
$ 	

 	
X_{affine 	
, 	
\etale 	
} 	
\subset 	
X_\etale 	
\subset 	
X_{spaces 	
, 	
\etale 	
} 	

 	
$ 	
$ 	

 	
whose 	
objects 	
are 	
$ 	
U 	
/ 	
X_n$ 	
with 	
$ 	
U$ 	
affine 	
, 	
respectively 	
a 	
scheme 	
. 	

 	
Endowing 	
these 	
categories 	
with 	
their 	
natural 	
topologies 	

 	
( 	
see 	

 	
Properties 	
of 	
Spaces 	
, 	
Lemma 	
\ref{spaces 	
- 	
properties 	
- 	
lemma 	
- 	
alternative 	
} 	
, 	

 	
Definition 	
\ref{spaces 	
- 	
properties 	
- 	
definition 	
- 	
etale 	
- 	
site 	
} 	
, 	
and 	

 	
Lemma 	
\ref{spaces 	
- 	
properties 	
- 	
lemma 	
- 	
compare 	
- 	
etale 	
- 	
sites 	
} 	
) 	

 	
these 	
inclusion 	
functors 	
define 	
equivalences 	
of 	
topoi 	

 	
$ 	
$ 	

 	
\Sh(X_{affine 	
, 	
\etale 	
} 	
) 	
= 	
\Sh(X_\etale 	
) 	
= 	
\Sh(X_{spaces 	
, 	
\etale 	
} 	
) 	

 	
$ 	
$ 	

 	
In 	
the 	
following 	
we 	
will 	
silently 	
identify 	
these 	
topoi 	
. 	

 	
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 	
$ 	
\mathcal{O}_{X_n}$. 	
This 	
follows 	
from 	

 	
Lemma 	
\ref{lemma 	
- 	
describe 	
- 	
sheaves 	
- 	
simplicial 	
- 	
site 	
- 	
site}. 	
We 	
will 	
say 	

 	
{ 	
\it 	
$ 	
\mathcal{O}$ 	
is 	
the 	
structure 	
sheaf 	
of 	
the 	

 	
simplicial 	
algebraic 	
space 	
$ 	
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 	TYPE
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}. 	

 	
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$. 	

 	
By 	
Lemma 	
\ref{lemma 	
- 	
morphism 	
- 	
simplicial 	
- 	
sites 	
} 	
applied 	
to 	
the 	
morphisms 	

 	
of 	
sites 	
$ 	
( 	
h_n)_{spaces 	
, 	
\etale 	
} 	
: 	
X_{spaces 	
, 	
\etale 	
} 	
\to 	
Y_{spaces 	
, 	
\etale}$ 	

 	
( 	
Properties 	
of 	
Spaces 	
, 	
Lemma 	

 	
\ref{spaces 	
- 	
properties 	
- 	
lemma 	
- 	
functoriality 	
- 	
etale 	
- 	
site 	
} 	
) 	

 	
we 	
obtain 	
a 	
morphism 	
of 	
small 	
\'etale 	
topoi 	

 	
$ 	
h_\etale 	
: 	
\Sh(X_\etale 	
) 	
\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 	TYPE
algebraic 	
space 	
with 	
structure 	
sheaf 	
$ 	
\mathcal{O}$. 	

 	
Let 	
$ 	
X_{-1}$ 	
be 	
an 	
algebraic 	TYPE
space 	
over 	
$ 	
S$ 	
and 	
let 	
$ 	
a_0 	
: 	
X_0 	
\to 	
X_{-1}$ 	

 	
be 	
an 	
augmentation 	TYPE
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 	
) 	
. 	
This 	
puts 	
us 	
in 	
the 	
situation 	

 	
discussed 	
in 	

 	
Section 	
\ref{section 	
- 	
cohomology 	
- 	
augmentation 	
- 	
ringed 	
- 	
simplicial 	
- 	
sites}. 	

 	
Therefore 	
we 	
obtain 	
a 	
morphism 	
of 	
ringed 	
topoi 	

 	
$ 	
$ 	

 	
a 	
: 	

 	
( 	
\Sh(X_\etale 	
) 	
, 	
\mathcal{O 	
} 	
) 	

 	
\longrightarrow 	

 	
( 	
\Sh(X_{-1 	
} 	
) 	
, 	
\mathcal{O}_{X_{-1 	
} 	
} 	
) 	

 	
$ 	
$ 	


 	
\medskip\noindent 	

 	
A 	
final 	
observation 	
is 	
the 	
following 	
. 	
Suppose 	
we 	
are 	
given 	
a 	
morphism 	

 	
$ 	
h 	
: 	
X 	
\to 	
Y$ 	
of 	
simplicial 	
algebraic 	
spaces 	
$ 	
X$ 	
and 	
$ 	
Y$ 	
with 	
structure 	
sheaves 	

 	
$ 	
\mathcal{O}_X$ 	
, 	
$ 	
\mathcal{O}_Y$ 	
, 	
augmentations 	

 	
$ 	
a_0 	
: 	
X_0 	
\to 	
X_{-1}$ 	
, 	
$ 	
b_0 	
: 	
Y_0 	
\to 	
Y_{-1}$ 	
and 	
a 	
morphism 	

 	
$ 	
h_{-1 	
} 	
: 	
X_{-1 	
} 	
\to 	
Y_{-1}$ 	
such 	
that 	

 	
$ 	
$ 	

 	
\xymatrix 	
{ 	

 	
X_0 	
\ar[r]_{h_0 	
} 	
\ar[d]_{a_0 	
} 	
& 	
Y_0 	
\ar[d]^{b_0 	
} 	
\\ 	

 	
X_{-1 	
} 	
\ar[r]^{h_{-1 	
} 	
} 	
& 	
Y_{-1 	
} 	

 	
} 	

 	
$ 	
$ 	

 	
commutes 	
. 	
Then 	
from 	
the 	
constructions 	
elucidated 	
above 	

 	
we 	
obtain 	
a 	
commutative 	
diagram 	
of 	
morphisms 	
of 	
ringed 	
topoi 	
as 	
follows 	

 	
$ 	
$ 	

 	
\xymatrix 	
{ 	

 	
( 	
\Sh(X_\etale 	
) 	
, 	
\mathcal{O}_X 	
) 	
\ar[r]_{h_\etale 	
} 	
\ar[d]_a 	
& 	

 	
( 	
\Sh(Y_\etale 	
) 	
, 	
\mathcal{O}_Y 	
) 	
\ar[d]^b 	
\\ 	

 	
( 	
\Sh(X_{-1 	
} 	
) 	
, 	
\mathcal{O}_{X_{-1 	
} 	
} 	
) 	
\ar[r]^{h_{-1 	
} 	
} 	
& 	

 	
( 	
\Sh(Y_{-1 	
} 	
) 	
, 	
\mathcal{O}_{Y_{-1 	
} 	
} 	
) 	

 	
} 	

 	
$ 	
$ 	





 	
\section{Fppf 	
hypercoverings 	
of 	
algebraic 	
spaces 	
} 	

 	
\label{section 	
- 	
fppf 	
- 	
hypercovering 	
} 	


 	
\noindent 	

 	
This 	
section 	
is 	
the 	
analogue 	
of 	
Section 	
\ref{section 	
- 	
proper 	
- 	
hypercovering 	
} 	

 	
for 	
the 	
case 	
of 	
algebraic 	
spaces 	
and 	
fppf 	
hypercoverings 	
. 	

 	
The 	
reader 	
who 	
wishes 	
to 	
do 	
so 	
, 	
can 	
replace 	
`` 	
algebraic 	
space 	
'' 	

 	
everywhere 	
with 	
`` 	
scheme 	
'' 	
and 	
get 	
equally 	
valid 	
results 	
. 	

 	
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 	TYPE
space 	
over 	
$ 	
S$ 	
and 	
let 	
$ 	
U$ 	
be 	
a 	
simplicial 	TYPE

 	
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 	
. 	

 	
$ 	
$ 	

 	
\fbox{Principle 	
: 	
Fppf 	
hypercoverings 	
can 	
be 	
used 	
to 	
compute 	
\'etale 	
cohomology 	
. 	
} 	

 	
$ 	
$ 	

 	
The 	
key 	
idea 	
behind 	
the 	
proof 	
of 	
the 	
principle 	
is 	
to 	
compare 	
the 	

 	
fppf 	
and 	
\'etale 	
topologies 	
on 	
the 	
category 	
$ 	
\textit{Spaces}/S$. 	

 	
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 	TYPE
. 	TYPE
Let 	
$ 	
X$ 	
be 	
an 	
algebraic 	TYPE
space 	
over 	
$ 	
S$. 	

 	
Let 	
$ 	
U$ 	
be 	
a 	
simplicial 	TYPE
algebraic 	
space 	
over 	
$ 	
S$. 	

 	
Let 	
$ 	
a 	
: 	
U 	
\to 	
X$ 	
be 	
an 	
augmentation 	TYPE
. 	
There 	
is 	
a 	
commutative 	
diagram 	

 	
$ 	
$ 	

 	
\xymatrix 	
{ 	

 	
\Sh((\textit{Spaces}/U)_{fppf 	
, 	
total 	
} 	
) 	
\ar[r]_-h 	
\ar[d]_{a_{fppf 	
} 	
} 	
& 	

 	
\Sh(U_\etale 	
) 	
\ar[d]^a 	
\\ 	

 	
\Sh((\textit{Spaces}/X)_{fppf 	
} 	
) 	
\ar[r]^-{h_{-1 	
} 	
} 	
& 	

 	
\Sh(X_\etale 	
) 	

 	
} 	

 	
$ 	
$ 	

 	
where 	
the 	
left 	
vertical 	
arrow 	
is 	
defined 	
in 	

 	
Section 	
\ref{section 	
- 	
hypercovering 	
} 	

 	
and 	
the 	
right 	
vertical 	
arrow 	
is 	
defined 	
in 	

 	
Section 	
\ref{section 	
- 	
simplicial 	
- 	
algebraic 	
- 	
spaces}. 	

 	
\end{lemma 	
} 	


 	
\begin{proof 	
} 	

 	
The 	
notation 	
$ 	
( 	
\textit{Spaces}/U)_{fppf 	
, 	
total}$ 	
indicates 	
that 	

 	
we 	
are 	
using 	
the 	
construction 	
of 	

 	
Section 	
\ref{section 	
- 	
hypercovering 	
} 	

 	
for 	
the 	
site 	
$ 	
( 	
\textit{Spaces}/S)_{fppf}$ 	
and 	
the 	

 	
simplicial 	
object 	
$ 	
U$ 	
of 	
this 	
site\footnote{We 	
could 	
also 	

 	
use 	
the 	
\'etale 	
topology 	
and 	
this 	
would 	
be 	
denoted 	

 	
$ 	
( 	
\textit{Spaces}/U)_{\etale 	
, 	
total}$.}. 	

 	
We 	
will 	
use 	
the 	
sites 	
$ 	
X_{spaces 	
, 	
\etale}$ 	
and 	
$ 	
U_{spaces 	
, 	
\etale}$ 	

 	
for 	
the 	
topoi 	
on 	
the 	
right 	
hand 	
side 	
; 	
this 	
is 	
permissible 	

 	
see 	
discussion 	
in 	
Section 	
\ref{section 	
- 	
simplicial 	
- 	
algebraic 	
- 	
spaces}. 	


 	
\medskip\noindent 	

 	
Observe 	
that 	
both 	
$ 	
( 	
\textit{Spaces}/U)_{fppf 	
, 	
total}$ 	
and 	

 	
$ 	
U_{spaces 	
, 	
\etale}$ 	

 	
fall 	
into 	
case 	
A 	
of 	
Situation 	
\ref{situation 	
- 	
simplicial 	
- 	
site}. 	

 	
This 	
is 	
immediate 	
from 	
the 	
construction 	
of 	

 	
$ 	
U_\etale$ 	
in 	
Section 	
\ref{section 	
- 	
simplicial 	
- 	
algebraic 	
- 	
spaces 	
} 	

 	
and 	
it 	
follows 	
from 	
Lemma 	
\ref{lemma 	
- 	
sr 	
- 	
when 	
- 	
fibre 	
- 	
products 	
} 	

 	
for 	
$ 	
( 	
\textit{Spaces}/U)_{fppf 	
, 	
total}$. 	

 	
Next 	
, 	
consider 	
the 	
functors 	

 	
$ 	
U_{n 	
, 	
spaces 	
, 	
\etale 	
} 	
\to 	
( 	
\textit{Spaces}/U_n)_{fppf}$ 	
, 	
$ 	
U 	
\mapsto 	
U 	
/ 	
U_n$ 	

 	
and 	

 	
$ 	
X_{spaces 	
, 	
\etale 	
} 	
\to 	
( 	
\textit{Spaces}/X)_{fppf}$ 	
, 	
$ 	
U 	
\mapsto 	
U 	
/ 	
X$. 	

 	
We 	
have 	
seen 	
that 	
these 	
define 	
morphisms 	
of 	
sites 	
in 	

 	
More 	
on 	
Cohomology 	
of 	
Spaces 	
, 	
Section 	

 	
\ref{spaces 	
- 	
more 	
- 	
cohomology 	
- 	
section 	
- 	
fppf 	
- 	
etale 	
} 	

 	
where 	
these 	
were 	
denoted 	
$ 	
a_{U_n 	
} 	
= 	
\epsilon_{U_n 	
} 	
\circ 	
\pi_{u_n}$ 	

 	
and 	
$ 	
a_X 	
= 	
\epsilon_X 	
\circ 	
\pi_X$. 	

 	
Thus 	
we 	
obtain 	
a 	
morphism 	
of 	
simplicial 	
sites 	
compatible 	
with 	

 	
augmentations 	
as 	
in 	
Remark 	
\ref{remark 	
- 	
morphism 	
- 	
augmentation 	
- 	
simplicial 	
- 	
sites 	
} 	

 	
and 	
we 	
may 	
apply 	

 	
Lemma 	
\ref{lemma 	
- 	
morphism 	
- 	
augmentation 	
- 	
simplicial 	
- 	
sites 	
} 	
to 	
conclude 	
. 	

 	
\end{proof 	
} 	


 	
\begin{lemma 	
} 	

 	
\label{lemma 	
- 	
descent 	
- 	
sheaves 	
- 	
for 	
- 	
fppf 	
- 	
hypercovering 	
} 	

 	
Let 	
$ 	
S$ 	
be 	
a 	
scheme 	TYPE
. 	TYPE
Let 	
$ 	
X$ 	
be 	
an 	
algebraic 	TYPE
space 	
over 	
$ 	
S$. 	

 	
Let 	
$ 	
U$ 	
be 	
a 	
simplicial 	TYPE
algebraic 	
space 	
over 	
$ 	
S$. 	
Let 	
$ 	
a 	
: 	
U 	
\to 	
X$ 	

 	
be 	
an 	
augmentation 	TYPE
. 	
If 	
$ 	
a 	
: 	
U 	
\to 	
X$ 	
is 	
an 	
fppf 	
hypercovering 	
of 	
$ 	
X$ 	
, 	

 	
then 	

 	
$ 	
$ 	

 	
a^{-1 	
} 	
: 	
\Sh(X_\etale 	
) 	
\to 	
\Sh(U_\etale 	
) 	

 	
\quad\text{and}\quad 	

 	
a^{-1 	
} 	
: 	
\textit{Ab}(X_\etale 	
) 	
\to 	
\textit{Ab}(U_\etale 	
) 	

 	
$ 	
$ 	

 	
are 	
fully 	
faithful 	
with 	
essential 	
image 	
the 	
cartesian 	
sheaves 	
and 	

 	
quasi 	
- 	
inverse 	
given 	
by 	
$ 	
a_*$. 	
Here 	
$ 	
a 	
: 	
\Sh(U_\etale 	
) 	
\to 	
\Sh(X_\etale)$ 	

 	
is 	
as 	
in 	
Section 	
\ref{section 	
- 	
simplicial 	
- 	
algebraic 	
- 	
spaces}. 	

 	
\end{lemma 	
} 	


 	
\begin{proof 	
} 	

 	
We 	
will 	
prove 	
the 	
statement 	
for 	
sheaves 	
of 	
sets 	
. 	
It 	
will 	
be 	
an 	

 	TYPE
almost 	
formal 	
consequence 	
of 	
results 	
already 	
established 	
. 	

 	
Consider 	
the 	
diagram 	
of 	

 	
Lemma 	
\ref{lemma 	
- 	
compare 	
- 	
simplicial 	
- 	
objects 	
- 	
fppf 	
- 	
etale}. 	

 	
In 	
the 	
proof 	
of 	
this 	
lemma 	
we 	
have 	
seen 	
that 	

 	
$ 	
h_{-1}$ 	
is 	
the 	
morphism 	
$ 	
a_X$ 	
of 	

 	
More 	
on 	
Cohomology 	
of 	
Spaces 	
, 	
Section 	

 	
\ref{spaces 	
- 	
more 	
- 	
cohomology 	
- 	
section 	
- 	
fppf 	
- 	
etale}. 	

 	
Thus 	
it 	
follows 	
from 	

 	
More 	
on 	
Cohomology 	
of 	
Spaces 	
, 	
Lemma 	

 	
\ref{spaces 	
- 	
more 	
- 	
cohomology 	
- 	
lemma 	
- 	
comparison 	
- 	
fppf 	
- 	
etale 	
} 	

 	
that 	
$ 	
( 	
h_{-1})^{-1}$ 	
is 	
fully 	
faithful 	
with 	
quasi 	
- 	
inverse 	
$ 	
h_{-1 	
, 	
* 	
} 	
$ 	
. 	

 	
The 	
same 	
holds 	
true 	
for 	
the 	
components 	
$ 	
h_n$ 	
of 	
$ 	
h$. 	

 	
By 	
the 	
description 	
of 	
the 	
functors 	
$ 	
h^{-1}$ 	
and 	
$ 	
h_*$ 	
of 	

 	
Lemma 	
\ref{lemma 	
- 	
morphism 	
- 	
simplicial 	
- 	
sites 	
} 	

 	
we 	
conclude 	
that 	
$ 	
h^{-1}$ 	
is 	
fully 	
faithful 	
with 	
quasi 	
- 	
inverse 	
$ 	
h_*$. 	

 	
Observe 	
that 	
$ 	
U$ 	
is 	
a 	
hypercovering 	
of 	
$ 	
X$ 	
in 	
$ 	
( 	
\textit{Spaces}/S)_{fppf}$ 	

 	
as 	
defined 	
in 	
Section 	
\ref{section 	
- 	
hypercovering}. 	

 	
By 	
Lemma 	
\ref{lemma 	
- 	
hypercovering 	
- 	
X 	
- 	
simple 	
- 	
descent 	
- 	
sheaves 	
} 	

 	
we 	
see 	
that 	
$ 	
a_{fppf}^{-1}$ 	
is 	
fully 	
faithful 	
with 	
quasi 	
- 	
inverse 	

 	
$ 	
a_{fppf 	
, 	
* 	
} 	
$ 	
and 	
with 	
essential 	
image 	
the 	
cartesian 	
sheaves 	

 	
on 	
$ 	
( 	
\textit{Spaces}/U)_{fppf 	
, 	
total}$. 	

 	
A 	
formal 	
argument 	
( 	
chasing 	
around 	
the 	
diagram 	
) 	
now 	
shows 	
that 	

 	
$ 	
a^{-1}$ 	
is 	
fully 	
faithful 	
. 	


 	
\medskip\noindent 	

 	
Finally 	
, 	
suppose 	
that 	
$ 	
\mathcal{G}$ 	
is 	
a 	
cartesian 	
sheaf 	
on 	
$ 	
U_\etale$. 	

 	
Then 	
$ 	
h^{-1}\mathcal{G}$ 	
is 	
a 	
cartesian 	
sheaf 	
on 	

 	
$ 	
( 	
\textit{Spaces}/U)_{fppf 	
, 	
total}$. 	
Hence 	

 	
$ 	
h^{-1}\mathcal{G 	
} 	
= 	
a_{fppf}^{-1}\mathcal{H}$ 	
for 	
some 	
sheaf 	

 	
$ 	
\mathcal{H}$ 	
on 	
$ 	
( 	
\textit{Spaces}/X)_{fppf}$. 	

 	
In 	
particular 	
we 	
find 	
that 	

 	
$ 	
h_0^{-1}\mathcal{G}_0 	
= 	
( 	
a_{0 	
, 	
big 	
, 	
fppf})^{-1}\mathcal{H}$. 	

 	
Recalling 	
that 	
$ 	
h_0 	
= 	
a_{U_0}$ 	
and 	
that 	
$ 	
U_0 	
\to 	
X$ 	
is 	

 	
flat 	
, 	
locally 	
of 	
finite 	
presentation 	
, 	
and 	
surjective 	
, 	
we 	

 	
find 	
from 	
More 	
on 	
Cohomology 	
of 	
Spaces 	
, 	

 	
Lemma 	
\ref{spaces 	
- 	
more 	
- 	
cohomology 	
- 	
lemma 	
- 	
descent 	
- 	
sheaf 	
- 	
fppf 	
- 	
etale 	
} 	

 	
that 	
there 	
exists 	
a 	
sheaf 	
$ 	
\mathcal{F}$ 	
on 	
$ 	
X_\etale$ 	
and 	
isomorphism 	

 	
$ 	
\mathcal{H 	
} 	
= 	
( 	
h_{-1})^{-1}\mathcal{F}$. 	

 	
Since 	
$ 	
a_{fppf}^{-1}\mathcal{H 	
} 	
= 	
h^{-1}\mathcal{G}$ 	

 	
we 	
deduce 	
that 	
$ 	
h^{-1}\mathcal{G 	
} 	
\cong 	
h^{-1}a^{-1}\mathcal{F}$. 	

 	
By 	
fully 	
faithfulness 	
of 	
$ 	
h^{-1}$ 	
we 	
conclude 	
that 	

 	
$ 	
a^{-1}\mathcal{F 	
} 	
\cong 	
\mathcal{G}$. 	


 	
\medskip\noindent 	

 	
Fix 	
an 	
isomorphism 	
$ 	
\theta 	
: 	
a^{-1}\mathcal{F 	
} 	
\to 	
\mathcal{G}$. 	

 	
To 	
finish 	
the 	
proof 	
we 	
have 	
to 	
show 	
$ 	
\mathcal{G 	
} 	
= 	
a^{-1}a_*\mathcal{G}$ 	

 	
( 	
in 	
order 	
to 	
show 	
that 	
the 	
quasi 	
- 	
inverse 	
is 	
given 	
by 	
$ 	
a_*$ 	
; 	
everything 	

 	
else 	
has 	
been 	
proven 	
above 	
) 	
. 	

 	
Because 	
$ 	
a^{-1}$ 	
is 	
fully 	
faithful 	
we 	
have 	
$ 	
\text{id 	
} 	
\cong 	
a_*a^{-1}$ 	
by 	

 	
Categories 	
, 	
Lemma 	
\ref{categories 	
- 	
lemma 	
- 	
adjoint 	
- 	
fully 	
- 	
faithful}. 	

 	
Thus 	
$ 	
\mathcal{F 	
} 	
\cong 	
a_*a^{-1}\mathcal{F}$ 	
and 	

 	
$ 	
a_*\theta 	
: 	
a_*a^{-1}\mathcal{F 	
} 	
\to 	
a_*\mathcal{G}$ 	

 	
combine 	
to 	
an 	
isomorphism 	
$ 	
\mathcal{F 	
} 	
\to 	
a_*\mathcal{G}$. 	

 	
Pulling 	
back 	
by 	
$ 	
a$ 	
and 	
precomposing 	
by 	
$ 	
\theta^{-1}$ 	

 	
we 	
find 	
the 	
desired 	
isomorphism 	
. 	

 	
\end{proof 	
} 	


 	
\begin{lemma 	
} 	

 	
\label{lemma 	
- 	
cohomological 	
- 	
descent 	
- 	
for 	
- 	
fppf 	
- 	
hypercovering 	
} 	

 	
Let 	
$ 	
S$ 	
be 	
a 	
scheme 	TYPE
. 	TYPE
Let 	
$ 	
X$ 	
be 	
an 	
algebraic 	TYPE
space 	
over 	
$ 	
S$. 	

 	
Let 	
$ 	
U$ 	
be 	
a 	
simplicial 	TYPE
algebraic 	
space 	
over 	
$ 	
S$. 	
Let 	
$ 	
a 	
: 	
U 	
\to 	
X$ 	

 	
be 	
an 	
augmentation 	TYPE
. 	
If 	
$ 	
a 	
: 	
U 	
\to 	
X$ 	
is 	
an 	
fppf 	
hypercovering 	
of 	
$ 	
X$ 	
, 	

 	
then 	
for 	
$ 	
K 	
\in 	
D^+(X_\etale)$ 	

 	
$ 	
$ 	

 	
K 	
\to 	
Ra_*(a^{-1}K 	
) 	

 	
$ 	
$ 	

 	
is 	
an 	
isomorphism 	
. 	
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 	TYPE
. 	TYPE
Let 	
$ 	
X$ 	
be 	
an 	
algebraic 	TYPE
space 	
over 	
$ 	
S$. 	

 	
Let 	
$ 	
U$ 	
be 	
a 	
simplicial 	TYPE
algebraic 	
space 	
over 	
$ 	
S$. 	
Let 	
$ 	
a 	
: 	
U 	
\to 	
X$ 	

 	
be 	
an 	
augmentation 	TYPE
. 	
If 	
$ 	
a 	
: 	
U 	
\to 	
X$ 	
is 	
an 	
fppf 	
hypercovering 	
of 	
$ 	
X$ 	
, 	
then 	

 	
$ 	
$ 	

 	
R\Gamma(X_\etale 	
, 	
K 	
) 	
= 	
R\Gamma(U_\etale 	
, 	
a^{-1}K 	
) 	

 	
$ 	
$ 	

 	
for 	
$ 	
K 	
\in 	
D^+(X_\etale)$. 	
Here 	
$ 	
a 	
: 	
\Sh(U_\etale 	
) 	
\to 	
\Sh(X_\etale)$ 	

 	
is 	
as 	
in 	
Section 	
\ref{section 	
- 	
simplicial 	
- 	
algebraic 	
- 	
spaces}. 	

 	
\end{lemma 	
} 	


 	
\begin{proof 	
} 	

 	
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 	TYPE
. 	TYPE
Let 	
$ 	
X$ 	
be 	
an 	
algebraic 	TYPE
space 	
over 	
$ 	
S$. 	

 	
Let 	
$ 	
U$ 	
be 	
a 	
simplicial 	TYPE
algebraic 	
space 	
over 	
$ 	
S$. 	
Let 	
$ 	
a 	
: 	
U 	
\to 	
X$ 	

 	
be 	
an 	
augmentation 	TYPE
. 	

 	
Let 	
$ 	
\mathcal{A 	
} 	
\subset 	
\textit{Ab}(U_\etale)$ 	

 	
denote 	
the 	
weak 	
Serre 	
subcategory 	
of 	
cartesian 	
abelian 	
sheaves 	
. 	

 	
If 	
$ 	
U$ 	
is 	
an 	
fppf 	
hypercovering 	
of 	
$ 	
X$ 	
, 	
then 	

 	
the 	
functor 	
$ 	
a^{-1}$ 	
defines 	
an 	
equivalence 	

 	
$ 	
$ 	

 	
D^+(X_\etale 	
) 	
\longrightarrow 	
D_\mathcal{A}^+(U_\etale 	
) 	

 	
$ 	
$ 	

 	
with 	
quasi 	
- 	
inverse 	
$ 	
Ra_*$. 	
Here 	
$ 	
a 	
: 	
\Sh(U_\etale 	
) 	
\to 	
\Sh(X_\etale)$ 	

 	
is 	
as 	
in 	
Section 	
\ref{section 	
- 	
simplicial 	
- 	
algebraic 	
- 	
spaces}. 	

 	
\end{lemma 	
} 	


 	
\begin{proof 	
} 	

 	
Observe 	
that 	
$ 	
\mathcal{A}$ 	
is 	
a 	
weak 	
Serre 	
subcategory 	
by 	

 	
Lemma 	
\ref{lemma 	
- 	
Serre 	
- 	
subcat 	
- 	
cartesian 	
- 	
modules}. 	

 	
The 	
equivalence 	
is 	
a 	

 	
formal 	
consequence 	
of 	
the 	
results 	
obtained 	
so 	
far 	
. 	
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 	TYPE
. 	TYPE
Let 	
$ 	
X$ 	
be 	
an 	
algebraic 	TYPE
space 	
over 	
$ 	
S$. 	

 	
Let 	
$ 	
U$ 	
be 	
a 	
simplicial 	TYPE
algebraic 	
space 	
over 	
$ 	
S$. 	
Let 	
$ 	
a 	
: 	
U 	
\to 	
X$ 	

 	
be 	
an 	
augmentation 	TYPE
. 	
Let 	
$ 	
\mathcal{F}$ 	
be 	
an 	
abelian 	TYPE
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 	

 	
We 	
continue 	
the 	
discussion 	
of 	
( 	
cohomological 	
) 	
descent 	
for 	
fppf 	
hypercoverings 	

 	
started 	
in 	
Section 	
\ref{section 	
- 	
fppf 	
- 	
hypercovering 	
} 	

 	
but 	
in 	
this 	
section 	
we 	
discuss 	
what 	
happens 	
for 	
sheaves 	
of 	
modules 	
. 	

 	
We 	
mainly 	
discuss 	
quasi 	
- 	
coherent 	
modules 	
and 	
it 	
turns 	
out 	
that 	

 	
we 	
can 	
do 	
unbounded 	
cohomological 	
descent 	
for 	
those 	
. 	


 	
\begin{lemma 	
} 	

 	
\label{lemma 	
- 	
compare 	
- 	
simplicial 	
- 	
objects 	
- 	
fppf 	
- 	
etale 	
- 	
modules 	
} 	

 	
Let 	
$ 	
S$ 	
be 	
a 	
scheme 	TYPE
. 	TYPE
Let 	
$ 	
X$ 	
be 	
an 	
algebraic 	TYPE
space 	
over 	
$ 	
S$. 	

 	
Let 	
$ 	
U$ 	
be 	
a 	
simplicial 	TYPE
algebraic 	
space 	
over 	
$ 	
S$. 	

 	
Let 	
$ 	
a 	
: 	
U 	
\to 	
X$ 	
be 	
an 	
augmentation 	TYPE
. 	
There 	
is 	
a 	
commutative 	
diagram 	

 	
$ 	
$ 	

 	
\xymatrix 	
{ 	

 	
( 	
\Sh((\textit{Spaces}/U)_{fppf 	
, 	
total 	
} 	
) 	
, 	
\mathcal{O}_{big 	
, 	
total 	
} 	
) 	

 	
\ar[r]_-h 	
\ar[d]_{a_{fppf 	
} 	
} 	
& 	

 	
( 	
\Sh(U_\etale 	
) 	
, 	
\mathcal{O}_U 	
) 	
\ar[d]^a 	
\\ 	

 	
( 	
\Sh((\textit{Spaces}/X)_{fppf 	
} 	
) 	
, 	
\mathcal{O}_{big 	
} 	
) 	
\ar[r]^-{h_{-1 	
} 	
} 	
& 	

 	
( 	
\Sh(X_\etale 	
) 	
, 	
\mathcal{O}_X 	
) 	

 	
} 	

 	
$ 	
$ 	

 	
of 	
ringed 	
topoi 	
where 	
the 	
left 	
vertical 	
arrow 	
is 	
defined 	
in 	

 	
Section 	
\ref{section 	
- 	
hypercovering 	
- 	
modules 	
} 	

 	
and 	
the 	
right 	
vertical 	
arrow 	
is 	
defined 	
in 	

 	
Section 	
\ref{section 	
- 	
simplicial 	
- 	
algebraic 	
- 	
spaces}. 	

 	
\end{lemma 	
} 	


 	
\begin{proof 	
} 	

 	
For 	
the 	
underlying 	
diagram 	
of 	
topoi 	
we 	
refer 	
to 	
the 	
discussion 	
in 	

 	
the 	
proof 	
of 	
Lemma 	
\ref{lemma 	
- 	
compare 	
- 	
simplicial 	
- 	
objects 	
- 	
fppf 	
- 	
etale}. 	

 	
The 	
sheaf 	
$ 	
\mathcal{O}_U$ 	
is 	
the 	
structure 	
sheaf 	
of 	
the 	

 	
simplicial 	
algebraic 	
space 	
$ 	
U$ 	
as 	
defined 	
in 	

 	
Section 	
\ref{section 	
- 	
simplicial 	
- 	
algebraic 	
- 	
spaces}. 	

 	
The 	
sheaf 	
$ 	
\mathcal{O}_X$ 	
is 	
the 	
usual 	
structure 	
sheaf 	
of 	
the 	
algebraic 	

 	
space 	
$ 	
X$. 	
The 	
sheaves 	
of 	
rings 	
$ 	
\mathcal{O}_{big 	
, 	
total}$ 	
and 	

 	
$ 	
\mathcal{O}_{big}$ 	
come 	
from 	
the 	
structure 	
sheaf 	
on 	

 	
$ 	
( 	
\textit{Spaces}/S)_{fppf}$ 	
in 	
the 	
manner 	
explained 	
in 	

 	
Section 	
\ref{section 	
- 	
hypercovering 	
- 	
modules 	
} 	

 	
which 	
also 	
constructs 	
$ 	
a_{fppf}$ 	
as 	
a 	
morphism 	
of 	
ringed 	
topoi 	
. 	

 	
The 	
component 	
morphisms 	
$ 	
h_n 	
= 	
a_{U_n}$ 	
and 	
$ 	
h_{-1 	
} 	
= 	
a_X$ 	

 	
are 	
morphisms 	
of 	
ringed 	
topoi 	
by 	

 	
More 	
on 	
Cohomology 	
of 	
Spaces 	
, 	
Section 	

 	
\ref{spaces 	
- 	
more 	
- 	
cohomology 	
- 	
section 	
- 	
fppf 	
- 	
etale 	
- 	
modules}. 	

 	
Finally 	
, 	
since 	
the 	
continuous 	
functor 	

 	
$ 	
u 	
: 	
U_{spaces 	
, 	
\etale 	
} 	
\to 	
( 	
\textit{Spaces}/U)_{fppf 	
, 	
total}$ 	

 	
used 	
to 	
define 	
$ 	
h$\footnote{This 	
happened 	
in 	
the 	
proof 	
of 	

 	
Lemma 	
\ref{lemma 	
- 	
compare 	
- 	
simplicial 	
- 	
objects 	
- 	
fppf 	
- 	
etale 	
} 	

 	
via 	
an 	
application 	
of 	

 	
Lemma 	
\ref{lemma 	
- 	
morphism 	
- 	
augmentation 	
- 	
simplicial 	
- 	
sites}. 	
} 	

 	
is 	
given 	
by 	
$ 	
V 	
/ 	
U_n 	
\mapsto 	
V 	
/ 	
U_n$ 	

 	
we 	
see 	
that 	
$ 	
h_*\mathcal{O}_{big 	
, 	
total 	
} 	
= 	
\mathcal{O}_U$ 	

 	
which 	
is 	
how 	
we 	
endow 	
$ 	
h$ 	
with 	
the 	
structure 	
of 	
a 	
morphism 	

 	
of 	
ringed 	
simplicial 	
sites 	
as 	
in 	

 	
Remark 	
\ref{remark 	
- 	
morphism 	
- 	
simplicial 	
- 	
sites 	
- 	
modules}. 	

 	
Then 	
we 	
obtain 	
$ 	
h$ 	
as 	
a 	
morphism 	
of 	
ringed 	
topoi 	

 	
by 	
Lemma 	
\ref{lemma 	
- 	
morphism 	
- 	
simplicial 	
- 	
sites 	
- 	
modules}. 	

 	
Please 	
observe 	
that 	
the 	
morphisms 	
$ 	
h_n$ 	
indeed 	
agree 	

 	
with 	
the 	
morphisms 	
$ 	
a_{U_n}$ 	
described 	
above 	
. 	

 	
We 	
omit 	
the 	
verification 	

 	
that 	
the 	
diagram 	
is 	
commutative 	
( 	
as 	
a 	
diagram 	
of 	

 	
ringed 	
topoi 	
-- 	
we 	
already 	
know 	
it 	
is 	
commutative 	

 	
as 	
a 	
diagram 	
of 	
topoi 	
) 	
. 	

 	
\end{proof 	
} 	


 	
\begin{lemma 	
} 	

 	
\label{lemma 	
- 	
descent 	
- 	
qcoh 	
- 	
for 	
- 	
fppf 	
- 	
hypercovering 	
} 	

 	
Let 	
$ 	
S$ 	
be 	
a 	
scheme 	TYPE
. 	TYPE
Let 	
$ 	
X$ 	
be 	
an 	
algebraic 	TYPE
space 	
over 	
$ 	
S$. 	

 	
Let 	
$ 	
U$ 	
be 	
a 	
simplicial 	TYPE
algebraic 	
space 	
over 	
$ 	
S$. 	
Let 	
$ 	
a 	
: 	
U 	
\to 	
X$ 	

 	
be 	
an 	
augmentation 	TYPE
. 	
If 	
$ 	
a 	
: 	
U 	
\to 	
X$ 	
is 	
an 	
fppf 	
hypercovering 	
of 	
$ 	
X$ 	
, 	

 	
then 	

 	
$ 	
$ 	

 	
a^ 	
* 	
: 	
\QCoh(\mathcal{O}_X 	
) 	
\to 	
\QCoh(\mathcal{O}_U 	
) 	

 	
$ 	
$ 	

 	
is 	
an 	
equivalence 	
fully 	
faithful 	
with 	
quasi 	
- 	
inverse 	
given 	
by 	
$ 	
a_*$. 	

 	
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 	
- 	
modules}. 	

 	
In 	
the 	
proof 	
of 	
this 	
lemma 	
we 	
have 	
seen 	
that 	

 	
$ 	
h_{-1}$ 	
is 	
the 	
morphism 	
$ 	
a_X$ 	
of 	

 	
More 	
on 	
Cohomology 	
of 	
Spaces 	
, 	
Section 	

 	
\ref{spaces 	
- 	
more 	
- 	
cohomology 	
- 	
section 	
- 	
fppf 	
- 	
etale 	
- 	
modules}. 	

 	
Thus 	
it 	
follows 	
from 	

 	
More 	
on 	
Cohomology 	
of 	
Spaces 	
, 	
Lemma 	

 	
\ref{spaces 	
- 	
more 	
- 	
cohomology 	
- 	
lemma 	
- 	
review 	
- 	
quasi 	
- 	
coherent 	
} 	

 	
that 	

 	
$ 	
$ 	

 	
( 	
h_{-1})^ 	
* 	
: 	

 	
\QCoh(\mathcal{O}_X 	
) 	

 	
\longrightarrow 	

 	
\QCoh(\mathcal{O}_{big 	
} 	
) 	

 	
$ 	
$ 	

 	
is 	
an 	
equivalence 	
with 	
quasi 	
- 	
inverse 	
$ 	
h_{-1 	
, 	
* 	
} 	
$ 	
. 	

 	
The 	
same 	
holds 	
true 	
for 	
the 	
components 	
$ 	
h_n$ 	
of 	
$ 	
h$. 	

 	
Recall 	
that 	
$ 	
\QCoh(\mathcal{O}_U)$ 	
and 	
$ 	
\QCoh(\mathcal{O}_{big 	
, 	
total})$ 	

 	
consist 	
of 	
cartesian 	
modules 	
whose 	
components 	
are 	
quasi 	
- 	
coherent 	
, 	
see 	

 	
Lemma 	
\ref{lemma 	
- 	
quasi 	
- 	
coherent 	
- 	
sheaf}. 	

 	
Since 	
the 	
functors 	
$ 	
h^*$ 	
and 	
$ 	
h_*$ 	
of 	

 	
Lemma 	
\ref{lemma 	
- 	
morphism 	
- 	
simplicial 	
- 	
sites 	
- 	
modules 	
} 	

 	
agree 	
with 	
the 	
functors 	
$ 	
h_n^*$ 	
and 	
$ 	
h_{n 	
, 	
* 	
} 	
$ 	
on 	
components 	

 	
we 	
conclude 	
that 	

 	
$ 	
$ 	

 	
h^ 	
* 	
: 	

 	
\QCoh(\mathcal{O}_U 	
) 	

 	
\longrightarrow 	

 	
\QCoh(\mathcal{O}_{big 	
, 	
total 	
} 	
) 	

 	
$ 	
$ 	

 	
is 	
an 	
equivalence 	
with 	
quasi 	
- 	
inverse 	
$ 	
h_*$. 	

 	
Observe 	
that 	
$ 	
U$ 	
is 	
a 	
hypercovering 	
of 	
$ 	
X$ 	
in 	
$ 	
( 	
\textit{Spaces}/S)_{fppf}$ 	

 	
as 	
defined 	
in 	
Section 	
\ref{section 	
- 	
hypercovering}. 	

 	
By 	
Lemma 	
\ref{lemma 	
- 	
hypercovering 	
- 	
X 	
- 	
simple 	
- 	
descent 	
- 	
modules 	
} 	

 	
we 	
see 	
that 	
$ 	
a_{fppf}^*$ 	
is 	
fully 	
faithful 	
with 	
quasi 	
- 	
inverse 	

 	
$ 	
a_{fppf 	
, 	
* 	
} 	
$ 	
and 	
with 	
essential 	
image 	
the 	
cartesian 	
sheaves 	

 	
of 	
$ 	
\mathcal{O}_{fppf 	
, 	
total}$-modules 	
. 	

 	
Thus 	
, 	
by 	
the 	
description 	
of 	
$ 	
\QCoh(\mathcal{O}_{big})$ 	
and 	

 	
$ 	
\QCoh(\mathcal{O}_{big 	
, 	
total})$ 	
of 	
Lemma 	
\ref{lemma 	
- 	
quasi 	
- 	
coherent 	
- 	
sheaf 	
} 	
, 	

 	
we 	
get 	
an 	
equivalence 	

 	
$ 	
$ 	

 	
a_{fppf}^ 	
* 	
: 	

 	
\QCoh(\mathcal{O}_{big 	
} 	
) 	

 	
\longrightarrow 	

 	
\QCoh(\mathcal{O}_{big 	
, 	
total 	
} 	
) 	

 	
$ 	
$ 	

 	
with 	
quasi 	
- 	
inverse 	
given 	
by 	
$ 	
a_{fppf 	
, 	
* 	
} 	
$ 	
. 	

 	
A 	
formal 	
argument 	
( 	
chasing 	
around 	
the 	
diagram 	
) 	
now 	
shows 	
that 	

 	
$ 	
a^*$ 	
is 	
fully 	
faithful 	
on 	
$ 	
\QCoh(\mathcal{O}_X)$ 	
and 	
has 	

 	
image 	
contained 	
in 	
$ 	
\QCoh(\mathcal{O}_U)$. 	


 	
\medskip\noindent 	

 	
Finally 	
, 	
suppose 	
that 	
$ 	
\mathcal{G}$ 	
is 	
in 	
$ 	
\QCoh(\mathcal{O}_U)$. 	

 	
Then 	
$ 	
h^*\mathcal{G}$ 	
is 	
in 	
$ 	
\QCoh(\mathcal{O}_{big 	
, 	
total})$. 	

 	
Hence 	
$ 	
h^*\mathcal{G 	
} 	
= 	
a_{fppf}^*\mathcal{H}$ 	
with 	

 	
$ 	
\mathcal{H 	
} 	
= 	
a_{fppf 	
, 	
* 	
} 	
h^*\mathcal{G}$ 	

 	
in 	
$ 	
\QCoh(\mathcal{O}_{big})$ 	
( 	
see 	
above 	
) 	
. 	

 	
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 	TYPE
. 	TYPE
Let 	
$ 	
X$ 	
be 	
an 	
algebraic 	TYPE
space 	
over 	
$ 	
S$. 	

 	
Let 	
$ 	
U$ 	
be 	
a 	
simplicial 	TYPE
algebraic 	
space 	
over 	
$ 	
S$. 	
Let 	
$ 	
a 	
: 	
U 	
\to 	
X$ 	

 	
be 	
an 	
augmentation 	TYPE
. 	
If 	
$ 	
a 	
: 	
U 	
\to 	
X$ 	
is 	
an 	
fppf 	
hypercovering 	
of 	
$ 	
X$ 	
, 	

 	
then 	
for 	
$ 	
\mathcal{F}$ 	
a 	
quasi 	
- 	
coherent 	
$ 	
\mathcal{O}_X$-module 	

 	
the 	
map 	

 	
$ 	
$ 	

 	
\mathcal{F 	
} 	
\to 	
Ra_*(a^*\mathcal{F 	
} 	
) 	

 	
$ 	
$ 	

 	
is 	
an 	
isomorphism 	
. 	
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 	
. 	

 	
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 	TYPE
. 	TYPE
Let 	
$ 	
X$ 	
be 	
an 	
algebraic 	TYPE
space 	
over 	
$ 	
S$. 	

 	
Let 	
$ 	
U$ 	
be 	
a 	
simplicial 	TYPE
algebraic 	
space 	
over 	
$ 	
S$. 	
Let 	
$ 	
a 	
: 	
U 	
\to 	
X$ 	

 	
be 	
an 	
augmentation 	TYPE
. 	
Assume 	
$ 	
a 	
: 	
U 	
\to 	
X$ 	
is 	
an 	
fppf 	
hypercovering 	
of 	
$ 	
X$. 	

 	
Then 	
$ 	
\QCoh(\mathcal{O}_U)$ 	
is 	
a 	
weak 	
Serre 	
subcategory 	
of 	

 	
$ 	
\textit{Mod}(\mathcal{O}_U)$ 	
and 	

 	
$ 	
$ 	

 	
a^ 	
* 	
: 	
D_\QCoh(\mathcal{O}_X 	
) 	
\longrightarrow 	
D_\QCoh(\mathcal{O}_U 	
) 	

 	
$ 	
$ 	

 	
is 	
an 	
equivalence 	
of 	
categories 	
with 	
quasi 	
- 	
inverse 	
given 	
by 	

 	
$ 	
Ra_*$. 	
Here 	
$ 	
a 	
: 	
\Sh(U_\etale 	
) 	
\to 	
\Sh(X_\etale)$ 	

 	
is 	
as 	
in 	
Section 	
\ref{section 	
- 	
simplicial 	
- 	
algebraic 	
- 	
spaces}. 	

 	
\end{lemma 	
} 	


 	
\begin{proof 	
} 	

 	
First 	
observe 	
that 	
the 	
maps 	
$ 	
a_n 	
: 	
U_n 	
\to 	
X$ 	
and 	
$ 	
d^n_i 	
: 	
U_n 	
\to 	
U_{n 	
- 	
1}$ 	

 	
are 	
flat 	
, 	
locally 	
of 	
finite 	
presentation 	
, 	
and 	
surjective 	
by 	

 	
Hypercoverings 	
, 	
Remark 	
\ref{hypercovering 	
- 	
remark 	
- 	
P 	
- 	
covering}. 	


 	
\medskip\noindent 	

 	
Recall 	
that 	
an 	
$ 	
\mathcal{O}_U$-module 	
$ 	
\mathcal{F}$ 	
is 	
quasi 	
- 	
coherent 	
if 	
and 	

 	
only 	
if 	
it 	
is 	
cartesian 	
and 	
$ 	
\mathcal{F}_n$ 	
is 	
quasi 	
- 	
coherent 	
for 	
all 	
$ 	
n$. 	

 	
See 	
Lemma 	
\ref{lemma 	
- 	
quasi 	
- 	
coherent 	
- 	
sheaf}. 	

 	
By 	
Lemma 	
\ref{lemma 	
- 	
Serre 	
- 	
subcat 	
- 	
cartesian 	
- 	
modules 	
} 	

 	
( 	
and 	
flatness 	
of 	
the 	
maps 	
$ 	
d^n_i 	
: 	
U_n 	
\to 	
U_{n 	
- 	
1}$ 	
shown 	
above 	
) 	

 	
the 	
cartesian 	
modules 	
for 	
a 	
weak 	
Serre 	
subcategory 	
of 	

 	
$ 	
\textit{Mod}(\mathcal{O}_U)$. 	
On 	
the 	
other 	
hand 	

 	
$ 	
\QCoh(\mathcal{O}_{U_n 	
} 	
) 	
\subset 	
\textit{Mod}(\mathcal{O}_{U_n})$ 	

 	
is 	
a 	
weak 	
Serre 	
subcategory 	
for 	
each 	
$ 	
n$ 	

 	
( 	
Properties 	
of 	
Spaces 	
, 	
Lemma 	

 	
\ref{spaces 	
- 	
properties 	
- 	
lemma 	
- 	
properties 	
- 	
quasi 	
- 	
coherent 	
} 	
) 	
. 	

 	
Combined 	
we 	
see 	
that 	

 	
$ 	
\QCoh(\mathcal{O}_U 	
) 	
\subset 	
\textit{Mod}(\mathcal{O}_U)$ 	

 	
is 	
a 	
weak 	
Serre 	
subcategory 	
. 	


 	
\medskip\noindent 	

 	
To 	
finish 	
the 	
proof 	
we 	
check 	
the 	
conditions 	
( 	
1 	
) 	
-- 	
( 	
5 	
) 	
of 	

 	
Lemma 	
\ref{lemma 	
- 	
equivalence 	
- 	
unbounded 	
- 	
one 	
} 	
one 	
by 	
one 	
. 	


 	
\medskip\noindent 	

 	
Ad 	
( 	
1 	
) 	
. 	
This 	
holds 	
since 	
$ 	
a_n$ 	
flat 	
( 	
seen 	
above 	
) 	
implies 	
$ 	
a$ 	
is 	
flat 	

 	
by 	
Lemma 	
\ref{lemma 	
- 	
flat 	
- 	
augmentation 	
- 	
modules}. 	


 	
\medskip\noindent 	

 	
Ad 	
( 	
2 	
) 	
. 	
This 	
is 	
the 	
content 	
of 	

 	
Lemma 	
\ref{lemma 	
- 	
descent 	
- 	
qcoh 	
- 	
for 	
- 	
fppf 	
- 	
hypercovering}. 	


 	
\medskip\noindent 	

 	
Ad 	
( 	
3 	
) 	
. 	
This 	
is 	
the 	
content 	
of 	

 	
Lemma 	
\ref{lemma 	
- 	
cohomological 	
- 	
descent 	
- 	
qcoh 	
- 	
for 	
- 	
fppf 	
- 	
hypercovering}. 	


 	
\medskip\noindent 	

 	
Ad 	
( 	
4 	
) 	
. 	
Recall 	
that 	
we 	
can 	
use 	
either 	
the 	
site 	
$ 	
U_\etale$ 	
or 	

 	
$ 	
U_{spaces 	
, 	
\etale}$ 	
to 	
define 	
the 	
small 	
\'etale 	
topos 	

 	
$ 	
\Sh(U_\etale)$ 	
, 	
see 	
Section 	
\ref{section 	
- 	
simplicial 	
- 	
algebraic 	
- 	
spaces}. 	

 	
The 	
assumption 	
of 	

 	
Cohomology 	
on 	
Sites 	
, 	
Situation 	
\ref{sites 	
- 	
cohomology 	
- 	
situation 	
- 	
olsson 	
- 	
laszlo 	
} 	

 	
holds 	
for 	
the 	
triple 	

 	
$ 	
( 	
U_{spaces 	
, 	
\etale 	
} 	
, 	
\mathcal{O}_U 	
, 	
\QCoh(\mathcal{O}_U))$ 	

 	
and 	
by 	
the 	
same 	
reasoning 	
for 	
the 	
triple 	

 	
$ 	
( 	
U_\etale 	
, 	
\mathcal{O}_U 	
, 	
\QCoh(\mathcal{O}_U))$. 	

 	
Namely 	
, 	
take 	

 	
$ 	
$ 	

 	
\mathcal{B 	
} 	
\subset 	
\Ob(U_\etale 	
) 	
\subset 	
\Ob(U_{spaces 	
, 	
\etale 	
} 	
) 	

 	
$ 	
$ 	

 	
to 	
be 	
the 	
set 	
of 	
affine 	
objects 	
. 	
For 	
$ 	
V 	
/ 	
U_n 	
\in 	
\mathcal{B}$ 	

 	
take 	
$ 	
d_{V 	
/ 	
U_n 	
} 	
= 	
0 	
$ 	
and 	
take 	
$ 	
\text{Cov}_{V 	
/ 	
U_n}$ 	
to 	
be 	
the 	
set 	
of 	

 	
\'etale 	
coverings 	
$ 	
\{V_i 	
\to 	
V\}$ 	
with 	
$ 	
V_i$ 	
affine 	
. 	

 	
Then 	
we 	
get 	
the 	
desired 	
vanishing 	
because 	
for 	

 	
$ 	
\mathcal{F 	
} 	
\in 	
\QCoh(\mathcal{O}_U)$ 	

 	
and 	
any 	
$ 	
V 	
/ 	
U_n 	
\in 	
\mathcal{B}$ 	
we 	
have 	

 	
$ 	
$ 	

 	
H^p(V 	
/ 	
U_n 	
, 	
\mathcal{F 	
} 	
) 	
= 	
H^p(V 	
, 	
\mathcal{F}_n 	
) 	

 	
$ 	
$ 	

 	
by 	
Lemma 	
\ref{lemma 	
- 	
sanity 	
- 	
check 	
- 	
modules}. 	
Here 	
on 	
the 	

 	
right 	
hand 	
side 	
we 	
have 	
the 	
cohomology 	
of 	
the 	
quasi 	
- 	
coherent 	

 	
sheaf 	
$ 	
\mathcal{F}_n$ 	
on 	
$ 	
U_n$ 	
over 	
the 	
affine 	
obect 	
$ 	
V$ 	

 	
of 	
$ 	
U_{n 	
, 	
\etale}$. 	
This 	
vanishes 	
for 	
$ 	
p 	
> 	
0 	
$ 	
by 	
the 	
discussion 	
in 	

 	
Cohomology 	
of 	
Spaces 	
, 	
Section 	

 	
\ref{spaces 	
- 	
cohomology 	
- 	
section 	
- 	
higher 	
- 	
direct 	
- 	
image 	
} 	
and 	

 	
Cohomology 	
of 	
Schemes 	
, 	
Lemma 	

 	
\ref{coherent 	
- 	
lemma 	
- 	
quasi 	
- 	
coherent 	
- 	
affine 	
- 	
cohomology 	
- 	
zero}. 	


 	
\medskip\noindent 	

 	
Ad 	
( 	
5 	
) 	
. 	
Follows 	
by 	
taking 	
$ 	
\mathcal{B 	
} 	
\subset 	
\Ob(X_{spaces 	
, 	
\etale})$ 	

 	
the 	
set 	
of 	
affine 	
objects 	
and 	
the 	
references 	
given 	
above 	
. 	

 	
\end{proof 	
} 	


 	
\begin{lemma 	
} 	

 	
\label{lemma 	
- 	
compute 	
- 	
via 	
- 	
fppf 	
- 	
hypercovering 	
- 	
modules 	
} 	

 	
Let 	
$ 	
S$ 	
be 	
a 	
scheme 	TYPE
. 	TYPE
Let 	
$ 	
X$ 	
be 	
an 	
algebraic 	TYPE
space 	
over 	
$ 	
S$. 	

 	
Let 	
$ 	
U$ 	
be 	
a 	
simplicial 	TYPE
algebraic 	
space 	
over 	
$ 	
S$. 	
Let 	
$ 	
a 	
: 	
U 	
\to 	
X$ 	

 	
be 	
an 	
augmentation 	TYPE
. 	
If 	
$ 	
a 	
: 	
U 	
\to 	
X$ 	
is 	
an 	
fppf 	
hypercovering 	
of 	
$ 	
X$ 	
, 	
then 	

 	
$ 	
$ 	

 	
R\Gamma(X_\etale 	
, 	
K 	
) 	
= 	
R\Gamma(U_\etale 	
, 	
a^*K 	
) 	

 	
$ 	
$ 	

 	
for 	
$ 	
K 	
\in 	
D_\QCoh(\mathcal{O}_X)$. 	
Here 	
$ 	
a 	
: 	
\Sh(U_\etale 	
) 	
\to 	
\Sh(X_\etale)$ 	

 	
is 	
as 	
in 	
Section 	
\ref{section 	
- 	
simplicial 	
- 	
algebraic 	
- 	
spaces}. 	

 	
\end{lemma 	
} 	


 	
\begin{proof 	
} 	

 	
This 	
follows 	
from 	

 	
Lemma 	
\ref{lemma 	
- 	
coh 	
- 	
descent 	
- 	
qcoh 	
- 	
unbounded 	
- 	
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 	
- 	
spectral 	
- 	
sequence 	
- 	
fppf 	
- 	
hypercovering 	
- 	
modules 	
} 	

 	
Let 	
$ 	
S$ 	
be 	
a 	
scheme 	TYPE
. 	TYPE
Let 	
$ 	
X$ 	
be 	
an 	
algebraic 	TYPE
space 	
over 	
$ 	
S$. 	

 	
Let 	
$ 	
U$ 	
be 	
a 	
simplicial 	TYPE
algebraic 	
space 	
over 	
$ 	
S$. 	
Let 	
$ 	
a 	
: 	
U 	
\to 	
X$ 	

 	
be 	
an 	
augmentation 	TYPE
. 	
Let 	
$ 	
\mathcal{F}$ 	
be 	
quasi 	
- 	
coherent 	

 	
$ 	
\mathcal{O}_X$-module 	
. 	
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 	

 	
In 	
this 	
section 	
we 	
pull 	
some 	
of 	
the 	
previously 	
shown 	

 	
results 	
together 	
for 	
fppf 	
coverings 	
of 	
algebraic 	
spaces 	

 	
and 	
derived 	
categories 	
of 	
quasi 	
- 	
coherent 	
modules 	
. 	


 	
\begin{lemma 	
} 	

 	
\label{lemma 	
- 	
fppf 	
- 	
neg 	
- 	
ext 	
- 	
zero 	
- 	
hom 	
} 	

 	
Let 	
$ 	
X$ 	
be 	
an 	
algebraic 	TYPE
space 	
over 	
a 	
scheme 	
$ 	
S$. 	

 	
Let 	
$ 	
K 	
, 	
E 	
\in 	
D_\QCoh(\mathcal{O}_X)$. 	

 	
Let 	
$ 	
a 	
: 	
U 	
\to 	
X$ 	
be 	
an 	
fppf 	
hypercovering 	
. 	

 	
Assume 	
that 	
for 	
all 	
$ 	
n 	
\geq 	
0 	
$ 	
we 	
have 	

 	
$ 	
$ 	

 	
\Ext_{\mathcal{O}_{U_n}}^i(La_n^*K 	
, 	
La_n^*E 	
) 	
= 	
0 	

 	
\text 	
{ 	
for 	
} 	
i 	
< 	
0 	

 	
$ 	
$ 	

 	
Then 	
we 	
have 	

 	
\begin{enumerate 	
} 	

 	
\item 	
$ 	
\Ext_{\mathcal{O}_X}^i(K 	
, 	
E 	
) 	
= 	
0 	
$ 	
for 	
$ 	
i 	
< 	
0 	
$ 	
, 	
and 	

 	
\item 	
there 	
is 	
an 	
exact 	
sequence 	

 	
$ 	
$ 	

 	
0 	

 	
\to 	

 	
\Hom_{\mathcal{O}_X}(K 	
, 	
E 	
) 	

 	
\to 	

 	
\Hom_{\mathcal{O}_{U_0}}(La_0^*K 	
, 	
La_0^*E 	
) 	

 	
\to 	

 	
\Hom_{\mathcal{O}_{U_1}}(La_1^*K 	
, 	
La_1^*E 	
) 	

 	
$ 	
$ 	

 	
\end{enumerate 	
} 	

 	
\end{lemma 	
} 	


 	
\begin{proof 	
} 	

 	
Write 	
$ 	
K_n 	
= 	
La_n^*K$ 	
and 	
$ 	
E_n 	
= 	
La_n^*E$. 	
Then 	
these 	
are 	
the 	

 	
simplicial 	
systems 	
of 	
the 	
derived 	
category 	
of 	
modules 	

 	
( 	
Definition 	
\ref{definition 	
- 	
cartesian 	
- 	
derived 	
- 	
modules 	
} 	
) 	

 	
associated 	
to 	
$ 	
La^*K$ 	
and 	
$ 	
La^*E$ 	

 	
( 	
Lemma 	
\ref{lemma 	
- 	
cartesian 	
- 	
objects 	
- 	
derived 	
- 	
modules 	
} 	
) 	

 	
where 	
$ 	
a 	
: 	
U_\etale 	
\to 	
X_\etale$ 	
is 	
as 	
in 	

 	
Section 	
\ref{section 	
- 	
simplicial 	
- 	
algebraic 	
- 	
spaces}. 	

 	
Let 	
us 	
prove 	
( 	
2 	
) 	
first 	
. 	
By 	

 	
Lemma 	
\ref{lemma 	
- 	
coh 	
- 	
descent 	
- 	
qcoh 	
- 	
unbounded 	
- 	
for 	
- 	
fppf 	
- 	
hypercovering 	
} 	

 	
we 	
have 	

 	
$ 	
$ 	

 	
\Hom_{\mathcal{O}_X}(K 	
, 	
E 	
) 	
= 	

 	
\Hom_{\mathcal{O}_U}(La^*K 	
, 	
La^*E 	
) 	

 	
$ 	
$ 	

 	
Thus 	
the 	
sequence 	
looks 	
like 	
this 	
: 	

 	
$ 	
$ 	

 	
0 	

 	
\to 	

 	
\Hom_{\mathcal{O}_U}(La^*K 	
, 	
La^*E 	
) 	

 	
\to 	

 	
\Hom_{\mathcal{O}_{U_0}}(K_0 	
, 	
E_0 	
) 	

 	
\to 	

 	
\Hom_{\mathcal{O}_{U_1}}(K_1 	
, 	
E_1 	
) 	

 	
$ 	
$ 	

 	
The 	
first 	
arrow 	
  	
is 	
injective 	
by 	

 	
Lemma 	
\ref{lemma 	
- 	
nullity 	
- 	
cartesian 	
- 	
modules 	
- 	
derived}. 	

 	
The 	
image 	
of 	
this 	
arrow 	
is 	
the 	
kernel 	
of 	
the 	
second 	

 	
by 	
Lemma 	
\ref{lemma 	
- 	
hom 	
- 	
cartesian 	
- 	
modules 	
- 	
derived}. 	

 	
This 	
finishes 	
the 	
proof 	
of 	
( 	
2 	
) 	
. 	

 	
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 	TYPE
space 	
over 	
a 	
scheme 	
$ 	
S$. 	

 	
Let 	
$ 	
a 	
: 	
U 	
\to 	
X$ 	
be 	
an 	
fppf 	
hypercovering 	
. 	

 	
Suppose 	
given 	
$ 	
K_0 	
\in 	
D_\QCoh(U_0)$ 	
and 	
an 	
isomorphism 	

 	
$ 	
$ 	

 	
\alpha 	
: 	

 	
L(f_{\delta_1 	
^ 	
1})^*K_0 	

 	
\longrightarrow 	

 	
L(f_{\delta_0 	
^ 	
1})^*K_0 	

 	
$ 	
$ 	

 	
satisfying 	
the 	
cocycle 	
condition 	
on 	
$ 	
U_1$. 	
Set 	

 	
$ 	
\tau^n_i 	
: 	
[ 	
0 	
] 	
\to 	
[ 	
n]$ 	
, 	
$ 	
0 	
\mapsto 	
i$ 	
and 	

 	
set 	
$ 	
K_n 	
= 	
Lf_{\tau^n_n}^*K_0$. 	

 	
Assume 	
$ 	
\Ext^i_{\mathcal{O}_{U_n}}(K_n 	
, 	
K_n 	
) 	
= 	
0 	
$ 	
for 	
$ 	
i 	
< 	
0$. 	

 	
Then 	
there 	
exists 	
an 	
object 	
$ 	
K 	
\in 	
D_\QCoh(\mathcal{O}_X)$ 	

 	
and 	
an 	
isomorphism 	
$ 	
La_0^*K 	
\to 	
K$ 	
compatible 	
with 	
$ 	
\alpha$. 	

 	
\end{lemma 	
} 	


 	
\begin{proof 	
} 	

 	
We 	
claim 	
that 	
the 	
objects 	
$ 	
K_n$ 	
form 	
the 	
members 	
of 	
a 	

 	
simplicial 	
system 	
of 	
the 	
derived 	
category 	
of 	
modules 	

 	
( 	
Definition 	
\ref{definition 	
- 	
cartesian 	
- 	
derived 	
- 	
modules 	
} 	
) 	

 	
of 	
the 	
ringed 	
simplicial 	
site 	
$ 	
U_\etale$ 	
of 	

 	
Section 	
\ref{section 	
- 	
simplicial 	
- 	
algebraic 	
- 	
spaces}. 	

 	
The 	
construction 	
is 	
analogous 	
to 	
the 	
construction 	
discussed 	
in 	

 	
Descent 	
, 	
Section 	
\ref{descent 	
- 	
section 	
- 	
descent 	
- 	
modules 	
} 	
from 	
which 	
we 	
borrow 	

 	
the 	
notation 	
$ 	
\tau^n_i 	
: 	
[ 	
0 	
] 	
\to 	
[ 	
n]$ 	
, 	
$ 	
0 	
\mapsto 	
i$ 	
and 	

 	
$ 	
\tau^n_{ij 	
} 	
: 	
[ 	
1 	
] 	
\to 	
[ 	
n]$ 	
, 	
$ 	
0 	
\mapsto 	
i$ 	
, 	
$ 	
1 	
\mapsto 	
j$. 	

 	
Given 	
$ 	
\varphi 	
: 	
[ 	
n 	
] 	
\to 	
[ 	
m]$ 	
we 	
define 	

 	
$ 	
K_\varphi 	
: 	
L(f_\varphi)^*K_n 	
\to 	
K_m$ 	

 	
using 	

 	
$ 	
$ 	

 	
\xymatrix 	
{ 	

 	
L(f_\varphi)^*K_n 	
\ar@{=}[r 	
] 	
& 	

 	
L(f_\varphi)^ 	
* 	
L(f_{\tau^n_n})^*K_0 	
\ar@{=}[r 	
] 	
& 	

 	
L(f_{\tau^m_{\varphi(n)}})^*K_0 	
\ar@{=}[r 	
] 	
& 	

 	
L(f_{\tau^m_{\varphi(n)m}})^ 	
* 	
L(f_{\delta^1_1})^*K_0 	

 	
\ar[d]_{L(f_{\tau^m_{\varphi(n)m}})^*\alpha 	
} 	
\\ 	

 	
& 	

 	
K_m 	
\ar@{=}[r 	
] 	
& 	

 	
L(f_{\tau^m_m})^*K_0 	
\ar@{=}[r 	
] 	
& 	

 	
L(f_{\tau^m_{\varphi(n)m}})^ 	
* 	
L(f_{\delta^1_0})^*K_0 	

 	
} 	

 	
$ 	
$ 	

 	
We 	
omit 	
the 	
verification 	
that 	
the 	
cocycle 	
condition 	

 	
implies 	
the 	
maps 	
compose 	
correctly 	
( 	
in 	
their 	
respective 	

 	
derived 	
categories 	
) 	
and 	
hence 	
give 	
rise 	
to 	
a 	

 	
simplicial 	
systems 	
of 	
the 	
derived 	
category 	
of 	
modules\footnote{This 	

 	
verification 	
is 	
the 	
same 	
as 	
that 	
done 	
in 	
the 	
proof 	

 	
of 	
Lemma 	
\ref{lemma 	
- 	
characterize 	
- 	
cartesian 	
} 	

 	
as 	
well 	
as 	
in 	
the 	
chapter 	
on 	
descent 	
referenced 	

 	
above 	
. 	
We 	
should 	
probably 	
write 	
this 	
as 	
a 	
general 	
lemma 	
about 	

 	
fibred 	
and 	
cofibred 	
categories 	
over 	
$ 	
\Delta$.}. 	

 	
Once 	
this 	
is 	
verified 	
, 	
we 	
obtain 	
an 	
object 	

 	
$ 	
K 	
' 	
\in 	
D_\QCoh(\mathcal{O}_{U_\etale})$ 	

 	
such 	
that 	
$ 	
( 	
K_n 	
, 	
K_\varphi)$ 	
is 	
the 	
system 	
deduced 	
from 	
$ 	
K'$ 	
, 	
see 	

 	
Lemma 	
\ref{lemma 	
- 	
cartesian 	
- 	
module 	
- 	
derived 	
- 	
from 	
- 	
simplicial}. 	

 	
Finally 	
, 	
we 	
apply 	

 	
Lemma 	
\ref{lemma 	
- 	
coh 	
- 	
descent 	
- 	
qcoh 	
- 	
unbounded 	
- 	
for 	
- 	
fppf 	
- 	
hypercovering 	
} 	

 	
to 	
see 	
that 	
$ 	
K 	
' 	
= 	
La^*K$ 	
for 	
some 	
$ 	
K 	
\in 	
D_\QCoh(\mathcal{O}_X)$ 	

 	
as 	
desired 	
. 	

 	
\end{proof 	
} 	












 	
\section{Proper 	
hypercoverings 	
of 	
algebraic 	
spaces 	
} 	

 	
\label{section 	
- 	
proper 	
- 	
hypercovering 	
- 	
spaces 	
} 	


 	
\noindent 	

 	
This 	
section 	
is 	
the 	
analogue 	
of 	
Section 	
\ref{section 	
- 	
proper 	
- 	
hypercovering 	
} 	

 	
for 	
the 	
case 	
of 	
algebraic 	
spaces 	
. 	

 	
The 	
reader 	
who 	
wishes 	
to 	
do 	
so 	
, 	
can 	
replace 	
`` 	
algebraic 	
space 	
'' 	

 	
everywhere 	
with 	
`` 	
scheme 	
'' 	
and 	
get 	
equally 	
valid 	
results 	
. 	

 	
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 	TYPE
space 	
over 	
$ 	
S$ 	
and 	
let 	
$ 	
U$ 	
be 	
a 	
simplicial 	TYPE

 	
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 	
. 	

 	
$ 	
$ 	

 	
\fbox{Principle 	
: 	
Proper 	
hypercoverings 	
can 	
be 	

 	
used 	
to 	
compute 	
\'etale 	
cohomology 	
. 	
} 	

 	
$ 	
$ 	

 	
The 	
key 	
idea 	
behind 	
the 	
proof 	
of 	
the 	
principle 	
is 	
to 	
compare 	
the 	

 	
ph 	
and 	
\'etale 	
topologies 	
on 	
the 	
category 	
$ 	
\textit{Spaces}/S$. 	

 	
Namely 	
, 	
the 	
ph 	
topology 	
is 	
stronger 	
than 	
the 	
\'etale 	
topology 	
and 	
we 	
have 	

 	
( 	
a 	
) 	
a 	
proper 	
surjective 	
map 	
defines 	
a 	
ph 	
covering 	
, 	
and 	

 	
( 	
b 	
) 	
ph 	
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 	
- 	
ph 	
- 	
etale}. 	


 	
\medskip\noindent 	

 	
All 	
results 	
in 	
this 	
section 	
generalize 	
to 	
the 	
case 	

 	
where 	
$ 	
U 	
\to 	
X$ 	
is 	
merely 	
a 	
`` 	
ph 	
hypercovering 	
'' 	
, 	
meaning 	
a 	

 	
hypercovering 	
of 	
$ 	
X$ 	
in 	
the 	
site 	
$ 	
( 	
\textit{Spaces}/S)_{ph}$ 	

 	
as 	
defined 	
in 	
Section 	
\ref{section 	
- 	
hypercovering}. 	
If 	
we 	
ever 	
need 	

 	
this 	
, 	
we 	
will 	
precisely 	
formulate 	
and 	
prove 	
this 	
here 	
. 	


 	
\begin{lemma 	
} 	

 	
\label{lemma 	
- 	
compare 	
- 	
simplicial 	
- 	
objects 	
- 	
ph 	
- 	
etale 	
} 	

 	
Let 	
$ 	
S$ 	
be 	
a 	
scheme 	TYPE
. 	TYPE
Let 	
$ 	
X$ 	
be 	
an 	
algebraic 	TYPE
space 	
over 	
$ 	
S$. 	

 	
Let 	
$ 	
U$ 	
be 	
a 	
simplicial 	TYPE
algebraic 	
space 	
over 	
$ 	
S$. 	

 	
Let 	
$ 	
a 	
: 	
U 	
\to 	
X$ 	
be 	
an 	
augmentation 	TYPE
. 	
There 	
is 	
a 	
commutative 	
diagram 	

 	
$ 	
$ 	

 	
\xymatrix 	
{ 	

 	
\Sh((\textit{Spaces}/U)_{ph 	
, 	
total 	
} 	
) 	
\ar[r]_-h 	
\ar[d]_{a_{ph 	
} 	
} 	
& 	

 	
\Sh(U_\etale 	
) 	
\ar[d]^a 	
\\ 	

 	
\Sh((\textit{Spaces}/X)_{ph 	
} 	
) 	
\ar[r]^-{h_{-1 	
} 	
} 	
& 	

 	
\Sh(X_\etale 	
) 	

 	
} 	

 	
$ 	
$ 	

 	
where 	
the 	
left 	
vertical 	
arrow 	
is 	
defined 	
in 	

 	
Section 	
\ref{section 	
- 	
hypercovering 	
} 	

 	
and 	
the 	
right 	
vertical 	
arrow 	
is 	
defined 	
in 	

 	
Section 	
\ref{section 	
- 	
simplicial 	
- 	
algebraic 	
- 	
spaces}. 	

 	
\end{lemma 	
} 	


 	
\begin{proof 	
} 	

 	
The 	
notation 	
$ 	
( 	
\textit{Spaces}/U)_{ph 	
, 	
total}$ 	
indicates 	
that 	

 	
we 	
are 	
using 	
the 	
construction 	
of 	

 	
Section 	
\ref{section 	
- 	
hypercovering 	
} 	

 	
for 	
the 	
site 	
$ 	
( 	
\textit{Spaces}/S)_{ph}$ 	
and 	
the 	

 	
simplicial 	
object 	
$ 	
U$ 	
of 	
this 	
site\footnote{To 	
distinguish 	
from 	

 	
$ 	
( 	
\textit{Spaces}/U)_{fppf 	
, 	
total}$ 	
defined 	
using 	
the 	
fppf 	

 	
topology 	
in 	
Section 	
\ref{section 	
- 	
fppf 	
- 	
hypercovering}.}. 	

 	
We 	
will 	
use 	
the 	
sites 	
$ 	
X_{spaces 	
, 	
\etale}$ 	
and 	
$ 	
U_{spaces 	
, 	
\etale}$ 	

 	
for 	
the 	
topoi 	
on 	
the 	
right 	
hand 	
side 	
; 	
this 	
is 	
permissible 	

 	
see 	
discussion 	
in 	
Section 	
\ref{section 	
- 	
simplicial 	
- 	
algebraic 	
- 	
spaces}. 	


 	
\medskip\noindent 	

 	
Observe 	
that 	
both 	
$ 	
( 	
\textit{Spaces}/U)_{ph 	
, 	
total}$ 	
and 	

 	
$ 	
U_{spaces 	
, 	
\etale}$ 	

 	
fall 	
into 	
case 	
A 	
of 	
Situation 	
\ref{situation 	
- 	
simplicial 	
- 	
site}. 	

 	
This 	
is 	
immediate 	
from 	
the 	
construction 	
of 	

 	
$ 	
U_\etale$ 	
in 	
Section 	
\ref{section 	
- 	
simplicial 	
- 	
algebraic 	
- 	
spaces 	
} 	

 	
and 	
it 	
follows 	
from 	
Lemma 	
\ref{lemma 	
- 	
sr 	
- 	
when 	
- 	
fibre 	
- 	
products 	
} 	

 	
for 	
$ 	
( 	
\textit{Spaces}/U)_{ph 	
, 	
total}$. 	

 	
Next 	
, 	
consider 	
the 	
functors 	

 	
$ 	
U_{n 	
, 	
spaces 	
, 	
\etale 	
} 	
\to 	
( 	
\textit{Spaces}/U_n)_{ph}$ 	
, 	
$ 	
U 	
\mapsto 	
U 	
/ 	
U_n$ 	

 	
and 	

 	
$ 	
X_{spaces 	
, 	
\etale 	
} 	
\to 	
( 	
\textit{Spaces}/X)_{ph}$ 	
, 	
$ 	
U 	
\mapsto 	
U 	
/ 	
X$. 	

 	
We 	
have 	
seen 	
that 	
these 	
define 	
morphisms 	
of 	
sites 	
in 	

 	
More 	
on 	
Cohomology 	
of 	
Spaces 	
, 	
Section 	

 	
\ref{spaces 	
- 	
more 	
- 	
cohomology 	
- 	
section 	
- 	
ph 	
- 	
etale 	
} 	

 	
where 	
these 	
were 	
denoted 	
$ 	
a_{U_n 	
} 	
= 	
\epsilon_{U_n 	
} 	
\circ 	
\pi_{u_n}$ 	

 	
and 	
$ 	
a_X 	
= 	
\epsilon_X 	
\circ 	
\pi_X$. 	

 	
Thus 	
we 	
obtain 	
a 	
morphism 	
of 	
simplicial 	
sites 	
compatible 	
with 	

 	
augmentations 	
as 	
in 	
Remark 	
\ref{remark 	
- 	
morphism 	
- 	
augmentation 	
- 	
simplicial 	
- 	
sites 	
} 	

 	
and 	
we 	
may 	
apply 	

 	
Lemma 	
\ref{lemma 	
- 	
morphism 	
- 	
augmentation 	
- 	
simplicial 	
- 	
sites 	
} 	
to 	
conclude 	
. 	

 	
\end{proof 	
} 	


 	
\begin{lemma 	
} 	

 	
\label{lemma 	
- 	
descent 	
- 	
sheaves 	
- 	
for 	
- 	
ph 	
- 	
hypercovering 	
} 	

 	
Let 	
$ 	
S$ 	
be 	
a 	
scheme 	TYPE
. 	TYPE
Let 	
$ 	
X$ 	
be 	
an 	
algebraic 	TYPE
space 	
over 	
$ 	
S$. 	

 	
Let 	
$ 	
U$ 	
be 	
a 	
simplicial 	TYPE
algebraic 	
space 	
over 	
$ 	
S$. 	
Let 	
$ 	
a 	
: 	
U 	
\to 	
X$ 	

 	
be 	
an 	
augmentation 	TYPE
. 	
If 	
$ 	
a 	
: 	
U 	
\to 	
X$ 	
is 	
a 	
proper 	
hypercovering 	
of 	
$ 	
X$ 	
, 	

 	
then 	

 	
$ 	
$ 	

 	
a^{-1 	
} 	
: 	
\Sh(X_\etale 	
) 	
\to 	
\Sh(U_\etale 	
) 	

 	
\quad\text{and}\quad 	

 	
a^{-1 	
} 	
: 	
\textit{Ab}(X_\etale 	
) 	
\to 	
\textit{Ab}(U_\etale 	
) 	

 	
$ 	
$ 	

 	
are 	
fully 	
faithful 	
with 	
essential 	
image 	
the 	
cartesian 	
sheaves 	
and 	

 	
quasi 	
- 	
inverse 	
given 	
by 	
$ 	
a_*$. 	
Here 	
$ 	
a 	
: 	
\Sh(U_\etale 	
) 	
\to 	
\Sh(X_\etale)$ 	

 	
is 	
as 	
in 	
Section 	
\ref{section 	
- 	
simplicial 	
- 	
algebraic 	
- 	
spaces}. 	

 	
\end{lemma 	
} 	


 	
\begin{proof 	
} 	

 	
We 	
will 	
prove 	
the 	
statement 	
for 	
sheaves 	
of 	
sets 	
. 	
It 	
will 	
be 	
an 	

 	TYPE
almost 	
formal 	
consequence 	
of 	
results 	
already 	
established 	
. 	

 	
Consider 	
the 	
diagram 	
of 	

 	
Lemma 	
\ref{lemma 	
- 	
compare 	
- 	
simplicial 	
- 	
objects 	
- 	
ph 	
- 	
etale}. 	

 	
In 	
the 	
proof 	
of 	
this 	
lemma 	
we 	
have 	
seen 	
that 	

 	
$ 	
h_{-1}$ 	
is 	
the 	
morphism 	
$ 	
a_X$ 	
of 	

 	
More 	
on 	
Cohomology 	
of 	
Spaces 	
, 	
Section 	

 	
\ref{spaces 	
- 	
more 	
- 	
cohomology 	
- 	
section 	
- 	
ph 	
- 	
etale}. 	

 	
Thus 	
it 	
follows 	
from 	

 	
More 	
on 	
Cohomology 	
of 	
Spaces 	
, 	
Lemma 	

 	
\ref{spaces 	
- 	
more 	
- 	
cohomology 	
- 	
lemma 	
- 	
comparison 	
- 	
ph 	
- 	
etale 	
} 	

 	
that 	
$ 	
( 	
h_{-1})^{-1}$ 	
is 	
fully 	
faithful 	
with 	
quasi 	
- 	
inverse 	
$ 	
h_{-1 	
, 	
* 	
} 	
$ 	
. 	

 	
The 	
same 	
holds 	
true 	
for 	
the 	
components 	
$ 	
h_n$ 	
of 	
$ 	
h$. 	

 	
By 	
the 	
description 	
of 	
the 	
functors 	
$ 	
h^{-1}$ 	
and 	
$ 	
h_*$ 	
of 	

 	
Lemma 	
\ref{lemma 	
- 	
morphism 	
- 	
simplicial 	
- 	
sites 	
} 	

 	
we 	
conclude 	
that 	
$ 	
h^{-1}$ 	
is 	
fully 	
faithful 	
with 	
quasi 	
- 	
inverse 	
$ 	
h_*$. 	

 	
Observe 	
that 	
$ 	
U$ 	
is 	
a 	
hypercovering 	
of 	
$ 	
X$ 	
in 	
$ 	
( 	
\textit{Spaces}/S)_{ph}$ 	

 	
as 	
defined 	
in 	
Section 	
\ref{section 	
- 	
hypercovering 	
} 	
since 	
a 	
surjective 	

 	
proper 	
morphism 	
gives 	
a 	
ph 	
covering 	
by 	
Topologies 	
on 	
Spaces 	
, 	
Lemma 	

 	
\ref{spaces 	
- 	
topologies 	
- 	
lemma 	
- 	
surjective 	
- 	
proper 	
- 	
ph}. 	

 	
By 	
Lemma 	
\ref{lemma 	
- 	
hypercovering 	
- 	
X 	
- 	
simple 	
- 	
descent 	
- 	
sheaves 	
} 	

 	
we 	
see 	
that 	
$ 	
a_{ph}^{-1}$ 	
is 	
fully 	
faithful 	
with 	
quasi 	
- 	
inverse 	

 	
$ 	
a_{ph 	
, 	
* 	
} 	
$ 	
and 	
with 	
essential 	
image 	
the 	
cartesian 	
sheaves 	

 	
on 	
$ 	
( 	
\textit{Spaces}/U)_{ph 	
, 	
total}$. 	

 	
A 	
formal 	
argument 	
( 	
chasing 	
around 	
the 	
diagram 	
) 	
now 	
shows 	
that 	

 	
$ 	
a^{-1}$ 	
is 	
fully 	
faithful 	
. 	


 	
\medskip\noindent 	

 	
Finally 	
, 	
suppose 	
that 	
$ 	
\mathcal{G}$ 	
is 	
a 	
cartesian 	
sheaf 	
on 	
$ 	
U_\etale$. 	

 	
Then 	
$ 	
h^{-1}\mathcal{G}$ 	
is 	
a 	
cartesian 	
sheaf 	
on 	

 	
$ 	
( 	
\textit{Spaces}/U)_{ph 	
, 	
total}$. 	

 	
Hence 	
$ 	
h^{-1}\mathcal{G 	
} 	
= 	
a_{ph}^{-1}\mathcal{H}$ 	
for 	
some 	
sheaf 	

 	
$ 	
\mathcal{H}$ 	
on 	
$ 	
( 	
\textit{Spaces}/X)_{ph}$. 	

 	
We 	
compute 	
using 	
somewhat 	
pedantic 	
notation 	

 	
\begin{align 	
* 	
} 	

 	
( 	
h_{-1})^{-1}(a_*\mathcal{G 	
} 	
) 	

 	
& 	
= 	

 	
( 	
h_{-1})^{-1 	
} 	

 	
\text{Eq 	
} 	
( 	

 	
\xymatrix 	
{ 	

 	
a_{0 	
, 	
small 	
, 	
* 	
} 	
\mathcal{G}_0 	

 	
\ar@<1ex>[r 	
] 	
\ar@<-1ex>[r 	
] 	
& 	

 	
a_{1 	
, 	
small 	
, 	
* 	
} 	
\mathcal{G}_1 	

 	
} 	

 	
) 	
\\ 	

 	
& 	
= 	

 	
\text{Eq 	
} 	
( 	

 	
\xymatrix 	
{ 	

 	
( 	
h_{-1})^{-1}a_{0 	
, 	
small 	
, 	
* 	
} 	
\mathcal{G}_0 	

 	
\ar@<1ex>[r 	
] 	
\ar@<-1ex>[r 	
] 	
& 	

 	
( 	
h_{-1})^{-1}a_{1 	
, 	
small 	
, 	
* 	
} 	
\mathcal{G}_1 	

 	
} 	

 	
) 	
\\ 	

 	
& 	
= 	

 	
\text{Eq 	
} 	
( 	

 	
\xymatrix 	
{ 	

 	
a_{0 	
, 	
big 	
, 	
ph 	
, 	
* 	
} 	
h_0^{-1}\mathcal{G}_0 	

 	
\ar@<1ex>[r 	
] 	
\ar@<-1ex>[r 	
] 	
& 	

 	
a_{1 	
, 	
big 	
, 	
ph 	
, 	
* 	
} 	
h_1^{-1}\mathcal{G}_1 	

 	
} 	

 	
) 	
\\ 	

 	
& 	
= 	

 	
\text{Eq 	
} 	
( 	

 	
\xymatrix 	
{ 	

 	
a_{0 	
, 	
big 	
, 	
ph 	
, 	
* 	
} 	
( 	
a_{0 	
, 	
big 	
, 	
ph})^{-1}\mathcal{H 	
} 	

 	
\ar@<1ex>[r 	
] 	
\ar@<-1ex>[r 	
] 	
& 	

 	
a_{1 	
, 	
big 	
, 	
ph 	
, 	
* 	
} 	
( 	
a_{1 	
, 	
big 	
, 	
ph})^{-1}\mathcal{H 	
} 	

 	
} 	

 	
) 	
\\ 	

 	
& 	
= 	

 	
a_{ph 	
, 	
* 	
} 	
a_{ph}^{-1}\mathcal{H 	
} 	
\\ 	

 	
& 	
= 	

 	
\mathcal{H 	
} 	

 	
\end{align 	
* 	
} 	

 	
Here 	
the 	
first 	
equality 	
follows 	
from 	
Lemma 	
\ref{lemma 	
- 	
augmentation 	
- 	
site 	
} 	
, 	

 	
the 	
second 	
equality 	
follows 	
as 	
$ 	
( 	
h_{-1})^{-1}$ 	
is 	
an 	
exact 	
functor 	
, 	

 	
the 	
third 	
equality 	
follows 	
from 	

 	
More 	
on 	
Cohomology 	
of 	
Spaces 	
, 	
Lemma 	

 	
\ref{spaces 	
- 	
more 	
- 	
cohomology 	
- 	
lemma 	
- 	
proper 	
- 	
push 	
- 	
pull 	
- 	
ph 	
- 	
etale 	
} 	

 	
( 	
here 	
we 	
use 	
that 	
$ 	
a_0 	
: 	
U_0 	
\to 	
X$ 	
and 	
$ 	
a_1 	
: 	
U_1 	
\to 	
X$ 	
are 	
proper 	
) 	
, 	

 	
the 	
fourth 	
follows 	
from 	
$ 	
a_{ph}^{-1}\mathcal{H 	
} 	
= 	
h^{-1}\mathcal{G}$ 	
, 	

 	
the 	
fifth 	
from 	
Lemma 	
\ref{lemma 	
- 	
augmentation 	
- 	
site 	
} 	
, 	
and 	
the 	

 	
sixth 	
we 	
've 	
seen 	
above 	
. 	
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 	TYPE
. 	TYPE
Let 	
$ 	
X$ 	
be 	
an 	
algebraic 	TYPE
space 	
over 	
$ 	
S$. 	

 	
Let 	
$ 	
U$ 	
be 	
a 	
simplicial 	TYPE
algebraic 	
space 	
over 	
$ 	
S$. 	
Let 	
$ 	
a 	
: 	
U 	
\to 	
X$ 	

 	
be 	
an 	
augmentation 	TYPE
. 	
If 	
$ 	
a 	
: 	
U 	
\to 	
X$ 	
is 	
a 	
proper 	
hypercovering 	
of 	
$ 	
X$ 	
, 	

 	
then 	
for 	
$ 	
K 	
\in 	
D^+(X_\etale)$ 	

 	
$ 	
$ 	

 	
K 	
\to 	
Ra_*(a^{-1}K 	
) 	

 	
$ 	
$ 	

 	
is 	
an 	
isomorphism 	
. 	
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 	TYPE
. 	TYPE
Let 	
$ 	
X$ 	
be 	
an 	
algebraic 	TYPE
space 	
over 	
$ 	
S$. 	

 	
Let 	
$ 	
U$ 	
be 	
a 	
simplicial 	TYPE
algebraic 	
space 	
over 	
$ 	
S$. 	
Let 	
$ 	
a 	
: 	
U 	
\to 	
X$ 	

 	
be 	
an 	
augmentation 	TYPE
. 	
If 	
$ 	
a 	
: 	
U 	
\to 	
X$ 	
is 	
a 	
proper 	
hypercovering 	
of 	
$ 	
X$ 	
, 	
then 	

 	
$ 	
$ 	

 	
R\Gamma(X_\etale 	
, 	
K 	
) 	
= 	
R\Gamma(U_\etale 	
, 	
a^{-1}K 	
) 	

 	
$ 	
$ 	

 	
for 	
$ 	
K 	
\in 	
D^+(X_\etale)$. 	
Here 	
$ 	
a 	
: 	
\Sh(U_\etale 	
) 	
\to 	
\Sh(X_\etale)$ 	

 	
is 	
as 	
in 	
Section 	
\ref{section 	
- 	
simplicial 	
- 	
algebraic 	
- 	
spaces}. 	

 	
\end{lemma 	
} 	


 	
\begin{proof 	
} 	

 	
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 	TYPE
. 	TYPE
Let 	
$ 	
X$ 	
be 	
an 	
algebraic 	TYPE
space 	
over 	
$ 	
S$. 	

 	
Let 	
$ 	
U$ 	
be 	
a 	
simplicial 	TYPE
algebraic 	
space 	
over 	
$ 	
S$. 	
Let 	
$ 	
a 	
: 	
U 	
\to 	
X$ 	

 	
be 	
an 	
augmentation 	TYPE
. 	

 	
Let 	
$ 	
\mathcal{A 	
} 	
\subset 	
\textit{Ab}(U_\etale)$ 	

 	
denote 	
the 	
weak 	
Serre 	
subcategory 	
of 	
cartesian 	
abelian 	
sheaves 	
. 	

 	
If 	
$ 	
U$ 	
is 	
a 	
proper 	
hypercovering 	
of 	
$ 	
X$ 	
, 	
then 	

 	
the 	
functor 	
$ 	
a^{-1}$ 	
defines 	
an 	
equivalence 	

 	
$ 	
$ 	

 	
D^+(X_\etale 	
) 	
\longrightarrow 	
D_\mathcal{A}^+(U_\etale 	
) 	

 	
$ 	
$ 	

 	
with 	
quasi 	
- 	
inverse 	
$ 	
Ra_*$. 	
Here 	
$ 	
a 	
: 	
\Sh(U_\etale 	
) 	
\to 	
\Sh(X_\etale)$ 	

 	
is 	
as 	
in 	
Section 	
\ref{section 	
- 	
simplicial 	
- 	
algebraic 	
- 	
spaces}. 	

 	
\end{lemma 	
} 	


 	
\begin{proof 	
} 	

 	
Observe 	
that 	
$ 	
\mathcal{A}$ 	
is 	
a 	
weak 	
Serre 	
subcategory 	
by 	

 	
Lemma 	
\ref{lemma 	
- 	
Serre 	
- 	
subcat 	
- 	
cartesian 	
- 	
modules}. 	

 	
The 	
equivalence 	
is 	
a 	

 	
formal 	
consequence 	
of 	
the 	
results 	
obtained 	
so 	
far 	
. 	
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 	TYPE
. 	TYPE
Let 	
$ 	
X$ 	
be 	
an 	
algebraic 	TYPE
space 	
over 	
$ 	
S$. 	

 	
Let 	
$ 	
U$ 	
be 	
a 	
simplicial 	TYPE
algebraic 	
space 	
over 	
$ 	
S$. 	
Let 	
$ 	
a 	
: 	
U 	
\to 	
X$ 	

 	
be 	
an 	
augmentation 	TYPE
. 	
Let 	
$ 	
\mathcal{F}$ 	
be 	
an 	
abelian 	TYPE
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 	
} 	











 	
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} 	


 	
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} 	

 	
\bibliographystyle{amsalpha 	
} 	


 	
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} 	

 	

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