In [3]:

    

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 [4]:

    

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


    

In [5]:

    

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 [6]:

    

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 [23]:

    

annotated_data=[]

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


    

    




file:  intersection.tex
file:  spaces-simplicial.tex
file:  stacks-sheaves.tex
file:  cotangent.tex
file:  stacks-more-morphisms.tex
file:  formal-defos.tex
file:  spaces-more-cohomology.tex
file:  divisors.tex
file:  more-morphisms.tex

In [29]:

    

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


    

    




9
("\\input{preamble}\n\n% OK, start here.\n%\n\\begin{document}\n\n\\title{Sheaves on Algebraic Stacks}\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\nThere is a myriad of ways to think about sheaves on algebraic stacks.\nIn this chapter we discuss one approach, which is particularly well\nadapted to our foundations for algebraic stacks. Whenever we introduce\na type of sheaves we will indicate the precise relationship with\nsimilar notions in the literature.\nThe goal of this chapter is to state those results\nthat are either obviously true or straightforward to prove\nand leave more intricate constructions till later.\n\n\\medskip\\noindent\nIn fact, it turns out that to develop a fully fledged theory of\nconstructible \\'etale sheaves and/or an adequate discussion of\nderived categories of complexes $\\mathcal{O}$-modules whose\ncohomology sheaves are quasi-coherent takes a significant amount of work, see\n\\cite{olsson_sheaves}. We will return to this in\nCohomology of Stacks, Section \\ref{stacks-cohomology-section-introduction}.\n\n\\medskip\\noindent\nIn the literature and in research papers on sheaves on algebraic stacks\nthe lisse-\\'etale site of an algebraic stack often plays a prominent role.\nHowever, it is a problematic beast, because it turns out that a morphism of\nalgebraic stacks does not induce a morphism of lisse-\\'etale topoi. We have\ntherefore made the design decision to avoid any mention of the lisse-\\'etale\nsite as long as possible. Arguments that traditionally use the lisse-\\'etale\nsite will be replaced by an argument using a {\\v C}ech covering\nin the site $\\mathcal{X}_{smooth}$ defined below.\n\n\\medskip\\noindent\nSome of the notation, conventions and terminology in this chapter is awkward\nand may seem backwards to the more experienced reader. This is intentional.\nPlease see Quot, Section \\ref{quot-section-conventions} for an\nexplanation.\n\n\n\n\n\\section{Conventions}\n\\label{section-conventions}\n\n\\noindent\nThe conventions we use in this chapter are the same as those in the\nchapter on algebraic stacks, see\nAlgebraic Stacks, Section \\ref{algebraic-section-conventions}.\nFor convenience we repeat them here.\n\n\\medskip\\noindent\nWe work in a suitable big fppf site $\\Sch_{fppf}$ as in\nTopologies, Definition \\ref{topologies-definition-big-fppf-site}.\nSo, if not explicitly stated otherwise all schemes will be objects\nof $\\Sch_{fppf}$. We record what changes if you change the big\nfppf site elsewhere (insert future reference here).\n\n\\medskip\\noindent\nWe will always work relative to a base $S$ contained in $\\Sch_{fppf}$.\nAnd we will then work with the big fppf site $(\\Sch/S)_{fppf}$, see\nTopologies, Definition \\ref{topologies-definition-big-small-fppf}.\nThe absolute case can be recovered by taking\n$S = \\Spec(\\mathbf{Z})$.\n\n\n\n\n\n\\section{Presheaves}\n\\label{section-presheaves}\n\n\\noindent\nIn this section we define presheaves on categories fibred in groupoids\nover $(\\Sch/S)_{fppf}$, but most of the discussion works\nfor categories over any base category. This section also serves to\nintroduce the notation we will use later on.\n\n\\begin{definition}\n\\label{definition-presheaves}\nLet $p : \\mathcal{X} \\to (\\Sch/S)_{fppf}$ be a category fibred in\ngroupoids.\n\\begin{enumerate}\n\\item A {\\it presheaf on $\\mathcal{X}$} is a presheaf on the\nunderlying category of $\\mathcal{X}$.\n\\item A {\\it morphism of presheaves on $\\mathcal{X}$} is a morphism of\npresheaves on the underlying category of $\\mathcal{X}$.\n\\end{enumerate}\nWe denote $\\textit{PSh}(\\mathcal{X})$ the category of presheaves on\n$\\mathcal{X}$.\n\\end{definition}\n\n\\noindent\nThis defines presheaves of sets. Of course we can also talk about\npresheaves of pointed sets, abelian groups, groups, monoids, rings,\nmodules over a fixed ring, and lie algebras over a fixed field, etc.\nThe category of {\\it abelian presheaves}, i.e., presheaves of abelian\ngroups, is denoted $\\textit{PAb}(\\mathcal{X})$.\n\n\\medskip\\noindent\nLet $f : \\mathcal{X} \\to \\mathcal{Y}$ be a $1$-morphism of categories\nfibred in groupoids over $(\\Sch/S)_{fppf}$. Recall that this\nmeans just that $f$ is a functor over $(\\Sch/S)_{fppf}$.\nThe material in\nSites, Section \\ref{sites-section-more-functoriality-PSh}\nprovides us with a pair of adjoint functors\\footnote{These functors\nwill be denoted $f^{-1}$ and $f_*$ after\nLemma \\ref{lemma-functoriality-sheaves}\nhas been proved.}\n\\begin{equation}\n\\label{equation-pushforward-pullback}\nf^p : \\textit{PSh}(\\mathcal{Y}) \\longrightarrow \\textit{PSh}(\\mathcal{X})\n\\quad\\text{and}\\quad\n{}_pf : \\textit{PSh}(\\mathcal{X}) \\longrightarrow \\textit{PSh}(\\mathcal{Y}).\n\\end{equation}\nThe adjointness is\n$$\n\\Mor_{\\textit{PSh}(\\mathcal{X})}(f^p\\mathcal{G}, \\mathcal{F})\n=\n\\Mor_{\\textit{PSh}(\\mathcal{Y})}(\\mathcal{G}, {}_pf\\mathcal{F})\n$$\nwhere $\\mathcal{F} \\in \\Ob(\\textit{PSh}(\\mathcal{X}))$ and\n$\\mathcal{G} \\in \\Ob(\\textit{PSh}(\\mathcal{Y}))$. We call\n$f^p\\mathcal{G}$ the {\\it pullback} of $\\mathcal{G}$. It follows\nfrom the definitions that\n$$\nf^p\\mathcal{G}(x) = \\mathcal{G}(f(x))\n$$\nfor any $x \\in \\Ob(\\mathcal{X})$. The presheaf ${}_pf\\mathcal{F}$\nis called the {\\it pushforward} of $\\mathcal{F}$. It is described\nby the formula\n$$\n({}_pf\\mathcal{F})(y) = \\lim_{f(x) \\to y} \\mathcal{F}(x).\n$$\nThe rest of this section should probably be moved to the chapter\non sites and in any case should be skipped on a first reading.\n\n\\begin{lemma}\n\\label{lemma-1-morphisms-presheaves}\nLet $f : \\mathcal{X} \\to \\mathcal{Y}$ and $g : \\mathcal{Y} \\to \\mathcal{Z}$\nbe $1$-morphisms of categories fibred in groupoids over\n$(\\Sch/S)_{fppf}$. Then $(g \\circ f)^p = f^p \\circ g^p$ and\nthere is a canonical isomorphism\n${}_p(g \\circ f) \\to {}_pg \\circ {}_pf$\ncompatible with with adjointness of $(f^p, {}_pf)$, $(g^p, {}_pg)$, and\n$((g \\circ f)^p, {}_p(g \\circ f))$.\n\\end{lemma}\n\n\\begin{proof}\nLet $\\mathcal{H}$ be a presheaf on $\\mathcal{Z}$. Then\n$(g \\circ f)^p\\mathcal{H} = f^p (g^p\\mathcal{H})$ is given\nby the equalities\n$$\n(g \\circ f)^p\\mathcal{H}(x) = \\mathcal{H}((g \\circ f)(x))\n= \\mathcal{H}(g(f(x))) = f^p (g^p\\mathcal{H})(x).\n$$\nWe omit the verification that this is compatible with restriction maps.\n\n\\medskip\\noindent\nNext, we define the transformation ${}_p(g \\circ f) \\to {}_pg \\circ {}_pf$.\nLet $\\mathcal{F}$ be a presheaf on $\\mathcal{X}$.\nIf $z$ is an object of $\\mathcal{Z}$ then we get a\ncategory $\\mathcal{J}$ of quadruples\n$(x, f(x) \\to y, y, g(y) \\to z)$ and a category $\\mathcal{I}$\nof pairs $(x, g(f(x)) \\to z)$. There is a canonical functor\n$\\mathcal{J} \\to \\mathcal{I}$ sending the object\n$(x, \\alpha : f(x) \\to y, y, \\beta : g(y) \\to z)$ to\n$(x, \\beta \\circ f(\\alpha) : g(f(x)) \\to z)$. This gives the arrow in\n\\begin{align*}\n({}_p(g \\circ f)\\mathcal{F})(z) & =\n\\lim_{g(f(x)) \\to z} \\mathcal{F}(x) \\\\\n& = \\lim_\\mathcal{I} \\mathcal{F} \\\\\n& \\to \\lim_\\mathcal{J} \\mathcal{F} \\\\\n& = \\lim_{g(y) \\to z}\n\\Big(\\lim_{f(x) \\to y} \\mathcal{F}(x)\\Big) \\\\\n& =\n({}_pg \\circ {}_pf\\mathcal{F})(x)\n\\end{align*}\nby\nCategories, Lemma \\ref{categories-lemma-functorial-limit}.\nWe omit the verification that this is compatible with restriction maps.\nAn alternative to this direct construction is to define\n${}_p(g \\circ f) \\cong {}_pg \\circ {}_pf$\nas the unique map compatible with the adjointness properties. This also\nhas the advantage that one does not need to prove the compatibility.\n\n\\medskip\\noindent\nCompatibility with adjointness of $(f^p, {}_pf)$, $(g^p, {}_pg)$, and\n$((g \\circ f)^p, {}_p(g \\circ f))$ means that given presheaves\n$\\mathcal{H}$ and $\\mathcal{F}$ as above we have a commutative diagram\n$$\n\\xymatrix{\n\\Mor_{\\textit{PSh}(\\mathcal{X})}(f^pg^p\\mathcal{H}, \\mathcal{F})\n\\ar@{=}[r] \\ar@{=}[d] &\n\\Mor_{\\textit{PSh}(\\mathcal{Y})}(g^p\\mathcal{H}, {}_pf\\mathcal{F})\n\\ar@{=}[r] &\n\\Mor_{\\textit{PSh}(\\mathcal{Y})}(\\mathcal{H}, {}_pg{}_pf\\mathcal{F})\n\\\\\n\\Mor_{\\textit{PSh}(\\mathcal{X})}((g \\circ f)^p\\mathcal{G}, \\mathcal{F})\n\\ar@{=}[rr] & &\n\\Mor_{\\textit{PSh}(\\mathcal{Y})}(\\mathcal{G}, {}_p(g \\circ f)\\mathcal{F})\n\\ar[u]\n}\n$$\nProof omitted.\n\\end{proof}\n\n\\begin{lemma}\n\\label{lemma-2-morphisms-presheaves}\nLet $f, g : \\mathcal{X} \\to \\mathcal{Y}$ be $1$-morphisms of categories\nfibred in groupoids over $(\\Sch/S)_{fppf}$. Let $t : f \\to g$\nbe a $2$-morphism of categories fibred in groupoids over\n$(\\Sch/S)_{fppf}$. Assigned to $t$ there are canonical\nisomorphisms of functors\n$$\nt^p : g^p \\longrightarrow f^p\n\\quad\\text{and}\\quad\n{}_pt : {}_pf \\longrightarrow {}_pg\n$$\nwhich compatible with adjointness of $(f^p, {}_pf)$ and\n$(g^p, {}_pg)$ and with\nvertical and horizontal composition of $2$-morphisms.\n\\end{lemma}\n\n\\begin{proof}\nLet $\\mathcal{G}$ be a presheaf on $\\mathcal{Y}$. Then\n$t^p : g^p\\mathcal{G} \\to f^p\\mathcal{G}$ is given by the family\nof maps\n$$\ng^p\\mathcal{G}(x) = \\mathcal{G}(g(x))\n\\xrightarrow{\\mathcal{G}(t_x)}\n\\mathcal{G}(f(x)) = f^p\\mathcal{G}(x)\n$$\nparametrized by $x \\in \\Ob(\\mathcal{X})$. This makes sense as\n$t_x : f(x) \\to g(x)$ and $\\mathcal{G}$ is a contravariant functor.\nWe omit the verification that this is compatible with restriction\nmappings.\n\n\\medskip\\noindent\nTo define the transformation ${}_pt$ for $y \\in \\Ob(\\mathcal{Y})$\ndefine ${}_y^f\\mathcal{I}$, resp.\\ ${}_y^g\\mathcal{I}$ to be the category\nof pairs $(x, \\psi : f(x) \\to y)$, resp.\\ $(x, \\psi : g(x) \\to y)$, see\nSites, Section \\ref{sites-section-more-functoriality-PSh}.\nNote that $t$ defines a functor\n${}_yt : {}_y^g\\mathcal{I} \\to {}_y^f\\mathcal{I}$\ngiven by the rule\n$$\n(x, g(x) \\to y) \\longmapsto (x, f(x) \\xrightarrow{t_x} g(x) \\to y).\n$$\nNote that for $\\mathcal{F}$ a presheaf on $\\mathcal{X}$ the composition\nof ${}_yt$ with $\\mathcal{F} : {}_y^f\\mathcal{I}^{opp} \\to \\textit{Sets}$,\n$(x, f(x) \\to y) \\mapsto \\mathcal{F}(x)$ is equal to\n$\\mathcal{F} : {}_y^g\\mathcal{I}^{opp} \\to \\textit{Sets}$. Hence by\nCategories, Lemma \\ref{categories-lemma-functorial-limit}\nwe get for every $y \\in \\Ob(\\mathcal{Y})$ a canonical map\n$$\n({}_pf\\mathcal{F})(y) = \\lim_{{}_y^f\\mathcal{I}} \\mathcal{F}\n\\longrightarrow\n\\lim_{{}_y^g\\mathcal{I}} \\mathcal{F} = ({}_pg\\mathcal{F})(y)\n$$\nWe omit the verification that this is compatible with restriction\nmappings. An alternative to this direct construction is to define\n${}_pt$ as the unique map compatible with the adjointness properties\nof the pairs $(f^p, {}_pf)$ and $(g^p, {}_pg)$ (see below). This also\nhas the advantage that one does not need to prove the compatibility.\n\n\\medskip\\noindent\nCompatibility with adjointness of $(f^p, {}_pf)$ and $(g^p, {}_pg)$ means\nthat given presheaves $\\mathcal{G}$ and $\\mathcal{F}$ as above we have\na commutative diagram\n$$\n\\xymatrix{\n\\Mor_{\\textit{PSh}(\\mathcal{X})}(f^p\\mathcal{G}, \\mathcal{F})\n\\ar@{=}[r] \\ar[d]_{- \\circ t^p} &\n\\Mor_{\\textit{PSh}(\\mathcal{Y})}(\\mathcal{G}, {}_pf\\mathcal{F})\n\\ar[d]^{{}_pt \\circ -} \\\\\n\\Mor_{\\textit{PSh}(\\mathcal{X})}(g^p\\mathcal{G}, \\mathcal{F})\n\\ar@{=}[r] &\n\\Mor_{\\textit{PSh}(\\mathcal{Y})}(\\mathcal{G}, {}_pg\\mathcal{F})\n}\n$$\nProof omitted. Hint: Work through the proof of\nSites, Lemma \\ref{sites-lemma-adjoints-pu}\nand observe the compatibility from the explicit description of the\nhorizontal and vertical maps in the diagram.\n\n\\medskip\\noindent\nWe omit the verification that this is compatible with vertical and horizontal\ncompositions. Hint: The proof of this for $t^p$ is straightforward and\none can conclude that this holds for the ${}_pt$ maps using compatibility\nwith adjointness.\n\\end{proof}\n\n\n\n\n\n\n\n\\section{Sheaves}\n\\label{section-sheaves}\n\n\\noindent\nWe first make an observation that is important and trivial\n(especially for those readers who do not worry about set theoretical\nissues).\n\n\\medskip\\noindent\nConsider a big fppf site $\\Sch_{fppf}$ as in\nTopologies, Definition \\ref{topologies-definition-big-fppf-site}\nand denote its underlying category $\\Sch_\\alpha$.\nBesides being the underlying category of a fppf site,\nthe category $\\Sch_\\alpha$ can also can serve as the underlying\ncategory for a big Zariski site, a big \\'etale site, a big smooth site,\nand a big syntomic site, see\nTopologies, Remark \\ref{topologies-remark-choice-sites}.\nWe denote these sites $\\Sch_{Zar}$, $\\Sch_\\etale$,\n$\\Sch_{smooth}$, and $\\Sch_{syntomic}$.\nIn this situation, since we have defined\nthe big Zariski site $(\\Sch/S)_{Zar}$ of $S$,\nthe big \\'etale site $(\\Sch/S)_\\etale$ of $S$,\nthe big smooth site $(\\Sch/S)_{smooth}$ of $S$,\nthe big syntomic site $(\\Sch/S)_{syntomic}$ of $S$, and\nthe big fppf site $(\\Sch/S)_{fppf}$ of $S$\nas the localizations (see\nSites, Section \\ref{sites-section-localize})\n$\\Sch_{Zar}/S$, $\\Sch_\\etale/S$,\n$\\Sch_{smooth}/S$, $\\Sch_{syntomic}/S$, and\n$\\Sch_{fppf}/S$\nof these (absolute) big sites we see that all of these have the\nsame underlying category, namely $\\Sch_\\alpha/S$.\n\n\\medskip\\noindent\nIt follows that if we have a category\n$p : \\mathcal{X} \\to (\\Sch/S)_{fppf}$ fibred in groupoids, then\n$\\mathcal{X}$ inherits a Zariski, \\'etale, smooth, syntomic, and\nfppf topology, see\nStacks, Definition \\ref{stacks-definition-topology-inherited}.\n\n\\begin{definition}\n\\label{definition-inherited-topologies}\nLet $\\mathcal{X}$ be a category fibred in groupoids over\n$(\\Sch/S)_{fppf}$.\n\\begin{enumerate}\n\\item The {\\it associated Zariski site}, denoted $\\mathcal{X}_{Zar}$,\nis the structure of site on $\\mathcal{X}$ inherited from\n$(\\Sch/S)_{Zar}$.\n\\item The {\\it associated \\'etale site}, denoted $\\mathcal{X}_\\etale$,\nis the structure of site on $\\mathcal{X}$ inherited from\n$(\\Sch/S)_\\etale$.\n\\item The {\\it associated smooth site}, denoted $\\mathcal{X}_{smooth}$,\nis the structure of site on $\\mathcal{X}$ inherited from\n$(\\Sch/S)_{smooth}$.\n\\item The {\\it associated syntomic site}, denoted $\\mathcal{X}_{syntomic}$,\nis the structure of site on $\\mathcal{X}$ inherited from\n$(\\Sch/S)_{syntomic}$.\n\\item The {\\it associated fppf site}, denoted $\\mathcal{X}_{fppf}$,\nis the structure of site on $\\mathcal{X}$ inherited from\n$(\\Sch/S)_{fppf}$.\n\\end{enumerate}\n\\end{definition}\n\n\\noindent\nThis definition makes sense by the discussion above. If $\\mathcal{X}$\nis an algebraic stack, the literature calls $\\mathcal{X}_{fppf}$ (or a\nsite equivalent to it) the {\\it big fppf site} of $\\mathcal{X}$ and similarly\nfor the other ones. We may occasionally use this terminology to\ndistinguish this construction from others.\n\n\\begin{remark}\n\\label{remark-ambiguity}\nWe only use this notation when the symbol $\\mathcal{X}$ refers to a\ncategory fibred in groupoids, and not a scheme, an algebraic space, etc.\nIn this way we will avoid confusion with the small \\'etale site of a\nscheme, or algebraic space which is denoted $X_\\etale$ (in which\ncase we use a roman capital instead of a calligraphic one).\n\\end{remark}\n\n\\noindent\nNow that we have these topologies defined we can say what it means\nto have a sheaf on $\\mathcal{X}$, i.e., define the corresponding topoi.\n\n\\begin{definition}\n\\label{definition-sheaves}\nLet $\\mathcal{X}$ be a category fibred in groupoids over\n$(\\Sch/S)_{fppf}$. Let $\\mathcal{F}$ be a presheaf on $\\mathcal{X}$.\n\\begin{enumerate}\n\\item We say $\\mathcal{F}$ is a {\\it Zariski sheaf}, or a\n{\\it sheaf for the Zariski topology} if $\\mathcal{F}$\nis a sheaf on the associated Zariski site $\\mathcal{X}_{Zar}$.\n\\item We say $\\mathcal{F}$ is an {\\it \\'etale sheaf}, or a\n{\\it sheaf for the \\'etale topology} if $\\mathcal{F}$\nis a sheaf on the associated \\'etale site $\\mathcal{X}_\\etale$.\n\\item We say $\\mathcal{F}$ is a {\\it smooth sheaf}, or a\n{\\it sheaf for the smooth topology} if $\\mathcal{F}$\nis a sheaf on the associated smooth site $\\mathcal{X}_{smooth}$.\n\\item We say $\\mathcal{F}$ is a {\\it syntomic sheaf}, or a\n{\\it sheaf for the syntomic topology} if $\\mathcal{F}$\nis a sheaf on the associated syntomic site $\\mathcal{X}_{syntomic}$.\n\\item We say $\\mathcal{F}$ is an {\\it fppf sheaf}, or a {\\it sheaf},\nor a {\\it sheaf for the fppf topology} if $\\mathcal{F}$\nis a sheaf on the associated fppf site $\\mathcal{X}_{fppf}$.\n\\end{enumerate}\nA morphism of sheaves is just a morphism of presheaves. We denote\nthese categories of sheaves\n$\\Sh(\\mathcal{X}_{Zar})$,\n$\\Sh(\\mathcal{X}_\\etale)$,\n$\\Sh(\\mathcal{X}_{smooth})$,\n$\\Sh(\\mathcal{X}_{syntomic})$, and\n$\\Sh(\\mathcal{X}_{fppf})$.\n\\end{definition}\n\n\\noindent\nOf course we can also talk about sheaves of pointed sets, abelian groups,\ngroups, monoids, rings, modules over a fixed ring, and lie algebras over\na fixed field, etc. The category of {\\it abelian sheaves}, i.e., sheaves\nof abelian groups, is denoted $\\textit{Ab}(\\mathcal{X}_{fppf})$\nand similarly for the other topologies. If $\\mathcal{X}$ is an algebraic\nstack, then $\\Sh(\\mathcal{X}_{fppf})$ is equivalent (modulo\nset theoretical problems) to what in the literature would be termed\nthe {\\it category of sheaves on the big fppf site of $\\mathcal{X}$}. Similar\nfor other topologies. We may occasionally use this terminology to\ndistinguish this construction from others.\n\n\\medskip\\noindent\nSince the topologies are listed in increasing order of strength we have\nthe following strictly full inclusions\n$$\n\\Sh(\\mathcal{X}_{fppf}) \\subset\n\\Sh(\\mathcal{X}_{syntomic}) \\subset\n\\Sh(\\mathcal{X}_{smooth}) \\subset\n\\Sh(\\mathcal{X}_\\etale) \\subset\n\\Sh(\\mathcal{X}_{Zar}) \\subset \\textit{PSh}(\\mathcal{X})\n$$\nWe sometimes write\n$\\Sh(\\mathcal{X}_{fppf}) = \\Sh(\\mathcal{X})$\nand\n$\\textit{Ab}(\\mathcal{X}_{fppf}) = \\textit{Ab}(\\mathcal{X})$\nin accordance with our terminology that a sheaf on $\\mathcal{X}$\nis an fppf sheaf on $\\mathcal{X}$.\n\n\\medskip\\noindent\nWith this setup functoriality of these topoi is straightforward, and\nmoreover, is compatible with the inclusion functors above.\n\n\\begin{lemma}\n\\label{lemma-functoriality-sheaves}\nLet $f : \\mathcal{X} \\to \\mathcal{Y}$ be a $1$-morphism of categories\nfibred in groupoids over $(\\Sch/S)_{fppf}$. Let\n$\\tau \\in \\{Zar, \\etale, smooth, syntomic, fppf\\}$.\nThe functors ${}_pf$ and $f^p$ of (\\ref{equation-pushforward-pullback})\ntransform $\\tau$ sheaves into $\\tau$ sheaves and define a morphism\nof topoi\n$f : \\Sh(\\mathcal{X}_\\tau) \\to \\Sh(\\mathcal{Y}_\\tau)$.\n\\end{lemma}\n\n\\begin{proof}\nThis follows immediately from\nStacks, Lemma \\ref{stacks-lemma-topology-inherited-functorial}.\n\\end{proof}\n\n\\noindent\nIn other words, pushforward and pullback of presheaves as defined in\nSection \\ref{section-presheaves}\nalso produces {\\it pushforward} and {\\it pullback} of $\\tau$-sheaves.\nHaving said all of the above we see that we can write $f^p = f^{-1}$\nand ${}_pf = f_*$ without any possibility of confusion.\n\n\\begin{definition}\n\\label{definition-morphism}\nLet $f : \\mathcal{X} \\to \\mathcal{Y}$ be a morphism of categories\nfibred in groupoids over $(\\Sch/S)_{fppf}$. We denote\n$$\nf = (f^{-1}, f_*) :\n\\Sh(\\mathcal{X}_{fppf})\n\\longrightarrow\n\\Sh(\\mathcal{Y}_{fppf})\n$$\nthe {\\it associated morphism of fppf topoi} constructed above.\nSimilarly for the associated Zariski, \\'etale, smooth, and syntomic topoi.\n\\end{definition}\n\n\\noindent\nAs discussed in\nSites, Section \\ref{sites-section-sheaves-algebraic-structures}\nthe same formula (on the underlying sheaf of sets) defines\npushforward and pullback for sheaves (for one of our topologies)\nof pointed sets, abelian groups, groups, monoids, rings, modules\nover a fixed ring, and lie algebras over a fixed field, etc.\n\n\n\n\n\n\n\n\n\\section{Computing pushforward}\n\\label{section-pushforward}\n\n\\noindent\nLet $f : \\mathcal{X} \\to \\mathcal{Y}$ be a $1$-morphism of categories\nfibred in groupoids over $(\\Sch/S)_{fppf}$. Let $\\mathcal{F}$\nbe a presheaf on $\\mathcal{X}$. Let $y \\in \\Ob(\\mathcal{Y})$.\nWe can compute $f_*\\mathcal{F}(y)$ in the following way. Suppose that\n$y$ lies over the scheme $V$ and using the $2$-Yoneda lemma think\nof $y$ as a $1$-morphism. Consider the projection\n$$\n\\text{pr} :\n(\\Sch/V)_{fppf} \\times_{y, \\mathcal{Y}} \\mathcal{X}\n\\longrightarrow\n\\mathcal{X}\n$$\nThen we have a canonical identification\n\\begin{equation}\n\\label{equation-pushforward}\nf_*\\mathcal{F}(y) = \\Gamma\\Big(\n(\\Sch/V)_{fppf} \\times_{y, \\mathcal{Y}} \\mathcal{X},\n\\ \\text{pr}^{-1}\\mathcal{F}\\Big)\n\\end{equation}\nNamely, objects of the $2$-fibre product are triples\n$(h : U \\to V, x, f(x) \\to h^*y)$. Dropping the $h$ from the\nnotation we see that this is equivalent to the data of an object\n$x$ of $\\mathcal{X}$ and a morphism $\\alpha : f(x) \\to y$ of $\\mathcal{Y}$.\nSince $f_*\\mathcal{F}(y) = \\lim_{f(x) \\to y} \\mathcal{F}(x)$ by definition\nthe equality follows.\n\n\\medskip\\noindent\nAs a consequence we have the following ``base change'' result for\npushforwards. This result is trivial and hinges on the fact that\nwe are using ``big'' sites.\n\n\\begin{lemma}\n\\label{lemma-base-change}\nLet $S$ be a scheme. Let\n$$\n\\xymatrix{\n\\mathcal{Y}' \\times_\\mathcal{Y} \\mathcal{X} \\ar[r]_{g'} \\ar[d]_{f'} &\n\\mathcal{X} \\ar[d]^f \\\\\n\\mathcal{Y}' \\ar[r]^g & \\mathcal{Y}\n}\n$$\nbe a $2$-cartesian diagram of categories fibred in groupoids over $S$.\nThen we have a canonical isomorphism\n$$\ng^{-1}f_*\\mathcal{F} \\longrightarrow f'_*(g')^{-1}\\mathcal{F}\n$$\nfunctorial in the presheaf $\\mathcal{F}$ on $\\mathcal{X}$.\n\\end{lemma}\n\n\\begin{proof}\nGiven an object $y'$ of $\\mathcal{Y}'$ over $V$\nthere is an equivalence\n$$\n(\\Sch/V)_{fppf} \\times_{g(y'), \\mathcal{Y}} \\mathcal{X}\n=\n(\\Sch/V)_{fppf} \\times_{y', \\mathcal{Y}'}\n(\\mathcal{Y}' \\times_\\mathcal{Y} \\mathcal{X})\n$$\nHence by (\\ref{equation-pushforward}) a bijection\n$g^{-1}f_*\\mathcal{F}(y') \\to f'_*(g')^{-1}\\mathcal{F}(y')$.\nWe omit the verification that this is compatible with restriction\nmappings.\n\\end{proof}\n\n\\noindent\nIn the case of a representable morphism of categories fibred in groupoids\nthis formula (\\ref{equation-pushforward}) simplifies. We suggest the\nreader skip the rest of this section.\n\n\\begin{lemma}\n\\label{lemma-representable}\nLet $f : \\mathcal{X} \\to \\mathcal{Y}$ be a $1$-morphism of categories\nfibred in groupoids over $(\\Sch/S)_{fppf}$. The following are\nequivalent\n\\begin{enumerate}\n\\item $f$ is representable, and\n\\item for every $y \\in \\Ob(\\mathcal{Y})$ the functor\n$\\mathcal{X}^{opp} \\to \\textit{Sets}$,\n$x \\mapsto \\Mor_\\mathcal{Y}(f(x), y)$\nis representable.\n\\end{enumerate}\n\\end{lemma}\n\n\\begin{proof}\nAccording to the discussion in\nAlgebraic Stacks, Section \\ref{algebraic-section-representable-morphism}\nwe see that $f$ is representable if and only if\nfor every $y \\in \\Ob(\\mathcal{Y})$\nlying over $U$ the $2$-fibre product\n$(\\Sch/U)_{fppf} \\times_{y, \\mathcal{Y}} \\mathcal{X}$\nis representable, i.e., of the form $(\\Sch/V_y)_{fppf}$ for some\nscheme $V_y$ over $U$. Objects in this $2$-fibre products are triples\n$(h : V \\to U, x, \\alpha : f(x) \\to h^*y)$ where $\\alpha$ lies\nover $\\text{id}_V$. Dropping the $h$ from the notation we see that this\nis equivalent to the data of an object $x$ of $\\mathcal{X}$ and a\nmorphism $f(x) \\to y$. Hence the $2$-fibre product is\nrepresentable by $V_y$ and $f(x_y) \\to y$ where $x_y$ is an object\nof $\\mathcal{X}$ over $V_y$ if and only if the functor in (2) is representable\nby $x_y$ with universal object a map $f(x_y) \\to y$.\n\\end{proof}\n\n\\noindent\nLet\n$$\n\\xymatrix{\n\\mathcal{X} \\ar[rr]_f \\ar[rd]_p & &  \\mathcal{Y} \\ar[ld]^q \\\\\n& (\\Sch/S)_{fppf}\n}\n$$\nbe a $1$-morphism of categories fibred in groupoids. Assume $f$ is\nrepresentable. For every $y \\in \\Ob(\\mathcal{Y})$ we choose\nan object $u(y) \\in \\Ob(\\mathcal{X})$ representing the functor\n$x \\mapsto \\Mor_\\mathcal{Y}(f(x), y)$ of\nLemma \\ref{lemma-representable}\n(this is possible by the axiom of choice).\nThe objects come with canonical morphisms $f(u(y)) \\to y$ by\nconstruction.\nFor every morphism $\\beta : y' \\to y$ in $\\mathcal{Y}$ we obtain a unique\nmorphism $u(\\beta) : u(y') \\to u(y)$ in $\\mathcal{X}$ such that the diagram\n$$\n\\xymatrix{\nf(u(y')) \\ar[d] \\ar[rr]_{f(u(\\beta))} & & f(u(y)) \\ar[d] \\\\\ny' \\ar[rr] & & y\n}\n$$\ncommutes. In other words, $u : \\mathcal{Y} \\to \\mathcal{X}$ is a functor.\nIn fact, we can say a little bit more. Namely, suppose that\n$V' = q(y')$, $V = q(y)$, $U' = p(u(y'))$ and $U = p(u(y))$. Then\n$$\n\\xymatrix{\nU' \\ar[rr]_{p(u(\\beta))} \\ar[d] & & U \\ar[d] \\\\\nV' \\ar[rr]^{q(\\beta)} & & V\n}\n$$\nis a fibre product square. This is true because $U' \\to U$ represents\nthe base change\n$(\\Sch/V')_{fppf} \\times_{y', \\mathcal{Y}} \\mathcal{X} \\to\n(\\Sch/V)_{fppf} \\times_{y, \\mathcal{Y}} \\mathcal{X}$\nof $V' \\to V$.\n\n\\begin{lemma}\n\\label{lemma-representable-pushforward}\nLet $f : \\mathcal{X} \\to \\mathcal{Y}$ be a representable $1$-morphism of\ncategories fibred in groupoids over $(\\Sch/S)_{fppf}$. Let\n$\\tau \\in \\{Zar, \\etale, smooth, syntomic, fppf\\}$.\nThen the functor $u : \\mathcal{Y}_\\tau \\to \\mathcal{X}_\\tau$ is continuous\nand defines a morphism of sites $\\mathcal{X}_\\tau \\to \\mathcal{Y}_\\tau$\nwhich induces the same morphism of topoi\n$\\Sh(\\mathcal{X}_\\tau) \\to \\Sh(\\mathcal{Y}_\\tau)$\nas the morphism $f$ constructed in\nLemma \\ref{lemma-functoriality-sheaves}.\nMoreover, $f_*\\mathcal{F}(y) = \\mathcal{F}(u(y))$ for any presheaf\n$\\mathcal{F}$ on $\\mathcal{X}$.\n\\end{lemma}\n\n\\begin{proof}\nLet $\\{y_i \\to y\\}$ be a $\\tau$-covering in $\\mathcal{Y}$. By definition\nthis simply means that $\\{q(y_i) \\to q(y)\\}$ is a $\\tau$-covering of\nschemes. By the final remark above the lemma we see that\n$\\{p(u(y_i)) \\to p(u(y))\\}$ is the base change of the $\\tau$-covering\n$\\{q(y_i) \\to q(y)\\}$ by $p(u(y)) \\to q(y)$, hence is itself a\n$\\tau$-covering by the axioms of a site. Hence $\\{u(y_i) \\to u(y)\\}$\nis a $\\tau$-covering of $\\mathcal{X}$. This proves that $u$ is\ncontinuous.\n\n\\medskip\\noindent\nLet's use the notation $u_p, u_s, u^p, u^s$ of\nSites, Sections \\ref{sites-section-functoriality-PSh} and\n\\ref{sites-section-continuous-functors}.\nIf we can show the final assertion of the lemma, then we see that\n$f_* = u^p = u^s$ (by continuity of $u$ seen above) and hence by adjointness\n$f^{-1} = u_s$ which will prove $u_s$ is exact, hence that $u$ determines\na morphism of sites, and the equality will be clear as well.\nTo see that $f_*\\mathcal{F}(y) = \\mathcal{F}(u(y))$ note that by\ndefinition\n$$\nf_*\\mathcal{F}(y) = ({}_pf\\mathcal{F})(y) =\n\\lim_{f(x) \\to y} \\mathcal{F}(x).\n$$\nSince $u(y)$ is a final object in the category the limit is taken\nover we conclude.\n\\end{proof}\n\n\n\n\n\n\\section{The structure sheaf}\n\\label{section-structure-sheaf}\n\n\\noindent\nLet $\\tau \\in \\{Zar, \\etale, smooth, syntomic, fppf\\}$.\nLet $p : \\mathcal{X} \\to (\\Sch/S)_{fppf}$ be a category\nfibred in groupoids. The 2-category of categories fibred in groupoids over\n$(\\Sch/S)_{fppf}$ has a final object, namely,\n$\\text{id} : (\\Sch/S)_{fppf} \\to (\\Sch/S)_{fppf}$\nand $p$ is a $1$-morphism from $\\mathcal{X}$ to this final object.\nHence any presheaf $\\mathcal{G}$ on $(\\Sch/S)_{fppf}$ gives a\npresheaf $p^{-1}\\mathcal{G}$ on $\\mathcal{X}$ defined by the rule\n$p^{-1}\\mathcal{G}(x) = \\mathcal{G}(p(x))$. Moreover, the discussion in\nSection \\ref{section-sheaves}\nshows that $p^{-1}\\mathcal{G}$ is a $\\tau$ sheaf whenever\n$\\mathcal{G}$ is a $\\tau$-sheaf.\n\n\\medskip\\noindent\nRecall that the site $(\\Sch/S)_{fppf}$ is a ringed site\nwith structure sheaf $\\mathcal{O}$ defined by the rule\n$$\n(\\Sch/S)^{opp} \\longrightarrow \\textit{Rings},\n\\quad\nU/S \\longmapsto \\Gamma(U, \\mathcal{O}_U)\n$$\nsee\nDescent, Definition \\ref{descent-definition-structure-sheaf}.\n\n\\begin{definition}\n\\label{definition-structure-sheaf}\nLet $p : \\mathcal{X} \\to (\\Sch/S)_{fppf}$ be a category\nfibred in groupoids. The\n{\\it structure sheaf of $\\mathcal{X}$} is the sheaf of rings\n$\\mathcal{O}_\\mathcal{X} = p^{-1}\\mathcal{O}$.\n\\end{definition}\n\n\\noindent\nFor an object $x$ of $\\mathcal{X}$ lying over $U$ we have\n$\\mathcal{O}_\\mathcal{X}(x) = \\mathcal{O}(U) = \\Gamma(U, \\mathcal{O}_U)$.\nNeedless to say $\\mathcal{O}_\\mathcal{X}$ is also a Zariski, \\'etale,\nsmooth, and syntomic sheaf, and hence each of the sites\n$\\mathcal{X}_{Zar}$, $\\mathcal{X}_\\etale$, $\\mathcal{X}_{smooth}$,\n$\\mathcal{X}_{syntomic}$, and $\\mathcal{X}_{fppf}$ is a ringed site.\nThis construction is functorial as well.\n\n\\begin{lemma}\n\\label{lemma-functoriality-structure-sheaf}\nLet $f : \\mathcal{X} \\to \\mathcal{Y}$ be a $1$-morphism of categories\nfibred in groupoids over $(\\Sch/S)_{fppf}$. Let\n$\\tau \\in \\{Zar, \\etale, smooth, syntomic, fppf\\}$.\nThere is a canonical identification\n$f^{-1}\\mathcal{O}_\\mathcal{X} = \\mathcal{O}_\\mathcal{Y}$\nwhich turns\n$f : \\Sh(\\mathcal{X}_\\tau) \\to \\Sh(\\mathcal{Y}_\\tau)$\ninto a morphism of ringed topoi.\n\\end{lemma}\n\n\\begin{proof}\nDenote $p : \\mathcal{X} \\to (\\Sch/S)_{fppf}$ and\n$q : \\mathcal{Y} \\to (\\Sch/S)_{fppf}$ the structural functors.\nThen $q = p \\circ f$, hence $q^{-1} = f^{-1} \\circ p^{-1}$ by\nLemma \\ref{lemma-1-morphisms-presheaves}.\nThe result follows.\n\\end{proof}\n\n\\begin{remark}\n\\label{remark-flat}\nIn the situation of\nLemma \\ref{lemma-functoriality-structure-sheaf}\nthe morphism of ringed topoi\n$f : \\Sh(\\mathcal{X}_\\tau) \\to \\Sh(\\mathcal{Y}_\\tau)$\nis flat as is clear from the equality\n$f^{-1}\\mathcal{O}_\\mathcal{X} = \\mathcal{O}_\\mathcal{Y}$.\nThis is a bit counter intuitive, for example because a closed\nimmersion of algebraic stacks is typically not flat (as a morphism of\nalgebraic stacks).\nHowever, exactly the same thing happens when taking a closed\nimmersion $i : X \\to Y$ of schemes: in this case the associated\nmorphism of big $\\tau$-sites\n$i : (\\Sch/X)_\\tau \\to (\\Sch/Y)_\\tau$\nalso is flat.\n\\end{remark}\n\n\n\n\n\\section{Sheaves of modules}\n\\label{section-modules}\n\n\\noindent\nSince we have a structure sheaf we have modules.\n\n\\begin{definition}\n\\label{definition-modules}\nLet $\\mathcal{X}$ be a category fibred in groupoids over\n$(\\Sch/S)_{fppf}$.\n\\begin{enumerate}\n\\item A {\\it presheaf of modules on $\\mathcal{X}$} is a\npresheaf of $\\mathcal{O}_\\mathcal{X}$-modules. The category of\npresheaves of modules is denoted $\\textit{PMod}(\\mathcal{O}_\\mathcal{X})$.\n\\item We say a presheaf of modules $\\mathcal{F}$ is an\n{\\it $\\mathcal{O}_\\mathcal{X}$-module}, or more precisely a\n{\\it sheaf of $\\mathcal{O}_\\mathcal{X}$-modules} if $\\mathcal{F}$\nis an fppf sheaf. The category of $\\mathcal{O}_\\mathcal{X}$-modules\nis denoted $\\textit{Mod}(\\mathcal{O}_\\mathcal{X})$.\n\\end{enumerate}\n\\end{definition}\n\n\\noindent\nThese (pre)sheaves of modules occur in the literature as {\\it (pre)sheaves\nof $\\mathcal{O}_\\mathcal{X}$-modules on the big fppf site of $\\mathcal{X}$}.\nWe will occasionally use this terminology if we want to distinguish these\ncategories from others. We will also encounter presheaves of modules which\nare sheaves in the Zariski, \\'etale, smooth, or syntomic topologies\n(without necessarily being sheaves). If need be these will be denoted\n$\\textit{Mod}(\\mathcal{X}_\\etale, \\mathcal{O}_\\mathcal{X})$\nand similarly for the other topologies.\n\n\\medskip\\noindent\nNext, we address functoriality -- first for presheaves of modules. Let\n$$\n\\xymatrix{\n\\mathcal{X} \\ar[rr]_f \\ar[rd]_p & &  \\mathcal{Y} \\ar[ld]^q \\\\\n& (\\Sch/S)_{fppf}\n}\n$$\nbe a $1$-morphism of categories fibred in groupoids.\nThe functors $f^{-1}$, $f_*$ on abelian presheaves extend to functors\n\\begin{equation}\n\\label{equation-functoriality-presheaves-modules}\nf^{-1} :\n\\textit{PMod}(\\mathcal{O}_\\mathcal{Y})\n\\longrightarrow\n\\textit{PMod}(\\mathcal{O}_\\mathcal{X})\n\\quad\\text{and}\\quad\nf_* :\n\\textit{PMod}(\\mathcal{O}_\\mathcal{Y})\n\\longrightarrow\n\\textit{PMod}(\\mathcal{O}_\\mathcal{X})\n\\end{equation}\nThis is immediate for $f^{-1}$ because\n$f^{-1}\\mathcal{G}(x) = \\mathcal{G}(f(x))$ which is a module over\n$\\mathcal{O}_\\mathcal{Y}(f(x)) = \\mathcal{O}(q(f(x))) = \\mathcal{O}(p(x)) =\n\\mathcal{O}_\\mathcal{X}(x)$. Alternatively it follows because\n$f^{-1}\\mathcal{O}_\\mathcal{Y} = \\mathcal{O}_\\mathcal{X}$\nand because $f^{-1}$ commutes with limits (on presheaves).\nSince $f_*$ is a right adjoint it commutes with all limits\n(on presheaves) in particular products. Hence we can extend\n$f_*$ to a functor on presheaves of modules as in the proof of\nModules on Sites, Lemma \\ref{sites-modules-lemma-pushforward-module}.\nWe claim that the functors (\\ref{equation-functoriality-presheaves-modules})\nform an adjoint pair of functors:\n$$\n\\Mor_{\\textit{PMod}(\\mathcal{O}_\\mathcal{X})}(\nf^{-1}\\mathcal{G}, \\mathcal{F})\n=\n\\Mor_{\\textit{PMod}(\\mathcal{O}_\\mathcal{Y})}(\n\\mathcal{G}, f_*\\mathcal{F}).\n$$\nAs $f^{-1}\\mathcal{O}_\\mathcal{Y} = \\mathcal{O}_\\mathcal{X}$\nthis follows from\nModules on Sites, Lemma \\ref{sites-modules-lemma-adjoint-push-pull-modules}\nby endowing $\\mathcal{X}$ and $\\mathcal{Y}$ with the chaotic\ntopology.\n\n\\medskip\\noindent\nNext, we discuss functoriality for modules, i.e., for sheaves of modules\nin the fppf topology. Denote by $f$ also the induced morphism of ringed\ntopoi, see\nLemma \\ref{lemma-functoriality-structure-sheaf}\n(for the fppf topologies right now). Note that the functors\n$f^{-1}$ and $f_*$ of (\\ref{equation-functoriality-presheaves-modules})\npreserve the subcategories of sheaves of modules, see\nLemma \\ref{lemma-functoriality-sheaves}.\nHence it follows immediately that\n\\begin{equation}\n\\label{equation-functoriality-sheaves-modules}\nf^{-1} :\n\\textit{Mod}(\\mathcal{O}_\\mathcal{Y})\n\\longrightarrow\n\\textit{Mod}(\\mathcal{O}_\\mathcal{X})\n\\quad\\text{and}\\quad\nf_* :\n\\textit{Mod}(\\mathcal{O}_\\mathcal{Y})\n\\longrightarrow\n\\textit{Mod}(\\mathcal{O}_\\mathcal{X})\n\\end{equation}\nform an adjoint pair of functors:\n$$\n\\Mor_{\\textit{Mod}(\\mathcal{O}_\\mathcal{X})}(\nf^{-1}\\mathcal{G}, \\mathcal{F})\n=\n\\Mor_{\\textit{Mod}(\\mathcal{O}_\\mathcal{Y})}(\n\\mathcal{G}, f_*\\mathcal{F}).\n$$\nBy uniqueness of adjoints we conclude that\n$f^* = f^{-1}$ where $f^*$ is as defined in\nModules on Sites, Section \\ref{sites-modules-section-functoriality-modules}\nfor the morphism of ringed topoi $f$ above. Of course we could have\nseen this directly because\n$f^*(-) = f^{-1}(-) \\otimes_{f^{-1}\\mathcal{O}_\\mathcal{Y}}\n\\mathcal{O}_\\mathcal{X}$ and because\n$f^{-1}\\mathcal{O}_\\mathcal{Y} = \\mathcal{O}_\\mathcal{X}$.\n\n\\medskip\\noindent\nSimilarly for sheaves of modules in the Zariski, \\'etale, smooth, syntomic\ntopology.\n\n\n\n\\section{Representable categories}\n\\label{section-representable}\n\n\\noindent\nIn this short section we compare our definitions with what happens\nin case the algebraic stacks in question are representable.\n\n\\begin{lemma}\n\\label{lemma-compare-with-scheme}\nLet $S$ be a scheme. Let $\\mathcal{X}$ be a category fibred\nin groupoids over $(\\Sch/S)$. Assume $\\mathcal{X}$ is representable\nby a scheme $X$. For $\\tau \\in \\{Zar,\\linebreak[0] \\etale,\\linebreak[0]\nsmooth,\\linebreak[0] syntomic,\\linebreak[0] fppf\\}$\nthere is a canonical equivalence\n$$\n(\\mathcal{X}_\\tau, \\mathcal{O}_\\mathcal{X}) =\n((\\Sch/X)_\\tau, \\mathcal{O}_X)\n$$\nof ringed sites.\n\\end{lemma}\n\n\\begin{proof}\nThis follows by choosing an equivalence\n$(\\Sch/X)_\\tau \\to \\mathcal{X}$ of categories fibred in groupoids\nover $(\\Sch/S)_{fppf}$ and using the functoriality of\nthe construction $\\mathcal{X} \\leadsto \\mathcal{X}_\\tau$.\n\\end{proof}\n\n\\begin{lemma}\n\\label{lemma-compare-with-morphism-of-schemes}\nLet $S$ be a scheme. Let $f : \\mathcal{X} \\to \\mathcal{Y}$ be a morphism\nof categories fibred in groupoids over $S$.\nAssume $\\mathcal{X}$, $\\mathcal{Y}$ are representable by schemes\n$X$, $Y$. Let $f : X \\to Y$ be the morphism of schemes corresponding\nto $f$. For $\\tau \\in \\{Zar,\\linebreak[0] \\etale,\\linebreak[0]\nsmooth,\\linebreak[0] syntomic,\\linebreak[0] fppf\\}$\nthe morphism of ringed topoi\n$f : (\\Sh(\\mathcal{X}_\\tau), \\mathcal{O}_\\mathcal{X}) \\to\n(\\Sh(\\mathcal{Y}_\\tau), \\mathcal{O}_\\mathcal{Y})$\nagrees with the morphism of ringed topoi\n$f : (\\Sh((\\Sch/X)_\\tau), \\mathcal{O}_X) \\to \n(\\Sh((\\Sch/Y)_\\tau), \\mathcal{O}_Y)$ via the identifications of\nLemma \\ref{lemma-compare-with-scheme}.\n\\end{lemma}\n\n\\begin{proof}\nFollows by unwinding the definitions.\n\\end{proof}\n\n\n\n\n\\section{Restriction}\n\\label{section-restriction}\n\n\n\\noindent\nA trivial but useful observation is that the localization\nof a category fibred in groupoids at an object\nis equivalent to the big site of the scheme it lies over.\n\n\\begin{lemma}\n\\label{lemma-localizing}\nLet $p : \\mathcal{X} \\to (\\Sch/S)_{fppf}$ be a category fibred\nin groupoids. Let $\\tau \\in \\{Zar, \\etale, smooth, syntomic, fppf\\}$.\nLet $x \\in \\Ob(\\mathcal{X})$ lying over $U = p(x)$.\nThe functor $p$ induces an equivalence of sites\n$\\mathcal{X}_\\tau/x \\to (\\Sch/U)_\\tau$.\n\\end{lemma}\n\n\\begin{proof}\nSpecial case of Stacks, Lemma \\ref{stacks-lemma-localizing}.\n\\end{proof}\n\n\\noindent\nWe use the lemma above to talk about the pullback and the restriction\nof a (pre)sheaf to a scheme.\n\n\\begin{definition}\n\\label{definition-pullback}\nLet $p : \\mathcal{X} \\to (\\Sch/S)_{fppf}$ be a category fibred\nin groupoids. Let $x \\in \\Ob(\\mathcal{X})$ lying over $U = p(x)$.\nLet $\\mathcal{F}$ be a presheaf on $\\mathcal{X}$.\n\\begin{enumerate}\n\\item The {\\it pullback $x^{-1}\\mathcal{F}$ of $\\mathcal{F}$} is the\nrestriction $\\mathcal{F}|_{(\\mathcal{X}/x)}$ viewed as a presheaf on\n$(\\Sch/U)_{fppf}$ via the equivalence\n$\\mathcal{X}/x \\to (\\Sch/U)_{fppf}$ of\nLemma \\ref{lemma-localizing}.\n\\item The {\\it restriction of $\\mathcal{F}$ to $U_\\etale$}\nis $x^{-1}\\mathcal{F}|_{U_\\etale}$, abusively written\n$\\mathcal{F}|_{U_\\etale}$.\n\\end{enumerate}\n\\end{definition}\n\n\\noindent\nThis notation makes sense because to the object $x$ the $2$-Yoneda lemma, see\nAlgebraic Stacks, Section \\ref{algebraic-section-2-yoneda}\nassociates a $1$-morphism $x : (\\Sch/U)_{fppf} \\to \\mathcal{X}/x$\nwhich is quasi-inverse to $p : \\mathcal{X}/x \\to (\\Sch/U)_{fppf}$.\nHence $x^{-1}\\mathcal{F}$ truly is the pullback of $\\mathcal{F}$ via this\n$1$-morphism. In particular, by the material above, if $\\mathcal{F}$\nis a sheaf (or a Zariski, \\'etale, smooth, syntomic sheaf), then\n$x^{-1}\\mathcal{F}$ is a sheaf on $(\\Sch/U)_{fppf}$ (or on\n$(\\Sch/U)_{Zar}$, $(\\Sch/U)_\\etale$,\n$(\\Sch/U)_{smooth}$, $(\\Sch/U)_{syntomic}$).\n\n\\medskip\\noindent\nLet $p : \\mathcal{X} \\to (\\Sch/S)_{fppf}$ be a category fibred\nin groupoids. Let $\\varphi : x \\to y$ be a morphism of $\\mathcal{X}$\nlying over the morphism of schemes $a : U \\to V$.\nRecall that $a$ induces a morphism of small \\'etale\nsites $a_{small} : U_\\etale \\to V_\\etale$, see\n\\'Etale Cohomology, Section \\ref{etale-cohomology-section-functoriality}.\nLet $\\mathcal{F}$ be a presheaf on $\\mathcal{X}$.\nLet $\\mathcal{F}|_{U_\\etale}$ and\n$\\mathcal{F}|_{V_\\etale}$ be the restrictions of $\\mathcal{F}$\nvia $x$ and $y$. There is a natural {\\it comparison} map\n\\begin{equation}\n\\label{equation-comparison-push}\nc_\\varphi :\n\\mathcal{F}|_{V_\\etale}\n\\longrightarrow\na_{small, *}(\\mathcal{F}|_{U_\\etale})\n\\end{equation}\nof presheaves on $U_\\etale$. Namely, if $V' \\to V$ is \\'etale,\nset $U' = V' \\times_V U$ and define $c_\\varphi$ on sections over $V'$\nvia\n$$\n\\xymatrix{\na_{small, *}(\\mathcal{F}|_{U_\\etale})(V') &\n\\mathcal{F}|_{U_\\etale}(U') \\ar@{=}[l] &\n\\mathcal{F}(x') \\ar@{=}[l] \\\\\n\\mathcal{F}|_{V_\\etale}(V') \\ar@{=}[rr] \\ar[u]^{c_\\varphi} &\n&\n\\mathcal{F}(y') \\ar[u]_{\\mathcal{F}(\\varphi')}\n}\n$$\nHere $\\varphi' : x' \\to y'$ is a morphism of $\\mathcal{X}$\nfitting into a commutative diagram\n$$\n\\vcenter{\n\\xymatrix{\nx' \\ar[r] \\ar[d]_{\\varphi'} & x \\ar[d]^\\varphi \\\\\ny' \\ar[r] & y\n}\n}\n\\quad\\text{lying over}\\quad\n\\vcenter{\n\\xymatrix{\nU' \\ar[r] \\ar[d] & U \\ar[d]^a \\\\\nV' \\ar[r] & V\n}\n}\n$$\nThe existence and uniqueness of $\\varphi'$ follow from the axioms\nof a category fibred in groupoids.\nWe omit the verification that $c_\\varphi$ so defined is indeed a map\nof presheaves (i.e., compatible with restriction mappings) and that\nit is functorial in $\\mathcal{F}$. In case $\\mathcal{F}$ is a sheaf for the\n\\'etale topology we obtain a {\\it comparison} map\n\\begin{equation}\n\\label{equation-comparison}\nc_\\varphi : a_{small}^{-1}(\\mathcal{F}|_{V_\\etale})\n\\longrightarrow\n\\mathcal{F}|_{U_\\etale}\n\\end{equation}\nwhich is also denoted $c_\\varphi$ as indicated (this is the customary abuse\nof notation in not distinguishing between adjoint maps).\n\n\\begin{lemma}\n\\label{lemma-comparison}\nLet $\\mathcal{F}$ be an \\'etale sheaf on $\\mathcal{X} \\to (\\Sch/S)_{fppf}$.\n\\begin{enumerate}\n\\item If $\\varphi : x \\to y$ and $\\psi : y \\to z$\nare morphisms of $\\mathcal{X}$ lying over $a : U \\to V$ and\n$b : V \\to W$, then the composition\n$$\na_{small}^{-1}(b_{small}^{-1} (\\mathcal{F}|_{W_\\etale}))\n\\xrightarrow{a_{small}^{-1}c_\\psi}\na_{small}^{-1}(\\mathcal{F}|_{V_\\etale})\n\\xrightarrow{c_\\varphi}\n\\mathcal{F}|_{U_\\etale}\n$$\nis equal to $c_{\\psi \\circ \\varphi}$ via the identification\n$$\n(b \\circ a)_{small}^{-1}(\\mathcal{F}|_{W_\\etale}) =\na_{small}^{-1}(b_{small}^{-1} (\\mathcal{F}|_{W_\\etale})).\n$$\n\\item If $\\varphi : x \\to y$ lies over an \\'etale morphism of schemes\n$a : U \\to V$, then (\\ref{equation-comparison}) is an isomorphism.\n\\item Suppose $f : \\mathcal{Y} \\to \\mathcal{X}$ is a $1$-morphism of\ncategories fibred in groupoids over $(\\Sch/S)_{fppf}$ and $y$ is\nan object of $\\mathcal{Y}$ lying over the scheme $U$ with image\n$x = f(y)$. Then there is a canonical identification\n$f^{-1}\\mathcal{F}|_{U_\\etale} = \\mathcal{F}|_{U_\\etale}$.\n\\item Moreover, given $\\psi : y' \\to y$ in $\\mathcal{Y}$ lying over\n$a : U' \\to U$ the comparison map\n$c_\\psi : a_{small}^{-1}(F^{-1}\\mathcal{F}|_{U_\\etale}) \\to\nF^{-1}\\mathcal{F}|_{U'_\\etale}$ is equal to the\ncomparison map $c_{f(\\psi)} : a_{small}^{-1}\\mathcal{F}|_{U_\\etale}\n\\to \\mathcal{F}|_{U'_\\etale}$ via the identifications in (3).\n\\end{enumerate}\n\\end{lemma}\n\n\\begin{proof}\nThe verification of these properties is omitted.\n\\end{proof}\n\n\\noindent\nNext, we turn to the restriction of (pre)sheaves of modules.\n\n\\begin{lemma}\n\\label{lemma-localizing-structure-sheaf}\nLet $p : \\mathcal{X} \\to (\\Sch/S)_{fppf}$ be a category fibred\nin groupoids. Let $\\tau \\in \\{Zar, \\etale, smooth, syntomic, fppf\\}$.\nLet $x \\in \\Ob(\\mathcal{X})$ lying over $U = p(x)$.\nThe equivalence of\nLemma \\ref{lemma-localizing}\nextends to an equivalence of ringed sites\n$(\\mathcal{X}_\\tau/x, \\mathcal{O}_\\mathcal{X}|_x) \\to\n((\\Sch/U)_\\tau, \\mathcal{O})$.\n\\end{lemma}\n\n\\begin{proof}\nThis is immediate from the construction of the structure sheaves.\n\\end{proof}\n\n\\noindent\nLet $\\mathcal{X}$ be a category fibred in groupoids over $(\\Sch/S)_{fppf}$.\nLet $\\mathcal{F}$ be a (pre)sheaf of modules on $\\mathcal{X}$ as in\nDefinition \\ref{definition-modules}.\nLet $x$ be an object of $\\mathcal{X}$ lying over $U$. Then\nLemma \\ref{lemma-localizing-structure-sheaf}\nguarantees that the restriction\n$x^{-1}\\mathcal{F}$ is a (pre)sheaf of modules on $(\\Sch/U)_{fppf}$.\nWe will sometimes write $x^*\\mathcal{F} = x^{-1}\\mathcal{F}$ in this case.\nSimilarly, if $\\mathcal{F}$ is a sheaf for the Zariski, \\'etale, smooth,\nor syntomic topology, then $x^{-1}\\mathcal{F}$ is as well. Moreover, the\nrestriction\n$\\mathcal{F}|_{U_\\etale} = x^{-1}\\mathcal{F}|_{U_\\etale}$\nto $U$ is a presheaf of $\\mathcal{O}_{U_\\etale}$-modules.\nIf $\\mathcal{F}$ is a sheaf for the \\'etale topology, then\n$\\mathcal{F}|_{U_\\etale}$ is a sheaf of modules. Moreover,\nif $\\varphi : x \\to y$ is a morphism of $\\mathcal{X}$ lying over\n$a : U \\to V$ then the comparison map (\\ref{equation-comparison})\nis compatible with $a_{small}^\\sharp$ (see\nDescent, Remark \\ref{descent-remark-change-topologies-ringed})\nand induces a {\\it comparison} map\n\\begin{equation}\n\\label{equation-comparison-modules}\nc_\\varphi : a_{small}^*(\\mathcal{F}|_{V_\\etale})\n\\longrightarrow\n\\mathcal{F}|_{U_\\etale}\n\\end{equation}\nof $\\mathcal{O}_{U_\\etale}$-modules.\nNote that the properties (1), (2), (3), and (4) of\nLemma \\ref{lemma-comparison}\nhold in the setting of \\'etale sheaves of modules as well.\nWe will use this in the following without further mention.\n\n\\begin{lemma}\n\\label{lemma-enough-points}\nLet $p : \\mathcal{X} \\to (\\Sch/S)_{fppf}$ be a category fibred\nin groupoids. Let $\\tau \\in \\{Zar, \\etale, smooth, syntomic, fppf\\}$.\nThe site $\\mathcal{X}_\\tau$ has enough points.\n\\end{lemma}\n\n\\begin{proof}\nBy\nSites, Lemma \\ref{sites-lemma-enough-points-local}\nwe have to show that there exists a family of objects $x$ of $\\mathcal{X}$\nsuch that $\\mathcal{X}_\\tau/x$ has enough points and such that the sheaves\n$h_x^\\#$ cover the final object of the category of sheaves.\nBy\nLemma \\ref{lemma-localizing}\nand\n\\'Etale Cohomology, Lemma \\ref{etale-cohomology-lemma-points-fppf}\nwe see that $\\mathcal{X}_\\tau/x$ has enough points for every object\n$x$ and we win.\n\\end{proof}\n\n\n\n\n\n\n\n\\section{Restriction to algebraic spaces}\n\\label{section-restriction-algebraic-spaces}\n\n\\noindent\nIn this section we consider sheaves on categories representable by\nalgebraic spaces. The following lemma is the analogue of\nTopologies, Lemma \\ref{topologies-lemma-at-the-bottom-etale}\nfor algebraic spaces.\n\n\\begin{lemma}\n\\label{lemma-compare}\nLet $S$ be a scheme. Let $\\mathcal{X} \\to (\\Sch/S)_{fppf}$ be a category\nfibred in groupoids. Assume $\\mathcal{X}$ is representable by an algebraic\nspace $F$. Then there exists a continuous and cocontinuous functor\n$\nF_\\etale \\to \\mathcal{X}_\\etale\n$\nwhich induces a morphism of ringed sites\n$$\n\\pi_F :\n(\\mathcal{X}_\\etale, \\mathcal{O}_\\mathcal{X})\n\\longrightarrow\n(F_\\etale, \\mathcal{O}_F)\n$$\nand a morphism of ringed topoi\n$$\ni_F :\n(\\Sh(F_\\etale), \\mathcal{O}_F)\n\\longrightarrow\n(\\Sh(\\mathcal{X}_\\etale), \\mathcal{O}_\\mathcal{X})\n$$\nsuch that $\\pi_F \\circ i_F = \\text{id}$. Moreover $\\pi_{F, *} = i_F^{-1}$.\n\\end{lemma}\n\n\\begin{proof}\nChoose an equivalence $j : \\mathcal{S}_F \\to \\mathcal{X}$, see\nAlgebraic Stacks, Sections \\ref{algebraic-section-split} and\n\\ref{algebraic-section-representable-by-algebraic-spaces}.\nAn object of $F_\\etale$ is a scheme $U$ together with an\n\\'etale morphism $\\varphi : U \\to F$. Then $\\varphi$ is an object\nof $\\mathcal{S}_F$ over $U$. Hence $j(\\varphi)$ is an object of\n$\\mathcal{X}$ over $U$. In this way $j$ induces a functor\n$u : F_\\etale \\to \\mathcal{X}$. It is clear that\n$u$ is continuous and cocontinuous for the \\'etale topology on\n$\\mathcal{X}$. Since $j$ is an equivalence, the functor $u$ is fully\nfaithful. Also, fibre products and equalizers exist in $F_\\etale$\nand $u$ commutes with them because these are computed on the level\nof underlying schemes in $F_\\etale$. Thus\nSites, Lemmas \\ref{sites-lemma-when-shriek},\n\\ref{sites-lemma-preserve-equalizers}, and\n\\ref{sites-lemma-back-and-forth}\napply. In particular $u$ defines a morphism of topoi\n$i_F : \\Sh(F_\\etale) \\to \\Sh(\\mathcal{X}_\\etale)$\nand there exists a left adjoint $i_{F, !}$ of $i_F^{-1}$ which commutes\nwith fibre products and equalizers.\n\n\\medskip\\noindent\nWe claim that $i_{F, !}$ is exact. If this is true, then we can define\n$\\pi_F$ by the rules $\\pi_F^{-1} = i_{F, !}$ and $\\pi_{F, *} = i_F^{-1}$\nand everything is clear. To prove the claim, note that we already know\nthat $i_{F, !}$\nis right exact and preserves fibre products. Hence it suffices to show\nthat $i_{F, !}* = *$ where $*$ indicates the final object in the category\nof sheaves of sets. Let $U$ be a scheme and let\n$\\varphi : U \\to F$ be surjective and \\'etale. Set $R = U \\times_F U$.\nThen\n$$\n\\xymatrix{\nh_R \\ar@<1ex>[r] \\ar@<-1ex>[r] & h_U \\ar[r] & {*}\n}\n$$\nis a coequalizer diagram in $\\Sh(F_\\etale)$. Using the\nright exactness of $i_{F, !}$, using $i_{F, !} = (u_p\\ )^\\#$, and using\nSites, Lemma \\ref{sites-lemma-pullback-representable-presheaf}\nwe see that\n$$\n\\xymatrix{\nh_{u(R)} \\ar@<1ex>[r] \\ar@<-1ex>[r] & h_{u(U)} \\ar[r] & i_{F, !}{*}\n}\n$$\nis a coequalizer diagram in $\\Sh(F_\\etale)$. Using that\n$j$ is an equivalence and that $F = U/R$ it follows that\nthe coequalizer in $\\Sh(\\mathcal{X}_\\etale)$ of the\ntwo maps $h_{u(R)} \\to h_{u(U)}$ is $*$. We omit the proof that\nthese morphisms are compatible with structure sheaves.\n\\end{proof}\n\n\\noindent\nAssume $\\mathcal{X}$ is an algebraic stack represented by the\nalgebraic space $F$.\nLet $j : \\mathcal{S}_F \\to \\mathcal{X}$ be an equivalence and denote\n$u : F_\\etale \\to \\mathcal{X}_\\etale$ the\nfunctor of the proof of Lemma \\ref{lemma-compare} above.\nGiven a sheaf $\\mathcal{F}$ on $\\mathcal{X}_\\etale$ we have\n$$\n\\pi_{F, *}\\mathcal{F}(U) = i_F^{-1}\\mathcal{F}(U) = \\mathcal{F}(u(U)).\n$$\nThis is why we often think of $i_F^{-1}$ as a {\\it restriction functor}\nsimilarly to\nDefinition \\ref{definition-pullback}\nand to the restriction of a sheaf on the big \\'etale site of\na scheme to the small \\'etale site of a scheme. We often use the notation\n\\begin{equation}\n\\label{equation-restrict}\n\\mathcal{F}|_{F_\\etale} = i_F^{-1}\\mathcal{F} = \\pi_{F, *}\\mathcal{F}\n\\end{equation}\nin this situation.\n\n\\begin{lemma}\n\\label{lemma-compare-morphism}\nLet $S$ be a scheme. Let $f : \\mathcal{X} \\to \\mathcal{Y}$ be a morphism\nof categories fibred in groupoids over $(\\Sch/S)_{fppf}$. Assume\n$\\mathcal{X}$, $\\mathcal{Y}$ are representable by algebraic spaces $F$, $G$.\nDenote $f : F \\to G$ the induced morphism of algebraic spaces, and\n$f_{small} : F_\\etale \\to G_\\etale$\nthe corresponding morphism of ringed topoi. Then\n$$\n\\xymatrix{\n(\\Sh(\\mathcal{X}_\\etale), \\mathcal{O}_\\mathcal{X})\n\\ar[d]_{\\pi_F} \\ar[rr]_f & &\n(\\Sh(\\mathcal{Y}_\\etale), \\mathcal{O}_\\mathcal{Y}) \\ar[d]^{\\pi_G} \\\\\n(\\Sh(F_\\etale), \\mathcal{O}_F) \\ar[rr]^{f_{small}} & &\n(\\Sh(G_\\etale), \\mathcal{O}_G)\n}\n$$\nis a commutative diagram of ringed topoi.\n\\end{lemma}\n\n\\begin{proof}\nThis is similar to\nTopologies, Lemma \\ref{topologies-lemma-morphism-big-small-etale} (3)\nbut there is a small snag due to the fact that $F \\to G$ may not be\nrepresentable by schemes. In particular we don't get a commutative diagram\nof ringed sites, but only a commutative diagram of ringed topoi.\n\n\\medskip\\noindent\nBefore we start the proof proper, we choose equivalences\n$j : \\mathcal{S}_F \\to \\mathcal{X}$ and\n$j' : \\mathcal{S}_G \\to \\mathcal{Y}$ which induce functors\n$u : F_\\etale \\to \\mathcal{X}$ and\n$u' : G_\\etale \\to \\mathcal{Y}$ as in the proof of\nLemma \\ref{lemma-compare}. Because of the 2-functoriality of\nsheaves on categories fibred in groupoids over $\\Sch_{fppf}$\n(see discussion in Section \\ref{section-presheaves})\nwe may assume that $\\mathcal{X} = \\mathcal{S}_F$ and\n$\\mathcal{Y} = \\mathcal{S}_G$ and that $f : \\mathcal{S}_F \\to \\mathcal{S}_G$\nis the functor associated to the morphism $f : F \\to G$. Correspondingly\nwe will omit $u$ and $u'$ from the notation, i.e., given an object\n$U \\to F$ of $F_\\etale$ we denote $U/F$\nthe corresponding object of $\\mathcal{X}$. Similarly for $G$.\n\n\\medskip\\noindent\nLet $\\mathcal{G}$ be a sheaf on $\\mathcal{X}_\\etale$.\nTo prove (2) we compute $\\pi_{G, *}f_*\\mathcal{G}$ and\n$f_{small, *}\\pi_{F, *}\\mathcal{G}$. To do this let $V \\to G$ be an object\nof $G_\\etale$. Then\n$$\n\\pi_{G, *}f_*\\mathcal{G}(V) = f_*\\mathcal{G}(V/G) =\n\\Gamma\\Big(\n(\\Sch/V)_{fppf} \\times_{\\mathcal{Y}} \\mathcal{X},\n\\ \\text{pr}^{-1}\\mathcal{G}\\Big)\n$$\nsee (\\ref{equation-pushforward}). The fibre product in the formula is\n$$\n(\\Sch/V)_{fppf} \\times_{\\mathcal{Y}} \\mathcal{X} =\n(\\Sch/V)_{fppf} \\times_{\\mathcal{S}_G} \\mathcal{S}_F =\n\\mathcal{S}_{V \\times_G F}\n$$\ni.e., it is the split category fibred in groupoids associated to the\nalgebraic space $V \\times_G F$. And $\\text{pr}^{-1}\\mathcal{G}$ is a\nsheaf on $\\mathcal{S}_{V \\times_G F}$ for the \\'etale topology.\n\n\\medskip\\noindent\nIn particular, if $V \\times_G F$ is representable, i.e., if it is a scheme,\nthen $\\pi_{G, *}f_*\\mathcal{G}(V) = \\mathcal{G}(V \\times_G F/F)$ and\nalso\n$$\nf_{small, *}\\pi_{F, *}\\mathcal{G}(V) =\n\\pi_{F, *}\\mathcal{G}(V \\times_G F) =\n\\mathcal{G}(V \\times_G F/F)\n$$\nwhich proves the desired equality in this special case.\n\n\\medskip\\noindent\nIn general, choose a scheme $U$ and a surjective \\'etale morphism\n$U \\to V \\times_G F$. Set $R = U \\times_{V \\times_G F} U$. Then\n$U/V \\times_G F$ and $R/V \\times_G F$ are objects of the fibre\nproduct category above. Since $\\text{pr}^{-1}\\mathcal{G}$ is a\nsheaf for the \\'etale topology on $\\mathcal{S}_{V \\times_G F}$\nthe diagram\n$$\n\\xymatrix{\n\\Gamma\\Big(\n(\\Sch/V)_{fppf} \\times_{\\mathcal{Y}} \\mathcal{X},\n\\ \\text{pr}^{-1}\\mathcal{G}\\Big)\n\\ar[r] &\n\\text{pr}^{-1}\\mathcal{G}(U/V \\times_G F) \\ar@<1ex>[r] \\ar@<-1ex>[r] &\n\\text{pr}^{-1}\\mathcal{G}(R/V \\times_G F)\n}\n$$\nis an equalizer diagram. Note that\n$\\text{pr}^{-1}\\mathcal{G}(U/V \\times_G F) = \\mathcal{G}(U/F)$\nand $\\text{pr}^{-1}\\mathcal{G}(R/V \\times_G F) = \\mathcal{G}(R/F)$\nby the definition of pullbacks. Moreover, by the material in\nProperties of Spaces, Section \\ref{spaces-properties-section-etale-site}\n(especially,\nProperties of Spaces,\nRemark \\ref{spaces-properties-remark-explain-equivalence} and\nLemma \\ref{spaces-properties-lemma-functoriality-etale-site})\nwe see that there is an equalizer diagram\n$$\n\\xymatrix{\nf_{small, *}\\pi_{F, *}\\mathcal{G}(V)\n\\ar[r] &\n\\pi_{F, *}\\mathcal{G}(U/F) \\ar@<1ex>[r] \\ar@<-1ex>[r] &\n\\pi_{F, *}\\mathcal{G}(R/F)\n}\n$$\nSince we also have $\\pi_{F, *}\\mathcal{G}(U/F) = \\mathcal{G}(U/F)$\nand $\\pi_{F, *}\\mathcal{G}(U/F) = \\mathcal{G}(U/F)$\nwe obtain a canonical identification\n$f_{small, *}\\pi_{F, *}\\mathcal{G}(V) = \\pi_{G, *}f_*\\mathcal{G}(V)$.\nWe omit the proof that this is compatible with restriction mappings\nand that it is functorial in $\\mathcal{G}$.\n\\end{proof}\n\n\\noindent\nLet $f : \\mathcal{X} \\to \\mathcal{Y}$ and $f : F \\to G$ be as in the\nsecond part of the lemma above. A consequence of the lemma, using\n(\\ref{equation-restrict}), is that\n\\begin{equation}\n\\label{equation-compare-big-small}\n(f_*\\mathcal{F})|_{G_\\etale} =\nf_{small, *}(\\mathcal{F}|_{F_\\etale})\n\\end{equation}\nfor any sheaf $\\mathcal{F}$ on $\\mathcal{X}_\\etale$.\nMoreover, if $\\mathcal{F}$ is a sheaf of $\\mathcal{O}$-modules, then\n(\\ref{equation-compare-big-small}) is an isomorphism of\n$\\mathcal{O}_G$-modules on $G_\\etale$.\n\n\\medskip\\noindent\nFinally, suppose that we have a $2$-commutative diagram\n$$\n\\xymatrix{\n\\mathcal{U} \\ar[r]^a \\ar[dr]_f \\drtwocell<\\omit>{<-2>\\varphi} &\n\\mathcal{V} \\ar[d]^g \\\\\n& \\mathcal{X}\n}\n$$\nof $1$-morphisms of categories fibred in groupoids over $(\\Sch/S)_{fppf}$,\nthat $\\mathcal{F}$ is a sheaf on $\\mathcal{X}_\\etale$,\nand that $\\mathcal{U}, \\mathcal{V}$ are representable by algebraic\nspaces $U, V$. Then we obtain a comparison map\n\\begin{equation}\n\\label{equation-comparison-algebraic-spaces}\nc_\\varphi : a_{small}^{-1}(g^{-1}\\mathcal{F}|_{V_\\etale})\n\\longrightarrow\nf^{-1}\\mathcal{F}|_{U_\\etale}\n\\end{equation}\nwhere $a : U \\to V$ denotes the morphism of algebraic spaces corresponding\nto $a$. This is the analogue of (\\ref{equation-comparison}). We define\n$c_\\varphi$ as the adjoint to the map\n$$\ng^{-1}\\mathcal{F}|_{V_\\etale}\n\\longrightarrow\na_{small, *}(f^{-1}\\mathcal{F}|_{U_\\etale}) =\n(a_*f^{-1}\\mathcal{F})|_{V_\\etale}\n$$\n(equality by (\\ref{equation-compare-big-small}))\nwhich is the restriction to $V$ (\\ref{equation-restrict}) of the map\n$$\ng^{-1}\\mathcal{F} \\to a_*a^{-1}g^{-1}\\mathcal{F} = a_*f^{-1}\\mathcal{F}\n$$\nwhere the last equality uses the $2$-commutativity of the diagram above.\nIn case $\\mathcal{F}$ is a sheaf of $\\mathcal{O}_\\mathcal{X}$-modules\n$c_\\varphi$ induces a {\\it comparison} map\n\\begin{equation}\n\\label{equation-comparison-algebraic-spaces-modules}\nc_\\varphi : a_{small}^*(g^*\\mathcal{F}|_{V_\\etale})\n\\longrightarrow\nf^*\\mathcal{F}|_{U_\\etale}\n\\end{equation}\nof $\\mathcal{O}_{U_\\etale}$-modules.\nNote that the properties (1), (2), (3), and (4) of\nLemma \\ref{lemma-comparison}\nhold in this setting as well.\n\n\n\n\n\n\n\n\n\n\n\n\n\\section{Quasi-coherent modules}\n\\label{section-quasi-coherent}\n\n\\noindent\nAt this point we can apply the general definition of a quasi-coherent\nmodule to the situation discussed in this chapter.\n\n\\begin{definition}\n\\label{definition-quasi-coherent}\nLet $p : \\mathcal{X} \\to (\\Sch/S)_{fppf}$ be a category fibred\nin groupoids. A {\\it quasi-coherent module on $\\mathcal{X}$}, or a\n{\\it quasi-coherent $\\mathcal{O}_\\mathcal{X}$-module} is a\nquasi-coherent module on the ringed site\n$(\\mathcal{X}_{fppf}, \\mathcal{O}_\\mathcal{X})$ as in\nModules on Sites, Definition \\ref{sites-modules-definition-site-local}.\nThe category of quasi-coherent sheaves on $\\mathcal{X}$\nis denoted $\\QCoh(\\mathcal{O}_\\mathcal{X})$.\n\\end{definition}\n\n\\noindent\nIf $\\mathcal{X}$ is an algebraic stack, then this definition agrees with all\ndefinitions in the literature in the sense that $\\QCoh(\\mathcal{O}_\\mathcal{X})$\nis equivalent (modulo set theoretic issues) to any variant of this category\ndefined in the literature. For example, we will match our definition with\nthe definition in \\cite[Definition 6.1]{olsson_sheaves} in\nCohomology on Stacks, Lemma \\ref{lemma-quasi-coherent}.\nWe will also see alternative constructions of this category later on.\n\n\\medskip\\noindent\nIn general (as is the case for morphisms of schemes) the pushforward\nof quasi-coherent sheaf along a $1$-morphism is not quasi-coherent.\nPullback does preserve quasi-coherence.\n\n\\begin{lemma}\n\\label{lemma-pullback-quasi-coherent}\nLet $f : \\mathcal{X} \\to \\mathcal{Y}$ be a $1$-morphism of categories\nfibred in groupoids over $(\\Sch/S)_{fppf}$.\nThe pullback functor\n$f^* = f^{-1} : \\textit{Mod}(\\mathcal{O}_\\mathcal{Y}) \\to\n\\textit{Mod}(\\mathcal{O}_\\mathcal{X})$\npreserves quasi-coherent sheaves.\n\\end{lemma}\n\n\\begin{proof}\nThis is a general fact, see\nModules on Sites, Lemma \\ref{sites-modules-lemma-local-pullback}.\n\\end{proof}\n\n\\noindent\nIt turns out that quasi-coherent sheaves have a very simple\ncharacterization in terms of their pullbacks. See also\nLemma \\ref{lemma-quasi-coherent}\nfor a characterization in terms of restrictions.\n\n\\begin{lemma}\n\\label{lemma-characterize-quasi-coherent}\nLet $p : \\mathcal{X} \\to (\\Sch/S)_{fppf}$ be a category\nfibred in groupoids. Let $\\mathcal{F}$\nbe a sheaf of $\\mathcal{O}_\\mathcal{X}$-modules. Then $\\mathcal{F}$\nis quasi-coherent if and only if $x^*\\mathcal{F}$ is a quasi-coherent\nsheaf on $(\\Sch/U)_{fppf}$ for every object $x$ of\n$\\mathcal{X}$ with $U = p(x)$.\n\\end{lemma}\n\n\\begin{proof}\nBy\nLemma \\ref{lemma-pullback-quasi-coherent}\nthe condition is necessary. Conversely, since $x^*\\mathcal{F}$\nis just the restriction to $\\mathcal{X}_{fppf}/x$ we see that it\nis sufficient directly from the definition of a quasi-coherent sheaf\n(and the fact that the notion of being quasi-coherent is an intrinsic\nproperty of sheaves of modules, see\nModules on Sites, Section \\ref{sites-modules-section-intrinsic}).\n\\end{proof}\n\n\\noindent\nAlthough there is a variant for the Zariski topology, it seems\nthat the \\'etale topology is the natural topology to use in the\nfollowing definition.\n\n\\begin{definition}\n\\label{definition-locally-quasi-coherent}\nLet $p : \\mathcal{X} \\to (\\Sch/S)_{fppf}$ be a category\nfibred in groupoids. Let $\\mathcal{F}$\nbe a presheaf of $\\mathcal{O}_\\mathcal{X}$-modules.\nWe say $\\mathcal{F}$ is {\\it locally quasi-coherent}\\footnote{This is\nnonstandard notation.} if\n$\\mathcal{F}$ is a sheaf for the \\'etale topology and\nfor every object $x$ of $\\mathcal{X}$ the restriction\n$x^*\\mathcal{F}|_{U_\\etale}$ is a quasi-coherent\nsheaf. Here $U = p(x)$.\n\\end{definition}\n\n\\noindent\nWe use $\\textit{LQCoh}(\\mathcal{O}_\\mathcal{X})$ to indicate the category of\nlocally quasi-coherent modules. We now have the following diagram\nof categories of modules\n$$\n\\xymatrix{\n\\QCoh(\\mathcal{O}_\\mathcal{X}) \\ar[r] \\ar[d] &\n\\textit{Mod}(\\mathcal{O}_\\mathcal{X}) \\ar[d] \\\\\n\\textit{LQCoh}(\\mathcal{O}_\\mathcal{X}) \\ar[r] &\n\\textit{Mod}(\\mathcal{X}_\\etale, \\mathcal{O}_\\mathcal{X})\n}\n$$\nwhere the arrows are strictly full embeddings.\nIt turns out that many results for quasi-coherent sheaves have a\ncounter part for locally quasi-coherent modules. Moreover, from many\npoints of view (as we shall see later) this is a natural category to consider.\nFor example the quasi-coherent sheaves are exactly those\nlocally quasi-coherent modules that are ``cartesian'', i.e., satisfy\nthe second condition of the lemma below.\n\n\\begin{lemma}\n\\label{lemma-quasi-coherent}\nLet $p : \\mathcal{X} \\to (\\Sch/S)_{fppf}$ be a category\nfibred in groupoids. Let $\\mathcal{F}$\nbe a presheaf of $\\mathcal{O}_\\mathcal{X}$-modules. Then $\\mathcal{F}$\nis quasi-coherent if and only if the following two conditions hold\n\\begin{enumerate}\n\\item $\\mathcal{F}$ is locally quasi-coherent, and\n\\item for any morphism $\\varphi : x \\to y$ of $\\mathcal{X}$ lying over\n$f : U \\to V$ the comparison map\n$c_\\varphi : f_{small}^*\\mathcal{F}|_{V_\\etale} \\to\n\\mathcal{F}|_{U_\\etale}$ of\n(\\ref{equation-comparison-modules}) is an isomorphism.\n\\end{enumerate}\n\\end{lemma}\n\n\\begin{proof}\nAssume $\\mathcal{F}$ is quasi-coherent. Then $\\mathcal{F}$ is a sheaf\nfor the fppf topology, hence a sheaf for the \\'etale topology. Moreover,\nany pullback of $\\mathcal{F}$ to a ringed topos is quasi-coherent, hence\nthe restrictions $x^*\\mathcal{F}|_{U_\\etale}$ are quasi-coherent.\nThis proves $\\mathcal{F}$ is locally quasi-coherent.\nLet $y$ be an object of $\\mathcal{X}$ with $V = p(y)$.\nWe have seen that $\\mathcal{X}/y = (\\Sch/V)_{fppf}$. By\nDescent, Proposition \\ref{descent-proposition-equivalence-quasi-coherent}\nit follows that $y^*\\mathcal{F}$ is the quasi-coherent module\nassociated to a (usual) quasi-coherent module $\\mathcal{F}_V$ on\nthe scheme $V$. Hence certainly the comparison maps\n(\\ref{equation-comparison-modules}) are isomorphisms.\n\n\\medskip\\noindent\nConversely, suppose that $\\mathcal{F}$ satisfies (1) and (2).\nLet $y$ be an object of $\\mathcal{X}$ with $V = p(y)$. Denote\n$\\mathcal{F}_V$ the quasi-coherent module on\nthe scheme $V$ corresponding to the restriction\n$y^*\\mathcal{F}|_{V_\\etale}$ which is quasi-coherent by\nassumption (1), see\nDescent, Proposition \\ref{descent-proposition-equivalence-quasi-coherent}.\nCondition (2) now signifies that the restrictions\n$x^*\\mathcal{F}|_{U_\\etale}$ for $x$ over $y$ are each\nisomorphic to the (\\'etale sheaf associated to the) pullback of $\\mathcal{F}_V$\nvia the corresponding morphism of schemes $U \\to V$.\nHence $y^*\\mathcal{F}$ is the sheaf on $(\\Sch/V)_{fppf}$\nassociated to $\\mathcal{F}_V$. Hence it is quasi-coherent (by\nDescent, Proposition \\ref{descent-proposition-equivalence-quasi-coherent}\nagain) and we see that $\\mathcal{F}$ is quasi-coherent on $\\mathcal{X}$ by\nLemma \\ref{lemma-characterize-quasi-coherent}.\n\\end{proof}\n\n\\begin{lemma}\n\\label{lemma-pullback-lqc}\nLet $f : \\mathcal{X} \\to \\mathcal{Y}$ be a $1$-morphism of categories\nfibred in groupoids over $(\\Sch/S)_{fppf}$. The pullback functor\n$f^* = f^{-1} :\n\\textit{Mod}(\\mathcal{Y}_\\etale, \\mathcal{O}_\\mathcal{Y})\n\\to\n\\textit{Mod}(\\mathcal{X}_\\etale, \\mathcal{O}_\\mathcal{X})$\npreserves locally quasi-coherent sheaves.\n\\end{lemma}\n\n\\begin{proof}\nLet $\\mathcal{G}$ be locally quasi-coherent on $\\mathcal{Y}$.\nChoose an object $x$ of $\\mathcal{X}$ lying over the scheme $U$.\nThe restriction $x^*f^*\\mathcal{G}|_{U_\\etale}$ equals\n$(f \\circ x)^*\\mathcal{G}|_{U_\\etale}$\nhence is a quasi-coherent sheaf by assumption on $\\mathcal{G}$.\n\\end{proof}\n\n\\begin{lemma}\n\\label{lemma-lqc-colimits}\nLet $p : \\mathcal{X} \\to (\\Sch/S)_{fppf}$ be a category fibred in\ngroupoids.\n\\begin{enumerate}\n\\item The category $\\textit{LQCoh}(\\mathcal{O}_\\mathcal{X})$\nhas colimits and they agree with colimits in the category\n$\\textit{Mod}(\\mathcal{X}_\\etale, \\mathcal{O}_\\mathcal{X})$.\n\\item The category $\\textit{LQCoh}(\\mathcal{O}_\\mathcal{X})$\nis abelian with kernels and cokernels computed in\n$\\textit{Mod}(\\mathcal{X}_\\etale, \\mathcal{O}_\\mathcal{X})$,\nin other words the inclusion functor is exact.\n\\item Given a short exact sequence\n$0 \\to \\mathcal{F}_1 \\to \\mathcal{F}_2 \\to \\mathcal{F}_3 \\to 0$ of\n$\\textit{Mod}(\\mathcal{X}_\\etale, \\mathcal{O}_\\mathcal{X})$\nif two out of three are locally quasi-coherent so is the third.\n\\item Given $\\mathcal{F}, \\mathcal{G}$ in\n$\\textit{LQCoh}(\\mathcal{O}_\\mathcal{X})$\nthe tensor product $\\mathcal{F} \\otimes_{\\mathcal{O}_\\mathcal{X}} \\mathcal{G}$\nin $\\textit{Mod}(\\mathcal{X}_\\etale, \\mathcal{O}_\\mathcal{X})$\nis an object of $\\textit{LQCoh}(\\mathcal{O}_\\mathcal{X})$.\n\\item Given $\\mathcal{F}, \\mathcal{G}$ in\n$\\textit{LQCoh}(\\mathcal{O}_\\mathcal{X})$\nwith $\\mathcal{F}$ locally of finite presentation on\n$\\mathcal{X}_\\etale$ the sheaf\n$\\SheafHom_{\\mathcal{O}_\\mathcal{X}}(\\mathcal{F}, \\mathcal{G})$\nin $\\textit{Mod}(\\mathcal{X}_\\etale, \\mathcal{O}_\\mathcal{X})$\nis an object of $\\textit{LQCoh}(\\mathcal{O}_\\mathcal{X})$.\n\\end{enumerate}\n\\end{lemma}\n\n\\begin{proof}\nEach of these statements follows from the corresponding statement of\nDescent, Lemma \\ref{descent-lemma-equivalence-quasi-coherent-limits}.\nFor example, suppose that\n$\\mathcal{I} \\to \\textit{LQCoh}(\\mathcal{O}_\\mathcal{X})$,\n$i \\mapsto \\mathcal{F}_i$ is a diagram.\nConsider the object $\\mathcal{F} = \\colim_i \\mathcal{F}_i$ of\n$\\textit{Mod}(\\mathcal{X}_\\etale, \\mathcal{O}_\\mathcal{X})$.\nFor any object $x$ of $\\mathcal{X}$ with $U = p(x)$ the pullback functor\n$x^*$ commutes with all colimits as it is a left adjoint. Hence\n$x^*\\mathcal{F} = \\colim_i x^*\\mathcal{F}_i$. Similarly we have\n$x^*\\mathcal{F}|_{U_\\etale} =\n\\colim_i x^*\\mathcal{F}_i|_{U_\\etale}$.\nNow by assumption each $x^*\\mathcal{F}_i|_{U_\\etale}$\nis quasi-coherent, hence the colimit is quasi-coherent by the\naforementioned\nDescent, Lemma \\ref{descent-lemma-equivalence-quasi-coherent-limits}.\nThis proves (1).\n\n\\medskip\\noindent\nIt follows from (1) that cokernels exist in\n$\\textit{LQCoh}(\\mathcal{O}_\\mathcal{X})$ and agree with the cokernels computed\nin $\\textit{Mod}(\\mathcal{X}_\\etale, \\mathcal{O}_\\mathcal{X})$.\nLet $\\varphi : \\mathcal{F} \\to \\mathcal{G}$ be a morphism of\n$\\textit{LQCoh}(\\mathcal{O}_\\mathcal{X})$ and let\n$\\mathcal{K} = \\Ker(\\varphi)$ computed in\n$\\textit{Mod}(\\mathcal{X}_\\etale, \\mathcal{O}_\\mathcal{X})$.\nIf we can show that $\\mathcal{K}$ is a locally quasi-coherent module,\nthen the proof of (2) is complete. To see this, note that kernels\nare computed in the category of presheaves (no sheafification necessary).\nHence $\\mathcal{K}|_{U_\\etale}$ is the kernel of the map\n$\\mathcal{F}|_{U_\\etale} \\to \\mathcal{G}|_{U_\\etale}$,\ni.e., is the kernel of a map of quasi-coherent sheaves on $U_\\etale$\nwhence quasi-coherent by\nDescent, Lemma \\ref{descent-lemma-equivalence-quasi-coherent-limits}.\nThis proves (2).\n\n\\medskip\\noindent\nParts (3), (4), and (5) follow in exactly the same way. Details omitted.\n\\end{proof}\n\n\\noindent\nIn the generality discussed here the category of quasi-coherent sheaves\nis not abelian. See Examples, Section \\ref{examples-section-nonabelian-QCoh}.\nHere is what we can prove without any further work.\n\n\\begin{lemma}\n\\label{lemma-qc-colimits}\nLet $p : \\mathcal{X} \\to (\\Sch/S)_{fppf}$ be a category\nfibred in groupoids.\n\\begin{enumerate}\n\\item The category $\\QCoh(\\mathcal{O}_\\mathcal{X})$\nhas colimits and they agree with colimits in the category\n$\\textit{Mod}(\\mathcal{O}_\\mathcal{X})$ as well as with colimits\nin the category $\\textit{LQCoh}(\\mathcal{O}_\\mathcal{X})$.\n\\item Given $\\mathcal{F}, \\mathcal{G}$ in\n$\\QCoh(\\mathcal{O}_\\mathcal{X})$\nthe tensor product $\\mathcal{F} \\otimes_{\\mathcal{O}_\\mathcal{X}} \\mathcal{G}$\nin $\\textit{Mod}(\\mathcal{O}_\\mathcal{X})$\nis an object of $\\QCoh(\\mathcal{O}_\\mathcal{X})$.\n\\item Given $\\mathcal{F}, \\mathcal{G}$ in\n$\\QCoh(\\mathcal{O}_\\mathcal{X})$\nwith $\\mathcal{F}$ locally of finite presentation on\n$\\mathcal{X}_{fppf}$ the sheaf\n$\\SheafHom_{\\mathcal{O}_\\mathcal{X}}(\\mathcal{F}, \\mathcal{G})$\nin $\\textit{Mod}(\\mathcal{O}_\\mathcal{X})$\nis an object of $\\QCoh(\\mathcal{O}_\\mathcal{X})$.\n\\end{enumerate}\n\\end{lemma}\n\n\\begin{proof}\nLet $\\mathcal{I} \\to \\QCoh(\\mathcal{O}_\\mathcal{X})$,\n$i \\mapsto \\mathcal{F}_i$ be a diagram.\nConsider the object $\\mathcal{F} = \\colim_i \\mathcal{F}_i$ of\n$\\textit{Mod}(\\mathcal{O}_\\mathcal{X})$.\nFor any object $x$ of $\\mathcal{X}$ with $U = p(x)$ the pullback functor\n$x^*$ commutes with all colimits as it is a left adjoint. Hence\n$x^*\\mathcal{F} = \\colim_i x^*\\mathcal{F}_i$ in\n$\\textit{Mod}((\\Sch/U)_{fppf}, \\mathcal{O})$. We conclude from\nDescent, Lemma \\ref{descent-lemma-equivalence-quasi-coherent-limits}\nthat $x^*\\mathcal{F}$ is quasi-coherent, hence $\\mathcal{F}$\nis quasi-coherent, see\nLemma \\ref{lemma-characterize-quasi-coherent}.\nThus we see that $\\QCoh(\\mathcal{O}_\\mathcal{X})$\nhas colimits and they agree with colimits in the category\n$\\textit{Mod}(\\mathcal{O}_\\mathcal{X})$. In particular the (fppf) sheaf\n$\\mathcal{F}$ is also the colimit of the diagram in\n$\\textit{Mod}(\\mathcal{X}_\\etale, \\mathcal{O}_\\mathcal{X})$,\nhence $\\mathcal{F}$ is also the colimit in\n$\\textit{LQCoh}(\\mathcal{O}_\\mathcal{X})$. This proves (1).\n\n\\medskip\\noindent\nParts (2) and (3) are proved in the same way.\nDetails omitted.\n\\end{proof}\n\n\n\n\n\n\n\\section{Stackification and sheaves}\n\\label{section-stackification}\n\n\\noindent\nIt turns out that the category of sheaves on a category fibred in\ngroupoids only ``knows about'' the stackification.\n\n\\begin{lemma}\n\\label{lemma-stackification}\nLet $f : \\mathcal{X} \\to \\mathcal{Y}$ be a $1$-morphism of categories\nfibred in groupoids over $(\\Sch/S)_{fppf}$. If\n$f$ induces an equivalence of stackifications, then the morphism\nof topoi\n$f : \\Sh(\\mathcal{X}_{fppf}) \\to \\Sh(\\mathcal{Y}_{fppf})$\nis an equivalence.\n\\end{lemma}\n\n\\begin{proof}\nWe may assume $\\mathcal{Y}$ is the stackification of $\\mathcal{X}$.\nWe claim that $f : \\mathcal{X} \\to \\mathcal{Y}$ is a special cocontinuous\nfunctor, see\nSites, Definition \\ref{sites-definition-special-cocontinuous-functor}\nwhich will prove the lemma. By\nStacks, Lemma \\ref{stacks-lemma-topology-inherited-functorial}\nthe functor $f$ is continuous and cocontinuous. By\nStacks, Lemma \\ref{stacks-lemma-stackify}\nwe see that conditions (3), (4), and (5) of\nSites, Lemma \\ref{sites-lemma-equivalence}\nhold.\n\\end{proof}\n\n\\begin{lemma}\n\\label{lemma-stackification-quasi-coherent}\nLet $f : \\mathcal{X} \\to \\mathcal{Y}$ be a $1$-morphism of categories\nfibred in groupoids over $(\\Sch/S)_{fppf}$. If\n$f$ induces an equivalence of stackifications, then $f^*$\ninduces equivalences\n$\\textit{Mod}(\\mathcal{O}_\\mathcal{X}) \\to\n\\textit{Mod}(\\mathcal{O}_\\mathcal{Y})$\nand\n$\\QCoh(\\mathcal{O}_\\mathcal{X}) \\to\n\\QCoh(\\mathcal{O}_\\mathcal{Y})$.\n\\end{lemma}\n\n\\begin{proof}\nWe may assume $\\mathcal{Y}$ is the stackification of $\\mathcal{X}$.\nThe first assertion is clear from\nLemma \\ref{lemma-stackification}\nand\n$\\mathcal{O}_\\mathcal{X} = f^{-1}\\mathcal{O}_\\mathcal{Y}$.\nPullback of quasi-coherent sheaves are quasi-coherent, see\nLemma \\ref{lemma-pullback-quasi-coherent}.\nHence it suffices to show that if $f^*\\mathcal{G}$ is\nquasi-coherent, then $\\mathcal{G}$ is.\nTo see this, let $y$ be an object of $\\mathcal{Y}$.\nTranslating the condition that $\\mathcal{Y}$ is the stackification\nof $\\mathcal{X}$ we see there exists an fppf covering $\\{y_i \\to y\\}$\nin $\\mathcal{Y}$ such that $y_i \\cong f(x_i)$ for some\n$x_i$ object of $\\mathcal{X}$. Say $x_i$ and $y_i$ lie over the scheme $U_i$.\nThen $f^*\\mathcal{G}$ being quasi-coherent, means that $x_i^*f^*\\mathcal{G}$\nis quasi-coherent. Since $x_i^*f^*\\mathcal{G}$ is isomorphic to\n$y_i^*\\mathcal{G}$ (as sheaves on $(\\Sch/U_i)_{fppf}$ we\nsee that $y_i^*\\mathcal{G}$ is quasi-coherent.\nIt follows from\nModules on Sites, Lemma \\ref{sites-modules-lemma-local-final-object}\nthat the restriction of $\\mathcal{G}$ to $\\mathcal{Y}/y$ is\nquasi-coherent. Hence $\\mathcal{G}$ is quasi-coherent by\nLemma \\ref{lemma-characterize-quasi-coherent}.\n\\end{proof}\n\n\n\n\n\n\\section{Quasi-coherent sheaves and presentations}\n\\label{section-quasi-coherent-presentation}\n\n\\noindent\nIn\nGroupoids in Spaces, Definition\n\\ref{spaces-groupoids-definition-groupoid-module}\nwe have the defined the notion of a quasi-coherent module\non an arbitrary groupoid. The following (formal) proposition tells us\nthat we can study quasi-coherent sheaves on quotient stacks in\nterms of quasi-coherent modules on presentations.\n\n\\begin{proposition}\n\\label{proposition-quasi-coherent}\nLet $(U, R, s, t, c)$ be a groupoid in algebraic spaces over $S$.\nLet $\\mathcal{X} = [U/R]$ be the quotient stack.\nThe category of quasi-coherent modules on $\\mathcal{X}$\nis equivalent to the category of quasi-coherent modules\non $(U, R, s, t, c)$.\n\\end{proposition}\n\n\\begin{proof}\nDenote $\\QCoh(U, R, s, t, c)$ the category of quasi-coherent modules\non the groupoid $(U, R, s, t, c)$. We will construct quasi-inverse functors\n$$\n\\QCoh(\\mathcal{O}_\\mathcal{X})\n\\longleftrightarrow\n\\QCoh(U, R, s, t, c).\n$$\nAccording to\nLemma \\ref{lemma-stackification-quasi-coherent}\nthe stackification map $[U/_{\\!p}R] \\to [U/R]$ (see\nGroupoids in Spaces, Definition\n\\ref{spaces-groupoids-definition-quotient-stack})\ninduces an equivalence of categories of quasi-coherent sheaves.\nThus it suffices to prove the lemma with $\\mathcal{X} = [U/_{\\!p}R]$.\n\n\\medskip\\noindent\nRecall that an object $x = (T, u)$ of $\\mathcal{X} = [U/_{\\!p}R]$\nis given by a scheme $T$ and a morphism $u : T \\to U$. A morphism\n$(T, u) \\to (T', u')$ is given by a pair $(f, r)$ where $f : T \\to T'$\nand $r : T \\to R$ with $s \\circ r = u$ and $t \\circ r = u' \\circ f$.\nLet us call a {\\it special morphism} any morphism of the form\n$(f, e \\circ u' \\circ f) : (T, u' \\circ f) \\to (T', u')$.\nThe category of $(T, u)$ with special morphisms is just the\ncategory of schemes over $U$.\n\n\\medskip\\noindent\nLet $\\mathcal{F}$ be a quasi-coherent sheaf on $\\mathcal{X}$.\nThen we obtain for every $x = (T, u)$ a quasi-coherent sheaf\n$\\mathcal{F}_{(T, u)} = x^*\\mathcal{F}|_{T_\\etale}$ on $T$.\nMoreover, for any morphism $(f, r) : x = (T, u) \\to (T', u') = x'$\nwe obtain a comparison isomorphism\n$$\nc_{(f, r)} :\nf_{small}^*\\mathcal{F}_{(T', u')}\n\\longrightarrow\n\\mathcal{F}_{(T, u)}\n$$\nsee Lemma \\ref{lemma-quasi-coherent}. Moreover, these isomorphisms are\ncompatible with compositions, see\nLemma \\ref{lemma-comparison}.\nIf $U$, $R$ are schemes, then we can\nconstruct the quasi-coherent sheaf on the groupoid as follows: First\nthe object $(U, \\text{id})$ corresponds to a quasi-coherent sheaf\n$\\mathcal{F}_{(U, \\text{id})}$ on $U$. Next, the isomorphism\n$\\alpha : t_{small}^*\\mathcal{F}_{(U, \\text{id})} \\to\ns_{small}^*\\mathcal{F}_{(U, \\text{id})}$\ncomes from\n\\begin{enumerate}\n\\item the morphism $(R, \\text{id}_R) : (R, s) \\to (R, t)$\nin the category $[U/_{\\!p}R]$ which produces an isomorphism\n$\\mathcal{F}_{(R, t)} \\to \\mathcal{F}_{(R, s)}$,\n\\item the special morphism $(R, s) \\to (U, \\text{id})$ which produces an\nisomorphism\n$s_{small}^*\\mathcal{F}_{(U, \\text{id})} \\to \\mathcal{F}_{(R, s)}$, and\n\\item the special morphism $(R, t) \\to (U, \\text{id})$ which produces an\nisomorphism $t_{small}^*\\mathcal{F}_{(U, \\text{id})} \\to \\mathcal{F}_{(R, t)}$.\n\\end{enumerate}\nThe cocycle condition for $\\alpha$ follows from the condition\nthat $(U, R, s, t, c)$ is groupoid, i.e., that composition is\nassociative (details omitted).\n\n\\medskip\\noindent\nTo do this in general, i.e., when $U$ and $R$ are algebraic spaces,\nit suffices to explain how to associate to an algebraic space $(W, u)$ over\n$U$ a quasi-coherent sheaf $\\mathcal{F}_{(W, u)}$ and to construct the\ncomparison maps for morphisms between these. We set\n$\\mathcal{F}_{(W, u)} = x^*\\mathcal{F}|_{W_\\etale}$\nwhere $x$ is the $1$-morphism\n$\\mathcal{S}_W \\to \\mathcal{S}_U \\to [U/_{\\!p}R]$ and the comparison\nmaps are explained in (\\ref{equation-comparison-algebraic-spaces-modules}).\n\n\\medskip\\noindent\nConversely, suppose that $(\\mathcal{G}, \\alpha)$ is a quasi-coherent\nmodule on $(U, R, s, t, c)$. We are going to define a presheaf of modules\n$\\mathcal{F}$ on $\\mathcal{X}$ as follows. Given an object\n$(T, u)$ of $[U/_{\\!p}R]$ we set\n$$\n\\mathcal{F}(T, u) : = \\Gamma(T, u_{small}^*\\mathcal{G}).\n$$\nGiven a morphism $(f, r) : (T, u) \\to (T', u')$ we get a map\n\\begin{align*}\n\\mathcal{F}(T', u') & = \\Gamma(T', (u')_{small}^*\\mathcal{G}) \\\\\n& \\to \\Gamma(T, f_{small}^*(u')_{small}^*\\mathcal{G}) =\n\\Gamma(T, (u' \\circ f)_{small}^*\\mathcal{G}) \\\\\n& = \\Gamma(T, (t \\circ r)_{small}^*\\mathcal{G}) =\n\\Gamma(T, r_{small}^*t_{small}^*\\mathcal{G}) \\\\\n& \\to \\Gamma(T, r_{small}^*s_{small}^*\\mathcal{G}) =\n\\Gamma(T, (s \\circ r)_{small}^*\\mathcal{G}) \\\\\n& = \\Gamma(T, u_{small}^*\\mathcal{G}) \\\\\n& = \\mathcal{F}(T, u)\n\\end{align*}\nwhere the first arrow is pullback along $f$ and the second arrow is\n$\\alpha$. Note that if $(T, r)$ is a special morphism, then this\nmap is just pullback along $f$ as $e_{small}^*\\alpha = \\text{id}$ by\nthe axioms of a sheaf of quasi-coherent modules on a groupoid.\nThe cocycle condition implies that $\\mathcal{F}$ is a presheaf\nof modules (details omitted). It is immediate from the definition that\n$\\mathcal{F}$ is quasi-coherent when pulled back to\n$(\\Sch/T)_{fppf}$ (by the simple description of the\nrestriction maps of $\\mathcal{F}$ in case of a special morphism).\n\n\\medskip\\noindent\nWe omit the verification that the functors constructed above are\nquasi-inverse to each other.\n\\end{proof}\n\n\\noindent\nWe finish this section with a technical lemma on maps out of quasi-coherent\nsheaves. It is an analogue of\nSchemes, Lemma \\ref{schemes-lemma-compare-constructions}.\nWe will see later (Criteria for Representability, Theorem\n\\ref{criteria-theorem-flat-groupoid-gives-algebraic-stack})\nthat the assumptions on the groupoid imply that $\\mathcal{X}$ is\nan algebraic stack.\n\n\\begin{lemma}\n\\label{lemma-map-from-quasi-coherent}\nLet $(U, R, s, t, c)$ be a groupoid in algebraic spaces over $S$.\nAssume $s, t$ are flat and locally of finite presentation.\nLet $\\mathcal{X} = [U/R]$ be the quotient stack. Denote\n$\\pi : \\mathcal{S}_U \\to \\mathcal{X}$ the quotient map.\nLet $\\mathcal{F}$ be a quasi-coherent\n$\\mathcal{O}_\\mathcal{X}$-module, and let $\\mathcal{H}$ be any object\nof $\\textit{Mod}(\\mathcal{O}_\\mathcal{X})$.\nThe map\n$$\n\\Hom_{\\mathcal{O}_\\mathcal{X}}(\\mathcal{F}, \\mathcal{H})\n\\longrightarrow\n\\Hom_{\\mathcal{O}_U}(x^*\\mathcal{F}|_{U_\\etale},\nx^*\\mathcal{H}|_{U_\\etale}),\n\\quad\n\\phi \\longmapsto x^*\\phi|_{U_\\etale}\n$$\nis injective and its image consists of exactly those\n$\\varphi : x^*\\mathcal{F}|_{U_\\etale} \\to\nx^*\\mathcal{H}|_{U_\\etale}$ which give rise to a commutative\ndiagram\n$$\n\\xymatrix{\ns_{small}^*(x^*\\mathcal{F}|_{U_\\etale})\n\\ar[r] \\ar[d]^{s_{small}^*\\varphi} &\n(x \\circ s)^*\\mathcal{F}|_{R_\\etale} =\n(x \\circ t)^*\\mathcal{F}|_{R_\\etale} &\nt_{small}^*(x^*\\mathcal{F}|_{U_\\etale})\n\\ar[l] \\ar[d]_{t_{small}^*\\varphi} \\\\\ns_{small}^*(x^*\\mathcal{H}|_{U_\\etale})\n\\ar[r] &\n(x \\circ s)^*\\mathcal{H}|_{R_\\etale} =\n(x \\circ t)^*\\mathcal{H}|_{R_\\etale} &\nt_{small}^*(x^*\\mathcal{H}|_{U_\\etale})\n\\ar[l]\n}\n$$\nof modules on $R_\\etale$\nwhere the horizontal arrows are the comparison maps\n(\\ref{equation-comparison-algebraic-spaces-modules}).\n\\end{lemma}\n\n\\begin{proof}\nAccording to\nLemma \\ref{lemma-stackification-quasi-coherent}\nthe stackification map $[U/_{\\!p}R] \\to [U/R]$ (see\nGroupoids in Spaces, Definition\n\\ref{spaces-groupoids-definition-quotient-stack})\ninduces an equivalence of categories of quasi-coherent sheaves\nand of fppf $\\mathcal{O}$-modules.\nThus it suffices to prove the lemma with $\\mathcal{X} = [U/_{\\!p}R]$.\nBy Proposition \\ref{proposition-quasi-coherent}\nand its proof there exists a quasi-coherent module\n$(\\mathcal{G}, \\alpha)$ on $(U, R, s, t, c)$ such that\n$\\mathcal{F}$ is given by the rule\n$\\mathcal{F}(T, u) = \\Gamma(T, u^*\\mathcal{G})$.\nIn particular $x^*\\mathcal{F}|_{U_\\etale} = \\mathcal{G}$\nand it is clear that the map of the statement of the\nlemma is injective. Moreover, given a map\n$\\varphi : \\mathcal{G} \\to x^*\\mathcal{H}|_{U_\\etale}$\nand given any object\n$y = (T, u)$ of $[U/_{\\!p}R]$ we can consider the map\n$$\n\\mathcal{F}(y) = \\Gamma(T, u^*\\mathcal{G})\n\\xrightarrow{u_{small}^*\\varphi}\n\\Gamma(T, u_{small}^*x^*\\mathcal{H}|_{U_\\etale})\n\\rightarrow\n\\Gamma(T, y^*\\mathcal{H}|_{T_\\etale}) = \\mathcal{H}(y)\n$$\nwhere the second arrow is the comparison map\n(\\ref{equation-comparison-modules}) for the sheaf $\\mathcal{H}$.\nThis assignment is compatible with the restriction mappings of the\nsheaves $\\mathcal{F}$ and $\\mathcal{G}$ for morphisms of\n$[U/_{\\!p}R]$ if the cocycle condition of\nthe lemma is satisfied. Proof omitted. Hint: the restriction maps\nof $\\mathcal{F}$ are made explicit in terms of $(\\mathcal{G}, \\alpha)$\nin the proof of\nProposition \\ref{proposition-quasi-coherent}.\n\\end{proof}\n\n\n\n\n\n\n\n\\section{Quasi-coherent sheaves on algebraic stacks}\n\\label{section-quasi-coherent-algebraic-stacks}\n\n\\noindent\nLet $\\mathcal{X}$ be an algebraic stack over $S$. By\nAlgebraic Stacks, Lemma \\ref{algebraic-lemma-stack-presentation}\nwe can find an equivalence $[U/R] \\to \\mathcal{X}$\nwhere $(U, R, s, t, c)$ is a smooth groupoid in algebraic spaces.\nThen\n$$\n\\QCoh(\\mathcal{O}_\\mathcal{X})\n\\cong\n\\QCoh(\\mathcal{O}_{[U/R]})\n\\cong\n\\QCoh(U, R, s, t, c)\n$$\nwhere the second equivalence is\nProposition \\ref{proposition-quasi-coherent}.\nHence the category of quasi-coherent sheaves on an algebraic stack\nis equivalent to the category of quasi-coherent modules on a smooth\ngroupoid in algebraic spaces. In particular, by\nGroupoids in Spaces, Lemma \\ref{spaces-groupoids-lemma-abelian}\nwe see that $\\QCoh(\\mathcal{O}_\\mathcal{X})$ is abelian!\n\n\\medskip\\noindent\nThere is something slightly disconcerting about our current setup.\nIt is that the fully faithful embedding\n$$\n\\QCoh(\\mathcal{O}_\\mathcal{X})\n\\longrightarrow\n\\textit{Mod}(\\mathcal{O}_\\mathcal{X})\n$$\nis in general {\\bf not} exact. However, exactly the same thing happens\nfor schemes: for most schemes $X$ the embedding\n$$\n\\QCoh(\\mathcal{O}_X) \\cong\n\\QCoh((\\Sch/X)_{fppf}, \\mathcal{O}_X) \\longrightarrow\n\\textit{Mod}((\\Sch/X)_{fppf}, \\mathcal{O}_X)\n$$\nisn't exact, see\nDescent, Lemma \\ref{descent-lemma-equivalence-quasi-coherent-limits}.\nParenthetically, the example in the proof of\nDescent, Lemma \\ref{descent-lemma-equivalence-quasi-coherent-limits}\nshows that in general the strictly full embedding\n$\\QCoh(\\mathcal{O}_\\mathcal{X}) \\to\n\\textit{LQCoh}(\\mathcal{O}_\\mathcal{X})$ isn't exact either.\n\n\\medskip\\noindent\nWe collect all the positive results obtained so far in a single statement.\n\n\\begin{lemma}\n\\label{lemma-quasi-coherent-algebraic-stack}\nLet $\\mathcal{X}$ be an algebraic stack over $S$.\n\\begin{enumerate}\n\\item If $[U/R] \\to \\mathcal{X}$ is a presentation of $\\mathcal{X}$\nthen there is a canonical equivalence\n$\\QCoh(\\mathcal{O}_\\mathcal{X}) \\cong\n\\QCoh(U, R, s, t, c)$.\n\\item The category $\\QCoh(\\mathcal{O}_\\mathcal{X})$ is abelian.\n\\item The category $\\QCoh(\\mathcal{O}_\\mathcal{X})$\nhas colimits and they agree with colimits in the category\n$\\textit{Mod}(\\mathcal{O}_\\mathcal{X})$.\n\\item Given $\\mathcal{F}, \\mathcal{G}$ in\n$\\QCoh(\\mathcal{O}_\\mathcal{X})$\nthe tensor product $\\mathcal{F} \\otimes_{\\mathcal{O}_\\mathcal{X}} \\mathcal{G}$\nin $\\textit{Mod}(\\mathcal{O}_\\mathcal{X})$\nis an object of $\\QCoh(\\mathcal{O}_\\mathcal{X})$.\n\\item Given $\\mathcal{F}, \\mathcal{G}$ in\n$\\QCoh(\\mathcal{O}_\\mathcal{X})$\nwith $\\mathcal{F}$ locally of finite presentation on\n$\\mathcal{X}_{fppf}$ the sheaf\n$\\SheafHom_{\\mathcal{O}_\\mathcal{X}}(\\mathcal{F}, \\mathcal{G})$\nin $\\textit{Mod}(\\mathcal{O}_\\mathcal{X})$\nis an object of $\\QCoh(\\mathcal{O}_\\mathcal{X})$.\n\\end{enumerate}\n\\end{lemma}\n\n\\begin{proof}\nProperties (3), (4), and (5) were proven in\nLemma \\ref{lemma-qc-colimits}.\nPart (1) is\nProposition \\ref{proposition-quasi-coherent}.\nPart (2) follows from\nGroupoids in Spaces, Lemma \\ref{spaces-groupoids-lemma-abelian}\nas discussed above.\n\\end{proof}\n\n\\begin{proposition}\n\\label{proposition-coherator}\nLet $\\mathcal{X}$ be an algebraic stack over $S$.\n\\begin{enumerate}\n\\item The category $\\QCoh(\\mathcal{O}_\\mathcal{X})$ is a Grothendieck\nabelian category. Consequently, $\\QCoh(\\mathcal{O}_\\mathcal{X})$\nhas enough injectives and all limits.\n\\item The inclusion functor\n$\\QCoh(\\mathcal{O}_\\mathcal{X}) \\to\n\\textit{Mod}(\\mathcal{O}_\\mathcal{X})$ has a right adjoint\\footnote{This\nfunctor is sometimes called the {\\it coherator}.}\n$$\nQ :\n\\textit{Mod}(\\mathcal{O}_\\mathcal{X})\n\\longrightarrow\n\\QCoh(\\mathcal{O}_\\mathcal{X})\n$$\nsuch that for every quasi-coherent sheaf $\\mathcal{F}$ the adjunction mapping\n$Q(\\mathcal{F}) \\to \\mathcal{F}$ is an isomorphism.\n\\end{enumerate}\n\\end{proposition}\n\n\\begin{proof}\nThis proof is a repeat of the proof in the case of schemes, see\nProperties, Proposition \\ref{properties-proposition-coherator}\nand the case of algebraic spaces, see\nProperties of Spaces, Proposition\n\\ref{spaces-properties-proposition-coherator}.\nWe advise the reader to read either of those proofs first.\n\n\\medskip\\noindent\nPart (1) means $\\QCoh(\\mathcal{O}_\\mathcal{X})$ (a) has all colimits,\n(b) filtered colimits are exact, and (c) has a generator, see\nInjectives, Section \\ref{injectives-section-grothendieck-conditions}.\nBy Lemma \\ref{lemma-quasi-coherent-algebraic-stack}\ncolimits in $\\QCoh(\\mathcal{O}_X)$ exist and agree\nwith colimits in $\\textit{Mod}(\\mathcal{O}_X)$. By\nModules on Sites, Lemma \\ref{sites-modules-lemma-limits-colimits}\nfiltered colimits are exact. Hence (a) and (b) hold.\n\n\\medskip\\noindent\nChoose a presentation $\\mathcal{X} = [U/R]$ so that $(U, R, s, t, c)$\nis a smooth groupoid in algebraic spaces and in particular $s$ and $t$\nare flat morphisms of algebraic spaces. By\nLemma \\ref{lemma-quasi-coherent-algebraic-stack}\nabove we have\n$\\QCoh(\\mathcal{O}_\\mathcal{X}) = \\QCoh(U, R, s, t, c)$.\nBy Groupoids in Spaces, Lemma \\ref{spaces-groupoids-lemma-set-generators}\nthere exists a set $T$ and a family $(\\mathcal{F}_t)_{t \\in T}$ of\nquasi-coherent sheaves on $\\mathcal{X}$ such that every quasi-coherent\nsheaf on $\\mathcal{X}$ is the directed colimit of its subsheaves\nwhich are isomorphic to one of the $\\mathcal{F}_t$.\nThus $\\bigoplus_t \\mathcal{F}_t$ is\na generator of $\\QCoh(\\mathcal{O}_X)$ and we conclude that (c) holds.\nThe assertions on limits and injectives hold in any\nGrothendieck abelian category, see\nInjectives, Theorem\n\\ref{injectives-theorem-injective-embedding-grothendieck} and\nLemma \\ref{injectives-lemma-grothendieck-products}.\n\n\\medskip\\noindent\nProof of (2). To construct $Q$ we use the following general procedure.\nGiven an object $\\mathcal{F}$ of $\\textit{Mod}(\\mathcal{O}_\\mathcal{X})$\nwe consider the functor\n$$\n\\QCoh(\\mathcal{O}_\\mathcal{X})^{opp}\n\\longrightarrow\n\\textit{Sets},\n\\quad\n\\mathcal{G}\n\\longmapsto\n\\Hom_\\mathcal{X}(\\mathcal{G}, \\mathcal{F})\n$$\nThis functor transforms colimits into limits,\nhence is representable, see\nInjectives, Lemma \\ref{injectives-lemma-grothendieck-brown}.\nThus there exists a quasi-coherent sheaf $Q(\\mathcal{F})$\nand a functorial isomorphism\n$\\Hom_\\mathcal{X}(\\mathcal{G}, \\mathcal{F}) =\n\\Hom_\\mathcal{X}(\\mathcal{G}, Q(\\mathcal{F}))$\nfor $\\mathcal{G}$ in $\\QCoh(\\mathcal{O}_\\mathcal{X})$.\nBy the Yoneda lemma\n(Categories, Lemma \\ref{categories-lemma-yoneda})\nthe construction $\\mathcal{F} \\leadsto Q(\\mathcal{F})$ is\nfunctorial in $\\mathcal{F}$. By construction $Q$ is a right\nadjoint to the inclusion functor.\nThe fact that $Q(\\mathcal{F}) \\to \\mathcal{F}$ is an isomorphism\nwhen $\\mathcal{F}$ is quasi-coherent is a formal consequence of the fact\nthat the inclusion functor\n$\\QCoh(\\mathcal{O}_\\mathcal{X}) \\to\n\\textit{Mod}(\\mathcal{O}_\\mathcal{X})$\nis fully faithful.\n\\end{proof}\n\n\n\n\n\n\n\n\n\\section{Cohomology}\n\\label{section-cohomology-general}\n\n\\noindent\nLet $S$ be a scheme and let $\\mathcal{X}$ be a category fibred in groupoids\nover $(\\Sch/S)_{fppf}$. For any $\\tau \\in \\{Zariski, \\etale, smooth,\nsyntomic, fppf\\}$ the categories $\\textit{Ab}(\\mathcal{X}_\\tau)$ and\n$\\textit{Mod}(\\mathcal{X}_\\tau, \\mathcal{O}_\\mathcal{X})$ have\nenough injectives, see\nInjectives, Theorems \\ref{injectives-theorem-sheaves-injectives} and\n\\ref{injectives-theorem-sheaves-modules-injectives}.\nThus we can use the machinery of\nCohomology on Sites, Section \\ref{sites-cohomology-section-cohomology-sheaves}\nto define the cohomology groups\n$$\nH^p(\\mathcal{X}_\\tau, \\mathcal{F}) = H^p_\\tau(\\mathcal{X}, \\mathcal{F})\n\\quad\\text{and}\\quad\nH^p(x, \\mathcal{F}) = H^p_\\tau(x, \\mathcal{F})\n$$\nfor any $x \\in \\Ob(\\mathcal{X})$ and any object $\\mathcal{F}$ of\n$\\textit{Ab}(\\mathcal{X}_\\tau)$ or\n$\\textit{Mod}(\\mathcal{X}_\\tau, \\mathcal{O}_\\mathcal{X})$. Moreover, if\n$f : \\mathcal{X} \\to \\mathcal{Y}$ is a $1$-morphism of categories fibred\nin groupoids over $(\\Sch/S)_{fppf}$, then we obtain the higher direct\nimages $R^if_*\\mathcal{F}$ in $\\textit{Ab}(\\mathcal{Y}_\\tau)$ or\n$\\textit{Mod}(\\mathcal{Y}_\\tau, \\mathcal{O}_\\mathcal{Y})$.\nOf course, as explained in\nCohomology on Sites, Section \\ref{sites-cohomology-section-derived-functors}\nthere are also derived versions of $H^p(-)$ and $R^if_*$.\n\n\\begin{lemma}\n\\label{lemma-cohomology-restriction}\nLet $S$ be a scheme. Let $\\mathcal{X}$ be a category fibred in groupoids\nover $(\\Sch/S)_{fppf}$. Let $\\tau \\in \\{Zariski, \\etale, smooth,\nsyntomic, fppf\\}$. Let $x \\in \\Ob(\\mathcal{X})$ be an object lying\nover the scheme $U$. Let $\\mathcal{F}$ be\nan object of $\\textit{Ab}(\\mathcal{X}_\\tau)$ or\n$\\textit{Mod}(\\mathcal{X}_\\tau, \\mathcal{O}_\\mathcal{X})$. Then\n$$\nH^p_\\tau(x, \\mathcal{F}) = H^p((\\Sch/U)_\\tau, x^{-1}\\mathcal{F})\n$$\nand if $\\tau = \\etale$, then we also have\n$$\nH^p_\\etale(x, \\mathcal{F}) =\nH^p(U_\\etale, \\mathcal{F}|_{U_\\etale}).\n$$\n\\end{lemma}\n\n\\begin{proof}\nThe first statement follows from\nCohomology on Sites, Lemma \\ref{sites-cohomology-lemma-cohomology-of-open}\nand the equivalence of \nLemma \\ref{lemma-localizing-structure-sheaf}.\nThe second statement follows from the first combined with\n\\'Etale Cohomology, Lemma\n\\ref{etale-cohomology-lemma-compare-cohomology-big-small}.\n\\end{proof}\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\\section{Injective sheaves}\n\\label{section-lower-shriek}\n\n\\noindent\nThe pushforward of an injective abelian sheaf or module is injective.\n\n\\begin{lemma}\n\\label{lemma-pushforward-injective}\nLet $f : \\mathcal{X} \\to \\mathcal{Y}$ be a $1$-morphism of categories\nfibred in groupoids over $(\\Sch/S)_{fppf}$. Let\n$\\tau \\in \\{Zar, \\etale, smooth, syntomic, fppf\\}$.\n\\begin{enumerate}\n\\item $f_*\\mathcal{I}$ is injective in $\\textit{Ab}(\\mathcal{Y}_\\tau)$\nfor $\\mathcal{I}$ injective in $\\textit{Ab}(\\mathcal{X}_\\tau)$, and\n\\item $f_*\\mathcal{I}$ is injective in\n$\\textit{Mod}(\\mathcal{Y}_\\tau, \\mathcal{O}_\\mathcal{Y})$\nfor $\\mathcal{I}$ injective in\n$\\textit{Mod}(\\mathcal{X}_\\tau, \\mathcal{O}_\\mathcal{X})$.\n\\end{enumerate}\n\\end{lemma}\n\n\\begin{proof}\nThis follows formally from the fact that $f^{-1}$ is an exact\nleft adjoint of $f_*$, see\nHomology, Lemma \\ref{homology-lemma-adjoint-preserve-injectives}.\n\\end{proof}\n\n\\noindent\nIn the rest of this section we prove that pullback $f^{-1}$ has a left\nadjoint $f_!$ on abelian sheaves and modules. If $f$ is representable (by\nschemes or by algebraic spaces), then it will turn out that $f_!$ is exact\nand $f^{-1}$ will preserve injectives. We first prove a few\npreliminary lemmas about fibre products and equalizers in categories\nfibred in groupoids and their behaviour with respect to morphisms.\n\n\\begin{lemma}\n\\label{lemma-fibre-products}\nLet $p : \\mathcal{X} \\to (\\Sch/S)_{fppf}$\nbe a category fibred in groupoids.\n\\begin{enumerate}\n\\item The category $\\mathcal{X}$ has fibre products.\n\\item If the $\\mathit{Isom}$-presheaves of $\\mathcal{X}$\nare representable by algebraic spaces, then $\\mathcal{X}$ has equalizers.\n\\item If $\\mathcal{X}$ is an algebraic stack (or more generally\na quotient stack), then $\\mathcal{X}$ has equalizers.\n\\end{enumerate}\n\\end{lemma}\n\n\\begin{proof}\nPart (1) follows\nCategories, Lemma \\ref{categories-lemma-fibred-groupoids-fibre-product-goes-up}\nas $(\\Sch/S)_{fppf}$ has fibre products.\n\n\\medskip\\noindent\nLet $a, b : x \\to y$ be morphisms of $\\mathcal{X}$.\nSet $U = p(x)$ and $V = p(y)$. The category of schemes has equalizers\nhence we can let $W \\to U$ be the equalizer of $p(a)$ and $p(b)$.\nDenote $c : z \\to x$ a morphism of $\\mathcal{X}$ lying over $W \\to U$.\nThe equalizer of $a$ and $b$, if it exists, is the equalizer of $a \\circ c$\nand $b \\circ c$. Thus we may assume that $p(a) = p(b) = f : U \\to V$.\nAs $\\mathcal{X}$ is fibred in groupoids, there exists a unique automorphism\n$i : x \\to x$ in the fibre category of $\\mathcal{X}$ over $U$ such that\n$a \\circ i = b$. Again the equalizer of $a$ and $b$ is the equalizer\nof $\\text{id}_x$ and $i$. Recall that the $\\mathit{Isom}_\\mathcal{X}(x)$\nis the presheaf on $(\\Sch/U)_{fppf}$ which to\n$V/U$ associates the set of automorphisms of $x|_V$ in the fibre category\nof $\\mathcal{X}$ over $V$, see\nStacks, Definition \\ref{stacks-definition-mor-presheaf}.\nIf $\\mathit{Isom}_\\mathcal{X}(x)$ is representable by an algebraic space\n$G \\to U$, then we see that $\\text{id}_x$ and $i$ define morphisms\n$e, i : U \\to G$ over $U$. Set $V = U \\times_{e, G, i} U$, which by\nMorphisms of Spaces, Lemma \\ref{spaces-morphisms-lemma-section-immersion}\nis a scheme. Then it is clear that $x|_V \\to x$ is the equalizer of\nthe maps $\\text{id}_x$ and $i$ in $\\mathcal{X}$.\nThis proves (2).\n\n\\medskip\\noindent\nIf $\\mathcal{X} = [U/R]$ for some groupoid in algebraic spaces\n$(U, R, s, t, c)$ over $S$, then the hypothesis of (2) holds by\nBootstrap, Lemma \\ref{bootstrap-lemma-quotient-stack-isom}.\nIf $\\mathcal{X}$ is an algebraic stack, then we can choose a\npresentation $[U/R] \\cong \\mathcal{X}$ by\nAlgebraic Stacks, Lemma \\ref{algebraic-lemma-stack-presentation}.\n\\end{proof}\n\n\\begin{lemma}\n\\label{lemma-fibre-products-morphism}\nLet $f : \\mathcal{X} \\to \\mathcal{Y}$ be a $1$-morphism of categories\nfibred in groupoids over $(\\Sch/S)_{fppf}$.\n\\begin{enumerate}\n\\item The functor $f$ transforms fibre products into fibre products.\n\\item If $f$ is faithful, then $f$ transforms equalizers into equalizers.\n\\end{enumerate}\n\\end{lemma}\n\n\\begin{proof}\nBy\nCategories, Lemma \\ref{categories-lemma-fibred-groupoids-fibre-product-goes-up}\nwe see that a fibre product in $\\mathcal{X}$ is any commutative square lying\nover a fibre product diagram in $(\\Sch/S)_{fppf}$. Similarly for\n$\\mathcal{Y}$. Hence (1) is clear.\n\n\\medskip\\noindent\nLet $x \\to x'$ be the equalizer of two morphisms $a, b : x' \\to x''$\nin $\\mathcal{X}$. We will show that $f(x) \\to f(x')$ is the equalizer\nof $f(a)$ and $f(b)$. Let $y \\to f(x)$ be a morphism of $\\mathcal{Y}$\nequalizing $f(a)$ and $f(b)$. Say $x, x', x''$ lie over the schemes\n$U, U', U''$ and $y$ lies over $V$. Denote $h : V \\to U'$ the image\nof $y \\to f(x)$ in the category of schemes. The morphism\n$y \\to f(x)$ is isomorphic to $f(h^*x') \\to f(x')$ by the axioms of\nfibred categories. Hence, as $f$ is faithful, we see that\n$h^*x' \\to x'$ equalizes $a$ and $b$. Thus we obtain a unique morphism\n$h^*x' \\to x$ whose image $y = f(h^*x') \\to f(x)$ is the desired morphism\nin $\\mathcal{Y}$.\n\\end{proof}\n\n\\begin{lemma}\n\\label{lemma-fibre-products-preserve-properties}\nLet $f : \\mathcal{X} \\to \\mathcal{Y}$, $g : \\mathcal{Z} \\to \\mathcal{Y}$\nbe faithful $1$-morphisms of categories\nfibred in groupoids over $(\\Sch/S)_{fppf}$.\n\\begin{enumerate}\n\\item the functor $\\mathcal{X} \\times_\\mathcal{Y} \\mathcal{Z} \\to \\mathcal{Y}$\nis faithful, and\n\\item if $\\mathcal{X}, \\mathcal{Z}$ have equalizers, so does\n$\\mathcal{X} \\times_\\mathcal{Y} \\mathcal{Z}$.\n\\end{enumerate}\n\\end{lemma}\n\n\\begin{proof}\nWe think of objects in $\\mathcal{X} \\times_\\mathcal{Y} \\mathcal{Z}$ as\nquadruples $(U, x, z, \\alpha)$ where $\\alpha : f(x) \\to g(z)$ is an\nisomorphism over $U$, see\nCategories, Lemma \\ref{categories-lemma-2-product-categories-over-C}.\nA morphism $(U, x, z, \\alpha) \\to (U', x', z', \\alpha')$ is a\npair of morphisms $a : x \\to x'$ and $b : z \\to z'$ compatible\nwith $\\alpha$ and $\\alpha'$. Thus it is clear that if $f$ and\n$g$ are faithful, so is the functor\n$\\mathcal{X} \\times_\\mathcal{Y} \\mathcal{Z} \\to \\mathcal{Y}$.\nNow, suppose that\n$(a, b), (a', b') : (U, x, z, \\alpha) \\to (U', x', z', \\alpha')$\nare two morphisms of the $2$-fibre product. Then consider the equalizer\n$x'' \\to x$ of $a$ and $a'$ and the equalizer $z'' \\to z$ of $b$ and $b'$.\nSince $f$ commutes with equalizers (by\nLemma \\ref{lemma-fibre-products-morphism})\nwe see that $f(x'') \\to f(x)$ is the equalizer of $f(a)$ and $f(a')$.\nSimilarly, $g(z'') \\to g(z)$ is the equalizer of $g(b)$ and $g(b')$.\nPicture\n$$\n\\xymatrix{\nf(x'') \\ar[r] \\ar@{..>}[d]_{\\alpha''}&\nf(x) \\ar[d]_\\alpha\n\\ar@<0.5ex>[r]^{f(a)}\n\\ar@<-0.5ex>[r]_{f(a')}\n &\nf(x') \\ar[d]^{\\alpha'} \\\\\ng(z'') \\ar[r] &\ng(z)\n\\ar@<0.5ex>[r]^{g(b)}\n\\ar@<-0.5ex>[r]_{g(b')}\n &\ng(z')\n}\n$$\nIt is clear that the dotted arrow exists and is an isomorphism.\nHowever, it is not a priori the case that the image of $\\alpha''$\nin the category of schemes is the identity of its source. On the other\nhand, the existence of $\\alpha''$ means that we can assume that $x''$\nand $z''$ are defined over the same scheme and that the morphisms\n$x'' \\to x$ and $z'' \\to z$ have the same image in the category of schemes.\nRedoing the diagram above we see that the dotted arrow now does\nproject to an identity morphism and we win. Some details omitted.\n\\end{proof}\n\n\\noindent\nAs we are working with big sites we have the following somewhat\ncounter intuitive result (which also holds for morphisms of big sites\nof schemes). Warning: This result isn't true if we drop the hypothesis\nthat $f$ is faithful.\n\n\\begin{lemma}\n\\label{lemma-pullback-injective}\nLet $f : \\mathcal{X} \\to \\mathcal{Y}$ be a $1$-morphism of categories\nfibred in groupoids over $(\\Sch/S)_{fppf}$. Let\n$\\tau \\in \\{Zar, \\etale, smooth, syntomic, fppf\\}$.\nThe functor\n$f^{-1} : \\textit{Ab}(\\mathcal{Y}_\\tau) \\to \\textit{Ab}(\\mathcal{X}_\\tau)$\nhas a left adjoint\n$f_! : \\textit{Ab}(\\mathcal{X}_\\tau) \\to \\textit{Ab}(\\mathcal{Y}_\\tau)$.\nIf $f$ is faithful and $\\mathcal{X}$ has equalizers, then\n\\begin{enumerate}\n\\item $f_!$ is exact, and\n\\item $f^{-1}\\mathcal{I}$ is injective in $\\textit{Ab}(\\mathcal{X}_\\tau)$\nfor $\\mathcal{I}$ injective in $\\textit{Ab}(\\mathcal{Y}_\\tau)$.\n\\end{enumerate}\n\\end{lemma}\n\n\\begin{proof}\nBy\nStacks, Lemma \\ref{stacks-lemma-topology-inherited-functorial}\nthe functor $f$ is continuous and cocontinuous. Hence by\nModules on Sites, Lemma \\ref{sites-modules-lemma-g-shriek-adjoint}\nthe functor\n$f^{-1} : \\textit{Ab}(\\mathcal{Y}_\\tau) \\to \\textit{Ab}(\\mathcal{X}_\\tau)$\nhas a left adjoint\n$f_! : \\textit{Ab}(\\mathcal{X}_\\tau) \\to \\textit{Ab}(\\mathcal{Y}_\\tau)$.\nTo see (1) we apply\nModules on Sites, Lemma \\ref{sites-modules-lemma-exactness-lower-shriek}\nand to see that the hypotheses of that lemma are satisfied use\nLemmas \\ref{lemma-fibre-products} and\n\\ref{lemma-fibre-products-morphism}\nabove. Part (2) follows from this formally, see\nHomology, Lemma \\ref{homology-lemma-adjoint-preserve-injectives}.\n\\end{proof}\n\n\\begin{lemma}\n\\label{lemma-pullback-injective-modules}\nLet $f : \\mathcal{X} \\to \\mathcal{Y}$ be a $1$-morphism of categories\nfibred in groupoids over $(\\Sch/S)_{fppf}$. Let\n$\\tau \\in \\{Zar, \\etale, smooth, syntomic, fppf\\}$.\nThe functor\n$f^* : \\textit{Mod}(\\mathcal{Y}_\\tau, \\mathcal{O}_\\mathcal{Y}) \\to\n\\textit{Mod}(\\mathcal{X}_\\tau, \\mathcal{O}_\\mathcal{X})$\nhas a left adjoint\n$f_! : \\textit{Mod}(\\mathcal{X}_\\tau, \\mathcal{O}_\\mathcal{X}) \\to\n\\textit{Mod}(\\mathcal{Y}_\\tau, \\mathcal{O}_\\mathcal{Y})$ which\nagrees with the functor $f_!$ of Lemma \\ref{lemma-pullback-injective}\non underlying abelian sheaves.\nIf $f$ is faithful and $\\mathcal{X}$ has equalizers, then\n\\begin{enumerate}\n\\item $f_!$ is exact, and\n\\item $f^{-1}\\mathcal{I}$ is injective in\n$\\textit{Mod}(\\mathcal{X}_\\tau, \\mathcal{O}_\\mathcal{X})$\nfor $\\mathcal{I}$ injective in\n$\\textit{Mod}(\\mathcal{Y}_\\tau, \\mathcal{O}_\\mathcal{X})$.\n\\end{enumerate}\n\\end{lemma}\n\n\\begin{proof}\nRecall that $f$ is a continuous and cocontinuous functor of sites\nand that $f^{-1}\\mathcal{O}_\\mathcal{Y} = \\mathcal{O}_\\mathcal{X}$. Hence\nModules on Sites, Lemma \\ref{sites-modules-lemma-lower-shriek-modules}\nimplies $f^*$ has a left adjoint $f_!^{Mod}$.\nLet $x$ be an object of $\\mathcal{X}$ lying over the scheme $U$.\nThen $f$ induces an equivalence of ringed sites\n$$\n\\mathcal{X}/x \\longrightarrow \\mathcal{Y}/f(x)\n$$\nas both sides are equivalent to $(\\Sch/U)_\\tau$, see\nLemma \\ref{lemma-localizing-structure-sheaf}.\nModules on Sites, Remark \\ref{sites-modules-remark-when-shriek-equal}\nshows that $f_!$ agrees with the functor on abelian sheaves.\n\n\\medskip\\noindent\nAssume now that $\\mathcal{X}$ has equalizers and that $f$ is faithful.\nLemma \\ref{lemma-pullback-injective}\ntells us that $f_!$ is exact. Finally,\nHomology, Lemma \\ref{homology-lemma-adjoint-preserve-injectives}\nimplies the statement on pullbacks of injective modules.\n\\end{proof}\n\n\n\n\n\\section{The {\\v C}ech complex}\n\\label{section-cech}\n\n\\noindent\nTo compute the cohomology of a sheaf on an algebraic stack we compare\nit to the cohomology of the sheaf restricted to coverings of the\ngiven algebraic stack.\n\n\\medskip\\noindent\nThroughout this section the situation will be as follows. We are given\na $1$-morphism of categories fibred in groupoids\n\\begin{equation}\n\\label{equation-covering}\n\\vcenter{\n\\xymatrix{\n\\mathcal{U} \\ar[rr]_f \\ar[rd]_q & &  \\mathcal{X} \\ar[ld]^p \\\\\n& (\\Sch/S)_{fppf}\n}\n}\n\\end{equation}\nWe are going to think about $\\mathcal{U}$ as a ``covering'' of $\\mathcal{X}$.\nHence we want to consider the simplicial object\n$$\n\\xymatrix{\n\\mathcal{U} \\times_\\mathcal{X} \\mathcal{U} \\times_\\mathcal{X} \\mathcal{U}\n\\ar@<1ex>[r]\n\\ar@<0ex>[r]\n\\ar@<-1ex>[r]\n&\n\\mathcal{U} \\times_\\mathcal{X} \\mathcal{U}\n\\ar@<0.5ex>[r]\n\\ar@<-0.5ex>[r]\n&\n\\mathcal{U}\n}\n$$\nin the category of categories fibred in groupoids over\n$(\\Sch/S)_{fppf}$. However, since this is a $(2, 1)$-category and\nnot a category, we should say explicitly what we mean. Namely, we let\n$\\mathcal{U}_n$ be the category with objects\n$(u_0, \\ldots, u_n, x, \\alpha_0, \\ldots, \\alpha_n)$\nwhere $\\alpha_i : f(u_i) \\to x$ is an isomorphism in $\\mathcal{X}$.\nWe denote $f_n : \\mathcal{U}_n \\to \\mathcal{X}$ the $1$-morphism\nwhich assigns to $(u_0, \\ldots, u_n, x, \\alpha_0, \\ldots, \\alpha_n)$\nthe object $x$. Note that $\\mathcal{U}_0 = \\mathcal{U}$ and $f_0 = f$.\nGiven a map $\\varphi : [m] \\to [n]$ we consider the $1$-morphism\n$\\mathcal{U}_\\varphi : \\mathcal{U}_n \\longrightarrow \\mathcal{U}_n$\ngiven by\n$$\n(u_0, \\ldots, u_n, x, \\alpha_0, \\ldots, \\alpha_n)\n\\longmapsto\n(u_{\\varphi(0)}, \\ldots, u_{\\varphi(n)}, x,\n\\alpha_{\\varphi(0)}, \\ldots, \\alpha_{\\varphi(n)})\n$$\non objects. All of these $1$-morphisms compose correctly on the nose\n(no $2$-morphisms required) and all of these $1$-morphisms are $1$-morphisms\nover $\\mathcal{X}$. We denote $\\mathcal{U}_\\bullet$ this simplicial object.\nIf $\\mathcal{F}$ is a presheaf of sets on $\\mathcal{X}$, then we obtain a\ncosimplicial set\n$$\n\\xymatrix{\n\\Gamma(\\mathcal{U}_0, f_0^{-1}\\mathcal{F})\n\\ar@<0.5ex>[r]\n\\ar@<-0.5ex>[r]\n&\n\\Gamma(\\mathcal{U}_1, f_1^{-1}\\mathcal{F})\n\\ar@<1ex>[r]\n\\ar@<0ex>[r]\n\\ar@<-1ex>[r]\n&\n\\Gamma(\\mathcal{U}_2, f_2^{-1}\\mathcal{F})\n}\n$$\nHere the arrows are the pullback maps along the given morphisms of\nthe simplicial object.\nIf $\\mathcal{F}$ is a presheaf of abelian groups, this is a cosimplicial\nabelian group.\n\n\\medskip\\noindent\nLet $\\mathcal{U} \\to \\mathcal{X}$ be as above and let $\\mathcal{F}$\nbe an abelian presheaf on $\\mathcal{X}$.\nThe {\\it {\\v C}ech complex} associated to the situation is denoted\n$\\check{\\mathcal{C}}^\\bullet(\\mathcal{U} \\to \\mathcal{X}, \\mathcal{F})$.\nIt is the cochain complex associated to the cosimplicial abelian group\nabove, see\nSimplicial, Section \\ref{simplicial-section-dold-kan-cosimplicial}.\nIt has terms\n$$\n\\check{\\mathcal{C}}^n(\\mathcal{U} \\to \\mathcal{X}, \\mathcal{F})\n= \\Gamma(\\mathcal{U}_n, f_n^{-1}\\mathcal{F}).\n$$\nThe boundary maps are the maps\n$$\nd^n = \\sum\\nolimits_{i = 0}^{n + 1} (-1)^i \\delta^{n + 1}_i :\n\\Gamma(\\mathcal{U}_n, f_n^{-1}\\mathcal{F})\n\\longrightarrow\n\\Gamma(\\mathcal{U}_{n + 1}, f_{n + 1}^{-1}\\mathcal{F})\n$$\nwhere $\\delta^{n + 1}_i$ corresponds to the map\n$[n] \\to [n + 1]$ omitting the index $i$. Note that the map\n$\\Gamma(\\mathcal{X}, \\mathcal{F}) \\to\n\\Gamma(\\mathcal{U}_0, f_0^{-1}\\mathcal{F}_0)$\nis in the kernel of the differential $d^0$. Hence we define\nthe {\\it extended {\\v C}ech complex} to be the complex\n$$\n\\ldots \\to 0 \\to\n\\Gamma(\\mathcal{X}, \\mathcal{F}) \\to\n\\Gamma(\\mathcal{U}_0, f_0^{-1}\\mathcal{F}_0) \\to\n\\Gamma(\\mathcal{U}_1, f_1^{-1}\\mathcal{F}_1) \\to \\ldots\n$$\nwith $\\Gamma(\\mathcal{X}, \\mathcal{F})$ placed in degree $-1$.\nThe extended {\\v C}ech complex is acyclic if and only if the canonical map\n$$\n\\Gamma(\\mathcal{X}, \\mathcal{F})[0]\n\\longrightarrow\n\\check{\\mathcal{C}}^\\bullet(\\mathcal{U} \\to \\mathcal{X}, \\mathcal{F})\n$$\nis a quasi-isomorphism of complexes.\n\n\\begin{lemma}\n\\label{lemma-generalities}\nGeneralities on {\\v C}ech complexes.\n\\begin{enumerate}\n\\item If\n$$\n\\xymatrix{\n\\mathcal{V} \\ar[d]_g \\ar[r]_h & \\mathcal{U} \\ar[d]^f \\\\\n\\mathcal{Y} \\ar[r]^e & \\mathcal{X}\n}\n$$\nis $2$-commutative diagram of categories fibred in groupoids over\n$(\\Sch/S)_{fppf}$, then there is a morphism of {\\v C}ech complexes\n$$\n\\check{\\mathcal{C}}^\\bullet(\\mathcal{U} \\to \\mathcal{X}, \\mathcal{F})\n\\longrightarrow\n\\check{\\mathcal{C}}^\\bullet(\\mathcal{V} \\to \\mathcal{Y}, e^{-1}\\mathcal{F})\n$$\n\\item if $h$ and $e$ are equivalences, then the map of (1) is an isomorphism,\n\\item if $f, f' : \\mathcal{U} \\to \\mathcal{X}$ are $2$-isomorphic, then\nthe associated {\\v C}ech complexes are isomorphic,\n\\end{enumerate}\n\\end{lemma}\n\n\\begin{proof}\nIn the situation of (1) let $t : f \\circ h \\to e \\circ g$ be a $2$-morphism.\nThe map on complexes is given in degree $n$ by\npullback along the $1$-morphisms\n$\\mathcal{V}_n \\to \\mathcal{U}_n$ given by the rule\n$$\n(v_0, \\ldots, v_n, y, \\beta_0, \\ldots, \\beta_n)\n\\longmapsto\n(h(v_0), \\ldots, h(v_n), e(y),\ne(\\beta_0) \\circ t_{v_0}, \\ldots, e(\\beta_n) \\circ t_{v_n}).\n$$\nFor (2), note that pullback on global sections is an isomorphism\nfor any presheaf of sets when the pullback is along an equivalence\nof categories. Part (3) follows on combining (1) and (2).\n\\end{proof}\n\n\\begin{lemma}\n\\label{lemma-homotopy}\nIf there exists a $1$-morphism $s : \\mathcal{X} \\to \\mathcal{U}$\nsuch that $f \\circ s$ is $2$-isomorphic to $\\text{id}_\\mathcal{X}$\nthen the extended {\\v C}ech complex is homotopic to zero.\n\\end{lemma}\n\n\\begin{proof}\nSet $\\mathcal{U}' = \\mathcal{U} \\times_\\mathcal{X} \\mathcal{X}$\nequal to the fibre product as described in\nCategories, Lemma \\ref{categories-lemma-2-product-categories-over-C}.\nSet $f' : \\mathcal{U}' \\to \\mathcal{X}$ equal to the second projection.\nThen $\\mathcal{U} \\to \\mathcal{U}'$, $u \\mapsto (u, f(x), 1)$\nis an equivalence over $\\mathcal{X}$, hence we may replace\n$(\\mathcal{U}, f)$ by $(\\mathcal{U}', f')$ by\nLemma \\ref{lemma-generalities}.\nThe advantage of this is that now $f'$ has a section $s'$ such\nthat $f' \\circ s' = \\text{id}_\\mathcal{X}$ on the nose. Namely, if\n$t : s \\circ f \\to \\text{id}_\\mathcal{X}$ is a $2$-isomorphism\nthen we can set $s'(x) = (s(x), x, t_x)$. Thus we may assume that\n$f \\circ s = \\text{id}_\\mathcal{X}$.\n\n\\medskip\\noindent\nIn the case that $f \\circ s = \\text{id}_\\mathcal{X}$ the result follows\nfrom general principles. We give the homotopy explicitly. Namely,\nfor $n \\geq 0$ define $s_n : \\mathcal{U}_n \\to \\mathcal{U}_{n + 1}$\nto be the $1$-morphism defined by the rule on objects\n$$\n(u_0, \\ldots, u_n, x, \\alpha_0, \\ldots, \\alpha_n)\n\\longmapsto\n(u_0, \\ldots, u_n, s(x), x,\n\\alpha_0, \\ldots, \\alpha_n, \\text{id}_x).\n$$\nDefine\n$$\nh^{n + 1} :\n\\Gamma(\\mathcal{U}_{n + 1}, f_{n + 1}^{-1}\\mathcal{F})\n\\longrightarrow\n\\Gamma(\\mathcal{U}_n, f_n^{-1}\\mathcal{F})\n$$\nas pullback along $s_n$. We also set $s_{-1} = s$ and\n$h^0 : \\Gamma(\\mathcal{U}_0, f_0^{-1}\\mathcal{F}) \\to\n\\Gamma(\\mathcal{X}, \\mathcal{F})$ equal to pullback along $s_{-1}$.\nThen the family of maps $\\{h^n\\}_{n \\geq 0}$ is a homotopy between\n$1$ and $0$ on the extended {\\v C}ech complex.\n\\end{proof}\n\n\n\n\n\n\n\n\n\n\\section{The relative {\\v C}ech complex}\n\\label{section-sheaf-cech-complex}\n\n\\noindent\nLet $f : \\mathcal{U} \\to \\mathcal{X}$ be a $1$-morphism of categories\nfibred in groupoids over $(\\Sch/S)_{fppf}$ as in\n(\\ref{equation-covering}). Consider the associated simplicial\nobject $\\mathcal{U}_\\bullet$ and the maps\n$f_n : \\mathcal{U}_n \\to \\mathcal{X}$. Let\n$\\tau \\in \\{Zar, \\etale, smooth, syntomic, fppf\\}$.\nFinally, suppose that $\\mathcal{F}$ is a sheaf (of sets)\non $\\mathcal{X}_\\tau$. Then\n$$\n\\xymatrix{\nf_{0, *}f_0^{-1}\\mathcal{F}\n\\ar@<0.5ex>[r]\n\\ar@<-0.5ex>[r]\n&\nf_{1, *}f_1^{-1}\\mathcal{F}\n\\ar@<1ex>[r]\n\\ar@<0ex>[r]\n\\ar@<-1ex>[r]\n&\nf_{2, *}f_2^{-1}\\mathcal{F}\n}\n$$\nis a cosimplicial sheaf on $\\mathcal{X}_\\tau$ where we use the pullback maps\nintroduced in\nSites, Section \\ref{sites-section-pullback}.\nIf $\\mathcal{F}$ is an abelian sheaf, then $f_{n, *}f_n^{-1}\\mathcal{F}$\nform a cosimplicial abelian sheaf on $\\mathcal{X}_\\tau$.\nThe associated complex (see\nSimplicial, Section \\ref{simplicial-section-dold-kan-cosimplicial})\n$$\n\\ldots \\to 0 \\to\nf_{0, *}f_0^{-1}\\mathcal{F} \\to\nf_{1, *}f_1^{-1}\\mathcal{F} \\to\nf_{2, *}f_2^{-1}\\mathcal{F} \\to \\ldots\n$$\nis called the {\\it relative {\\v C}ech complex} associated to the situation.\nWe will denote this complex $\\mathcal{K}^\\bullet(f, \\mathcal{F})$.\nThe {\\it extended relative {\\v C}ech complex} is the complex\n$$\n\\ldots \\to 0 \\to\n\\mathcal{F} \\to\nf_{0, *}f_0^{-1}\\mathcal{F} \\to\nf_{1, *}f_1^{-1}\\mathcal{F} \\to\nf_{2, *}f_2^{-1}\\mathcal{F} \\to \\ldots\n$$\nwith $\\mathcal{F}$ in degree $-1$. The extended relative {\\v C}ech complex\nis acyclic if and only if the map\n$\\mathcal{F}[0] \\to \\mathcal{K}^\\bullet(f, \\mathcal{F})$\nis a quasi-isomorphism of complexes of sheaves.\n\n\\begin{remark}\n\\label{remark-cech-complex-presheaves}\nWe can define the complex $\\mathcal{K}^\\bullet(f, \\mathcal{F})$\nalso if $\\mathcal{F}$ is a presheaf, only we cannot use the reference to\nSites, Section \\ref{sites-section-pullback}\nto define the pullback maps. To explain the pullback maps, suppose\ngiven a commutative diagram\n$$\n\\xymatrix{\n\\mathcal{V} \\ar[rd]_g \\ar[rr]_h & &  \\mathcal{U} \\ar[ld]^f \\\\\n& \\mathcal{X}\n}\n$$\nof categories fibred in groupoids over $(\\Sch/S)_{fppf}$\nand a presheaf $\\mathcal{G}$ on $\\mathcal{U}$\nwe can define the pullback map $f_*\\mathcal{G} \\to g_*h^{-1}\\mathcal{G}$\nas the composition\n$$\nf_*\\mathcal{G} \\longrightarrow\nf_*h_*h^{-1}\\mathcal{G} = g_*h^{-1}\\mathcal{G}\n$$\nwhere the map comes from the adjunction map\n$\\mathcal{G} \\to h_*h^{-1}\\mathcal{G}$. This works because in our situation\nthe functors $h_*$ and $h^{-1}$ are adjoint in presheaves (and agree with\ntheir counter parts on sheaves). See\nSections \\ref{section-presheaves} and \\ref{section-sheaves}.\n\\end{remark}\n\n\\begin{lemma}\n\\label{lemma-generalities-sheafified}\nGeneralities on relative {\\v C}ech complexes.\n\\begin{enumerate}\n\\item If\n$$\n\\xymatrix{\n\\mathcal{V} \\ar[d]_g \\ar[r]_h & \\mathcal{U} \\ar[d]^f \\\\\n\\mathcal{Y} \\ar[r]^e & \\mathcal{X}\n}\n$$\nis $2$-commutative diagram of categories fibred in groupoids over\n$(\\Sch/S)_{fppf}$, then there is a morphism\n$e^{-1}\\mathcal{K}^\\bullet(f, \\mathcal{F}) \\to\n\\mathcal{K}^\\bullet(g, e^{-1}\\mathcal{F})$.\n\\item if $h$ and $e$ are equivalences, then the map of (1) is an isomorphism,\n\\item if $f, f' : \\mathcal{U} \\to \\mathcal{X}$ are $2$-isomorphic, then\nthe associated relative {\\v C}ech complexes are isomorphic,\n\\end{enumerate}\n\\end{lemma}\n\n\\begin{proof}\nLiterally the same as the proof of\nLemma \\ref{lemma-generalities}\nusing the pullback maps of\nRemark \\ref{remark-cech-complex-presheaves}.\n\\end{proof}\n\n\\begin{lemma}\n\\label{lemma-homotopy-sheafified}\nIf there exists a $1$-morphism $s : \\mathcal{X} \\to \\mathcal{U}$\nsuch that $f \\circ s$ is $2$-isomorphic to $\\text{id}_\\mathcal{X}$\nthen the extended relative {\\v C}ech complex is homotopic to zero.\n\\end{lemma}\n\n\\begin{proof}\nLiterally the same as the proof of\nLemma \\ref{lemma-homotopy}.\n\\end{proof}\n\n\\begin{remark}\n\\label{remark-cech-complex-sections}\nLet us ``compute'' the value of the relative {\\v C}ech complex on an\nobject $x$ of $\\mathcal{X}$. Say $p(x) = U$.\nConsider the $2$-fibre product diagram (which serves to introduce\nthe notation $g : \\mathcal{V} \\to \\mathcal{Y}$)\n$$\n\\xymatrix{\n\\mathcal{V} \\ar@{=}[r] \\ar[d]_g &\n(\\Sch/U)_{fppf} \\times_{x, \\mathcal{X}} \\mathcal{U} \\ar[r] \\ar[d] &\n\\mathcal{U} \\ar[d]^f \\\\\n\\mathcal{Y} \\ar@{=}[r] &\n(\\Sch/U)_{fppf} \\ar[r]^-x & \\mathcal{X}\n}\n$$\nNote that the morphism $\\mathcal{V}_n \\to \\mathcal{U}_n$ of the proof of\nLemma \\ref{lemma-generalities}\ninduces an equivalence\n$\\mathcal{V}_n =\n(\\Sch/U)_{fppf} \\times_{x, \\mathcal{X}} \\mathcal{U}_n$.\nHence we see from\n(\\ref{equation-pushforward})\nthat\n$$\n\\Gamma(x, \\mathcal{K}^\\bullet(f, \\mathcal{F})) =\n\\check{\\mathcal{C}}^\\bullet(\\mathcal{V} \\to \\mathcal{Y}, x^{-1}\\mathcal{F})\n$$\nIn words: The value of the relative {\\v C}ech complex on an object $x$ of\n$\\mathcal{X}$ is the {\\v C}ech complex of the base change of $f$ to\n$\\mathcal{X}/x \\cong (\\Sch/U)_{fppf}$. This implies for example that\nLemma \\ref{lemma-homotopy}\nimplies\nLemma \\ref{lemma-homotopy-sheafified}\nand more generally that results on the (usual) {\\v C}ech complex imply\nresults for the relative {\\v C}ech complex.\n\\end{remark}\n\n\\begin{lemma}\n\\label{lemma-base-change-cech-complex}\nLet\n$$\n\\xymatrix{\n\\mathcal{V} \\ar[d]_g \\ar[r]_h & \\mathcal{U} \\ar[d]^f \\\\\n\\mathcal{Y} \\ar[r]^e & \\mathcal{X}\n}\n$$\nbe a $2$-fibre product of categories fibred in groupoids over\n$(\\Sch/S)_{fppf}$ and let $\\mathcal{F}$ be an abelian presheaf\non $\\mathcal{X}$. Then the map\n$e^{-1}\\mathcal{K}^\\bullet(f, \\mathcal{F}) \\to\n\\mathcal{K}^\\bullet(g, e^{-1}\\mathcal{F})$\nof\nLemma \\ref{lemma-generalities-sheafified}\nis an isomorphism of complexes of abelian presheaves.\n\\end{lemma}\n\n\\begin{proof}\nLet $y$ be an object of $\\mathcal{Y}$ lying over the scheme $T$.\nSet $x = e(y)$. We are going to show that the map induces an isomorphism\non sections over $y$. Note that\n$$\n\\Gamma(y, e^{-1}\\mathcal{K}^\\bullet(f, \\mathcal{F})) =\n\\Gamma(x, \\mathcal{K}^\\bullet(f, \\mathcal{F})) =\n\\check{\\mathcal{C}}^\\bullet(\n(\\Sch/T)_{fppf} \\times_{x, \\mathcal{X}} \\mathcal{U} \\to\n(\\Sch/T)_{fppf}, x^{-1}\\mathcal{F})\n$$\nby\nRemark \\ref{remark-cech-complex-sections}. On the other hand,\n$$\n\\Gamma(y, \\mathcal{K}^\\bullet(g, e^{-1}\\mathcal{F})) =\n\\check{\\mathcal{C}}^\\bullet(\n(\\Sch/T)_{fppf} \\times_{y, \\mathcal{Y}} \\mathcal{V} \\to\n(\\Sch/T)_{fppf}, y^{-1}e^{-1}\\mathcal{F})\n$$\nalso by\nRemark \\ref{remark-cech-complex-sections}.\nNote that $y^{-1}e^{-1}\\mathcal{F} = x^{-1}\\mathcal{F}$\nand since the diagram is $2$-cartesian the $1$-morphism\n$$\n(\\Sch/T)_{fppf} \\times_{y, \\mathcal{Y}} \\mathcal{V} \\to\n(\\Sch/T)_{fppf} \\times_{x, \\mathcal{X}} \\mathcal{U}\n$$\nis an equivalence. Hence the map on sections over $y$ is an\nisomorphism by\nLemma \\ref{lemma-generalities}.\n\\end{proof}\n\n\\noindent\nExactness can be checked on a ``covering''.\n\n\\begin{lemma}\n\\label{lemma-check-exactness-covering}\nLet $f : \\mathcal{U} \\to \\mathcal{X}$ be a $1$-morphism of categories fibred\nin groupoids over $(\\Sch/S)_{fppf}$. Let\n$\\tau \\in \\{Zar, \\etale, smooth, syntomic, fppf\\}$.\nLet\n$$\n\\mathcal{F} \\to \\mathcal{G} \\to \\mathcal{H}\n$$\nbe a complex in $\\textit{Ab}(\\mathcal{X}_\\tau)$. Assume that\n\\begin{enumerate}\n\\item for every object $x$ of $\\mathcal{X}$ there exists a covering\n$\\{x_i \\to x\\}$ in $\\mathcal{X}_\\tau$ such that each $x_i$ is isomorphic\nto $f(u_i)$ for some object $u_i$ of $\\mathcal{U}$, and\n\\item $f^{-1}\\mathcal{F} \\to f^{-1}\\mathcal{G} \\to f^{-1}\\mathcal{H}$ is exact.\n\\end{enumerate}\nThen the sequence $\\mathcal{F} \\to \\mathcal{G} \\to \\mathcal{H}$\nis exact.\n\\end{lemma}\n\n\\begin{proof}\nLet $x$ be an object of $\\mathcal{X}$ lying over the scheme $T$.\nConsider the sequence\n$x^{-1}\\mathcal{F} \\to x^{-1}\\mathcal{G} \\to x^{-1}\\mathcal{H}$\nof abelian sheaves on $(\\Sch/T)_\\tau$. It suffices to show\nthis sequence is exact. By assumption there exists a $\\tau$-covering\n$\\{T_i \\to T\\}$ such that $x|_{T_i}$ is isomorphic to $f(u_i)$ for\nsome object $u_i$ of $\\mathcal{U}$ over $T_i$ and moreover the sequence\n$u_i^{-1}f^{-1}\\mathcal{F} \\to u_i^{-1}f^{-1}\\mathcal{G} \\to\nu_i^{-1}f^{-1}\\mathcal{H}$ of abelian sheaves on $(\\Sch/T_i)_\\tau$\nis exact. Since\n$u_i^{-1}f^{-1}\\mathcal{F} = x^{-1}\\mathcal{F}|_{(\\Sch/T_i)_\\tau}$\nwe conclude that the sequence\n$x^{-1}\\mathcal{F} \\to x^{-1}\\mathcal{G} \\to x^{-1}\\mathcal{H}$\nbecome exact after localizing at each of the members of a covering,\nhence the sequence is exact.\n\\end{proof}\n\n\\begin{proposition}\n\\label{proposition-exactness-cech-complex}\nLet $f : \\mathcal{U} \\to \\mathcal{X}$ be a $1$-morphism of categories fibred\nin groupoids over $(\\Sch/S)_{fppf}$. Let\n$\\tau \\in \\{Zar, \\etale, smooth, syntomic, fppf\\}$.\nIf\n\\begin{enumerate}\n\\item $\\mathcal{F}$ is an abelian sheaf on $\\mathcal{X}_\\tau$, and\n\\item for every object $x$ of $\\mathcal{X}$ there exists a covering\n$\\{x_i \\to x\\}$ in $\\mathcal{X}_\\tau$ such that each $x_i$ is isomorphic\nto $f(u_i)$ for some object $u_i$ of $\\mathcal{U}$,\n\\end{enumerate}\nthen the extended relative {\\v C}ech complex\n$$\n\\ldots \\to 0 \\to\n\\mathcal{F} \\to\nf_{0, *}f_0^{-1}\\mathcal{F} \\to\nf_{1, *}f_1^{-1}\\mathcal{F} \\to\nf_{2, *}f_2^{-1}\\mathcal{F} \\to \\ldots\n$$\nis exact in $\\textit{Ab}(\\mathcal{X}_\\tau)$.\n\\end{proposition}\n\n\\begin{proof}\nBy\nLemma \\ref{lemma-check-exactness-covering}\nit suffices to check exactness after pulling back to $\\mathcal{U}$.\nBy\nLemma \\ref{lemma-base-change-cech-complex}\nthe pullback of the extended relative {\\v C}ech complex is isomorphic\nto the extend relative {\\v C}ech complex for the morphism\n$\\mathcal{U} \\times_\\mathcal{X} \\mathcal{U} \\to \\mathcal{U}$\nand an abelian sheaf on $\\mathcal{U}_\\tau$. Since there is a section\n$\\Delta_{\\mathcal{U}/\\mathcal{X}} : \\mathcal{U} \\to\n\\mathcal{U} \\times_\\mathcal{X} \\mathcal{U}$ exactness follows from\nLemma \\ref{lemma-homotopy-sheafified}.\n\\end{proof}\n\n\\noindent\nUsing this we can construct the {\\v C}ech-to-cohomology spectral sequence\nas follows. We first give a technical, precise version. In the next section\nwe give a version that applies only to algebraic stacks.\n\n\\begin{lemma}\n\\label{lemma-cech-to-cohomology}\nLet $f : \\mathcal{U} \\to \\mathcal{X}$ be a $1$-morphism of categories fibred\nin groupoids over $(\\Sch/S)_{fppf}$. Let\n$\\tau \\in \\{Zar, \\etale, smooth, syntomic, fppf\\}$.\nAssume\n\\begin{enumerate}\n\\item $\\mathcal{F}$ is an abelian sheaf on $\\mathcal{X}_\\tau$,\n\\item for every object $x$ of $\\mathcal{X}$ there exists a covering\n$\\{x_i \\to x\\}$ in $\\mathcal{X}_\\tau$ such that each $x_i$ is isomorphic\nto $f(u_i)$ for some object $u_i$ of $\\mathcal{U}$,\n\\item the category $\\mathcal{U}$ has equalizers, and\n\\item the functor $f$ is faithful.\n\\end{enumerate}\nThen there is a first quadrant spectral sequence of abelian groups\n$$\nE_1^{p, q} = H^q((\\mathcal{U}_p)_\\tau, f_p^{-1}\\mathcal{F})\n\\Rightarrow\nH^{p + q}(\\mathcal{X}_\\tau, \\mathcal{F})\n$$\nconverging to the cohomology of $\\mathcal{F}$ in the $\\tau$-topology.\n\\end{lemma}\n\n\\begin{proof}\nBefore we start the proof we make some remarks. By\nLemma \\ref{lemma-fibre-products-preserve-properties}\n(and induction) all of the categories fibred in groupoids $\\mathcal{U}_p$\nhave equalizers and all of the morphisms $f_p : \\mathcal{U}_p \\to \\mathcal{X}$\nare faithful. Let $\\mathcal{I}$ be an injective object\nof $\\textit{Ab}(\\mathcal{X}_\\tau)$. By\nLemma \\ref{lemma-pullback-injective}\nwe see $f_p^{-1}\\mathcal{I}$ is an injective object of\n$\\textit{Ab}((\\mathcal{U}_p)_\\tau)$.\nHence $f_{p, *}f_p^{-1}\\mathcal{I}$ is an injective object of\n$\\textit{Ab}(\\mathcal{X}_\\tau)$ by\nLemma \\ref{lemma-pushforward-injective}.\nHence\nProposition \\ref{proposition-exactness-cech-complex}\nshows that the extended relative {\\v C}ech complex\n$$\n\\ldots \\to 0 \\to\n\\mathcal{I} \\to\nf_{0, *}f_0^{-1}\\mathcal{I} \\to\nf_{1, *}f_1^{-1}\\mathcal{I} \\to\nf_{2, *}f_2^{-1}\\mathcal{I} \\to \\ldots\n$$\nis an exact complex in $\\textit{Ab}(\\mathcal{X}_\\tau)$ all of whose\nterms are injective. Taking global sections of this complex is exact\nand we see that the {\\v C}ech complex\n$\\check{\\mathcal{C}}^\\bullet(\\mathcal{U} \\to \\mathcal{X}, \\mathcal{I})$\nis quasi-isomorphic to $\\Gamma(\\mathcal{X}_\\tau, \\mathcal{I})[0]$.\n\n\\medskip\\noindent\nWith these preliminaries out of the way consider the two spectral sequences\nassociated to the double complex (see\nHomology, Section \\ref{homology-section-double-complex})\n$$\n\\check{\\mathcal{C}}^\\bullet(\\mathcal{U} \\to \\mathcal{X}, \\mathcal{I}^\\bullet)\n$$\nwhere $\\mathcal{F} \\to \\mathcal{I}^\\bullet$ is an injective resolution\nin $\\textit{Ab}(\\mathcal{X}_\\tau)$.\nThe discussion above shows that\nHomology, Lemma \\ref{homology-lemma-double-complex-gives-resolution}\napplies which shows that\n$\\Gamma(\\mathcal{X}_\\tau, \\mathcal{I}^\\bullet)$\nis quasi-isomorphic to the total complex associated to the double complex.\nBy our remarks above the complex $f_p^{-1}\\mathcal{I}^\\bullet$ is an\ninjective resolution of $f_p^{-1}\\mathcal{F}$. Hence the other spectral\nsequence is as indicated in the lemma.\n\\end{proof}\n\n\\noindent\nTo be sure there is a version for modules as well.\n\n\\begin{lemma}\n\\label{lemma-cech-to-cohomology-modules}\nLet $f : \\mathcal{U} \\to \\mathcal{X}$ be a $1$-morphism of categories fibred\nin groupoids over $(\\Sch/S)_{fppf}$. Let\n$\\tau \\in \\{Zar, \\etale, smooth, syntomic, fppf\\}$.\nAssume\n\\begin{enumerate}\n\\item $\\mathcal{F}$ is an object of\n$\\textit{Mod}(\\mathcal{X}_\\tau, \\mathcal{O}_\\mathcal{X})$,\n\\item for every object $x$ of $\\mathcal{X}$ there exists a covering\n$\\{x_i \\to x\\}$ in $\\mathcal{X}_\\tau$ such that each $x_i$ is isomorphic\nto $f(u_i)$ for some object $u_i$ of $\\mathcal{U}$,\n\\item the category $\\mathcal{U}$ has equalizers, and\n\\item the functor $f$ is faithful.\n\\end{enumerate}\nThen there is a first quadrant spectral sequence of\n$\\Gamma(\\mathcal{O}_\\mathcal{X})$-modules\n$$\nE_1^{p, q} = H^q((\\mathcal{U}_p)_\\tau, f_p^*\\mathcal{F})\n\\Rightarrow\nH^{p + q}(\\mathcal{X}_\\tau, \\mathcal{F})\n$$\nconverging to the cohomology of $\\mathcal{F}$ in the $\\tau$-topology.\n\\end{lemma}\n\n\\begin{proof}\nThe proof of this lemma is identical to the proof of\nLemma \\ref{lemma-cech-to-cohomology}\nexcept that it uses an injective resolution in\n$\\textit{Mod}(\\mathcal{X}_\\tau, \\mathcal{O}_\\mathcal{X})$\nand it uses\nLemma \\ref{lemma-pullback-injective-modules}\ninstead of\nLemma \\ref{lemma-pullback-injective}.\n\\end{proof}\n\n\\noindent\nHere is a lemma that translates a more usual kind of covering in the\nkinds of coverings we have encountered above.\n\n\\begin{lemma}\n\\label{lemma-surjective-flat-locally-finite-presentation}\nLet $f : \\mathcal{X} \\to \\mathcal{Y}$ be a $1$-morphism of\ncategories fibred in groupoids over $(\\Sch/S)_{fppf}$.\n\\begin{enumerate}\n\\item Assume that $f$ is representable by algebraic spaces, surjective,\nflat, and locally of finite presentation. Then for any object $y$ of\n$\\mathcal{Y}$ there exists an fppf covering $\\{y_i \\to y\\}$ and objects\n$x_i$ of $\\mathcal{X}$ such that $f(x_i) \\cong y_i$ in $\\mathcal{Y}$.\n\\item Assume that $f$ is representable by algebraic spaces, surjective,\nand smooth. Then for any object $y$ of\n$\\mathcal{Y}$ there exists an \\'etale covering $\\{y_i \\to y\\}$ and objects\n$x_i$ of $\\mathcal{X}$ such that $f(x_i) \\cong y_i$ in $\\mathcal{Y}$.\n\\end{enumerate}\n\\end{lemma}\n\n\\begin{proof}\nProof of (1). Suppose that $y$ lies over the scheme $V$.\nWe may think of $y$ as a morphism $(\\Sch/V)_{fppf} \\to \\mathcal{Y}$.\nBy definition the $2$-fibre product\n$\\mathcal{X} \\times_\\mathcal{Y} (\\Sch/V)_{fppf}$\nis representable by an algebraic space $W$ and the morphism\n$W \\to V$ is surjective, flat, and locally of finite presentation.\nChoose a scheme $U$ and a surjective \\'etale morphism $U \\to W$.\nThen $U \\to V$ is also surjective, flat, and locally of finite presentation\n(see Morphisms of Spaces, Lemmas\n\\ref{spaces-morphisms-lemma-etale-flat},\n\\ref{spaces-morphisms-lemma-etale-locally-finite-presentation},\n\\ref{spaces-morphisms-lemma-composition-surjective},\n\\ref{spaces-morphisms-lemma-composition-finite-presentation}, and\n\\ref{spaces-morphisms-lemma-composition-flat}).\nHence $\\{U \\to V\\}$ is an fppf covering. Denote $x$ the object of\n$\\mathcal{X}$ over $U$ corresponding to the $1$-morphism\n$(\\Sch/U)_{fppf} \\to \\mathcal{X}$. Then $\\{f(x) \\to y\\}$ is\nthe desired fppf covering of $\\mathcal{Y}$.\n\n\\medskip\\noindent\nProof of (1). Suppose that $y$ lies over the scheme $V$.\nWe may think of $y$ as a morphism $(\\Sch/V)_{fppf} \\to \\mathcal{Y}$.\nBy definition the $2$-fibre product\n$\\mathcal{X} \\times_\\mathcal{Y} (\\Sch/V)_{fppf}$\nis representable by an algebraic space $W$ and the morphism\n$W \\to V$ is surjective and smooth.\nChoose a scheme $U$ and a surjective \\'etale morphism $U \\to W$.\nThen $U \\to V$ is also surjective and smooth\n(see Morphisms of Spaces, Lemmas\n\\ref{spaces-morphisms-lemma-etale-smooth},\n\\ref{spaces-morphisms-lemma-composition-surjective}, and\n\\ref{spaces-morphisms-lemma-composition-smooth}).\nHence $\\{U \\to V\\}$ is a smooth covering. By\nMore on Morphisms, Lemma \\ref{more-morphisms-lemma-etale-dominates-smooth}\nthere exists an \\'etale covering $\\{V_i \\to V\\}$ such that\neach $V_i \\to V$ factors through $U$. Denote $x_i$ the object of\n$\\mathcal{X}$ over $V_i$ corresponding to the $1$-morphism\n$$\n(\\Sch/V_i)_{fppf} \\to (\\Sch/U)_{fppf} \\to \\mathcal{X}.\n$$\nThen $\\{f(x_i) \\to y\\}$ is\nthe desired \\'etale covering of $\\mathcal{Y}$.\n\\end{proof}\n\n\\begin{lemma}\n\\label{lemma-cech-to-cohomology-relative}\nLet $f : \\mathcal{U} \\to \\mathcal{X}$ and\n$g : \\mathcal{X} \\to \\mathcal{Y}$\nbe composable $1$-morphisms of categories fibred\nin groupoids over $(\\Sch/S)_{fppf}$. Let\n$\\tau \\in \\{Zar, \\etale, smooth, syntomic, \\linebreak[0] fppf\\}$.\nAssume\n\\begin{enumerate}\n\\item $\\mathcal{F}$ is an abelian sheaf on $\\mathcal{X}_\\tau$,\n\\item for every object $x$ of $\\mathcal{X}$ there exists a covering\n$\\{x_i \\to x\\}$ in $\\mathcal{X}_\\tau$ such that each $x_i$ is isomorphic\nto $f(u_i)$ for some object $u_i$ of $\\mathcal{U}$,\n\\item the category $\\mathcal{U}$ has equalizers, and\n\\item the functor $f$ is faithful.\n\\end{enumerate}\nThen there is a first quadrant spectral sequence of abelian sheaves\non $\\mathcal{Y}_\\tau$\n$$\nE_1^{p, q} = R^q(g \\circ f_p)_*f_p^{-1}\\mathcal{F}\n\\Rightarrow\nR^{p + q}g_*\\mathcal{F}\n$$\nwhere all higher direct images are computed in the $\\tau$-topology.\n\\end{lemma}\n\n\\begin{proof}\nNote that the assumptions on $f : \\mathcal{U} \\to \\mathcal{X}$\nand $\\mathcal{F}$ are identical to those in\nLemma \\ref{lemma-cech-to-cohomology}.\nHence the preliminary remarks made in the proof of that lemma\nhold here also. These remarks imply in particular that\n$$\n0 \\to g_*\\mathcal{I} \\to\n(g \\circ f_0)_*f_0^{-1}\\mathcal{I} \\to\n(g \\circ f_1)_*f_1^{-1}\\mathcal{I} \\to \\ldots\n$$\nis exact if $\\mathcal{I}$ is an injective object of\n$\\textit{Ab}(\\mathcal{X}_\\tau)$.\nHaving said this, consider the two spectral sequences of\nHomology, Section \\ref{homology-section-double-complex}\nassociated to the double complex $\\mathcal{C}^{\\bullet, \\bullet}$ with terms\n$$\n\\mathcal{C}^{p, q} = (g \\circ f_p)_*\\mathcal{I}^q\n$$\nwhere $\\mathcal{F} \\to \\mathcal{I}^\\bullet$ is an injective resolution\nin $\\textit{Ab}(\\mathcal{X}_\\tau)$. The first spectral sequence implies, via\nHomology, Lemma \\ref{homology-lemma-double-complex-gives-resolution},\nthat $g_*\\mathcal{I}^\\bullet$ is quasi-isomorphic to the total complex\nassociated to $\\mathcal{C}^{\\bullet, \\bullet}$.\nSince $f_p^{-1}\\mathcal{I}^\\bullet$ is an injective resolution of\n$f_p^{-1}\\mathcal{F}$ (see\nLemma \\ref{lemma-pullback-injective})\nthe second spectral sequence has terms\n$E_1^{p, q} = R^q(g \\circ f_p)_*f_p^{-1}\\mathcal{F}$ as in the statement\nof the lemma.\n\\end{proof}\n\n\\begin{lemma}\n\\label{lemma-cech-to-cohomology-relative-modules}\nLet $f : \\mathcal{U} \\to \\mathcal{X}$ and\n$g : \\mathcal{X} \\to \\mathcal{Y}$\nbe composable $1$-morphisms of categories fibred\nin groupoids over $(\\Sch/S)_{fppf}$. Let\n$\\tau \\in \\{Zar, \\etale, smooth, syntomic, \\linebreak[0] fppf\\}$.\nAssume\n\\begin{enumerate}\n\\item $\\mathcal{F}$ is an object of\n$\\textit{Mod}(\\mathcal{X}_\\tau, \\mathcal{O}_\\mathcal{X})$,\n\\item for every object $x$ of $\\mathcal{X}$ there exists a covering\n$\\{x_i \\to x\\}$ in $\\mathcal{X}_\\tau$ such that each $x_i$ is isomorphic\nto $f(u_i)$ for some object $u_i$ of $\\mathcal{U}$,\n\\item the category $\\mathcal{U}$ has equalizers, and\n\\item the functor $f$ is faithful.\n\\end{enumerate}\nThen there is a first quadrant spectral sequence in\n$\\textit{Mod}(\\mathcal{Y}_\\tau, \\mathcal{O}_\\mathcal{Y})$\n$$\nE_1^{p, q} = R^q(g \\circ f_p)_*f_p^{-1}\\mathcal{F}\n\\Rightarrow\nR^{p + q}g_*\\mathcal{F}\n$$\nwhere all higher direct images are computed in the $\\tau$-topology.\n\\end{lemma}\n\n\\begin{proof}\nThe proof is identical to the proof of\nLemma \\ref{lemma-cech-to-cohomology-relative}\nexcept that it uses an injective resolution in\n$\\textit{Mod}(\\mathcal{X}_\\tau, \\mathcal{O}_\\mathcal{X})$\nand it uses\nLemma \\ref{lemma-pullback-injective-modules}\ninstead of\nLemma \\ref{lemma-pullback-injective}.\n\\end{proof}\n\n\n\n\n\n\n\\section{Cohomology on algebraic stacks}\n\\label{section-cohomology}\n\n\\noindent\nLet $\\mathcal{X}$ be an algebraic stack over $S$. In the sections above\nwe have seen how to define sheaves for the \\'etale, ..., fppf\ntopologies on $\\mathcal{X}$. In fact, we have constructed a site\n$\\mathcal{X}_\\tau$ for each $\\tau \\in \\{Zar, \\etale, smooth, syntomic,\nfppf\\}$. There is a notion of an abelian sheaf $\\mathcal{F}$ on these sites.\nIn the chapter on cohomology of sites we have explained how to define\ncohomology. Putting all of this together, let's define the\n{\\it derived global sections}\n$$\nR\\Gamma_{Zar}(\\mathcal{X}, \\mathcal{F}),\nR\\Gamma_\\etale(\\mathcal{X}, \\mathcal{F}), \\ldots,\nR\\Gamma_{fppf}(\\mathcal{X}, \\mathcal{F})\n$$\nas $\\Gamma(\\mathcal{X}_\\tau, \\mathcal{I}^\\bullet)$ where\n$\\mathcal{F} \\to \\mathcal{I}^\\bullet$ is an injective resolution\nin $\\textit{Ab}(\\mathcal{X}_\\tau)$. The $i$th cohomology group is the\n$i$th cohomology of the total derived cohomology. We will denote\nthis\n$$\nH^i_{Zar}(\\mathcal{X}, \\mathcal{F}),\nH^i_\\etale(\\mathcal{X}, \\mathcal{F}), \\ldots,\nH^i_{fppf}(\\mathcal{X}, \\mathcal{F}).\n$$\nIt will turn out that $H^i_\\etale = H^i_{smooth}$\nbecause of\nMore on Morphisms, Lemma \\ref{more-morphisms-lemma-etale-dominates-smooth}.\nIf $\\mathcal{F}$ is a presheaf of $\\mathcal{O}_\\mathcal{X}$-modules\nwhich is a sheaf in the $\\tau$-topology, then we use injective\nresolutions in $\\textit{Mod}(\\mathcal{X}_\\tau, \\mathcal{O}_\\mathcal{X})$\nto compute total derived global sections and cohomology groups; of course\nthe end result is quasi-isomorphic resp.\\ isomorphic by the general fact\nCohomology on Sites, Lemma\n\\ref{sites-cohomology-lemma-cohomology-modules-abelian-agree}.\n\n\\medskip\\noindent\nSo far our only tool to compute cohomology groups is the result on\n{\\v C}ech complexes proved above. We rephrase it here in the language\nof algebraic stacks for the \\'etale and the fppf topology. Let\n$f : \\mathcal{U} \\to \\mathcal{X}$ be a $1$-morphism of algebraic stacks.\nRecall that\n$$\nf_p : \\mathcal{U}_p =\n\\mathcal{U} \\times_\\mathcal{X} \\ldots \\times_\\mathcal{X} \\mathcal{U}\n\\longrightarrow\n\\mathcal{X}\n$$\nis the structure morphism where there are $(p + 1)$-factors. Also, recall\nthat a sheaf on $\\mathcal{X}$ is a sheaf for the fppf topology. Note\nthat if $\\mathcal{U}$ is an algebraic space, then\n$f : \\mathcal{U} \\to \\mathcal{X}$ is representable by algebraic spaces,\nsee\nAlgebraic Stacks, Lemma \\ref{algebraic-lemma-representable-diagonal}.\nThus the proposition applies in particular to a smooth cover of the\nalgebraic stack $\\mathcal{X}$ by a scheme.\n\n\\begin{proposition}\n\\label{proposition-smooth-covering-compute-cohomology}\nLet $f : \\mathcal{U} \\to \\mathcal{X}$ be a $1$-morphism of algebraic stacks.\n\\begin{enumerate}\n\\item Let $\\mathcal{F}$ be an abelian \\'etale sheaf on $\\mathcal{X}$.\nAssume that $f$ is representable by algebraic spaces, surjective, and smooth.\nThen there is a spectral sequence\n$$\nE_1^{p, q} = H^q_\\etale(\\mathcal{U}_p, f_p^{-1}\\mathcal{F})\n\\Rightarrow\nH^{p + q}_\\etale(\\mathcal{X}, \\mathcal{F})\n$$\n\\item Let $\\mathcal{F}$ be an abelian sheaf on $\\mathcal{X}$.\nAssume that $f$ is representable by algebraic spaces, surjective, flat,\nand locally of finite presentation. Then there is\na spectral sequence\n$$\nE_1^{p, q} = H^q_{fppf}(\\mathcal{U}_p, f_p^{-1}\\mathcal{F})\n\\Rightarrow\nH^{p + q}_{fppf}(\\mathcal{X}, \\mathcal{F})\n$$\n\\end{enumerate}\n\\end{proposition}\n\n\\begin{proof}\nTo see this we will check the hypotheses (1) -- (4) of\nLemma \\ref{lemma-cech-to-cohomology}.\nThe $1$-morphism $f$ is faithful by\nAlgebraic Stacks, Lemma\n\\ref{algebraic-lemma-characterize-representable-by-algebraic-spaces}.\nThis proves (4).\nHypothesis (3) follows from the fact that $\\mathcal{U}$ is an algebraic\nstack, see\nLemma \\ref{lemma-fibre-products}.\nTo see (2) apply\nLemma \\ref{lemma-surjective-flat-locally-finite-presentation}.\nCondition (1) is satisfied by fiat.\n\\end{proof}\n\n\n\n\n\n\n\n\n\n\n\\section{Higher direct images and algebraic stacks}\n\\label{section-higher-direct-images}\n\n\\noindent\nLet $g : \\mathcal{X} \\to \\mathcal{Y}$ be a $1$-morphism of algebraic stacks\nover $S$. In the sections above we have constructed a morphism of ringed\ntopoi $g : \\Sh(\\mathcal{X}_\\tau) \\to \\Sh(\\mathcal{Y}_\\tau)$\nfor each $\\tau \\in \\{Zar, \\etale, smooth, syntomic, fppf\\}$.\nIn the chapter on cohomology of sites we have explained how to\ndefine higher direct images. Hence the {\\it derived direct image}\n$Rg_*\\mathcal{F}$ is defined as $g_*\\mathcal{I}^\\bullet$ where\n$\\mathcal{F} \\to \\mathcal{I}^\\bullet$ is an injective resolution in\n$\\textit{Ab}(\\mathcal{X}_\\tau)$. The $i$th higher direct image\n$R^ig_*\\mathcal{F}$ is the $i$th cohomology of the derived direct image.\nImportant: it matters which topology $\\tau$ is used here!\n\n\\medskip\\noindent\nIf $\\mathcal{F}$ is a presheaf of $\\mathcal{O}_\\mathcal{X}$-modules\nwhich is a sheaf in the $\\tau$-topology, then we use injective\nresolutions in $\\textit{Mod}(\\mathcal{X}_\\tau, \\mathcal{O}_\\mathcal{X})$\nto compute derived direct image and higher direct images.\n\n\\medskip\\noindent\nSo far our only tool to compute the higher direct images of $g_*$\nis the result on {\\v C}ech complexes proved above. This requires the choice\nof a ``covering'' $f : \\mathcal{U} \\to \\mathcal{X}$. If $\\mathcal{U}$\nis an algebraic space, then $f : \\mathcal{U} \\to \\mathcal{X}$\nis representable by algebraic spaces, see\nAlgebraic Stacks, Lemma \\ref{algebraic-lemma-representable-diagonal}.\nThus the proposition applies in particular to a smooth cover of the\nalgebraic stack $\\mathcal{X}$ by a scheme.\n\n\\begin{proposition}\n\\label{proposition-smooth-covering-compute-direct-image}\nLet $f : \\mathcal{U} \\to \\mathcal{X}$ and $g : \\mathcal{X} \\to \\mathcal{Y}$\nbe composable $1$-morphisms of algebraic stacks.\n\\begin{enumerate}\n\\item Assume that $f$ is representable by algebraic spaces, surjective and\nsmooth.\n\\begin{enumerate}\n\\item If $\\mathcal{F}$ is in $\\textit{Ab}(\\mathcal{X}_\\etale)$\nthen there is a spectral sequence\n$$\nE_1^{p, q} = R^q(g \\circ f_p)_*f_p^{-1}\\mathcal{F}\n\\Rightarrow\nR^{p + q}g_*\\mathcal{F}\n$$\nin $\\textit{Ab}(\\mathcal{Y}_\\etale)$ with higher direct images\ncomputed in the \\'etale topology.\n\\item If $\\mathcal{F}$ is in\n$\\textit{Mod}(\\mathcal{X}_\\etale, \\mathcal{O}_\\mathcal{X})$ then\nthere is a spectral sequence\n$$\nE_1^{p, q} = R^q(g \\circ f_p)_*f_p^{-1}\\mathcal{F}\n\\Rightarrow\nR^{p + q}g_*\\mathcal{F}\n$$\nin $\\textit{Mod}(\\mathcal{Y}_\\etale, \\mathcal{O}_\\mathcal{Y})$.\n\\end{enumerate}\n\\item Assume that $f$ is representable by algebraic spaces, surjective,\nflat, and locally of finite presentation.\n\\begin{enumerate}\n\\item If $\\mathcal{F}$ is in $\\textit{Ab}(\\mathcal{X})$ then there is\na spectral sequence\n$$\nE_1^{p, q} = R^q(g \\circ f_p)_*f_p^{-1}\\mathcal{F}\n\\Rightarrow\nR^{p + q}g_*\\mathcal{F}\n$$\nin $\\textit{Ab}(\\mathcal{Y})$ with higher direct images\ncomputed in the fppf topology.\n\\item If $\\mathcal{F}$ is in $\\textit{Mod}(\\mathcal{O}_\\mathcal{X})$ then\nthere is a spectral sequence\n$$\nE_1^{p, q} = R^q(g \\circ f_p)_*f_p^{-1}\\mathcal{F}\n\\Rightarrow\nR^{p + q}g_*\\mathcal{F}\n$$\nin $\\textit{Mod}(\\mathcal{O}_\\mathcal{Y})$.\n\\end{enumerate}\n\\end{enumerate}\n\\end{proposition}\n\n\\begin{proof}\nTo see this we will check the hypotheses (1) -- (4) of\nLemma \\ref{lemma-cech-to-cohomology-relative} and\nLemma \\ref{lemma-cech-to-cohomology-relative-modules}.\nThe $1$-morphism $f$ is faithful by\nAlgebraic Stacks, Lemma\n\\ref{algebraic-lemma-characterize-representable-by-algebraic-spaces}.\nThis proves (4).\nHypothesis (3) follows from the fact that $\\mathcal{U}$ is an algebraic\nstack, see\nLemma \\ref{lemma-fibre-products}.\nTo see (2) apply\nLemma \\ref{lemma-surjective-flat-locally-finite-presentation}.\nCondition (1) is satisfied by fiat in all four cases.\n\\end{proof}\n\n\\noindent\nHere is a description of higher direct images for a\nmorphism of algebraic stacks.\n\n\\begin{lemma}\n\\label{lemma-pushforward-restriction}\nLet $S$ be a scheme. Let $f : \\mathcal{X} \\to \\mathcal{Y}$ be a\n$1$-morphism of algebraic stacks\\footnote{This result should hold\nfor any $1$-morphism of categories fibred in groupoids over\n$(\\Sch/S)_{fppf}$.} over $S$.\nLet $\\tau \\in \\{Zariski,\\linebreak[0] \\etale,\\linebreak[0]\nsmooth,\\linebreak[0] syntomic,\\linebreak[0] fppf\\}$.\nLet $\\mathcal{F}$ be\nan object of $\\textit{Ab}(\\mathcal{X}_\\tau)$ or\n$\\textit{Mod}(\\mathcal{X}_\\tau, \\mathcal{O}_\\mathcal{X})$.\nThen the sheaf $R^if_*\\mathcal{F}$ is the sheaf associated to the\npresheaf\n$$\ny \\longmapsto\nH^i_\\tau\\Big((\\Sch/V)_{fppf} \\times_{y, \\mathcal{Y}} \\mathcal{X},\n\\ \\text{pr}^{-1}\\mathcal{F}\\Big)\n$$\nHere $y$ is a typical object of $\\mathcal{Y}$ lying over the scheme $V$.\n\\end{lemma}\n\n\\begin{proof}\nChoose an injective resolution $\\mathcal{F}[0] \\to \\mathcal{I}^\\bullet$.\nBy the formula for pushforward (\\ref{equation-pushforward}) we see that\n$R^if_*\\mathcal{F}$ is the sheaf associated to the presheaf which associates\nto $y$ the cohomology of the complex\n$$\n\\begin{matrix}\n\\Gamma\\Big((\\Sch/V)_{fppf} \\times_{y, \\mathcal{Y}} \\mathcal{X},\n\\ \\text{pr}^{-1}\\mathcal{I}^{i - 1}\\Big) \\\\\n\\downarrow \\\\\n\\Gamma\\Big((\\Sch/V)_{fppf} \\times_{y, \\mathcal{Y}} \\mathcal{X},\n\\ \\text{pr}^{-1}\\mathcal{I}^i\\Big) \\\\\n\\downarrow \\\\\n\\Gamma\\Big((\\Sch/V)_{fppf} \\times_{y, \\mathcal{Y}} \\mathcal{X},\n\\ \\text{pr}^{-1}\\mathcal{I}^{i + 1}\\Big)\n\\end{matrix}\n$$\nSince $\\text{pr}^{-1}$ is exact, it suffices to show that\n$\\text{pr}^{-1}$ preserves injectives. This follows from\nLemmas \\ref{lemma-pullback-injective} and\n\\ref{lemma-pullback-injective-modules}\nas well as the fact that $\\text{pr}$ is a representable morphism of\nalgebraic stacks (so that $\\text{pr}$ is faithful by\nAlgebraic Stacks, Lemma\n\\ref{algebraic-lemma-characterize-representable-by-algebraic-spaces}\nand that\n$(\\Sch/V)_{fppf} \\times_{y, \\mathcal{Y}} \\mathcal{X}$\nhas equalizers by \nLemma \\ref{lemma-fibre-products}).\n\\end{proof}\n\n\\noindent\nHere is a trivial base change result.\n\n\\begin{lemma}\n\\label{lemma-base-change-higher-direct-images}\nLet $S$ be a scheme. Let\n$\\tau \\in \\{Zariski,\\linebreak[0] \\etale,\\linebreak[0]\nsmooth,\\linebreak[0] syntomic,\\linebreak[0] fppf\\}$. Let\n$$\n\\xymatrix{\n\\mathcal{Y}' \\times_\\mathcal{Y} \\mathcal{X} \\ar[r]_{g'} \\ar[d]_{f'} &\n\\mathcal{X} \\ar[d]^f \\\\\n\\mathcal{Y}' \\ar[r]^g & \\mathcal{Y}\n}\n$$\nbe a $2$-cartesian diagram of algebraic stacks over $S$. Then the base change\nmap is an isomorphism\n$$\ng^{-1}Rf_*\\mathcal{F} \\longrightarrow Rf'_*(g')^{-1}\\mathcal{F}\n$$\nfunctorial for $\\mathcal{F}$ in $\\textit{Ab}(\\mathcal{X}_\\tau)$\nor $\\mathcal{F}$ in $\\textit{Mod}(\\mathcal{X}_\\tau, \\mathcal{O}_\\mathcal{X})$.\n\\end{lemma}\n\n\\begin{proof}\nThe isomorphism $g^{-1}f_*\\mathcal{F} = f'_*(g')^{-1}\\mathcal{F}$ is\nLemma \\ref{lemma-base-change} (and it holds for arbitrary presheaves).\nFor the derived direct images, there is a base change map because the\nmorphisms $g$ and $g'$ are flat, see\nCohomology on Sites, Section \\ref{sites-cohomology-section-base-change-map}.\nTo see that this map is a quasi-isomorphism we can use that for\nan object $y'$ of $\\mathcal{Y}'$ over a scheme $V$ there is an equivalence\n$$\n(\\Sch/V)_{fppf} \\times_{g(y'), \\mathcal{Y}} \\mathcal{X}\n=\n(\\Sch/V)_{fppf} \\times_{y', \\mathcal{Y}'}\n(\\mathcal{Y}' \\times_\\mathcal{Y} \\mathcal{X})\n$$\nWe conclude that the induced map\n$g^{-1}R^if_*\\mathcal{F} \\to R^if'_*(g')^{-1}\\mathcal{F}$\nis an isomorphism by\nLemma \\ref{lemma-pushforward-restriction}.\n\\end{proof}\n\n\n\n\n\n\n\n\n\n\n\n\n\\section{Comparison}\n\\label{section-compare}\n\n\\noindent\nIn this section we collect some results on comparing cohomology defined\nusing stacks and using algebraic spaces.\n\n\\begin{lemma}\n\\label{lemma-compare-injectives}\nLet $S$ be a scheme. Let $\\mathcal{X}$ be an algebraic stack over $S$\nrepresentable by the algebraic space $F$.\n\\begin{enumerate}\n\\item If $\\mathcal{I}$ injective in $\\textit{Ab}(\\mathcal{X}_\\etale)$, then\n$\\mathcal{I}|_{F_\\etale}$ is injective in $\\textit{Ab}(F_\\etale)$,\n\\item If $\\mathcal{I}^\\bullet$ is a K-injective complex in\n$\\textit{Ab}(\\mathcal{X}_\\etale)$, then $\\mathcal{I}^\\bullet|_{F_\\etale}$\nis a K-injective complex in $\\textit{Ab}(F_\\etale)$.\n\\end{enumerate}\nThe same does not hold for modules.\n\\end{lemma}\n\n\\begin{proof}\nThis follows formally from the fact that the restriction functor\n$\\pi_{F, *} = i_F^{-1}$ (see Lemma \\ref{lemma-compare})\nis right adjoint to the exact functor $\\pi_F^{-1}$, see\nHomology, Lemma \\ref{homology-lemma-adjoint-preserve-injectives} and\nDerived Categories, Lemma \\ref{derived-lemma-adjoint-preserve-K-injectives}.\nTo see that the lemma does not hold for modules, we refer the\nreader to \\'Etale Cohomology, Lemma\n\\ref{etale-cohomology-lemma-compare-injectives}.\n\\end{proof}\n\n\\begin{lemma}\n\\label{lemma-compare-morphism-cohomology}\nLet $S$ be a scheme. Let $f : \\mathcal{X} \\to \\mathcal{Y}$ be a morphism\nof algebraic stacks over $S$. Assume $\\mathcal{X}$, $\\mathcal{Y}$ are\nrepresentable by algebraic spaces $F$, $G$. Denote $f : F \\to G$ the\ninduced morphism of algebraic spaces.\n\\begin{enumerate}\n\\item For any $\\mathcal{F} \\in \\textit{Ab}(\\mathcal{X}_\\etale)$\nwe have\n$$\n(Rf_*\\mathcal{F})|_{G_\\etale} =\nRf_{small, *}(\\mathcal{F}|_{F_\\etale})\n$$\nin $D(G_\\etale)$.\n\\item For any object $\\mathcal{F}$ of\n$\\textit{Mod}(\\mathcal{X}_\\etale, \\mathcal{O}_\\mathcal{X})$\nwe have\n$$\n(Rf_*\\mathcal{F})|_{G_\\etale} =\nRf_{small, *}(\\mathcal{F}|_{F_\\etale})\n$$\nin $D(\\mathcal{O}_G)$.\n\\end{enumerate}\n\\end{lemma}\n\n\\begin{proof}\nPart (1) follows immediately from\nLemma \\ref{lemma-compare-injectives}\nand (\\ref{equation-compare-big-small})\non choosing an injective resolution of $\\mathcal{F}$.\n\n\\medskip\\noindent\nPart (2) can be proved as follows. In Lemma \\ref{lemma-compare-morphism}\nwe have seen that $\\pi_G \\circ f = f_{small} \\circ \\pi_F$ as morphisms\nof ringed sites. Hence we obtain\n$R\\pi_{G, *} \\circ Rf_* = Rf_{small, *} \\circ R\\pi_{F, *}$\nby Cohomology on Sites, Lemma\n\\ref{sites-cohomology-lemma-derived-pushforward-composition}.\nSince the restriction functors $\\pi_{F, *}$ and $\\pi_{G, *}$\nare exact, we conclude.\n\\end{proof}\n\n\\begin{lemma}\n\\label{lemma-compare-representable-morphism-cohomology}\nLet $S$ be a scheme. Consider a $2$-fibre product square\n$$\n\\xymatrix{\n\\mathcal{X}' \\ar[r]_{g'} \\ar[d]_{f'} & \\mathcal{X} \\ar[d]^f \\\\\n\\mathcal{Y}' \\ar[r]^g & \\mathcal{Y}\n}\n$$\nof algebraic stacks over $S$. Assume that $f$ is representable by algebraic\nspaces and that $\\mathcal{Y}'$ is representable by an algebraic space $G'$.\nThen $\\mathcal{X}'$ is representable by an algebraic space $F'$ and\ndenoting $f' : F' \\to G'$ the induced morphism of algebraic spaces\nwe have\n$$\ng^{-1}(Rf_*\\mathcal{F})|_{G'_\\etale} =\nRf'_{small, *}((g')^{-1}\\mathcal{F}|_{F'_\\etale})\n$$\nfor any $\\mathcal{F}$ in $\\textit{Ab}(\\mathcal{X}_\\etale)$\nor in\n$\\textit{Mod}(\\mathcal{X}_\\etale, \\mathcal{O}_\\mathcal{X})$\n\\end{lemma}\n\n\\begin{proof}\nFollows formally on combining\nLemmas \\ref{lemma-base-change-higher-direct-images} and\n\\ref{lemma-compare-morphism-cohomology}.\n\\end{proof}\n\n\n\n\n\n\n\n\n\n\n\n\\section{Change of topology}\n\\label{section-change-topology}\n\n\\noindent\nHere is a technical lemma which tells us that the\nfppf cohomology of a locally quasi-coherent sheaf is equal to its\n\\'etale cohomology provided the comparison maps are isomorphisms\nfor morphisms of $\\mathcal{X}$ lying over flat morphisms.\n\n\\begin{lemma}\n\\label{lemma-lqc-flat-base-change-fppf-sheaf}\nLet $S$ be a scheme. Let $\\mathcal{X}$ be an algebraic stack over $S$.\nLet $\\mathcal{F}$ be a presheaf of $\\mathcal{O}_\\mathcal{X}$-modules.\nAssume\n\\begin{enumerate}\n\\item[(a)] $\\mathcal{F}$ is locally quasi-coherent, and\n\\item[(b)] for any morphism $\\varphi : x \\to y$ of $\\mathcal{X}$ which lies\nover a morphism of schemes $f : U \\to V$ which is flat and\nlocally of finite presentation the comparison map\n$c_\\varphi : f_{small}^*\\mathcal{F}|_{V_\\etale} \\to\n\\mathcal{F}|_{U_\\etale}$ of\n(\\ref{equation-comparison-modules}) is an isomorphism.\n\\end{enumerate}\nThen $\\mathcal{F}$ is a sheaf for the fppf topology.\n\\end{lemma}\n\n\\begin{proof}\nLet $\\{x_i \\to x\\}$ be an fppf covering of $\\mathcal{X}$ lying over the\nfppf covering $\\{f_i : U_i \\to U\\}$ of schemes over $S$.\nBy assumption the restriction $\\mathcal{G} = \\mathcal{F}|_{U_\\etale}$\nis quasi-coherent and the comparison maps\n$f_{i, small}^*\\mathcal{G} \\to \\mathcal{F}|_{U_{i, \\etale}}$\nare isomorphisms. Hence the sheaf condition for $\\mathcal{F}$\nand the covering $\\{x_i \\to x\\}$ is equivalent to the sheaf condition\nfor $\\mathcal{G}^a$ on $(\\Sch/U)_{fppf}$ and the covering $\\{U_i \\to U\\}$\nwhich holds by\nDescent, Lemma \\ref{descent-lemma-sheaf-condition-holds}.\n\\end{proof}\n\n\\begin{lemma}\n\\label{lemma-compare-fppf-etale}\nLet $S$ be a scheme. Let $\\mathcal{X}$ be an algebraic stack over $S$.\nLet $\\mathcal{F}$ be a presheaf $\\mathcal{O}_\\mathcal{X}$-module such that\n\\begin{enumerate}\n\\item[(a)] $\\mathcal{F}$ is locally quasi-coherent, and\n\\item[(b)] for any morphism $\\varphi : x \\to y$ of $\\mathcal{X}$ which lies\nover a morphism of schemes $f : U \\to V$ which is flat and\nlocally of finite presentation, the comparison map\n$c_\\varphi : f_{small}^*\\mathcal{F}|_{V_\\etale} \\to\n\\mathcal{F}|_{U_\\etale}$ of\n(\\ref{equation-comparison-modules}) is an isomorphism.\n\\end{enumerate}\nThen $\\mathcal{F}$ is an $\\mathcal{O}_\\mathcal{X}$-module and\nwe have the following\n\\begin{enumerate}\n\\item If $\\epsilon : \\mathcal{X}_{fppf} \\to \\mathcal{X}_\\etale$\nis the comparison morphism, then\n$R\\epsilon_*\\mathcal{F} = \\epsilon_*\\mathcal{F}$.\n\\item The cohomology groups $H^p_{fppf}(\\mathcal{X}, \\mathcal{F})$ are equal\nto the cohomology groups computed in the \\'etale topology on $\\mathcal{X}$.\nSimilarly for the cohomology groups $H^p_{fppf}(x, \\mathcal{F})$ and the\nderived versions $R\\Gamma(\\mathcal{X}, \\mathcal{F})$ and\n$R\\Gamma(x, \\mathcal{F})$.\n\\item If $f : \\mathcal{X} \\to \\mathcal{Y}$ is a $1$-morphism of\ncategories fibred in groupoids over $(\\Sch/S)_{fppf}$ then\n$R^if_*\\mathcal{F}$ is equal to the fppf-sheafification of the\nhigher direct image computed in the \\'etale cohomology.\nSimilarly for derived pushforward.\n\\end{enumerate}\n\\end{lemma}\n\n\\begin{proof}\nThe assertion that $\\mathcal{F}$ is an $\\mathcal{O}_\\mathcal{X}$-module\nfollows from\nLemma \\ref{lemma-lqc-flat-base-change-fppf-sheaf}.\nNote that $\\epsilon$ is a morphism of sites given by the identity\nfunctor on $\\mathcal{X}$. The sheaf $R^p\\epsilon_*\\mathcal{F}$ is therefore\nthe sheaf associated to the presheaf\n$x \\mapsto H^p_{fppf}(x, \\mathcal{F})$, see\nCohomology on Sites, Lemma \\ref{sites-cohomology-lemma-higher-direct-images}.\nTo prove (1) it suffices to show that\n$H^p_{fppf}(x, \\mathcal{F}) = 0$ for $p > 0$\nwhenever $x$ lies over an affine scheme $U$. By\nLemma \\ref{lemma-cohomology-restriction}\nwe have\n$H^p_{fppf}(x, \\mathcal{F}) = H^p((\\Sch/U)_{fppf}, x^{-1}\\mathcal{F})$.\nCombining\nDescent, Lemma \\ref{descent-lemma-quasi-coherent-and-flat-base-change}\nwith Cohomology of Schemes, Lemma\n\\ref{coherent-lemma-quasi-coherent-affine-cohomology-zero}\nwe see that these cohomology groups are zero.\n\n\\medskip\\noindent\nWe have seen above that $\\epsilon_*\\mathcal{F}$ and $\\mathcal{F}$ are the\nsheaves on $\\mathcal{X}_\\etale$ and $\\mathcal{X}_{fppf}$\ncorresponding to the same presheaf on $\\mathcal{X}$ (and this is true more\ngenerally for any sheaf in the fppf topology on $\\mathcal{X}$).\nWe often abusively identify $\\mathcal{F}$ and $\\epsilon_*\\mathcal{F}$\nand this is the sense in which parts (2) and (3) of the lemma should be\nunderstood. Thus part (2) follows formally from (1) and the Leray spectral\nsequence, see\nCohomology on Sites, Lemma \\ref{sites-cohomology-lemma-apply-Leray}.\n\n\\medskip\\noindent\nFinally we prove (3). The sheaf $R^if_*\\mathcal{F}$\n(resp.\\ $Rf_{\\etale, *}\\mathcal{F}$)\nis the sheaf associated to the presheaf\n$$\ny \\longmapsto\nH^i_\\tau\\Big((\\Sch/V)_{fppf} \\times_{y, \\mathcal{Y}} \\mathcal{X},\n\\ \\text{pr}^{-1}\\mathcal{F}\\Big)\n$$\nwhere $\\tau$ is $fppf$ (resp.\\ $\\etale$), see\nLemma \\ref{lemma-pushforward-restriction}.\nNote that $\\text{pr}^{-1}\\mathcal{F}$ satisfies properties (a) and (b)\nalso (by Lemmas \\ref{lemma-pullback-lqc} and \\ref{lemma-comparison}),\nhence these two presheaves are equal by (2).\nThis immediately implies (3).\n\\end{proof}\n\n\\noindent\nWe will use the following lemma to compare \\'etale cohomology of sheaves\non algebraic stacks with cohomology on the lisse-\\'etale topos.\n\n\\begin{lemma}\n\\label{lemma-cohomology-on-subcategory}\nLet $S$ be a scheme. Let $\\mathcal{X}$ be an algebraic stack over $S$.\nLet $\\tau = \\etale$ (resp.\\ $\\tau = fppf$). Let\n$\\mathcal{X}' \\subset \\mathcal{X}$ be a full subcategory with the\nfollowing properties\n\\begin{enumerate}\n\\item if $x \\to x'$ is a morphism of $\\mathcal{X}$ which lies over a\nsmooth (resp.\\ flat and locally finitely presented) morphism of\nschemes and $x' \\in \\Ob(\\mathcal{X}')$, then $x \\in \\Ob(\\mathcal{X}')$, and\n\\item there exists an object $x \\in \\Ob(\\mathcal{X}')$ lying over\na scheme $U$ such that the associated $1$-morphism\n$x : (\\Sch/U)_{fppf} \\to \\mathcal{X}$ is smooth and surjective.\n\\end{enumerate}\nWe get a site $\\mathcal{X}'_\\tau$ by declaring a covering of $\\mathcal{X}'$\nto be any family of morphisms $\\{x_i \\to x\\}$ in $\\mathcal{X}'$ which is a\ncovering in $\\mathcal{X}_\\tau$. Then the inclusion functor\n$\\mathcal{X}' \\to \\mathcal{X}_\\tau$ is fully faithful, cocontinuous, and\ncontinuous, whence defines a morphism of topoi\n$$\ng : \\Sh(\\mathcal{X}'_\\tau) \\longrightarrow \\Sh(\\mathcal{X}_\\tau)\n$$\nand $H^p(\\mathcal{X}'_\\tau, g^{-1}\\mathcal{F}) =\nH^p(\\mathcal{X}_\\tau, \\mathcal{F})$ for all $p \\geq 0$ and all\n$\\mathcal{F} \\in \\textit{Ab}(\\mathcal{X}_\\tau)$.\n\\end{lemma}\n\n\\begin{proof}\nNote that assumption (1) implies that if $\\{x_i \\to x\\}$ is a covering\nof $\\mathcal{X}_\\tau$ and $x \\in \\Ob(\\mathcal{X}')$, then we have\n$x_i \\in \\Ob(\\mathcal{X}')$. Hence we see that $\\mathcal{X}' \\to \\mathcal{X}$\nis continuous and cocontinuous as the coverings of objects of\n$\\mathcal{X}'_\\tau$ agree with their coverings seen as objects of\n$\\mathcal{X}_\\tau$. We obtain the morphism $g$ and the functor\n$g^{-1}$ is identified with the restriction functor, see\nSites, Lemma \\ref{sites-lemma-when-shriek}.\n\n\\medskip\\noindent\nIn particular, if $\\{x_i \\to x\\}$ is a covering in $\\mathcal{X}'_\\tau$,\nthen for any abelian sheaf $\\mathcal{F}$ on $\\mathcal{X}$ then\n$$\n\\check H^p(\\{x_i \\to x\\}, g^{-1}\\mathcal{F}) =\n\\check H^p(\\{x_i \\to x\\}, \\mathcal{F})\n$$\nThus if $\\mathcal{I}$ is an injective abelian sheaf on $\\mathcal{X}_\\tau$\nthen we see that the higher {\\v C}ech cohomology groups are zero\n(Cohomology on Sites,\nLemma \\ref{sites-cohomology-lemma-injective-trivial-cech}).\nHence $H^p(x, g^{-1}\\mathcal{I}) = 0$ for all objects $x$\nof $\\mathcal{X}'$\n(Cohomology on Sites,\nLemma \\ref{sites-cohomology-lemma-cech-vanish-collection}).\nIn other words injective abelian sheaves on $\\mathcal{X}_\\tau$\nare right acyclic for the functor $H^0(x, g^{-1}-)$.\nIt follows that $H^p(x, g^{-1}\\mathcal{F}) = H^p(x, \\mathcal{F})$\nfor all $\\mathcal{F} \\in \\textit{Ab}(\\mathcal{X})$ and all\n$x \\in \\Ob(\\mathcal{X}')$.\n\n\\medskip\\noindent\nChoose an object $x \\in \\mathcal{X}'$ lying over a scheme $U$\nas in assumption (2). In particular $\\mathcal{X}/x \\to \\mathcal{X}$\nis a morphism of algebraic stacks which representable by algebraic spaces,\nsurjective, and smooth. (Note that $\\mathcal{X}/x$ is equivalent to\n$(\\Sch/U)_{fppf}$, see Lemma \\ref{lemma-localizing}.)\nThe map of sheaves\n$$\nh_x \\longrightarrow *\n$$\nin $\\Sh(\\mathcal{X}_\\tau)$ is surjective. Namely, for any object $x'$\nof $\\mathcal{X}$ there exists a $\\tau$-covering $\\{x'_i \\to x'\\}$\nsuch that there exist morphisms $x'_i \\to x$, see\nLemma \\ref{lemma-surjective-flat-locally-finite-presentation}.\nSince $g$ is exact, the map of sheaves\n$$\ng^{-1}h_x \\longrightarrow * = g^{-1}*\n$$\nin $\\Sh(\\mathcal{X}'_\\tau)$ is surjective also. Let $h_{x, n}$ be\nthe $(n + 1)$-fold product $h_x \\times \\ldots \\times h_x$.\nThen we have spectral sequences\n\\begin{equation}\n\\label{equation-spectral-sequence-one}\nE_1^{p, q} = H^q(h_{x, p}, \\mathcal{F}) \\Rightarrow\nH^{p + q}(\\mathcal{X}_\\tau, \\mathcal{F})\n\\end{equation}\nand\n\\begin{equation}\n\\label{equation-spectral-sequence-two}\nE_1^{p, q} = H^q(g^{-1}h_{x, p}, g^{-1}\\mathcal{F}) \\Rightarrow\nH^{p + q}(\\mathcal{X}'_\\tau, g^{-1}\\mathcal{F})\n\\end{equation}\nsee Cohomology on Sites,\nLemma \\ref{sites-cohomology-lemma-cech-to-cohomology-sheaf-sets}.\n\n\\medskip\\noindent\nCase I: $\\mathcal{X}$ has a final object $x$ which is also an object of\n$\\mathcal{X}'$. This case follows immediately from the discussion\nin the second paragraph above.\n\n\\medskip\\noindent\nCase II: $\\mathcal{X}$ is representable by an algebraic space $F$.\nIn this case the sheaves $h_{x, n}$ are representable by an\nobject $x_n$ in $\\mathcal{X}$. (Namely, if $\\mathcal{S}_F = \\mathcal{X}$\nand $x : U \\to F$ is the given object, then $h_{x, n}$ is representable\nby the object $U \\times_F \\ldots \\times_F U \\to F$ of $\\mathcal{S}_F$.)\nIt follows that $H^q(h_{x, p}, \\mathcal{F}) = H^q(x_p, \\mathcal{F})$.\nThe morphisms $x_n \\to x$ lie over smooth morphisms of schemes, hence\n$x_n \\in \\mathcal{X}'$ for all $n$. Hence\n$H^q(g^{-1}h_{x, p}, g^{-1}\\mathcal{F}) = H^q(x_p, g^{-1}\\mathcal{F})$.\nThus in the two spectral sequences\n(\\ref{equation-spectral-sequence-one}) and\n(\\ref{equation-spectral-sequence-two}) above the $E_1^{p, q}$ terms agree\nby the discussion in the second paragraph. The lemma follows in Case II\nas well.\n\n\\medskip\\noindent\nCase III: $\\mathcal{X}$ is an algebraic stack. We claim that in this case\nthe cohomology groups $H^q(h_{x, p}, \\mathcal{F})$ and\n$H^q(g^{-1}h_{x, n}, g^{-1}\\mathcal{F})$ agree by Case II above.\nOnce we have proved this the result will follow as before.\n\n\\medskip\\noindent\nNamely, consider the category $\\mathcal{X}/h_{x, n}$, see\nSites, Lemma \\ref{sites-lemma-localize-topos-site}.\nSince $h_{x, n}$ is the $(n + 1)$-fold product of $h_x$ an\nobject of this category is an $(n + 2)$-tuple\n$(y, s_0, \\ldots, s_n)$ where $y$ is an object of $\\mathcal{X}$ and each\n$s_i : y \\to x$ is a morphism of $\\mathcal{X}$.\nThis is a category over $(\\Sch/S)_{fppf}$. There is an equivalence\n$$\n\\mathcal{X}/h_{x, n}\n\\longrightarrow\n(\\Sch/U)_{fppf} \\times_\\mathcal{X} \\ldots \\times_\\mathcal{X} (\\Sch/U)_{fppf}\n=: \\mathcal{U}_n\n$$\nover $(\\Sch/S)_{fppf}$. Namely, if $x : (\\Sch/U)_{fppf} \\to \\mathcal{X}$ also\ndenotes the $1$-morphism associated with $x$ and\n$p : \\mathcal{X} \\to (\\Sch/S)_{fppf}$ the structure functor,\nthen we can think of $(y, s_0, \\ldots, s_n)$ as\n$(y, f_0, \\ldots, f_n, \\alpha_0, \\ldots, \\alpha_n)$\nwhere $y$ is an object of $\\mathcal{X}$, $f_i : p(y) \\to p(x)$ is a\nmorphism of schemes, and $\\alpha_i : y \\to x(f_i)$ an isomorphism.\nThe category of $2n+3$-tuples\n$(y, f_0, \\ldots, f_n, \\alpha_0, \\ldots, \\alpha_n)$\nis an incarnation of the $(n + 1)$-fold fibred product $\\mathcal{U}_n$\nof algebraic stacks displayed above, as we discussed in\nSection \\ref{section-cech}.\nBy Cohomology on Sites, Lemma\n\\ref{sites-cohomology-lemma-cohomology-on-sheaf-sets}\nwe have\n$$\nH^p(\\mathcal{U}_n, \\mathcal{F}|_{\\mathcal{U}_n}) =\nH^p(\\mathcal{X}/h_{x, n}, \\mathcal{F}|_{\\mathcal{X}/h_{x, n}}) =\nH^p(h_{x, n}, \\mathcal{F}).\n$$\nFinally, we discuss the ``primed'' analogue of this. Namely,\n$\\mathcal{X}'/h_{x, n}$ corresponds, via the equivalence above\nto the full subcategory $\\mathcal{U}'_n \\subset \\mathcal{U}_n$\nconsisting of those tuples\n$(y, f_0, \\ldots, f_n, \\alpha_0, \\ldots, \\alpha_n)$\nwith $y \\in \\mathcal{X}'$. Hence certainly property (1) of the\nstatement of the lemma holds\nfor the inclusion $\\mathcal{U}'_n \\subset \\mathcal{U}_n$.\nTo see property (2) choose an object $\\xi = (y, s_0, \\ldots, s_n)$ which\nlies over a scheme $W$ such that $(\\Sch/W)_{fppf} \\to \\mathcal{U}_n$\nis smooth and surjective (this is possible as $\\mathcal{U}_n$ is\nan algebraic stack). Then\n$(\\Sch/W)_{fppf} \\to \\mathcal{U}_n \\to (\\Sch/U)_{fppf}$\nis smooth as a composition of base changes of the morphism\n$x : (\\Sch/U)_{fppf} \\to \\mathcal{X}$, see\nAlgebraic Stacks, Lemmas\n\\ref{algebraic-lemma-base-change-representable-transformations-property} and\n\\ref{algebraic-lemma-composition-representable-transformations-property}.\nThus axiom (1) for $\\mathcal{X}$ implies that $y$ is an object of\n$\\mathcal{X}'$ whence $\\xi$ is an object of $\\mathcal{U}'_n$.\nUsing again\n$$\nH^p(\\mathcal{U}'_n, \\mathcal{F}|_{\\mathcal{U}'_n}) =\nH^p(\\mathcal{X}'/h_{x, n}, \\mathcal{F}|_{\\mathcal{X}'/h_{x, n}}) =\nH^p(g^{-1}h_{x, n}, g^{-1}\\mathcal{F}).\n$$\nwe now can use Case II for \n$\\mathcal{U}'_n \\subset \\mathcal{U}_n$\nto conclude.\n\\end{proof}\n\n\n\n\n\n\n\n\n\n\\input{chapters}\n\n\\bibliography{my}\n\\bibliographystyle{amsalpha}\n\n\\end{document}\n", [(5839, 5850, 'VAR'), (5857, 5865, 'TYPE'), (6252, 6263, 'VAR'), (6270, 6278, 'TYPE'), (8596, 8607, 'VAR'), (8614, 8622, 'TYPE'), (13009, 13020, 'VAR'), (13027, 13035, 'TYPE'), (14801, 14812, 'VAR'), (14819, 14827, 'TYPE'), (14877, 14888, 'VAR'), (14895, 14903, 'TYPE'), (20463, 20464, 'VAR'), (20471, 20478, 'TYPE'), (24748, 24761, 'VAR'), (24768, 24783, 'TYPE'), (29190, 29201, 'VAR'), (29208, 29216, 'TYPE'), (33845, 33846, 'VAR'), (33853, 33860, 'TYPE'), (33866, 33877, 'VAR'), (33884, 33892, 'TYPE'), (34549, 34550, 'VAR'), (34557, 34564, 'TYPE'), (36248, 36259, 'VAR'), (36266, 36274, 'TYPE'), (37738, 37749, 'VAR'), (37756, 37764, 'TYPE'), (39456, 39467, 'VAR'), (39475, 39482, 'TYPE'), (41553, 41564, 'VAR'), (41571, 41579, 'TYPE'), (41629, 41640, 'VAR'), (41647, 41657, 'TYPE'), (41734, 41735, 'VAR'), (41743, 41749, 'TYPE'), (44132, 44133, 'VAR'), (44140, 44147, 'TYPE'), (44153, 44184, 'VAR'), (44191, 44199, 'TYPE'), (46300, 46301, 'VAR'), (46308, 46314, 'TYPE'), (47907, 47908, 'VAR'), (47915, 47922, 'TYPE'), (49721, 49732, 'VAR'), (49739, 49744, 'TYPE'), (49878, 49885, 'VAR'), (49893, 49899, 'TYPE'), (59964, 59965, 'VAR'), (59973, 59979, 'TYPE'), (60463, 60464, 'VAR'), (60472, 60478, 'TYPE'), (69587, 69588, 'VAR'), (69596, 69602, 'TYPE'), (72250, 72261, 'VAR'), (72268, 72282, 'TYPE'), (76476, 76487, 'VAR'), (76494, 76508, 'TYPE'), (79271, 79282, 'VAR'), (79290, 79299, 'TYPE'), (80960, 80971, 'VAR'), (80979, 80988, 'TYPE'), (82318, 82329, 'VAR'), (82337, 82346, 'TYPE'), (86070, 86071, 'VAR'), (86078, 86084, 'TYPE'), (86094, 86105, 'VAR'), (86112, 86120, 'TYPE'), (87435, 87436, 'VAR'), (87443, 87450, 'TYPE'), (87456, 87467, 'VAR'), (87474, 87482, 'TYPE'), (87592, 87614, 'VAR'), (87622, 87628, 'TYPE'), (92854, 92864, 'VAR'), (92871, 92879, 'TYPE'), (98490, 98491, 'VAR'), (98499, 98505, 'TYPE'), (111827, 111838, 'VAR'), (111846, 111853, 'TYPE'), (112115, 112116, 'VAR'), (112124, 112130, 'TYPE'), (113971, 113972, 'VAR'), (113980, 113986, 'TYPE'), (117562, 117573, 'VAR'), (117581, 117590, 'TYPE'), (127287, 127298, 'VAR'), (127306, 127315, 'TYPE'), (129954, 129965, 'VAR'), (129973, 129980, 'TYPE'), (130257, 130268, 'VAR'), (130276, 130283, 'TYPE'), (135069, 135070, 'VAR'), (135077, 135084, 'TYPE'), (137109, 137110, 'VAR'), (137117, 137124, 'TYPE'), (138746, 138747, 'VAR'), (138754, 138761, 'TYPE'), (138767, 138778, 'VAR'), (138786, 138795, 'TYPE'), (139823, 139824, 'VAR'), (139831, 139838, 'TYPE'), (141186, 141187, 'VAR'), (141194, 141201, 'TYPE'), (142425, 142426, 'VAR'), (142433, 142440, 'TYPE'), (142446, 142457, 'VAR'), (142465, 142474, 'TYPE'), (142496, 142507, 'VAR'), (142514, 142522, 'TYPE'), (143063, 143076, 'VAR'), (143084, 143088, 'TYPE'), (143704, 143705, 'VAR'), (143712, 143719, 'TYPE'), (143725, 143736, 'VAR'), (143744, 143753, 'TYPE'), (143775, 143786, 'VAR'), (143793, 143801, 'TYPE'), (147425, 147426, 'VAR'), (147433, 147440, 'TYPE'), (147446, 147457, 'VAR'), (147465, 147474, 'TYPE')])

In [38]:

    

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


    

In [35]:

    

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 [36]:

    

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


    

In [37]:

    

#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 [47]:

    

#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{More 	
on 	
Cohomology 	
of 	
Spaces 	
} 	


 	
\maketitle 	


 	
\phantomsection 	

 	
\label{section 	
- 	
phantom 	
} 	


 	
\tableofcontents 	





 	
\section{Introduction 	
} 	

 	
\label{section 	
- 	
introduction 	
} 	


 	
\noindent 	

 	
In 	
this 	
chapter 	
continues 	
the 	
discussion 	
started 	
in 	

 	
Cohomology 	
of 	
Spaces 	
, 	
Section 	
\ref{spaces 	
- 	
cohomology 	
- 	
section 	
- 	
introduction}. 	

 	
One 	
can 	
also 	
view 	
this 	
chapter 	
as 	
the 	
analogue 	
for 	
algebraic 	
spaces 	

 	
of 	
the 	
chapter 	
on 	
\'etale 	
cohomology 	
for 	
schemes 	
, 	
see 	

 	
\'Etale 	
Cohomology 	
, 	
Section 	
\ref{etale 	
- 	
cohomology 	
- 	
section 	
- 	
introduction}. 	


 	
\medskip\noindent 	

 	
In 	
fact 	
, 	
we 	
intend 	
this 	
chapter 	
to 	
be 	
mainly 	
a 	
translation 	
of 	
the 	

 	
results 	
already 	
proved 	
for 	
schemes 	
into 	
the 	
language 	
of 	
algebraic 	

 	
spaces 	
. 	
Some 	
of 	
our 	
results 	
can 	
be 	
found 	
in 	
\cite{Kn}. 	






 	
\section{Conventions 	
} 	

 	
\label{section 	
- 	
conventions 	
} 	


 	
\noindent 	

 	
The 	
standing 	
assumption 	
is 	
that 	
all 	
schemes 	
are 	
contained 	
in 	

 	
a 	
big 	
fppf 	
site 	
$ 	
\Sch_{fppf}$. 	
And 	
all 	
rings 	
$ 	
A$ 	
considered 	

 	
have 	
the 	
property 	
that 	
$ 	
\Spec(A)$ 	
is 	
( 	
isomorphic 	
) 	
to 	
an 	

 	
object 	
of 	
this 	
big 	
site 	
. 	


 	
\medskip\noindent 	

 	
Let 	
$ 	
S$ 	
be 	
a 	
scheme 	TYPE
and 	
let 	
$ 	
X$ 	
be 	
an 	
algebraic 	TYPE
space 	
over 	
$ 	
S$. 	

 	
In 	
this 	
chapter 	
and 	
the 	
following 	
we 	
will 	
write 	
$ 	
X 	
\times_S 	
X$ 	

 	
for 	
the 	
product 	
of 	
$ 	
X$ 	
with 	
itself 	
( 	
in 	
the 	
category 	
of 	
algebraic 	

 	
spaces 	
over 	
$ 	
S$ 	
) 	
, 	
instead 	
of 	
$ 	
X 	
\times 	
X$. 	








 	
\section{Transporting 	
results 	
from 	
schemes 	
} 	

 	
\label{section 	
- 	
api 	
} 	


 	
\noindent 	

 	
In 	
this 	
section 	
we 	
explain 	
briefly 	
how 	
results 	
for 	
schemes 	

 	
imply 	
results 	
for 	
( 	
representable 	
) 	
algebraic 	
spaces 	
and 	

 	
( 	
representable 	
) 	
morphisms 	
of 	
algebraic 	
spaces 	
. 	

 	
For 	
quasi 	
- 	
coherent 	
modules 	
more 	
is 	
true 	

 	
( 	
because 	
\'etale 	
cohomology 	
of 	
a 	
quasi 	
- 	
coherent 	
module 	

 	
over 	
a 	
scheme 	
agrees 	
with 	
Zariski 	
cohomology 	
) 	
and 	
this 	

 	
has 	
already 	
been 	
discussed 	
in 	
Cohomology 	
of 	
Spaces 	
, 	
Section 	

 	
\ref{spaces 	
- 	
cohomology 	
- 	
section 	
- 	
higher 	
- 	
direct 	
- 	
image}. 	


 	
\medskip\noindent 	

 	
Let 	
$ 	
S$ 	
be 	
a 	
scheme 	TYPE
. 	TYPE
Let 	
$ 	
X$ 	
be 	
an 	
algebraic 	TYPE
space 	
over 	
$ 	
S$. 	

 	
Now 	
suppose 	
that 	
$ 	
X$ 	
is 	
representable 	
by 	
the 	
scheme 	
$ 	
X_0 	
$ 	

 	
( 	
awkward 	
but 	
temporary 	
notation 	
; 	
we 	
usually 	
just 	
say 	
`` 	
$ 	
X$ 	

 	
is 	
a 	
scheme 	
'' 	
) 	
. 	
In 	
this 	
case 	
$ 	
X$ 	
and 	
$ 	
X_0 	
$ 	
have 	
the 	
same 	
small 	

 	
\'etale 	
sites 	
: 	

 	
$ 	
$ 	

 	
X_\etale 	
= 	
( 	
X_0)_\etale 	

 	
$ 	
$ 	

 	
This 	
is 	
pointed 	
out 	
in 	

 	
Properties 	
of 	
Spaces 	
, 	
Section 	
\ref{spaces 	
- 	
properties 	
- 	
section 	
- 	
etale 	
- 	
site}. 	

 	
Moreover 	
, 	
if 	
$ 	
f 	
: 	
X 	
\to 	
Y$ 	
is 	
a 	
morphism 	
of 	
representable 	
algebraic 	
spaces 	

 	
over 	
$ 	
S$ 	
and 	
if 	
$ 	
f_0 	
: 	
X_0 	
\to 	
Y_0 	
$ 	
is 	
a 	
morphism 	
of 	

 	
schemes 	
representing 	
$ 	
f$ 	
, 	
then 	
the 	
induced 	
morphisms 	
of 	
small 	

 	
\'etale 	
topoi 	
agree 	
: 	

 	
$ 	
$ 	

 	
\xymatrix 	
{ 	

 	
\Sh(X_\etale 	
) 	
\ar[rr]_{f_{small 	
} 	
} 	
\ar@{=}[d 	
] 	
& 	
& 	

 	
\Sh(Y_\etale 	
) 	
\ar@{=}[d 	
] 	
\\ 	

 	
\Sh((X_0)_\etale 	
) 	
\ar[rr]^{(f_0)_{small 	
} 	
} 	
& 	
& 	

 	
\Sh((Y_0)_\etale 	
) 	

 	
} 	

 	
$ 	
$ 	

 	
See 	
Properties 	
of 	
Spaces 	
, 	
Lemma 	

 	
\ref{spaces 	
- 	
properties 	
- 	
lemma 	
- 	
functoriality 	
- 	
etale 	
- 	
site 	
} 	
and 	

 	
Topologies 	
, 	
Lemma 	
\ref{topologies 	
- 	
lemma 	
- 	
morphism 	
- 	
big 	
- 	
small 	
- 	
etale}. 	


 	
\medskip\noindent 	

 	
Thus 	
there 	
is 	
absolutely 	
no 	
difference 	
between 	
\'etale 	
cohomology 	

 	
of 	
a 	
scheme 	
and 	
the 	
\'etale 	
cohomology 	
of 	
the 	
corresponding 	
algebraic 	
space 	
. 	

 	
Similarly 	
for 	
higher 	
direct 	
images 	
along 	
morphisms 	
of 	
schemes 	
. 	

 	
In 	
fact 	
, 	
if 	
$ 	
f 	
: 	
X 	
\to 	
Y$ 	
is 	
a 	
morphism 	
of 	
algebraic 	
spaces 	
over 	
$ 	
S$ 	

 	
which 	
is 	
representable 	
( 	
by 	
schemes 	
) 	
, 	
then 	
the 	
higher 	
direct 	
images 	

 	
$ 	
R^if_*\mathcal{F}$ 	
of 	
a 	
sheaf 	
$ 	
\mathcal{F}$ 	
on 	
$ 	
X_\etale$ 	

 	
can 	
be 	
computed 	
\'etale 	
locally 	
on 	
$ 	
Y$ 	
( 	
Cohomology 	
on 	
Sites 	
, 	

 	
Lemma 	
\ref{sites 	
- 	
cohomology 	
- 	
lemma 	
- 	
higher 	
- 	
direct 	
- 	
images 	
} 	
) 	

 	
hence 	
this 	
often 	
reduces 	
computations 	
and 	
proofs 	
to 	
the 	
case 	

 	
where 	
$ 	
Y$ 	
and 	
$ 	
X$ 	
are 	
schemes 	
. 	


 	
\medskip\noindent 	

 	
We 	
will 	
use 	
the 	
above 	
without 	
further 	
mention 	
in 	
this 	
chapter 	
. 	

 	
For 	
other 	
topologies 	
the 	
same 	
thing 	
is 	
true 	
; 	
we 	
state 	
it 	

 	
explicitly 	
as 	
a 	
lemma 	
for 	
cohomology 	
here 	
. 	


 	
\begin{lemma 	
} 	

 	
\label{lemma 	
- 	
compare 	
- 	
cohomology 	
- 	
other 	
- 	
topologies 	
} 	

 	
Let 	
$ 	
S$ 	
be 	
a 	
scheme 	TYPE
. 	TYPE
Let 	
$ 	
\tau 	
\in 	
\{\etale 	
, 	
fppf 	
, 	
ph\}$ 	
( 	
add 	
more 	
here 	
) 	
. 	

 	
The 	
inclusion 	
functor 	

 	
$ 	
$ 	

 	
( 	
\Sch 	
/ 	
S)_\tau 	
\longrightarrow 	
( 	
\textit{Spaces}/S)_\tau 	

 	
$ 	
$ 	

 	
is 	
a 	
special 	
cocontinuous 	
functor 	

 	
( 	
Sites 	
, 	
Definition 	
\ref{sites 	
- 	
definition 	
- 	
special 	
- 	
cocontinuous 	
- 	
functor 	
} 	
) 	

 	
and 	
hence 	
identifies 	
topoi 	
. 	

 	
\end{lemma 	
} 	


 	
\begin{proof 	
} 	

 	
The 	
conditions 	
of 	
Sites 	
, 	
Lemma 	
\ref{sites 	
- 	
lemma 	
- 	
equivalence 	
} 	

 	
are 	
immediately 	
verified 	
as 	
our 	
functor 	
is 	
fully 	
faithful 	

 	
and 	
as 	
every 	
algebraic 	
space 	
has 	
an 	
\'etale 	
covering 	
by 	
schemes 	
. 	

 	
\end{proof 	
} 	








 	
\section{Proper 	
base 	
change 	
} 	

 	
\label{section 	
- 	
proper 	
- 	
base 	
- 	
change 	
} 	


 	
\noindent 	

 	
The 	
proper 	
base 	
change 	
theorem 	
for 	
algebraic 	
spaces 	
follows 	
from 	

 	
the 	
proper 	
base 	
change 	
theorem 	
for 	
schemes 	
and 	
Chow 	
's 	
lemma 	

 	
with 	
a 	
little 	
bit 	
of 	
work 	
. 	


 	
\begin{lemma 	
} 	

 	
\label{lemma 	
- 	
surjective 	
- 	
proper 	
} 	

 	
Let 	
$ 	
S$ 	
be 	
a 	
scheme 	TYPE
. 	TYPE
Let 	
$ 	
f 	
: 	
Y 	
\to 	
X$ 	
be 	
a 	
surjective 	TYPE
proper 	
morphism 	

 	
of 	
algebraic 	
spaces 	
over 	
$ 	
S$. 	
Let 	
$ 	
\mathcal{F}$ 	
be 	
a 	
sheaf 	TYPE
on 	
$ 	
X_\etale$. 	

 	
Then 	
$ 	
\mathcal{F 	
} 	
\to 	
f_*f^{-1}\mathcal{F}$ 	
is 	
injective 	
with 	

 	
image 	
the 	
equalizer 	
of 	
the 	
two 	
maps 	

 	
$ 	
f_*f^{-1}\mathcal{F 	
} 	
\to 	
g_*g^{-1}\mathcal{F}$ 	
where 	

 	
$ 	
g$ 	
is 	
the 	
structure 	
morphism 	
$ 	
g 	
: 	
Y 	
\times_X 	
Y 	
\to 	
X$. 	

 	
\end{lemma 	
} 	


 	
\begin{proof 	
} 	

 	
For 	
any 	
surjective 	
morphism 	
$ 	
f 	
: 	
Y 	
\to 	
X$ 	
of 	
algebraic 	
spaces 	
over 	
$ 	
S$ 	
, 	

 	
the 	
map 	
$ 	
\mathcal{F 	
} 	
\to 	
f_*f^{-1}\mathcal{F}$ 	
is 	
injective 	
. 	

 	
Namely 	
, 	
if 	
$ 	
\overline{x}$ 	
is 	
a 	
geometric 	
point 	
of 	
$ 	
X$ 	
, 	
then 	
we 	

 	
choose 	
a 	
geometric 	
point 	
$ 	
\overline{y}$ 	
of 	
$ 	
Y$ 	
lying 	
over 	
$ 	
\overline{x}$ 	

 	
and 	
we 	
consider 	

 	
$ 	
$ 	

 	
\mathcal{F}_{\overline{x 	
} 	
} 	
\to 	

 	
( 	
f_*f^{-1}\mathcal{F})_{\overline{x 	
} 	
} 	
\to 	

 	
( 	
f^{-1}\mathcal{F})_{\overline{y 	
} 	
} 	
= 	
\mathcal{F}_{\overline{x 	
} 	
} 	

 	
$ 	
$ 	

 	
See 	
Properties 	
of 	
Spaces 	
, 	
Lemma 	
\ref{spaces 	
- 	
properties 	
- 	
lemma 	
- 	
stalk 	
- 	
pullback 	
} 	

 	
for 	
the 	
last 	
equality 	
. 	


 	
\medskip\noindent 	

 	
The 	
second 	
statement 	
is 	
local 	
on 	
$ 	
X$ 	
in 	
the 	
\'etale 	
topology 	
, 	
hence 	
we 	
may 	

 	
and 	
do 	
assume 	
$ 	
Y$ 	
is 	
an 	
affine 	
scheme 	
. 	


 	
\medskip\noindent 	

 	
Choose 	
a 	
surjective 	
proper 	
morphism 	
$ 	
Z 	
\to 	
Y$ 	
where 	
$ 	
Z$ 	
is 	
a 	
scheme 	
, 	
see 	

 	
Cohomology 	
of 	
Spaces 	
, 	
Lemma 	
\ref{spaces 	
- 	
cohomology 	
- 	
lemma 	
- 	
weak 	
- 	
chow}. 	

 	
The 	
result 	
for 	
$ 	
Z 	
\to 	
X$ 	
implies 	
the 	
result 	
for 	
$ 	
Y 	
\to 	
X$. 	

 	
Since 	
$ 	
Z 	
\to 	
X$ 	
is 	
a 	
surjective 	
proper 	
morphism 	
of 	
schemes 	

 	
and 	
hence 	
a 	
ph 	
covering 	

 	
( 	
Topologies 	
, 	
Lemma 	
\ref{topologies 	
- 	
lemma 	
- 	
surjective 	
- 	
proper 	
- 	
ph 	
} 	
) 	

 	
the 	
result 	
for 	
$ 	
Z 	
\to 	
X$ 	
follows 	
from 	

 	
\'Etale 	
Cohomology 	
, 	
Lemma 	
\ref{etale 	
- 	
cohomology 	
- 	
lemma 	
- 	
describe 	
- 	
pullback 	
- 	
pi 	
- 	
ph 	
} 	

 	
( 	
in 	
fact 	
it 	
is 	
in 	
some 	
sense 	
equivalent 	
to 	
this 	
lemma 	
) 	
. 	

 	
\end{proof 	
} 	


 	
\begin{lemma 	
} 	

 	
\label{lemma 	
- 	
h0-proper 	
- 	
over 	
- 	
henselian 	
- 	
pair 	
} 	

 	
Let 	
$ 	
( 	
A 	
, 	
I)$ 	
be 	
a 	
henselian 	TYPE
pair 	
. 	
Let 	
$ 	
X$ 	
be 	
an 	
algebraic 	TYPE
space 	
over 	
$ 	
A$ 	

 	
such 	
that 	
the 	
structure 	
morphism 	
$ 	
f 	
: 	
X 	
\to 	
\Spec(A)$ 	
is 	
proper 	
. 	

 	
Let 	
$ 	
i 	
: 	
X_0 	
\to 	
X$ 	
be 	
the 	
inclusion 	
of 	
$ 	
X 	
\times_{\Spec(A 	
) 	
} 	
\Spec(A 	
/ 	
I)$. 	

 	
For 	
any 	
sheaf 	
$ 	
\mathcal{F}$ 	
on 	
$ 	
X_\etale$ 	
we 	

 	
have 	
$ 	
\Gamma(X 	
, 	
\mathcal{F 	
} 	
) 	
= 	
\Gamma(Z 	
, 	
i^{-1}\mathcal{F})$. 	

 	
\end{lemma 	
} 	


 	
\begin{proof 	
} 	

 	
Choose 	
a 	
surjective 	
proper 	
morphism 	
$ 	
Y 	
\to 	
X$ 	
where 	
$ 	
Y$ 	
is 	
a 	
scheme 	
, 	
see 	

 	
Cohomology 	
of 	
Spaces 	
, 	
Lemma 	
\ref{spaces 	
- 	
cohomology 	
- 	
lemma 	
- 	
weak 	
- 	
chow}. 	

 	
Consider 	
the 	
diagram 	

 	
$ 	
$ 	

 	
\xymatrix 	
{ 	

 	
\Gamma(X_0 	
, 	
\mathcal{F}_0 	
) 	
\ar[r 	
] 	
\ar[d 	
] 	
& 	

 	
\Gamma(Y_0 	
, 	
\mathcal{G}_0 	
) 	
\ar@<1ex>[r 	
] 	
\ar@<-1ex>[r 	
] 	
\ar[d 	
] 	
& 	

 	
\Gamma((Y 	
\times_X 	
Y)_0 	
, 	
\mathcal{H}_0 	
) 	
\ar[d 	
] 	
\\ 	

 	
\Gamma(X 	
, 	
\mathcal{F 	
} 	
) 	
\ar[r 	
] 	
& 	

 	
\Gamma(Y 	
, 	
\mathcal{G 	
} 	
) 	
\ar@<1ex>[r 	
] 	
\ar@<-1ex>[r 	
] 	
& 	

 	
\Gamma(Y 	
\times_X 	
Y 	
, 	
\mathcal{H 	
} 	
) 	

 	
} 	

 	
$ 	
$ 	

 	
Here 	
$ 	
\mathcal{G}$ 	
, 	
resp.\ 	
$ 	
\mathcal{H}$ 	
is 	
the 	
pullbackf 	
or 	

 	
$ 	
\mathcal{F}$ 	
to 	
$ 	
Y$ 	
, 	
resp.\ 	
$ 	
Y 	
\times_X 	
Y$ 	
and 	
the 	
index 	
$ 	
0 	
$ 	

 	
indicates 	
base 	
change 	
to 	
$ 	
\Spec(A 	
/ 	
I)$. 	
By 	
the 	
case 	
of 	
schemes 	

 	
( 	
\'Etale 	
Cohomology 	
, 	
Lemma 	

 	
\ref{etale 	
- 	
cohomology 	
- 	
lemma 	
- 	
h0-proper 	
- 	
over 	
- 	
henselian 	
- 	
pair 	
} 	
) 	

 	
we 	
see 	
that 	
the 	
middle 	
and 	
right 	
vertical 	
arrows 	
are 	
bijective 	
. 	

 	
By 	
Lemma 	
\ref{lemma 	
- 	
surjective 	
- 	
proper 	
} 	
it 	
follows 	
that 	
the 	
left 	
one 	
is 	
too 	
. 	

 	
\end{proof 	
} 	


 	
\begin{lemma 	
} 	

 	
\label{lemma 	
- 	
h0-proper 	
- 	
over 	
- 	
henselian 	
- 	
local 	
} 	

 	
Let 	
$ 	
A$ 	
be 	
a 	
henselian 	TYPE
local 	
ring 	
. 	
Let 	
$ 	
X$ 	
be 	
an 	
algebraic 	TYPE
space 	

 	
over 	
$ 	
A$ 	
such 	
that 	
$ 	
f 	
: 	
X 	
\to 	
\Spec(A)$ 	

 	
be 	
a 	
proper 	
morphism 	
. 	
Let 	
$ 	
X_0 	
\subset 	
X$ 	
be 	
the 	
fibre 	
of 	

 	
$ 	
f$ 	
over 	
the 	
closed 	
point 	
. 	
For 	
any 	
sheaf 	
$ 	
\mathcal{F}$ 	
on 	
$ 	
X_\etale$ 	
we 	

 	
have 	
$ 	
\Gamma(X 	
, 	
\mathcal{F 	
} 	
) 	
= 	
\Gamma(X_0 	
, 	
\mathcal{F}|_{X_0})$. 	

 	
\end{lemma 	
} 	


 	
\begin{proof 	
} 	

 	
This 	
is 	
a 	
special 	
case 	
of 	
Lemma 	
\ref{lemma 	
- 	
h0-proper 	
- 	
over 	
- 	
henselian 	
- 	
pair}. 	

 	
\end{proof 	
} 	


 	
\begin{lemma 	
} 	

 	
\label{lemma 	
- 	
proper 	
- 	
base 	
- 	
change 	
- 	
f 	
- 	
star 	
} 	

 	
Let 	
$ 	
S$ 	
be 	
a 	
scheme 	TYPE
. 	TYPE
Let 	
$ 	
f 	
: 	
X 	
\to 	
Y$ 	
and 	
$ 	
g 	
: 	
Y 	
' 	
\to 	
Y$ 	

 	
be 	
a 	
morphisms 	TYPE
of 	
algebraic 	
spaces 	
over 	
$ 	
S$. 	
Assume 	
$ 	
f$ 	
is 	
proper 	
. 	

 	
Set 	
$ 	
X 	
' 	
= 	
Y 	
' 	
\times_Y 	
X$ 	
with 	
projections 	
$ 	
f 	
' 	
: 	
X 	
' 	
\to 	
Y'$ 	
and 	
$ 	
g 	
' 	
: 	
X 	
' 	
\to 	
X$. 	

 	
Let 	
$ 	
\mathcal{F}$ 	
be 	
any 	
sheaf 	
on 	
$ 	
X_\etale$. 	
Then 	

 	
$ 	
g^{-1}f_*\mathcal{F 	
} 	
= 	
f'_*(g')^{-1}\mathcal{F}$. 	

 	
\end{lemma 	
} 	


 	
\begin{proof 	
} 	

 	
The 	
question 	
is 	
\'etale 	
local 	
on 	
$ 	
Y'$. 	
Choose 	
a 	
scheme 	
$ 	
V$ 	
and 	
a 	
surjective 	

 	
\'etale 	
morphism 	
$ 	
V 	
\to 	
Y$. 	
Choose 	
a 	
scheme 	
$ 	
V'$ 	
and 	
a 	
surjective 	
\'etale 	

 	
morphism 	
$ 	
V 	
' 	
\to 	
V 	
\times_Y 	
Y'$. 	
Then 	
we 	
may 	
replace 	
$ 	
Y'$ 	
by 	
$ 	
V'$ 	
and 	

 	
$ 	
Y$ 	
by 	
$ 	
V$. 	
Hence 	
we 	
may 	
assume 	
$ 	
Y$ 	
and 	
$ 	
Y'$ 	
are 	
schemes 	
. 	

 	
Then 	
we 	
may 	
work 	
Zariski 	
locally 	
on 	
$ 	
Y$ 	
and 	
$ 	
Y'$ 	
and 	
hence 	
we 	
may 	

 	
assume 	
$ 	
Y$ 	
and 	
$ 	
Y'$ 	
are 	
affine 	
schemes 	
. 	


 	
\medskip\noindent 	

 	
Assume 	
$ 	
Y$ 	
and 	
$ 	
Y'$ 	
are 	
affine 	
schemes 	
. 	
Choose 	
a 	
surjective 	
proper 	
morphism 	

 	
$ 	
h_1 	
: 	
X_1 	
\to 	
X$ 	
where 	
$ 	
X_1 	
$ 	
is 	
a 	
scheme 	
, 	
see 	

 	
Cohomology 	
of 	
Spaces 	
, 	
Lemma 	
\ref{spaces 	
- 	
cohomology 	
- 	
lemma 	
- 	
weak 	
- 	
chow}. 	

 	
Set 	
$ 	
X_2 	
= 	
X_1 	
\times_X 	
X_1 	
$ 	
and 	
denote 	

 	
$ 	
h_2 	
: 	
X_2 	
\to 	
X$ 	
the 	
structure 	
morphism 	
. 	
Observe 	
this 	
is 	
a 	
scheme 	
. 	

 	
By 	
the 	
case 	
of 	
schemes 	

 	
( 	
\'Etale 	
Cohomology 	
, 	
Lemma 	

 	
\ref{etale 	
- 	
cohomology 	
- 	
lemma 	
- 	
proper 	
- 	
base 	
- 	
change 	
- 	
f 	
- 	
star 	
} 	
) 	

 	
we 	
know 	
the 	
lemma 	
is 	
true 	
for 	
the 	
cartesian 	
diagrams 	

 	
$ 	
$ 	

 	
\vcenter 	
{ 	

 	
\xymatrix 	
{ 	

 	
X'_1 	
\ar[r 	
] 	
\ar[d 	
] 	
& 	
X_1 	
\ar[d 	
] 	
\\ 	

 	
Y 	
' 	
\ar[r 	
] 	
& 	
Y 	

 	
} 	

 	
} 	

 	
\quad\text{and}\quad 	

 	
\vcenter 	
{ 	

 	
\xymatrix 	
{ 	

 	
X'_2 	
\ar[r 	
] 	
\ar[d 	
] 	
& 	
X_2 	
\ar[d 	
] 	
\\ 	

 	
Y 	
' 	
\ar[r 	
] 	
& 	
Y 	

 	
} 	

 	
} 	

 	
$ 	
$ 	

 	
and 	
the 	
sheaves 	
$ 	
\mathcal{F}_i 	
= 	
( 	
X_i 	
\to 	
X)^{-1}\mathcal{F}$. 	

 	
By 	
Lemma 	
\ref{lemma 	
- 	
surjective 	
- 	
proper 	
} 	
we 	
have 	
an 	
exact 	
sequence 	

 	
$ 	
0 	
\to 	
\mathcal{F 	
} 	
\to 	
h_{1 	
, 	
* 	
} 	
\mathcal{F}_1 	
\to 	
h_{2 	
, 	
* 	
} 	
\mathcal{F}_2 	
$ 	

 	
and 	
similarly 	
for 	
$ 	
( 	
g')^{-1}\mathcal{F}$ 	
because 	

 	
$ 	
X'_2 	
= 	
X'_1 	
\times_{X 	
' 	
} 	
X'_1$. 	
Hence 	
we 	
conlude 	
that 	
the 	

 	
lemma 	
is 	
true 	
( 	
some 	
details 	
omitted 	
) 	
. 	

 	
\end{proof 	
} 	


 	
\noindent 	

 	
Let 	
$ 	
S$ 	
be 	
a 	
scheme 	TYPE
. 	TYPE

 	
Let 	
$ 	
f 	
: 	
Y 	
\to 	
X$ 	
be 	
a 	
morphism 	
of 	
algebraic 	
spaces 	
over 	
$ 	
S$. 	
Let 	

 	
$ 	
\overline{x 	
} 	
: 	
\Spec(k 	
) 	
\to 	
S$ 	
be 	
a 	
geometric 	
point 	
. 	
The 	
fibre 	

 	
of 	
$ 	
f$ 	
at 	
$ 	
\overline{x}$ 	
is 	
the 	
algebraic 	
space 	

 	
$ 	
Y_{\overline{x 	
} 	
} 	
= 	
\Spec(k 	
) 	
\times_{\overline{x 	
} 	
, 	
X 	
} 	
Y$ 	
over 	
$ 	
\Spec(k)$. 	

 	
If 	
$ 	
\mathcal{F}$ 	
is 	
a 	
sheaf 	
on 	
$ 	
Y_\etale$ 	
, 	
then 	
denote 	

 	
$ 	
\mathcal{F}_{\overline{x 	
} 	
} 	
= 	
p^{-1}\mathcal{F}$ 	

 	
the 	
pullback 	
of 	
$ 	
\mathcal{F}$ 	
to 	
$ 	
( 	
Y_{\overline{x}})_\etale$. 	

 	
Here 	
$ 	
p 	
: 	
Y_{\overline{x 	
} 	
} 	
\to 	
Y$ 	
is 	
the 	
projection 	
. 	

 	
In 	
the 	
following 	
we 	
will 	
consider 	
the 	
set 	

 	
$ 	
\Gamma(Y_{\overline{x 	
} 	
} 	
, 	
\mathcal{F}_{\overline{x}})$. 	


 	
\begin{lemma 	
} 	

 	
\label{lemma 	
- 	
proper 	
- 	
pushforward 	
- 	
stalk 	
} 	

 	
Let 	
$ 	
S$ 	
be 	
a 	
scheme 	TYPE
. 	TYPE

 	
Let 	
$ 	
f 	
: 	
Y 	
\to 	
X$ 	
be 	
a 	
proper 	
morphism 	
of 	
algebraic 	
spaces 	
over 	
$ 	
S$. 	
Let 	

 	
$ 	
\overline{x 	
} 	
\to 	
X$ 	
be 	
a 	
geometric 	
point 	
. 	

 	
For 	
any 	
sheaf 	
$ 	
\mathcal{F}$ 	
on 	
$ 	
Y_\etale$ 	

 	
the 	
canonical 	
map 	

 	
$ 	
$ 	

 	
( 	
f_*\mathcal{F})_{\overline{x 	
} 	
} 	
\longrightarrow 	

 	
\Gamma(Y_{\overline{x 	
} 	
} 	
, 	
\mathcal{F}_{\overline{x 	
} 	
} 	
) 	

 	
$ 	
$ 	

 	
is 	
bijective 	
. 	

 	
\end{lemma 	
} 	


 	
\begin{proof 	
} 	

 	
This 	
is 	
a 	
special 	
case 	
of 	
Lemma 	
\ref{lemma 	
- 	
proper 	
- 	
base 	
- 	
change 	
- 	
f 	
- 	
star}. 	

 	
\end{proof 	
} 	


 	
\begin{theorem 	
} 	

 	
\label{theorem 	
- 	
proper 	
- 	
base 	
- 	
change 	
} 	

 	
Let 	
$ 	
S$ 	
be 	
a 	
scheme 	TYPE
. 	TYPE
Let 	

 	
$ 	
$ 	

 	
\xymatrix 	
{ 	

 	
X 	
' 	
\ar[r]_{g 	
' 	
} 	
\ar[d]_{f 	
' 	
} 	
& 	
X 	
\ar[d]^f 	
\\ 	

 	
Y 	
' 	
\ar[r]^g 	
& 	
Y 	

 	
} 	

 	
$ 	
$ 	

 	
be 	
a 	
cartesian 	TYPE
square 	
of 	
algebraic 	
spaces 	
over 	
$ 	
S$. 	

 	
Assume 	
$ 	
f$ 	
is 	
proper 	
. 	

 	
Let 	
$ 	
\mathcal{F}$ 	
be 	
an 	
abelian 	TYPE
torsion 	
sheaf 	
on 	
$ 	
X_\etale$. 	

 	
Then 	
the 	
base 	
change 	
map 	

 	
$ 	
$ 	

 	
g^{-1}Rf_*\mathcal{F 	
} 	
\longrightarrow 	
Rf'_*(g')^{-1}\mathcal{F 	
} 	

 	
$ 	
$ 	

 	
is 	
an 	
isomorphism 	
. 	

 	
\end{theorem 	
} 	


 	
\begin{proof 	
} 	

 	
This 	
proof 	
repeats 	
a 	
few 	
of 	
the 	
arguments 	
given 	
in 	
the 	
proof 	
of 	
the 	

 	
proper 	
base 	
change 	
theorem 	
for 	
schemes 	
. 	
See 	

 	
\'Etale 	
Cohomology 	
, 	
Section 	
\ref{etale 	
- 	
cohomology 	
- 	
section 	
- 	
proper 	
- 	
base 	
- 	
change 	
} 	

 	
for 	
more 	
details 	
. 	


 	
\medskip\noindent 	

 	
The 	
statement 	
is 	
\'etale 	
local 	
on 	
$ 	
Y'$ 	
and 	
$ 	
Y$ 	
, 	
hence 	
we 	
may 	
assume 	

 	
both 	
$ 	
Y$ 	
and 	
$ 	
Y'$ 	
are 	
affine 	
schemes 	
. 	
Observe 	
that 	
this 	
in 	
particular 	

 	
proves 	
the 	
theorem 	
in 	
case 	
$ 	
f$ 	
is 	
representable 	
( 	
we 	
will 	
use 	
this 	

 	
below 	
) 	
. 	


 	
\medskip\noindent 	

 	
For 	
every 	
$ 	
n 	
\geq 	
1 	
$ 	
let 	
$ 	
\mathcal{F}[n]$ 	
be 	
the 	
subsheaf 	
of 	
sections 	

 	
of 	
$ 	
\mathcal{F}$ 	
annihilated 	
by 	
$ 	
n$. 	
Then 	

 	
$ 	
\mathcal{F 	
} 	
= 	
\colim 	
\mathcal{F}[n]$. 	
By 	

 	
Cohomology 	
of 	
Spaces 	
, 	
Lemma 	
\ref{spaces 	
- 	
cohomology 	
- 	
lemma 	
- 	
colimit 	
- 	
cohomology 	
} 	

 	
the 	
functors 	
$ 	
g^{-1}R^pf_*$ 	
and 	
$ 	
R^pf'_*(g')^{-1}$ 	
commute 	

 	
with 	
filtered 	
colimits 	
. 	
Hence 	
it 	
suffices 	
to 	
prove 	
the 	
theorem 	

 	
if 	
$ 	
\mathcal{F}$ 	
is 	
killed 	
by 	
$ 	
n$. 	


 	
\medskip\noindent 	

 	
Let 	
$ 	
\mathcal{F 	
} 	
\to 	
\mathcal{I}^\bullet$ 	
be 	
a 	
resolution 	TYPE
by 	

 	
injective 	
sheaves 	
of 	
$ 	
\mathbf{Z}/n\mathbf{Z}$-modules 	
. 	

 	
Observe 	
that 	

 	
$ 	
g^{-1}f_*\mathcal{I}^\bullet 	
= 	
f'_*(g')^{-1}\mathcal{I}^\bullet$ 	

 	
by 	
Lemma 	
\ref{lemma 	
- 	
proper 	
- 	
base 	
- 	
change 	
- 	
f 	
- 	
star}. 	

 	
Applying 	
Leray 	
's 	
acyclicity 	
lemma 	

 	
( 	
Derived 	
Categories 	
, 	
Lemma 	
\ref{derived 	
- 	
lemma 	
- 	
leray 	
- 	
acyclicity 	
} 	
) 	

 	
we 	
conclude 	
it 	
suffices 	
to 	
prove 	

 	
$ 	
R^pf'_*(g')^{-1}\mathcal{I}^m 	
= 	
0 	
$ 	
for 	
$ 	
p 	
> 	
0 	
$ 	
and 	
$ 	
m 	
\in 	
\mathbf{Z}$. 	


 	
\medskip\noindent 	

 	
Choose 	
a 	
surjective 	
proper 	
morphism 	

 	
$ 	
h 	
: 	
Z 	
\to 	
X$ 	
where 	
$ 	
Z$ 	
is 	
a 	
scheme 	
, 	
see 	

 	
Cohomology 	
of 	
Spaces 	
, 	
Lemma 	
\ref{spaces 	
- 	
cohomology 	
- 	
lemma 	
- 	
weak 	
- 	
chow}. 	

 	
Choose 	
an 	
injective 	
map 	
$ 	
h^{-1}\mathcal{I}^m 	
\to 	
\mathcal{J}$ 	

 	
where 	
$ 	
\mathcal{J}$ 	
is 	
an 	
injective 	
sheaf 	
of 	

 	
$ 	
\mathbf{Z}/n\mathbf{Z}$-modules 	
on 	
$ 	
Z_\etale$. 	

 	
Since 	
$ 	
h$ 	
is 	
surjective 	
the 	
map 	
$ 	
\mathcal{I}^m 	
\to 	
h_*\mathcal{J}$ 	

 	
is 	
injective 	
( 	
see 	
Lemma 	
\ref{lemma 	
- 	
surjective 	
- 	
proper 	
} 	
) 	
. 	

 	
Since 	
$ 	
\mathcal{I}^m$ 	
is 	
injective 	
we 	
see 	
that 	
$ 	
\mathcal{I}^m$ 	

 	
is 	
a 	
direct 	
summand 	
of 	
$ 	
h_*\mathcal{J}$. 	
Thus 	
it 	
suffices 	

 	
to 	
prove 	
the 	
desired 	
vanishing 	
for 	
$ 	
h_*\mathcal{J}$. 	


 	
\medskip\noindent 	

 	
Denote 	
$ 	
h'$ 	
the 	
base 	
change 	
by 	
$ 	
g$ 	
and 	
denote 	
$ 	
g 	
'' 	
: 	
Z 	
' 	
\to 	
Z$ 	

 	
the 	
projection 	
. 	
There 	
is 	
a 	
spectral 	
sequence 	

 	
$ 	
$ 	

 	
E_2^{p 	
, 	
q 	
} 	
= 	
R^pf 	
' 	
_ 	
* 	
R^qh 	
' 	
_ 	
* 	
( 	
g'')^{-1}\mathcal{J 	
} 	

 	
$ 	
$ 	

 	
converging 	
to 	
$ 	
R^{p 	
+ 	
q}(f 	
' 	
\circ 	
h')_*(g'')^{-1}\mathcal{J}$. 	

 	
Since 	
$ 	
h$ 	
and 	
$ 	
f 	
\circ 	
h$ 	
are 	
representable 	
( 	
by 	
schemes 	
) 	

 	
we 	
know 	
the 	
result 	
we 	
want 	
holds 	
for 	
them 	
. 	
Thus 	
in 	
the 	

 	
spectral 	
sequence 	
we 	
see 	
that 	
$ 	
E_2^{p 	
, 	
q 	
} 	
= 	
0 	
$ 	
for 	
$ 	
q 	
> 	
0 	
$ 	

 	
and 	
$ 	
R^{p 	
+ 	
q}(f 	
' 	
\circ 	
h')_*(g'')^{-1}\mathcal{J 	
} 	
= 	
0 	
$ 	

 	
for 	
$ 	
p 	
+ 	
q 	
> 	
0$. 	
It 	
follows 	
that 	
$ 	
E_2^{p 	
, 	
0 	
} 	
= 	
0 	
$ 	
for 	
$ 	
p 	
> 	
0$. 	

 	
Now 	

 	
$ 	
$ 	

 	
E_2^{p 	
, 	
0 	
} 	
= 	
R^pf 	
' 	
_ 	
* 	
h 	
' 	
_ 	
* 	
( 	
g'')^{-1}\mathcal{J 	
} 	
= 	

 	
R^pf 	
' 	
_ 	
* 	
( 	
g')^{-1}h_*\mathcal{J 	
} 	

 	
$ 	
$ 	

 	
by 	
Lemma 	
\ref{lemma 	
- 	
proper 	
- 	
base 	
- 	
change 	
- 	
f 	
- 	
star}. 	
This 	
finishes 	
the 	
proof 	
. 	

 	
\end{proof 	
} 	


 	
\begin{lemma 	
} 	

 	
\label{lemma 	
- 	
proper 	
- 	
base 	
- 	
change 	
} 	

 	
Let 	
$ 	
S$ 	
be 	
a 	
scheme 	TYPE
. 	TYPE
Let 	

 	
$ 	
$ 	

 	
\xymatrix 	
{ 	

 	
X 	
' 	
\ar[r]_{g 	
' 	
} 	
\ar[d]_{f 	
' 	
} 	
& 	
X 	
\ar[d]^f 	
\\ 	

 	
Y 	
' 	
\ar[r]^g 	
& 	
Y 	

 	
} 	

 	
$ 	
$ 	

 	
be 	
a 	
cartesian 	TYPE
square 	
of 	
algebraic 	
spaces 	
over 	
$ 	
S$. 	
Assume 	
$ 	
f$ 	
is 	
proper 	
. 	

 	
Let 	
$ 	
E 	
\in 	
D^+(X_\etale)$ 	
have 	
torsion 	
cohomology 	
sheaves 	
. 	

 	
Then 	
the 	
base 	
change 	
map 	
$ 	
g^{-1}Rf_*E 	
\to 	
Rf'_*(g')^{-1}E$ 	

 	
is 	
an 	
isomorphism 	
. 	

 	
\end{lemma 	
} 	


 	
\begin{proof 	
} 	

 	
This 	
is 	
a 	
simple 	
consequence 	
of 	
the 	
proper 	
base 	
change 	
theorem 	

 	
( 	
Theorem 	
\ref{theorem 	
- 	
proper 	
- 	
base 	
- 	
change 	
} 	
) 	
using 	
the 	
spectral 	

 	
sequences 	

 	
$ 	
$ 	

 	
E_2^{p 	
, 	
q 	
} 	
= 	
R^pf_*H^q(E 	
) 	

 	
\quad\text{and}\quad 	

 	
{ 	
E'}_2^{p 	
, 	
q 	
} 	
= 	
R^pf'_*(g')^{-1}H^q(E 	
) 	

 	
$ 	
$ 	

 	
converging 	
to 	
$ 	
R^nf_*E$ 	
and 	
$ 	
R^nf'_*(g')^{-1}E$. 	

 	
The 	
spectral 	
sequences 	
are 	
constructed 	
in 	

 	
Derived 	
Categories 	
, 	
Lemma 	
\ref{derived 	
- 	
lemma 	
- 	
two 	
- 	
ss 	
- 	
complex 	
- 	
functor}. 	

 	
Some 	
details 	
omitted 	
. 	

 	
\end{proof 	
} 	


 	
\begin{lemma 	
} 	

 	
\label{lemma 	
- 	
proper 	
- 	
base 	
- 	
change 	
- 	
stalk 	
} 	

 	
Let 	
$ 	
S$ 	
be 	
a 	
scheme 	TYPE
. 	TYPE

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

 	
Let 	
$ 	
\overline{y 	
} 	
\to 	
Y$ 	
be 	
a 	
geometric 	
point 	
. 	

 	
\begin{enumerate 	
} 	

 	
\item 	
For 	
a 	
torsion 	TYPE
abelian 	
sheaf 	
$ 	
\mathcal{F}$ 	
on 	
$ 	
X_\etale$ 	
we 	
have 	

 	
$ 	
( 	
R^nf_*\mathcal{F})_{\overline{y 	
} 	
} 	
= 	

 	
H^n_\etale(X_{\overline{y 	
} 	
} 	
, 	
\mathcal{F}_{\overline{y}})$. 	

 	
\item 	
For 	
$ 	
E 	
\in 	
D^+(X_\etale)$ 	
with 	
torsion 	
cohomology 	
sheaves 	
we 	
have 	

 	
$ 	
( 	
R^nf_*E)_{\overline{y 	
} 	
} 	
= 	
H^n_\etale(X_{\overline{y 	
} 	
} 	
, 	
E_{\overline{y}})$. 	

 	
\end{enumerate 	
} 	

 	
\end{lemma 	
} 	


 	
\begin{proof 	
} 	

 	
In 	
the 	
statement 	
, 	
$ 	
\mathcal{F}_{\overline{y}}$ 	
denotes 	
the 	
pullback 	

 	
of 	
$ 	
\mathcal{F}$ 	
to 	
$ 	
X_{\overline{y 	
} 	
} 	
= 	
\overline{y 	
} 	
\times_Y 	
X$. 	

 	
Since 	
pulling 	
back 	
by 	
$ 	
\overline{y 	
} 	
\to 	
Y$ 	
produces 	
the 	

 	
stalk 	
of 	
$ 	
\mathcal{F}$ 	
, 	
the 	
first 	
statement 	
of 	
the 	
lemma 	

 	
is 	
a 	
special 	
case 	
of 	
Theorem 	
\ref{theorem 	
- 	
proper 	
- 	
base 	
- 	
change}. 	

 	
The 	
second 	
one 	
is 	
a 	
special 	
case 	
of 	
Lemma 	
\ref{lemma 	
- 	
proper 	
- 	
base 	
- 	
change}. 	

 	
\end{proof 	
} 	


 	
\begin{lemma 	
} 	

 	
\label{lemma 	
- 	
base 	
- 	
change 	
- 	
separably 	
- 	
closed 	
} 	

 	
Let 	
$ 	
k 	
\subset 	
k'$ 	
be 	
an 	
extension 	
of 	
separably 	
closed 	
fields 	
. 	

 	
Let 	
$ 	
X$ 	
be 	
a 	
proper 	TYPE
algebraic 	
space 	
over 	
$ 	
k$. 	

 	
Let 	
$ 	
\mathcal{F}$ 	
be 	
a 	
torsion 	TYPE
abelian 	
sheaf 	
on 	
$ 	
X$. 	

 	
Then 	
the 	
map 	
$ 	
H^q_\etale(X 	
, 	
\mathcal{F 	
} 	
) 	
\to 	

 	
H^q_\etale(X_{k 	
' 	
} 	
, 	
\mathcal{F}|_{X_{k'}})$ 	
is 	
an 	
isomorphism 	

 	
for 	
$ 	
q 	
\geq 	
0$. 	

 	
\end{lemma 	
} 	


 	
\begin{proof 	
} 	

 	
This 	
is 	
a 	
special 	
case 	
of 	
Theorem 	
\ref{theorem 	
- 	
proper 	
- 	
base 	
- 	
change}. 	

 	
\end{proof 	
} 	











 	
\section{Comparing 	
big 	
and 	
small 	
topoi 	
} 	

 	
\label{section 	
- 	
compare 	
} 	


 	
\noindent 	

 	
Let 	
$ 	
S$ 	
be 	
a 	
scheme 	TYPE
and 	
let 	
$ 	
X$ 	
be 	
an 	
algebraic 	TYPE
space 	
over 	
$ 	
S$. 	

 	
In 	
Topologies 	
on 	
Spaces 	
, 	
Lemma 	

 	
\ref{spaces 	
- 	
topologies 	
- 	
lemma 	
- 	
at 	
- 	
the 	
- 	
bottom 	
- 	
etale 	
} 	

 	
we 	
have 	
introduced 	
comparison 	
morphisms 	

 	
$ 	
\pi_X 	
: 	
( 	
\textit{Spaces}/X)_\etale 	
\to 	
X_{spaces 	
, 	
\etale}$ 	
and 	

 	
$ 	
i_X 	
: 	
\Sh(X_\etale 	
) 	
\to 	
\Sh((\textit{Spaces}/X)_\etale)$ 	

 	
with 	
$ 	
\pi_X 	
\circ 	
i_X 	
= 	
\text{id}$ 	
as 	
morphisms 	
of 	
topoi 	
and 	

 	
$ 	
\pi_{X 	
, 	
* 	
} 	
= 	
i_X^{-1}$. 	

 	
More 	
generally 	
, 	
if 	
$ 	
f 	
: 	
Y 	
\to 	
X$ 	
is 	
an 	
object 	
of 	
$ 	
( 	
\textit{Spaces}/X)_\etale$ 	
, 	

 	
then 	
there 	
is 	
a 	
morphism 	

 	
$ 	
i_f 	
: 	
\Sh(Y_\etale 	
) 	
\to 	
\Sh((\textit{Spaces}/X)_\etale)$ 	

 	
such 	
that 	
$ 	
f_{small 	
} 	
= 	
\pi_X 	
\circ 	
i_f$ 	
, 	
see 	

 	
Topologies 	
on 	
Spaces 	
, 	
Lemmas 	
\ref{spaces 	
- 	
topologies 	
- 	
lemma 	
- 	
put 	
- 	
in 	
- 	
T 	
- 	
etale 	
} 	
and 	

 	
\ref{spaces 	
- 	
topologies 	
- 	
lemma 	
- 	
morphism 	
- 	
big 	
- 	
small 	
- 	
etale}. 	
In 	

 	
Topologies 	
on 	
Spaces 	
, 	
Remark 	

 	
\ref{spaces 	
- 	
topologies 	
- 	
remark 	
- 	
change 	
- 	
topologies 	
- 	
ringed 	
} 	

 	
we 	
have 	
extended 	
these 	
to 	
a 	
morphism 	
of 	
ringed 	
sites 	

 	
$ 	
$ 	

 	
\pi_X 	
: 	

 	
( 	
( 	
\textit{Spaces}/X)_\etale 	
, 	
\mathcal{O 	
} 	
) 	

 	
\to 	

 	
( 	
X_{spaces 	
, 	
\etale 	
} 	
, 	
\mathcal{O}_X 	
) 	

 	
$ 	
$ 	

 	
and 	
morphisms 	
of 	
ringed 	
topoi 	

 	
$ 	
$ 	

 	
i_X 	
: 	

 	
( 	
\Sh(X_\etale 	
) 	
, 	
\mathcal{O}_X 	
) 	

 	
\to 	

 	
( 	
\Sh((\textit{Spaces}/X)_\etale 	
) 	
, 	
\mathcal{O 	
} 	
) 	

 	
$ 	
$ 	

 	
and 	

 	
$ 	
$ 	

 	
i_f 	
: 	

 	
( 	
\Sh(Y_\etale 	
) 	
, 	
\mathcal{O}_Y 	
) 	

 	
\to 	

 	
( 	
\Sh((\textit{Spaces}/X)_\etale 	
, 	
\mathcal{O 	
} 	
) 	
) 	

 	
$ 	
$ 	

 	
Note 	
that 	
the 	
restriction 	
$ 	
i_X^{-1 	
} 	
= 	
\pi_{X 	
, 	
* 	
} 	
$ 	
( 	
see 	

 	
Topologies 	
, 	
Definition 	
\ref{topologies 	
- 	
definition 	
- 	
restriction 	
- 	
small 	
- 	
etale 	
} 	
) 	

 	
transforms 	
$ 	
\mathcal{O}$ 	
into 	
$ 	
\mathcal{O}_X$. 	

 	
Similarly 	
, 	
$ 	
i_f^{-1}$ 	
transforms 	
$ 	
\mathcal{O}$ 	
into 	
$ 	
\mathcal{O}_Y$. 	

 	
See 	
Topologies 	
on 	
Spaces 	
, 	
Remark 	

 	
\ref{spaces 	
- 	
topologies 	
- 	
remark 	
- 	
change 	
- 	
topologies 	
- 	
ringed}. 	

 	
Hence 	
$ 	
i_X^*\mathcal{F 	
} 	
= 	
i_X^{-1}\mathcal{F}$ 	
and 	

 	
$ 	
i_f^*\mathcal{F 	
} 	
= 	
i_f^{-1}\mathcal{F}$ 	
for 	
any 	
$ 	
\mathcal{O}$-module 	

 	
$ 	
\mathcal{F}$ 	
on 	
$ 	
( 	
\textit{Spaces}/X)_\etale$. 	
In 	
particular 	
$ 	
i_X^*$ 	
and 	
$ 	
i_f^*$ 	

 	
are 	
exact 	
functors 	
. 	
The 	
functor 	
$ 	
i_X^*$ 	
is 	
often 	
denoted 	

 	
$ 	
\mathcal{F 	
} 	
\mapsto 	
\mathcal{F}|_{X_\etale}$ 	
( 	
and 	
this 	
does 	
not 	

 	
conflict 	
with 	
the 	
notation 	
in 	

 	
Topologies 	
on 	
Spaces 	
, 	
Definition 	

 	
\ref{spaces 	
- 	
topologies 	
- 	
definition 	
- 	
restriction 	
- 	
small 	
- 	
etale 	
} 	
) 	
. 	


 	
\begin{lemma 	
} 	

 	
\label{lemma 	
- 	
describe 	
- 	
pullback 	
} 	

 	
Let 	
$ 	
S$ 	
be 	
a 	
scheme 	TYPE
. 	TYPE
Let 	
$ 	
X$ 	
be 	
an 	
algebraic 	TYPE
space 	
over 	
$ 	
S$. 	

 	
Let 	
$ 	
\mathcal{F}$ 	
be 	
a 	
sheaf 	TYPE
on 	
$ 	
X_\etale$. 	
Then 	

 	
$ 	
\pi_X^{-1}\mathcal{F}$ 	
is 	
given 	
by 	
the 	
rule 	

 	
$ 	
$ 	

 	
( 	
\pi_X^{-1}\mathcal{F})(Y 	
) 	
= 	
\Gamma(Y_\etale 	
, 	
f_{small}^{-1}\mathcal{F 	
} 	
) 	

 	
$ 	
$ 	

 	
for 	
$ 	
f 	
: 	
Y 	
\to 	
X$ 	
in 	
$ 	
( 	
\textit{Spaces}/X)_\etale$. 	

 	
Moreover 	
, 	
$ 	
\pi_Y^{-1}\mathcal{F}$ 	
satisfies 	
the 	

 	
sheaf 	
condition 	
with 	
respect 	
to 	
smooth 	
, 	
syntomic 	
, 	
fppf 	
, 	
fpqc 	
, 	
and 	
ph 	
coverings 	
. 	

 	
\end{lemma 	
} 	


 	
\begin{proof 	
} 	

 	
Since 	
pullback 	
is 	
transitive 	
and 	
$ 	
f_{small 	
} 	
= 	
\pi_X 	
\circ 	
i_f$ 	

 	
( 	
see 	
above 	
) 	
we 	
see 	
that 	

 	
$ 	
i_f^{-1 	
} 	
\pi_X^{-1}\mathcal{F 	
} 	
= 	
f_{small}^{-1}\mathcal{F}$. 	

 	
This 	
shows 	
that 	
$ 	
\pi_X^{-1}$ 	
has 	
the 	
description 	
given 	
in 	
the 	
lemma 	
. 	


 	
\medskip\noindent 	

 	
To 	
prove 	
that 	
$ 	
\pi_X^{-1}\mathcal{F}$ 	
is 	
a 	
sheaf 	
for 	
the 	
ph 	
topology 	

 	
it 	
suffices 	
by 	
Topologies 	
on 	
Spaces 	
, 	
Lemma 	

 	
\ref{spaces 	
- 	
topologies 	
- 	
lemma 	
- 	
characterize 	
- 	
sheaf 	
} 	

 	
to 	
show 	
that 	
for 	
a 	
surjective 	
proper 	
morphism 	

 	
$ 	
V 	
\to 	
U$ 	
of 	
algebraic 	
spaces 	
over 	
$ 	
X$ 	
we 	
have 	

 	
$ 	
( 	
\pi_X^{-1}\mathcal{F})(U)$ 	
is 	
the 	
equalizer 	
of 	
the 	
two 	
maps 	

 	
$ 	
( 	
\pi_X^{-1}\mathcal{F})(V 	
) 	
\to 	
( 	
\pi_X^{-1}\mathcal{F})(V 	
\times_U 	
V)$. 	

 	
This 	
we 	
have 	
seen 	
in 	
Lemma 	
\ref{lemma 	
- 	
surjective 	
- 	
proper}. 	


 	
\medskip\noindent 	

 	
The 	
case 	
of 	
smooth 	
, 	
syntomic 	
, 	
fppf 	
coverings 	
follows 	
from 	
the 	
case 	

 	
of 	
ph 	
coverings 	
by 	
Topologies 	
on 	
Spaces 	
, 	
Lemma 	

 	
\ref{spaces 	
- 	
topologies 	
- 	
lemma 	
- 	
zariski 	
- 	
etale 	
- 	
smooth 	
- 	
syntomic 	
- 	
fppf 	
- 	
ph}. 	


 	
\medskip\noindent 	

 	
Let 	
$ 	
\mathcal{U 	
} 	
= 	
\{U_i 	
\to 	
U\}_{i 	
\in 	
I}$ 	
be 	
an 	
fpqc 	TYPE
covering 	
of 	
algebraic 	

 	
spaces 	
over 	
$ 	
X$. 	
Let 	
$ 	
s_i 	
\in 	
( 	
\pi_X^{-1}\mathcal{F})(U_i)$ 	
be 	
sections 	

 	
which 	
agree 	
over 	
$ 	
U_i 	
\times_U 	
U_j$. 	
We 	
have 	
to 	
prove 	
there 	
exists 	
a 	
unique 	

 	
$ 	
s 	
\in 	
( 	
\pi_X^{-1}\mathcal{F})(U)$ 	
restricting 	
to 	
$ 	
s_i$ 	
over 	
$ 	
U_i$. 	

 	
Case 	
I 	
: 	
$ 	
U$ 	
and 	
$ 	
U_i$ 	
are 	
schemes 	
. 	
This 	
case 	
follows 	
from 	

 	
\'Etale 	
Cohomology 	
, 	
Lemma 	
\ref{etale 	
- 	
cohomology 	
- 	
lemma 	
- 	
describe 	
- 	
pullback}. 	

 	
Case 	
II 	
: 	
$ 	
U$ 	
is 	
a 	
scheme 	
. 	
Here 	
we 	
choose 	
surjective 	
\'etale 	
morphisms 	

 	
$ 	
T_i 	
\to 	
U_i$ 	
where 	
$ 	
T_i$ 	
is 	
a 	
scheme 	
. 	
Then 	
$ 	
\mathcal{T 	
} 	
= 	
\{T_i 	
\to 	
U\}$ 	
is 	
an 	

 	
fpqc 	
covering 	
by 	
schemes 	
and 	
by 	
case 	
I 	
the 	
result 	
holds 	
for 	
$ 	
\mathcal{T}$. 	

 	
We 	
omit 	
the 	
verification 	
that 	
this 	
implies 	
the 	
result 	
for 	
$ 	
\mathcal{U}$. 	

 	
Case 	
III 	
: 	
general 	
case 	
. 	
Let 	
$ 	
W 	
\to 	
U$ 	
be 	
a 	
surjective 	
\'etale 	

 	
morphism 	
, 	
where 	
$ 	
W$ 	
is 	
a 	
scheme 	
. 	
Then 	
$ 	
\mathcal{W 	
} 	
= 	
\{U_i 	
\times_U 	
W 	
\to 	
W\}$ 	

 	
is 	
an 	
fpqc 	
covering 	
( 	
by 	
algebraic 	
spaces 	
) 	
of 	
the 	
scheme 	
$ 	
W$. 	

 	
By 	
case 	
II 	
the 	
result 	
hold 	
for 	
$ 	
\mathcal{W}$. 	

 	
We 	
omit 	
the 	
verification 	
that 	
this 	
implies 	
the 	
result 	
for 	
$ 	
\mathcal{U}$. 	

 	
\end{proof 	
} 	


 	
\begin{lemma 	
} 	

 	
\label{lemma 	
- 	
compare 	
- 	
injectives 	
} 	

 	
Let 	
$ 	
S$ 	
be 	
a 	
scheme 	TYPE
. 	TYPE

 	
Let 	
$ 	
Y 	
\to 	
X$ 	
be 	
a 	
morphism 	
of 	
$ 	
( 	
\textit{Spaces}/S)_\etale$. 	

 	
\begin{enumerate 	
} 	

 	
\item 	
If 	
$ 	
\mathcal{I}$ 	
is 	
injective 	
in 	

 	
$ 	
\textit{Ab}((\textit{Spaces}/X)_\etale)$ 	
, 	
then 	

 	
\begin{enumerate 	
} 	

 	
\item 	
$ 	
i_f^{-1}\mathcal{I}$ 	
is 	
injective 	
in 	
$ 	
\textit{Ab}(Y_\etale)$ 	
, 	

 	
\item 	
$ 	
\mathcal{I}|_{X_\etale}$ 	
is 	
injective 	
in 	
$ 	
\textit{Ab}(X_\etale)$ 	
, 	

 	
\end{enumerate 	
} 	

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

 	
in 	
$ 	
\textit{Ab}((\textit{Spaces}/X)_\etale)$ 	
, 	
then 	

 	
\begin{enumerate 	
} 	

 	
\item 	
$ 	
i_f^{-1}\mathcal{I}^\bullet$ 	
is 	
a 	
K 	
- 	
injective 	
complex 	
in 	

 	
$ 	
\textit{Ab}(Y_\etale)$ 	
, 	

 	
\item 	
$ 	
\mathcal{I}^\bullet|_{X_\etale}$ 	
is 	
a 	
K 	
- 	
injective 	
complex 	
in 	

 	
$ 	
\textit{Ab}(X_\etale)$ 	
, 	

 	
\end{enumerate 	
} 	

 	
\end{enumerate 	
} 	

 	
The 	
corresponding 	
statements 	
for 	
modules 	
do 	
not 	
hold 	
. 	

 	
\end{lemma 	
} 	


 	
\begin{proof 	
} 	

 	
Parts 	
( 	
1)(b 	
) 	
and 	
( 	
2)(b 	
) 	

 	
follow 	
formally 	
from 	
the 	
fact 	
that 	
the 	
restriction 	
functor 	

 	
$ 	
\pi_{X 	
, 	
* 	
} 	
= 	
i_X^{-1}$ 	
is 	
a 	
right 	
adjoint 	
of 	
the 	
exact 	
functor 	

 	
$ 	
\pi_X^{-1}$ 	
, 	
see 	

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

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


 	
\medskip\noindent 	

 	
Parts 	
( 	
1)(a 	
) 	
and 	
( 	
2)(a 	
) 	
can 	
be 	
seen 	
in 	
two 	
ways 	
. 	
First 	
proof 	
: 	
We 	
can 	
use 	

 	
that 	
$ 	
i_f^{-1}$ 	
is 	
a 	
right 	
adjoint 	
of 	
the 	
exact 	
functor 	
$ 	
i_{f 	
, 	
! 	
} 	
$ 	
. 	

 	
This 	
functor 	
is 	
constructed 	
in 	

 	
Topologies 	
, 	
Lemma 	
\ref{topologies 	
- 	
lemma 	
- 	
put 	
- 	
in 	
- 	
T 	
- 	
etale 	
} 	

 	
for 	
sheaves 	
of 	
sets 	
and 	
for 	
abelian 	
sheaves 	
in 	

 	
Modules 	
on 	
Sites 	
, 	
Lemma 	
\ref{sites 	
- 	
modules 	
- 	
lemma 	
- 	
g 	
- 	
shriek 	
- 	
adjoint}. 	

 	
It 	
is 	
shown 	
in 	
Modules 	
on 	
Sites 	
, 	
Lemma 	

 	
\ref{sites 	
- 	
modules 	
- 	
lemma 	
- 	
exactness 	
- 	
lower 	
- 	
shriek 	
} 	
that 	
it 	
is 	
exact 	
. 	

 	
Second 	
proof 	
. 	
We 	
can 	
use 	
that 	
$ 	
i_f 	
= 	
i_Y 	
\circ 	
f_{big}$ 	
as 	
is 	
shown 	

 	
in 	
Topologies 	
, 	
Lemma 	
\ref{topologies 	
- 	
lemma 	
- 	
morphism 	
- 	
big 	
- 	
small 	
- 	
etale}. 	

 	
Since 	
$ 	
f_{big}$ 	
is 	
a 	
localization 	
, 	
we 	
see 	
that 	
pullback 	
by 	
it 	

 	
preserves 	
injectives 	
and 	
K 	
- 	
injectives 	
, 	
see 	

 	
Cohomology 	
on 	
Sites 	
, 	
Lemmas 	
\ref{sites 	
- 	
cohomology 	
- 	
lemma 	
- 	
cohomology 	
- 	
of 	
- 	
open 	
} 	
and 	

 	
\ref{sites 	
- 	
cohomology 	
- 	
lemma 	
- 	
restrict 	
- 	
K 	
- 	
injective 	
- 	
to 	
- 	
open}. 	

 	
Then 	
we 	
apply 	
the 	
already 	
proved 	
parts 	
( 	
1)(b 	
) 	
and 	
( 	
2)(b 	
) 	

 	
to 	
the 	
functor 	
$ 	
i_Y^{-1}$ 	
to 	
conclude 	
. 	


 	
\medskip\noindent 	

 	
To 	
see 	
a 	
counter 	
example 	
for 	
the 	
case 	
of 	
modules 	
we 	
refer 	
to 	

 	
\'Etale 	
Cohomology 	
, 	
Lemma 	
\ref{etale 	
- 	
cohomology 	
- 	
lemma 	
- 	
compare 	
- 	
injectives}. 	

 	
\end{proof 	
} 	


 	
\noindent 	

 	
Let 	
$ 	
S$ 	
be 	
a 	
scheme 	TYPE
. 	TYPE

 	
Let 	
$ 	
f 	
: 	
Y 	
\to 	
X$ 	
be 	
a 	
morphism 	
of 	
algebraic 	
spaces 	
over 	
$ 	
S$. 	

 	
The 	
commutative 	
diagram 	
of 	

 	
Topologies 	
on 	
Spaces 	
, 	
Lemma 	

 	
\ref{spaces 	
- 	
topologies 	
- 	
lemma 	
- 	
morphism 	
- 	
big 	
- 	
small 	
- 	
etale 	
} 	
( 	
3 	
) 	

 	
leads 	
to 	
a 	
commutative 	
diagram 	
of 	
ringed 	
sites 	

 	
$ 	
$ 	

 	
\xymatrix 	
{ 	

 	
( 	
Y_{spaces 	
, 	
\etale 	
} 	
, 	
\mathcal{O}_Y 	
) 	
\ar[d]_{f_{spaces 	
, 	
\etale 	
} 	
} 	
& 	

 	
( 	
( 	
\textit{Spaces}/Y)_\etale 	
, 	
\mathcal{O 	
} 	
) 	
\ar[d]^{f_{big 	
} 	
} 	
\ar[l]^{\pi_Y 	
} 	
\\ 	

 	
( 	
X_{spaces 	
, 	
\etale 	
} 	
, 	
\mathcal{O}_X 	
) 	
& 	

 	
( 	
( 	
\textit{Spaces}/X)_\etale 	
, 	
\mathcal{O 	
} 	
) 	
\ar[l]_{\pi_X 	
} 	

 	
} 	

 	
$ 	
$ 	

 	
as 	
one 	
easily 	
sees 	
by 	
writing 	
out 	
the 	
definitions 	
of 	

 	
$ 	
f_{small}^\sharp$ 	
, 	
$ 	
f_{big}^\sharp$ 	
, 	
$ 	
\pi_X^\sharp$ 	
, 	
and 	
$ 	
\pi_Y^\sharp$. 	

 	
In 	
particular 	
this 	
means 	
that 	

 	
\begin{equation 	
} 	

 	
\label{equation 	
- 	
compare 	
- 	
big 	
- 	
small 	
} 	

 	
( 	
f_{big 	
, 	
* 	
} 	
\mathcal{F})|_{X_\etale 	
} 	
= 	

 	
f_{small 	
, 	
* 	
} 	
( 	
\mathcal{F}|_{Y_\etale 	
} 	
) 	

 	
\end{equation 	
} 	

 	
for 	
any 	
sheaf 	
$ 	
\mathcal{F}$ 	
on 	
$ 	
( 	
\textit{Spaces}/Y)_\etale$ 	
and 	
if 	

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

 	
( 	
\ref{equation 	
- 	
compare 	
- 	
big 	
- 	
small 	
} 	
) 	

 	
is 	
an 	
isomorphism 	
of 	
$ 	
\mathcal{O}_X$-modules 	
on 	
$ 	
X_\etale$. 	


 	
\begin{lemma 	
} 	

 	
\label{lemma 	
- 	
compare 	
- 	
higher 	
- 	
direct 	
- 	
image 	
} 	

 	
Let 	
$ 	
S$ 	
be 	
a 	
scheme 	TYPE
. 	TYPE

 	
Let 	
$ 	
f 	
: 	
Y 	
\to 	
X$ 	
be 	
a 	
morphism 	
of 	
algebraic 	
spaces 	
over 	
$ 	
S$. 	

 	
\begin{enumerate 	
} 	

 	
\item 	
For 	
$ 	
K$ 	
in 	
$ 	
D((\textit{Spaces}/Y)_\etale)$ 	
we 	
have 	

 	
$ 	

 	
( 	
Rf_{big 	
, 	
* 	
} 	
K)|_{X_\etale 	
} 	
= 	
Rf_{small 	
, 	
* 	
} 	
( 	
K|_{Y_\etale 	
} 	
) 	

 	
$ 	

 	
in 	
$ 	
D(X_\etale)$. 	

 	
\item 	
For 	
$ 	
K$ 	
in 	
$ 	
D((\textit{Spaces}/Y)_\etale 	
, 	
\mathcal{O})$ 	
we 	
have 	

 	
$ 	

 	
( 	
Rf_{big 	
, 	
* 	
} 	
K)|_{X_\etale 	
} 	
= 	
Rf_{small 	
, 	
* 	
} 	
( 	
K|_{Y_\etale 	
} 	
) 	

 	
$ 	

 	
in 	
$ 	
D(\textit{Mod}(X_\etale 	
, 	
\mathcal{O}_X))$. 	

 	
\end{enumerate 	
} 	

 	
More 	
generally 	
, 	
let 	
$ 	
g 	
: 	
X 	
' 	
\to 	
X$ 	
be 	
an 	
object 	TYPE
of 	

 	
$ 	
( 	
\textit{Spaces}/X)_\etale$. 	
Consider 	
the 	
fibre 	
product 	

 	
$ 	
$ 	

 	
\xymatrix 	
{ 	

 	
Y 	
' 	
\ar[r]_{g 	
' 	
} 	
\ar[d]_{f 	
' 	
} 	
& 	
Y 	
\ar[d]^f 	
\\ 	

 	
X 	
' 	
\ar[r]^g 	
& 	
X 	

 	
} 	

 	
$ 	
$ 	

 	
Then 	

 	
\begin{enumerate 	
} 	

 	
\item[(3 	
) 	
] 	
For 	
$ 	
K$ 	
in 	
$ 	
D((\textit{Spaces}/Y)_\etale)$ 	
we 	
have 	

 	
$ 	
i_g^{-1}(Rf_{big 	
, 	
* 	
} 	
K 	
) 	
= 	
Rf'_{small 	
, 	
* 	
} 	
( 	
i_{g'}^{-1}K)$ 	

 	
in 	
$ 	
D(X'_\etale)$. 	

 	
\item[(4 	
) 	
] 	
For 	
$ 	
K$ 	
in 	
$ 	
D((\textit{Spaces}/Y)_\etale 	
, 	
\mathcal{O})$ 	
we 	
have 	

 	
$ 	
i_g^*(Rf_{big 	
, 	
* 	
} 	
K 	
) 	
= 	
Rf'_{small 	
, 	
* 	
} 	
( 	
i_{g'}^*K)$ 	

 	
in 	
$ 	
D(\textit{Mod}(X'_\etale 	
, 	
\mathcal{O}_{X'}))$. 	

 	
\item[(5 	
) 	
] 	
For 	
$ 	
K$ 	
in 	
$ 	
D((\textit{Spaces}/Y)_\etale)$ 	
we 	
have 	

 	
$ 	
g_{big}^{-1}(Rf_{big 	
, 	
* 	
} 	
K 	
) 	
= 	
Rf'_{small 	
, 	
* 	
} 	
( 	
( 	
g'_{big})^{-1}K)$ 	

 	
in 	
$ 	
D((\textit{Spaces}/X')_\etale)$. 	

 	
\item[(6 	
) 	
] 	
For 	
$ 	
K$ 	
in 	
$ 	
D((\textit{Spaces}/Y)_\etale 	
, 	
\mathcal{O})$ 	
we 	
have 	

 	
$ 	
g_{big}^*(Rf_{big 	
, 	
* 	
} 	
K 	
) 	
= 	
Rf'_{small 	
, 	
* 	
} 	
( 	
( 	
g'_{big})^*K)$ 	

 	
in 	
$ 	
D(\textit{Mod}(X'_\etale 	
, 	
\mathcal{O}_{X'}))$. 	

 	
\end{enumerate 	
} 	

 	
\end{lemma 	
} 	


 	
\begin{proof 	
} 	

 	
Part 	
( 	
1 	
) 	
follows 	
from 	

 	
Lemma 	
\ref{lemma 	
- 	
compare 	
- 	
injectives 	
} 	

 	
and 	
( 	
\ref{equation 	
- 	
compare 	
- 	
big 	
- 	
small 	
} 	
) 	

 	
on 	
choosing 	
a 	
K 	
- 	
injective 	
complex 	
of 	
abelian 	
sheaves 	
representing 	
$ 	
K$. 	


 	
\medskip\noindent 	

 	
Part 	
( 	
3 	
) 	
follows 	
from 	
Lemma 	
\ref{lemma 	
- 	
compare 	
- 	
injectives 	
} 	

 	
and 	
Topologies 	
, 	
Lemma 	

 	
\ref{topologies 	
- 	
lemma 	
- 	
morphism 	
- 	
big 	
- 	
small 	
- 	
cartesian 	
- 	
diagram 	
- 	
etale 	
} 	

 	
on 	
choosing 	
a 	
K 	
- 	
injective 	
complex 	
of 	
abelian 	
sheaves 	
representing 	
$ 	
K$. 	


 	
\medskip\noindent 	

 	
Part 	
( 	
5 	
) 	
follows 	
from 	

 	
Cohomology 	
on 	
Sites 	
, 	
Lemmas 	
\ref{sites 	
- 	
cohomology 	
- 	
lemma 	
- 	
cohomology 	
- 	
of 	
- 	
open 	
} 	
and 	

 	
\ref{sites 	
- 	
cohomology 	
- 	
lemma 	
- 	
restrict 	
- 	
K 	
- 	
injective 	
- 	
to 	
- 	
open 	
} 	

 	
and 	
Topologies 	
, 	
Lemma 	

 	
\ref{topologies 	
- 	
lemma 	
- 	
morphism 	
- 	
big 	
- 	
small 	
- 	
cartesian 	
- 	
diagram 	
- 	
etale 	
} 	

 	
on 	
choosing 	
a 	
K 	
- 	
injective 	
complex 	
of 	
abelian 	
sheaves 	
representing 	
$ 	
K$. 	


 	
\medskip\noindent 	

 	
Part 	
( 	
6 	
) 	
: 	
Observe 	
that 	
$ 	
g_{big}$ 	
and 	
$ 	
g'_{big}$ 	
are 	
localizations 	

 	
and 	
hence 	
$ 	
g_{big}^{-1 	
} 	
= 	
g_{big}^*$ 	
and 	
$ 	
( 	
g'_{big})^{-1 	
} 	
= 	
( 	
g'_{big})^*$ 	

 	
are 	
the 	
restriction 	
functors 	
. 	
Hence 	
( 	
6 	
) 	
follows 	
from 	

 	
Cohomology 	
on 	
Sites 	
, 	
Lemmas 	
\ref{sites 	
- 	
cohomology 	
- 	
lemma 	
- 	
cohomology 	
- 	
of 	
- 	
open 	
} 	
and 	

 	
\ref{sites 	
- 	
cohomology 	
- 	
lemma 	
- 	
restrict 	
- 	
K 	
- 	
injective 	
- 	
to 	
- 	
open 	
} 	

 	
and 	
Topologies 	
, 	
Lemma 	

 	
\ref{topologies 	
- 	
lemma 	
- 	
morphism 	
- 	
big 	
- 	
small 	
- 	
cartesian 	
- 	
diagram 	
- 	
etale 	
} 	

 	
on 	
choosing 	
a 	
K 	
- 	
injective 	
complex 	
of 	
modules 	
representing 	
$ 	
K$. 	


 	
\medskip\noindent 	

 	
Part 	
( 	
2 	
) 	
can 	
be 	
proved 	
as 	
follows 	
. 	
Above 	
we 	
have 	
seen 	

 	
that 	
$ 	
\pi_X 	
\circ 	
f_{big 	
} 	
= 	
f_{small 	
} 	
\circ 	
\pi_Y$ 	
as 	
morphisms 	

 	
of 	
ringed 	
sites 	
. 	
Hence 	
we 	
obtain 	

 	
$ 	
R\pi_{X 	
, 	
* 	
} 	
\circ 	
Rf_{big 	
, 	
* 	
} 	
= 	
Rf_{small 	
, 	
* 	
} 	
\circ 	
R\pi_{Y 	
, 	
* 	
} 	
$ 	

 	
by 	
Cohomology 	
on 	
Sites 	
, 	
Lemma 	

 	
\ref{sites 	
- 	
cohomology 	
- 	
lemma 	
- 	
derived 	
- 	
pushforward 	
- 	
composition}. 	

 	
Since 	
the 	
restriction 	
functors 	
$ 	
\pi_{X 	
, 	
* 	
} 	
$ 	
and 	
$ 	
\pi_{Y 	
, 	
* 	
} 	
$ 	

 	
are 	
exact 	
, 	
we 	
conclude 	
. 	


 	
\medskip\noindent 	

 	
Part 	
( 	
4 	
) 	
follows 	
from 	
part 	
( 	
6 	
) 	
and 	
part 	
( 	
2 	
) 	
applied 	
to 	
$ 	
f 	
' 	
: 	
Y 	
' 	
\to 	
X'$. 	

 	
\end{proof 	
} 	


 	
\noindent 	

 	
Let 	
$ 	
S$ 	
be 	
a 	
scheme 	TYPE
. 	TYPE
Let 	
$ 	
X$ 	
be 	
an 	
algebraic 	TYPE
space 	
over 	
$ 	
S$. 	

 	
Let 	
$ 	
\mathcal{H}$ 	
be 	
an 	
abelian 	TYPE
sheaf 	
on 	

 	
$ 	
( 	
\textit{Spaces}/X)_\etale$. 	
Recall 	
that 	
$ 	
H^n_\etale(U 	
, 	
\mathcal{H})$ 	

 	
denotes 	
the 	
cohomology 	
of 	
$ 	
\mathcal{H}$ 	
over 	
an 	
object 	

 	
$ 	
U$ 	
of 	
$ 	
( 	
\textit{Spaces}/X)_\etale$. 	


 	
\begin{lemma 	
} 	

 	
\label{lemma 	
- 	
compare 	
- 	
cohommology 	
} 	

 	
Let 	
$ 	
S$ 	
be 	
a 	
scheme 	TYPE
. 	TYPE

 	
Let 	
$ 	
f 	
: 	
Y 	
\to 	
X$ 	
be 	
a 	
morphism 	
of 	
algebraic 	
spaces 	
over 	
$ 	
S$. 	
Then 	

 	
\begin{enumerate 	
} 	

 	
\item 	
For 	
$ 	
K$ 	
in 	
$ 	
D(X_\etale)$ 	
we 	
have 	

 	
$ 	
H^n_\etale(X 	
, 	
\pi_X^{-1}K 	
) 	
= 	
H^n(X_\etale 	
, 	
K)$. 	

 	
\item 	
For 	
$ 	
K$ 	
in 	
$ 	
D(X_\etale 	
, 	
\mathcal{O}_X)$ 	
we 	
have 	

 	
$ 	
H^n_\etale(X 	
, 	
L\pi_X^*K 	
) 	
= 	
H^n(X_\etale 	
, 	
K)$. 	

 	
\item 	
For 	
$ 	
K$ 	
in 	
$ 	
D(X_\etale)$ 	
we 	
have 	

 	
$ 	
H^n_\etale(Y 	
, 	
\pi_X^{-1}K 	
) 	
= 	
H^n(Y_\etale 	
, 	
f_{small}^{-1}K)$. 	

 	
\item 	
For 	
$ 	
K$ 	
in 	
$ 	
D(X_\etale 	
, 	
\mathcal{O}_X)$ 	
we 	
have 	

 	
$ 	
H^n_\etale(Y 	
, 	
L\pi_X^*K 	
) 	
= 	
H^n(Y_\etale 	
, 	
Lf_{small}^*K)$. 	

 	
\item 	
For 	
$ 	
M$ 	
in 	
$ 	
D((\textit{Spaces}/X)_\etale)$ 	
we 	
have 	

 	
$ 	
H^n_\etale(Y 	
, 	
M 	
) 	
= 	
H^n(Y_\etale 	
, 	
i_f^{-1}M)$. 	

 	
\item 	
For 	
$ 	
M$ 	
in 	
$ 	
D((\textit{Spaces}/X)_\etale 	
, 	
\mathcal{O})$ 	
we 	
have 	

 	
$ 	
H^n_\etale(Y 	
, 	
M 	
) 	
= 	
H^n(Y_\etale 	
, 	
i_f^*M)$. 	

 	
\end{enumerate 	
} 	

 	
\end{lemma 	
} 	


 	
\begin{proof 	
} 	

 	
To 	
prove 	
( 	
5 	
) 	
represent 	
$ 	
M$ 	
by 	
a 	
K 	
- 	
injective 	
complex 	
of 	
abelian 	
sheaves 	

 	
and 	
apply 	
Lemma 	
\ref{lemma 	
- 	
compare 	
- 	
injectives 	
} 	

 	
and 	
work 	
out 	
the 	
definitions 	
. 	
Part 	
( 	
3 	
) 	
follows 	
from 	

 	
this 	
as 	
$ 	
i_f^{-1}\pi_X^{-1 	
} 	
= 	
f_{small}^{-1}$. 	
Part 	
( 	
1 	
) 	
is 	
a 	
special 	

 	
case 	
of 	
( 	
3 	
) 	
. 	


 	
\medskip\noindent 	

 	
Part 	
( 	
6 	
) 	
follows 	
from 	
the 	
very 	
general 	
Cohomology 	
on 	
Sites 	
, 	
Lemma 	

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

 	
( 	
4 	
) 	
follows 	
because 	
$ 	
Lf_{small}^ 	
* 	
= 	
i_f^ 	
* 	
\circ 	
L\pi_X^*$. 	

 	
Part 	
( 	
2 	
) 	
is 	
a 	
special 	
case 	
of 	
( 	
4 	
) 	
. 	

 	
\end{proof 	
} 	


 	
\begin{lemma 	
} 	

 	
\label{lemma 	
- 	
cohomological 	
- 	
descent 	
- 	
etale 	
} 	

 	
Let 	
$ 	
S$ 	
be 	
a 	
scheme 	TYPE
. 	TYPE
Let 	
$ 	
X$ 	
be 	
an 	
algebraic 	TYPE
space 	
over 	
$ 	
S$. 	

 	
For 	
$ 	
K 	
\in 	
D(X_\etale)$ 	
the 	
map 	

 	
$ 	
$ 	

 	
K 	
\longrightarrow 	
R\pi_{X 	
, 	
* 	
} 	
\pi_X^{-1}K 	

 	
$ 	
$ 	

 	
is 	
an 	
isomorphism 	
where 	

 	
$ 	
\pi_X 	
: 	
\Sh((\textit{Spaces}/X)_\etale 	
) 	
\to 	
\Sh(X_\etale)$ 	
is 	
as 	
above 	
. 	

 	
\end{lemma 	
} 	


 	
\begin{proof 	
} 	

 	
This 	
is 	
true 	
because 	
both 	
$ 	
\pi_X^{-1}$ 	
and 	
$ 	
\pi_{X 	
, 	
* 	
} 	
= 	
i_X^{-1}$ 	

 	
are 	
exact 	
functors 	
and 	
the 	
composition 	
$ 	
\pi_{X 	
, 	
* 	
} 	
\circ 	
\pi_X^{-1}$ 	

 	
is 	
the 	
identity 	
functor 	
. 	

 	
\end{proof 	
} 	


 	
\begin{lemma 	
} 	

 	
\label{lemma 	
- 	
compare 	
- 	
higher 	
- 	
direct 	
- 	
image 	
- 	
proper 	
} 	

 	
Let 	
$ 	
S$ 	
be 	
a 	
scheme 	TYPE
. 	TYPE

 	
Let 	
$ 	
f 	
: 	
Y 	
\to 	
X$ 	
be 	
a 	
proper 	
morphism 	
of 	
algebraic 	
spaces 	
over 	
$ 	
S$. 	

 	
Then 	
we 	
have 	

 	
\begin{enumerate 	
} 	

 	
\item 	
$ 	
\pi_X^{-1 	
} 	
\circ 	
f_{small 	
, 	
* 	
} 	
= 	
f_{big 	
, 	
* 	
} 	
\circ 	
\pi_Y^{-1}$ 	

 	
as 	
functors 	
$ 	
\Sh(Y_\etale 	
) 	
\to 	
\Sh((\textit{Spaces}/X)_\etale)$ 	
, 	

 	
\item 	
$ 	
\pi_X^{-1}Rf_{small 	
, 	
* 	
} 	
K 	
= 	
Rf_{big 	
, 	
* 	
} 	
\pi_Y^{-1}K$ 	

 	
for 	
$ 	
K$ 	
in 	
$ 	
D^+(Y_\etale)$ 	
whose 	
cohomology 	
sheaves 	
are 	
torsion 	
, 	
and 	

 	
\item 	
$ 	
\pi_X^{-1}Rf_{small 	
, 	
* 	
} 	
K 	
= 	
Rf_{big 	
, 	
* 	
} 	
\pi_Y^{-1}K$ 	

 	
for 	
all 	
$ 	
K$ 	
in 	
$ 	
D(Y_\etale)$ 	
if 	
$ 	
f$ 	
is 	
finite 	
. 	

 	
\end{enumerate 	
} 	

 	
\end{lemma 	
} 	


 	
\begin{proof 	
} 	

 	
Proof 	
of 	
( 	
1 	
) 	
. 	
Let 	
$ 	
\mathcal{F}$ 	
be 	
a 	
sheaf 	TYPE
on 	
$ 	
Y_\etale$. 	

 	
Let 	
$ 	
g 	
: 	
X 	
' 	
\to 	
X$ 	
be 	
an 	
object 	
of 	
$ 	
( 	
\textit{Spaces}/X)_\etale$. 	

 	
Consider 	
the 	
fibre 	
product 	

 	
$ 	
$ 	

 	
\xymatrix 	
{ 	

 	
Y 	
' 	
\ar[r]_{f 	
' 	
} 	
\ar[d]_{g 	
' 	
} 	
& 	
X 	
' 	
\ar[d]^g 	
\\ 	

 	
Y 	
\ar[r]^f 	
& 	
X 	

 	
} 	

 	
$ 	
$ 	

 	
Then 	
we 	
have 	

 	
$ 	
$ 	

 	
( 	
f_{big 	
, 	
* 	
} 	
\pi_Y^{-1}\mathcal{F})(X 	
' 	
) 	
= 	

 	
( 	
\pi_Y^{-1}\mathcal{F})(Y 	
' 	
) 	
= 	

 	
( 	
( 	
g'_{small})^{-1}\mathcal{F})(Y 	
' 	
) 	
  	
= 	

 	
( 	
f'_{small 	
, 	
* 	
} 	
( 	
g'_{small})^{-1}\mathcal{F})(X 	
' 	
) 	

 	
$ 	
$ 	

 	
the 	
second 	
equality 	
by 	
Lemma 	
\ref{lemma 	
- 	
describe 	
- 	
pullback}. 	

 	
On 	
the 	
other 	
hand 	

 	
$ 	
$ 	

 	
( 	
\pi_X^{-1}f_{small 	
, 	
* 	
} 	
\mathcal{F})(X 	
' 	
) 	
= 	

 	
( 	
g_{small}^{-1}f_{small 	
, 	
* 	
} 	
\mathcal{F})(X 	
' 	
) 	

 	
$ 	
$ 	

 	
again 	
by 	
Lemma 	
\ref{lemma 	
- 	
describe 	
- 	
pullback}. 	

 	
Hence 	
by 	
proper 	
base 	
change 	
for 	
sheaves 	
of 	
sets 	

 	
( 	
Lemma 	
\ref{lemma 	
- 	
proper 	
- 	
base 	
- 	
change 	
- 	
f 	
- 	
star 	
} 	
) 	

 	
we 	
conclude 	
the 	
two 	
sets 	
are 	
canonically 	
isomorphic 	
. 	

 	
The 	
isomorphism 	
is 	
compatible 	
with 	
restriction 	
mappings 	

 	
and 	
defines 	
an 	
isomorphism 	

 	
$ 	
\pi_X^{-1}f_{small 	
, 	
* 	
} 	
\mathcal{F 	
} 	
= 	
f_{big 	
, 	
* 	
} 	
\pi_Y^{-1}\mathcal{F}$. 	

 	
Thus 	
an 	
isomorphism 	
of 	
functors 	

 	
$ 	
\pi_X^{-1 	
} 	
\circ 	
f_{small 	
, 	
* 	
} 	
= 	
f_{big 	
, 	
* 	
} 	
\circ 	
\pi_Y^{-1}$. 	


 	
\medskip\noindent 	

 	
Proof 	
of 	
( 	
2 	
) 	
. 	
There 	
is 	
a 	
canonical 	
base 	
change 	
map 	

 	
$ 	
\pi_X^{-1}Rf_{small 	
, 	
* 	
} 	
K 	
\to 	
Rf_{big 	
, 	
* 	
} 	
\pi_Y^{-1}K$ 	

 	
for 	
any 	
$ 	
K$ 	
in 	
$ 	
D(Y_\etale)$ 	
, 	
see 	

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

 	
To 	
prove 	
it 	
is 	
an 	
isomorphism 	
, 	
it 	
suffices 	
to 	
prove 	
the 	
pull 	
back 	
of 	

 	
the 	
base 	
change 	
map 	
by 	
$ 	
i_g 	
: 	
\Sh(X'_\etale 	
) 	
\to 	
\Sh((\Sch 	
/ 	
X)_\etale)$ 	

 	
is 	
an 	
isomorphism 	
for 	
any 	
object 	
$ 	
g 	
: 	
X 	
' 	
\to 	
X$ 	
of 	
$ 	
( 	
\Sch 	
/ 	
X)_\etale$. 	

 	
Let 	
$ 	
T 	
' 	
, 	
g 	
' 	
, 	
f'$ 	
be 	
as 	
in 	
the 	
previous 	
paragraph 	
. 	

 	
The 	
pullback 	
of 	
the 	
base 	
change 	
map 	
is 	

 	
\begin{align 	
* 	
} 	

 	
g_{small}^{-1}Rf_{small 	
, 	
* 	
} 	
K 	

 	
& 	
= 	

 	
i_g^{-1}\pi_X^{-1}Rf_{small 	
, 	
* 	
} 	
K 	
\\ 	

 	
& 	
\to 	

 	
i_g^{-1}Rf_{big 	
, 	
* 	
} 	
\pi_Y^{-1}K 	
\\ 	

 	
& 	
= 	

 	
Rf'_{small 	
, 	
* 	
} 	
( 	
i_{g'}^{-1}\pi_Y^{-1}K 	
) 	
\\ 	

 	
& 	
= 	

 	
Rf'_{small 	
, 	
* 	
} 	
( 	
( 	
g'_{small})^{-1}K 	
) 	

 	
\end{align 	
* 	
} 	

 	
where 	
we 	
have 	
used 	
$ 	
\pi_X 	
\circ 	
i_g 	
= 	
g_{small}$ 	
, 	

 	
$ 	
\pi_Y 	
\circ 	
i_{g 	
' 	
} 	
= 	
g'_{small}$ 	
, 	
and 	

 	
Lemma 	
\ref{lemma 	
- 	
compare 	
- 	
higher 	
- 	
direct 	
- 	
image}. 	

 	
This 	
map 	
is 	
an 	
isomorphism 	
by 	
the 	
proper 	
base 	
change 	
theorem 	

 	
( 	
Lemma 	
\ref{lemma 	
- 	
proper 	
- 	
base 	
- 	
change 	
} 	
) 	
provided 	
$ 	
K$ 	
is 	
bounded 	

 	
below 	
and 	
the 	
cohomology 	
sheaves 	
of 	
$ 	
K$ 	
are 	
torsion 	
. 	


 	
\medskip\noindent 	

 	
Proof 	
of 	
( 	
3 	
) 	
. 	
If 	
$ 	
f$ 	
is 	
finite 	
, 	
then 	
the 	
functors 	

 	
$ 	
f_{small 	
, 	
* 	
} 	
$ 	
and 	
$ 	
f_{big 	
, 	
* 	
} 	
$ 	
are 	
exact 	
. 	
This 	
follows 	

 	
from 	
Cohomology 	
of 	
Spaces 	
, 	
Lemma 	

 	
\ref{spaces 	
- 	
cohomology 	
- 	
lemma 	
- 	
finite 	
- 	
higher 	
- 	
direct 	
- 	
image 	
- 	
zero 	
} 	

 	
for 	
$ 	
f_{small}$. 	
Since 	
any 	
base 	
change 	
$ 	
f'$ 	
of 	
$ 	
f$ 	
is 	
finite 	
too 	
, 	

 	
we 	
conclude 	
from 	
Lemma 	
\ref{lemma 	
- 	
compare 	
- 	
higher 	
- 	
direct 	
- 	
image 	
} 	
part 	
( 	
3 	
) 	

 	
that 	
$ 	
f_{big 	
, 	
* 	
} 	
$ 	
is 	
exact 	
too 	
( 	
as 	
the 	
higher 	
derived 	
functors 	
are 	
zero 	
) 	
. 	

 	
Thus 	
this 	
case 	
follows 	
from 	
part 	
( 	
1 	
) 	
. 	

 	
\end{proof 	
} 	








 	
\section{Comparing 	
fppf 	
and 	
\'etale 	
topologies 	
} 	

 	
\label{section 	
- 	
fppf 	
- 	
etale 	
} 	


 	
\noindent 	

 	
This 	
section 	
is 	
the 	
analogue 	
of 	

 	
\'Etale 	
Cohomology 	
, 	
Section 	
\ref{etale 	
- 	
cohomology 	
- 	
section 	
- 	
fppf 	
- 	
etale}. 	


 	
\medskip\noindent 	

 	
Let 	
$ 	
S$ 	
be 	
a 	
scheme 	TYPE
. 	TYPE
Let 	
$ 	
X$ 	
be 	
an 	
algebraic 	TYPE
space 	
over 	
$ 	
S$. 	

 	
On 	
the 	
category 	
$ 	
\textit{Spaces}/X$ 	
we 	
consider 	
the 	
fppf 	

 	
and 	
\'etale 	
topologies 	
. 	
The 	
identity 	
functor 	

 	
$ 	
( 	
\textit{Spaces}/X)_\etale 	
\to 	
( 	
\textit{Spaces}/X)_{fppf}$ 	

 	
is 	
continuous 	
and 	
defines 	
a 	
morphism 	
of 	
sites 	

 	
$ 	
$ 	

 	
\epsilon_X 	
: 	

 	
( 	
\textit{Spaces}/X)_{fppf 	
} 	
\longrightarrow 	
( 	
\textit{Spaces}/X)_\etale 	

 	
$ 	
$ 	

 	
by 	
an 	
application 	
of 	
Sites 	
, 	
Proposition 	
\ref{sites 	
- 	
proposition 	
- 	
get 	
- 	
morphism}. 	

 	
Please 	
note 	
that 	
$ 	
\epsilon_{X 	
, 	
* 	
} 	
$ 	
is 	
the 	
identity 	
functor 	
on 	
underlying 	

 	
presheaves 	
and 	
that 	
$ 	
\epsilon_X^{-1}$ 	
associates 	
to 	
an 	
\'etale 	
sheaf 	
the 	

 	
fppf 	
sheafification 	
. 	

 	
Consider 	
the 	
morphism 	
of 	
sites 	

 	
$ 	
$ 	

 	
\pi_X 	
: 	
( 	
\textit{Spaces}/X)_\etale 	
\longrightarrow 	
X_{spaces 	
, 	
\etale 	
} 	

 	
$ 	
$ 	

 	
comparing 	
big 	
and 	
small 	
\'etale 	
sites 	
, 	
see 	
Section 	
\ref{section 	
- 	
compare}. 	

 	
The 	
composition 	
determines 	
a 	
morphism 	
of 	
sites 	

 	
$ 	
$ 	

 	
a_X 	
= 	
\pi_X 	
\circ 	
\epsilon_X 	
: 	

 	
( 	
\textit{Spaces}/X)_{fppf 	
} 	

 	
\longrightarrow 	

 	
X_{spaces 	
, 	
\etale 	
} 	

 	
$ 	
$ 	

 	
If 	
$ 	
\mathcal{H}$ 	
is 	
an 	
abelian 	
sheaf 	
on 	
$ 	
( 	
\textit{Spaces}/X)_{fppf}$ 	
, 	

 	
then 	
we 	
will 	
write 	
$ 	
H^n_{fppf}(U 	
, 	
\mathcal{H})$ 	
for 	
the 	
cohomology 	

 	
of 	
$ 	
\mathcal{H}$ 	
over 	
an 	
object 	
$ 	
U$ 	
of 	
$ 	
( 	
\textit{Spaces}/X)_{fppf}$. 	


 	
\begin{lemma 	
} 	

 	
\label{lemma 	
- 	
comparison 	
- 	
fppf 	
- 	
etale 	
} 	

 	
Let 	
$ 	
S$ 	
be 	
a 	
scheme 	TYPE
. 	TYPE
Let 	
$ 	
X$ 	
be 	
an 	
algebraic 	TYPE
space 	
over 	
$ 	
S$. 	

 	
\begin{enumerate 	
} 	

 	
\item 	
For 	
$ 	
\mathcal{F 	
} 	
\in 	
\Sh(X_\etale)$ 	
we 	
have 	

 	
$ 	
\epsilon_{X 	
, 	
* 	
} 	
a_X^{-1}\mathcal{F 	
} 	
= 	
\pi_X^{-1}\mathcal{F}$ 	

 	
and 	
$ 	
a_{X 	
, 	
* 	
} 	
a_X^{-1}\mathcal{F 	
} 	
= 	
\mathcal{F}$. 	

 	
\item 	
For 	
$ 	
\mathcal{F 	
} 	
\in 	
\textit{Ab}(X_\etale)$ 	
we 	
have 	

 	
$ 	
R^i\epsilon_{X 	
, 	
* 	
} 	
( 	
a_X^{-1}\mathcal{F 	
} 	
) 	
= 	
0 	
$ 	
for 	
$ 	
i 	
> 	
0$. 	

 	
\end{enumerate 	
} 	

 	
\end{lemma 	
} 	


 	
\begin{proof 	
} 	

 	
We 	
have 	
$ 	
a_X^{-1}\mathcal{F 	
} 	
= 	
\epsilon_X^{-1 	
} 	
\pi_X^{-1}\mathcal{F}$. 	

 	
By 	
Lemma 	
\ref{lemma 	
- 	
describe 	
- 	
pullback 	
} 	
the 	
\'etale 	
sheaf 	

 	
$ 	
\pi_X^{-1}\mathcal{F}$ 	
is 	
a 	
sheaf 	
for 	
the 	
fppf 	
topology 	

 	
and 	
therefore 	
is 	
equal 	
to 	
$ 	
a_X^{-1}\mathcal{F}$ 	
( 	
as 	
pulling 	

 	
back 	
by 	
$ 	
\epsilon_X$ 	
is 	
given 	
by 	
fppf 	
sheafification 	
) 	
. 	

 	
Recall 	
moreover 	
that 	
$ 	
\epsilon_{X 	
, 	
* 	
} 	
$ 	
is 	
the 	
identity 	

 	
on 	
underlying 	
presheaves 	
. 	

 	
Now 	
part 	
( 	
1 	
) 	
is 	
immediate 	
from 	
the 	
explicit 	
description 	
of 	
$ 	
\pi_X^{-1}$ 	

 	
in 	
Lemma 	
\ref{lemma 	
- 	
describe 	
- 	
pullback}. 	


 	
\medskip\noindent 	

 	
We 	
will 	
prove 	
part 	
( 	
2 	
) 	
by 	
reducing 	
it 	
to 	
the 	
case 	
of 	
schemes 	
-- 	

 	
see 	
part 	
( 	
1 	
) 	
of 	

 	
\'Etale 	
Cohomology 	
, 	
Lemma 	
\ref{etale 	
- 	
cohomology 	
- 	
lemma 	
- 	
V 	
- 	
C 	
- 	
all 	
- 	
n 	
- 	
etale 	
- 	
fppf}. 	

 	
This 	
will 	
`` 	
clearly 	
work 	
'' 	
as 	
every 	
algebraic 	
space 	
is 	

 	
\'etale 	
locally 	
a 	
scheme 	
. 	
The 	
details 	
are 	
given 	
below 	
but 	
we 	
urge 	

 	
the 	
reader 	
to 	
skip 	
the 	
proof 	
. 	


 	
\medskip\noindent 	

 	
For 	
an 	
abelian 	
sheaf 	
$ 	
\mathcal{H}$ 	
on 	
$ 	
( 	
\textit{Spaces}/X)_{fppf}$ 	
the 	

 	
higher 	
direct 	
image 	
$ 	
R^p\epsilon_{X 	
, 	
* 	
} 	
\mathcal{H}$ 	
is 	
the 	
sheaf 	

 	
associated 	
to 	
the 	
presheaf 	
$ 	
U 	
\mapsto 	
H^p_{fppf}(U 	
, 	
\mathcal{H})$ 	

 	
on 	
$ 	
( 	
\textit{Spaces}/X)_\etale$. 	
See 	

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

 	
Since 	
every 	
object 	
of 	
$ 	
( 	
\textit{Spaces}/X)_\etale$ 	
has 	
a 	
covering 	

 	
by 	
schemes 	
, 	
it 	
suffices 	
to 	
prove 	
that 	
given 	
$ 	
U 	
/ 	
X$ 	
a 	
scheme 	
and 	

 	
$ 	
\xi 	
\in 	
H^p_{fppf}(U 	
, 	
a_X^{-1}\mathcal{F})$ 	
we 	
can 	
find 	

 	
an 	
\'etale 	
covering 	
$ 	
\{U_i 	
\to 	
U\}$ 	
such 	
that 	
$ 	
\xi$ 	

 	
restricts 	
to 	
zero 	
on 	
$ 	
U_i$. 	
We 	
have 	

 	
\begin{align 	
* 	
} 	

 	
H^p_{fppf}(U 	
, 	
a_X^{-1}\mathcal{F 	
} 	
) 	

 	
& 	
= 	

 	
H^p((\textit{Spaces}/U)_{fppf 	
} 	
, 	
( 	
a_X^{-1}\mathcal{F})|_{\textit{Spaces}/U 	
} 	
) 	
\\ 	

 	
& 	
= 	

 	
H^p((\Sch 	
/ 	
U)_{fppf 	
} 	
, 	
( 	
a_X^{-1}\mathcal{F})|_{\Sch 	
/ 	
U 	
} 	
) 	

 	
\end{align 	
* 	
} 	

 	
where 	
the 	
second 	
identification 	
is 	

 	
Lemma 	
\ref{lemma 	
- 	
compare 	
- 	
cohomology 	
- 	
other 	
- 	
topologies 	
} 	

 	
and 	
the 	
first 	
is 	
a 	
general 	
fact 	
about 	
restriction 	

 	
( 	
Cohomology 	
on 	
Sites 	
, 	
Lemma 	
\ref{sites 	
- 	
cohomology 	
- 	
lemma 	
- 	
cohomology 	
- 	
of 	
- 	
open 	
} 	
) 	
. 	

 	
Looking 	
at 	
the 	
first 	
paragraph 	
and 	
the 	
corresponding 	
result 	
in 	
the 	

 	
case 	
of 	
schemes 	
( 	
\'Etale 	
Cohomology 	
, 	
Lemma 	

 	
\ref{etale 	
- 	
cohomology 	
- 	
lemma 	
- 	
describe 	
- 	
pullback 	
- 	
pi 	
- 	
fppf 	
} 	
) 	

 	
we 	
conclude 	
that 	
the 	
sheaf 	
$ 	
( 	
a_X^{-1}\mathcal{F})|_{\Sch 	
/ 	
U}$ 	

 	
matches 	
the 	
pullback 	
by 	
the 	
`` 	
schemes 	
version 	
of 	
$ 	
a_U$ 	
'' 	
. 	

 	
Therefore 	
we 	
can 	
find 	
an 	
\'etale 	
covering 	

 	
$ 	
\{U_i 	
\to 	
U\}$ 	
such 	
that 	
our 	
class 	
dies 	
in 	

 	
$ 	
H^p((\Sch 	
/ 	
U_i)_{fppf 	
} 	
, 	
( 	
a_X^{-1}\mathcal{F})|_{\Sch 	
/ 	
U_i})$ 	

 	
for 	
each 	
$ 	
i$ 	
, 	
see 	

 	
\'Etale 	
Cohomology 	
, 	
Lemma 	
\ref{etale 	
- 	
cohomology 	
- 	
lemma 	
- 	
V 	
- 	
C 	
- 	
all 	
- 	
n 	
- 	
etale 	
- 	
fppf 	
} 	

 	
( 	
the 	
precise 	
statement 	
one 	
should 	
use 	
here 	
is 	
that 	
$ 	
V_n$ 	
holds 	
for 	
all 	
$ 	
n$ 	

 	
which 	
is 	
the 	
statement 	
of 	
part 	
( 	
2 	
) 	
for 	
the 	
case 	
of 	
schemes 	
) 	
. 	

 	
Transporting 	
back 	
( 	
using 	
the 	
same 	
formulas 	
as 	
above 	
but 	
now 	
for 	

 	
$ 	
U_i$ 	
) 	
we 	
conclude 	
$ 	
\xi$ 	
restricts 	
to 	
zero 	
over 	
$ 	
U_i$ 	
as 	
desired 	
. 	

 	
\end{proof 	
} 	


 	
\noindent 	

 	
The 	
hard 	
work 	
done 	
in 	
the 	
case 	
of 	
schemes 	
now 	
tells 	
us 	
that 	

 	
\'etale 	
and 	
fppf 	
cohomology 	
agree 	
for 	
sheaves 	
coming 	
from 	
the 	

 	
small 	
\'etale 	
site 	
. 	


 	
\begin{lemma 	
} 	

 	
\label{lemma 	
- 	
cohomological 	
- 	
descent 	
- 	
etale 	
- 	
fppf 	
} 	

 	
Let 	
$ 	
S$ 	
be 	
a 	
scheme 	TYPE
. 	TYPE
Let 	
$ 	
X$ 	
be 	
an 	
algebraic 	TYPE
space 	
over 	
$ 	
S$. 	

 	
For 	
$ 	
K 	
\in 	
D^+(X_\etale)$ 	
the 	
maps 	

 	
$ 	
$ 	

 	
\pi_X^{-1}K 	
\longrightarrow 	
R\epsilon_{X 	
, 	
* 	
} 	
a_X^{-1}K 	

 	
\quad\text{and}\quad 	

 	
K 	
\longrightarrow 	
Ra_{X 	
, 	
* 	
} 	
a_X^{-1}K 	

 	
$ 	
$ 	

 	
are 	
isomorphisms 	
with 	

 	
$ 	
a_X 	
: 	
\Sh((\textit{Spaces}/X)_{fppf 	
} 	
) 	
\to 	
\Sh(X_\etale)$ 	
as 	
above 	
. 	

 	
\end{lemma 	
} 	


 	
\begin{proof 	
} 	

 	
We 	
only 	
prove 	
the 	
second 	
statement 	
; 	
the 	
first 	
is 	
easier 	
and 	
proved 	
in 	
exactly 	

 	
the 	
same 	
manner 	
. 	

 	
There 	
is 	
an 	
immediate 	
reduction 	
to 	
the 	
case 	
where 	

 	
$ 	
K$ 	
is 	
given 	
by 	
a 	
single 	
abelian 	
sheaf 	
. 	
Namely 	
, 	
represent 	
$ 	
K$ 	

 	
by 	
a 	
bounded 	
below 	
complex 	
$ 	
\mathcal{F}^\bullet$. 	
By 	
the 	
case 	
of 	
a 	

 	
sheaf 	
we 	
see 	
that 	

 	
$ 	
\mathcal{F}^n 	
= 	
a_{X 	
, 	
* 	
} 	
a_X^{-1 	
} 	
\mathcal{F}^n$ 	

 	
and 	
that 	
the 	
sheaves 	
$ 	
R^qa_{X 	
, 	
* 	
} 	
a_X^{-1}\mathcal{F}^n$ 	

 	
are 	
zero 	
for 	
$ 	
q 	
> 	
0$. 	
By 	
Leray 	
's 	
acyclicity 	
lemma 	

 	
( 	
Derived 	
Categories 	
, 	
Lemma 	
\ref{derived 	
- 	
lemma 	
- 	
leray 	
- 	
acyclicity 	
} 	
) 	

 	
applied 	
to 	
$ 	
a_X^{-1}\mathcal{F}^\bullet$ 	

 	
and 	
the 	
functor 	
$ 	
a_{X 	
, 	
* 	
} 	
$ 	
we 	
conclude 	
. 	
From 	
now 	
on 	
assume 	
$ 	
K 	
= 	
\mathcal{F}$. 	


 	
\medskip\noindent 	

 	
By 	
Lemma 	
\ref{lemma 	
- 	
comparison 	
- 	
fppf 	
- 	
etale 	
} 	
we 	
have 	

 	
$ 	
a_{X 	
, 	
* 	
} 	
a_X^{-1}\mathcal{F 	
} 	
= 	
\mathcal{F}$. 	
Thus 	
it 	
suffices 	
to 	
show 	
that 	

 	
$ 	
R^qa_{X 	
, 	
* 	
} 	
a_X^{-1}\mathcal{F 	
} 	
= 	
0 	
$ 	
for 	
$ 	
q 	
> 	
0$. 	

 	
For 	
this 	
we 	
can 	
use 	
$ 	
a_X 	
= 	
\epsilon_X 	
\circ 	
\pi_X$ 	
and 	

 	
the 	
Leray 	
spectral 	
sequence 	

 	
( 	
Cohomology 	
on 	
Sites 	
, 	
Lemma 	
\ref{sites 	
- 	
cohomology 	
- 	
lemma 	
- 	
relative 	
- 	
Leray 	
} 	
) 	
. 	

 	
By 	
Lemma 	
\ref{lemma 	
- 	
comparison 	
- 	
fppf 	
- 	
etale 	
} 	

 	
we 	
have 	
$ 	
R^i\epsilon_{X 	
, 	
* 	
} 	
( 	
a_X^{-1}\mathcal{F 	
} 	
) 	
= 	
0 	
$ 	
for 	
$ 	
i 	
> 	
0$. 	

 	
We 	
have 	

 	
$ 	
\epsilon_{X 	
, 	
* 	
} 	
a_X^{-1}\mathcal{F 	
} 	
= 	
\pi_X^{-1}\mathcal{F}$ 	

 	
and 	
by 	
Lemma 	
\ref{lemma 	
- 	
cohomological 	
- 	
descent 	
- 	
etale 	
} 	
we 	
have 	

 	
$ 	
R^j\pi_{X 	
, 	
* 	
} 	
( 	
\pi_X^{-1}\mathcal{F 	
} 	
) 	
= 	
0 	
$ 	
for 	
$ 	
j 	
> 	
0$. 	

 	
This 	
concludes 	
the 	
proof 	
. 	

 	
\end{proof 	
} 	


 	
\begin{lemma 	
} 	

 	
\label{lemma 	
- 	
compare 	
- 	
cohomology 	
- 	
etale 	
- 	
fppf 	
} 	

 	
Let 	
$ 	
S$ 	
be 	
a 	
scheme 	TYPE
and 	
let 	
$ 	
X$ 	
be 	
an 	
algebraic 	TYPE
space 	
over 	
$ 	
S$. 	

 	
With 	
$ 	
a_X 	
: 	
\Sh((\textit{Spaces}/X)_{fppf 	
} 	
) 	
\to 	
\Sh(X_\etale)$ 	

 	
as 	
above 	
: 	

 	
\begin{enumerate 	
} 	

 	
\item 	
$ 	
H^q(X_\etale 	
, 	
\mathcal{F 	
} 	
) 	
= 	
H^q_{fppf}(X 	
, 	
a_X^{-1}\mathcal{F})$ 	

 	
for 	
an 	
abelian 	
sheaf 	
$ 	
\mathcal{F}$ 	
on 	
$ 	
X_\etale$ 	
, 	

 	
\item 	
$ 	
H^q(X_\etale 	
, 	
K 	
) 	
= 	
H^q_{fppf}(X 	
, 	
a_X^{-1}K)$ 	
for 	
$ 	
K 	
\in 	
D^+(X_\etale)$. 	

 	
\end{enumerate 	
} 	

 	
Example 	
: 	
if 	
$ 	
A$ 	
is 	
an 	
abelian 	
group 	
, 	
then 	

 	
$ 	
H^q_\etale(X 	
, 	
\underline{A 	
} 	
) 	
= 	
H^q_{fppf}(X 	
, 	
\underline{A})$. 	

 	
\end{lemma 	
} 	


 	
\begin{proof 	
} 	

 	
This 	
follows 	
from 	
Lemma 	
\ref{lemma 	
- 	
cohomological 	
- 	
descent 	
- 	
etale 	
- 	
fppf 	
} 	

 	
by 	
Cohomology 	
on 	
Sites 	
, 	
Remark 	
\ref{sites 	
- 	
cohomology 	
- 	
remark 	
- 	
before 	
- 	
Leray}. 	

 	
\end{proof 	
} 	


 	
\begin{lemma 	
} 	

 	
\label{lemma 	
- 	
push 	
- 	
pull 	
- 	
fppf 	
- 	
etale 	
} 	

 	
Let 	
$ 	
S$ 	
be 	
a 	
scheme 	TYPE
. 	TYPE

 	
Let 	
$ 	
f 	
: 	
X 	
\to 	
Y$ 	
be 	
a 	
morphism 	
of 	
algebraic 	
spaces 	
over 	
$ 	
S$. 	

 	
Then 	
there 	
are 	
commutative 	
diagrams 	
of 	
topoi 	

 	
$ 	
$ 	

 	
\xymatrix 	
{ 	

 	
\Sh((\textit{Spaces}/X)_{fppf 	
} 	
) 	
\ar[rr]_{f_{big 	
, 	
fppf 	
} 	
} 	
\ar[d]_{\epsilon_X 	
} 	
& 	
& 	

 	
\Sh((\textit{Spaces}/Y)_{fppf 	
} 	
) 	
\ar[d]^{\epsilon_Y 	
} 	
\\ 	

 	
\Sh((\textit{Spaces}/X)_\etale 	
) 	
\ar[rr]^{f_{big 	
, 	
\etale 	
} 	
} 	
& 	
& 	

 	
\Sh((\textit{Spaces}/Y)_\etale 	
) 	

 	
} 	

 	
$ 	
$ 	

 	
and 	

 	
$ 	
$ 	

 	
\xymatrix 	
{ 	

 	
\Sh((\textit{Spaces}/X)_{fppf 	
} 	
) 	
\ar[rr]_{f_{big 	
, 	
fppf 	
} 	
} 	
\ar[d]_{a_X 	
} 	
& 	
& 	

 	
\Sh((\textit{Spaces}/Y)_{fppf 	
} 	
) 	
\ar[d]^{a_Y 	
} 	
\\ 	

 	
\Sh(X_\etale 	
) 	
\ar[rr]^{f_{small 	
} 	
} 	
& 	
& 	

 	
\Sh(Y_\etale 	
) 	

 	
} 	

 	
$ 	
$ 	

 	
with 	
$ 	
a_X 	
= 	
\pi_X 	
\circ 	
\epsilon_X$ 	
and 	
$ 	
a_Y 	
= 	
\pi_X 	
\circ 	
\epsilon_X$. 	

 	
\end{lemma 	
} 	


 	
\begin{proof 	
} 	

 	
This 	
follows 	
immediately 	
from 	
working 	
out 	
the 	
definitions 	
of 	
the 	

 	
morphisms 	
involved 	
, 	
see 	

 	
Topologies 	
on 	
Spaces 	
, 	
Section 	
\ref{spaces 	
- 	
topologies 	
- 	
section 	
- 	
fppf 	
} 	

 	
and 	
Section 	
\ref{section 	
- 	
compare}. 	

 	
\end{proof 	
} 	


 	
\begin{lemma 	
} 	

 	
\label{lemma 	
- 	
proper 	
- 	
push 	
- 	
pull 	
- 	
fppf 	
- 	
etale 	
} 	

 	
In 	
Lemma 	
\ref{lemma 	
- 	
push 	
- 	
pull 	
- 	
fppf 	
- 	
etale 	
} 	
if 	
$ 	
f$ 	
is 	
proper 	
, 	
then 	
we 	
have 	

 	
\begin{enumerate 	
} 	

 	
\item 	
$ 	
a_Y^{-1 	
} 	
\circ 	
f_{small 	
, 	
* 	
} 	
= 	
f_{big 	
, 	
fppf 	
, 	
* 	
} 	
\circ 	
a_X^{-1}$ 	
, 	
and 	

 	
\item 	

 	
$ 	
a_Y^{-1}(Rf_{small 	
, 	
* 	
} 	
K 	
) 	
= 	
Rf_{big 	
, 	
fppf 	
, 	
* 	
} 	
( 	
a_X^{-1}K)$ 	

 	
for 	
$ 	
K$ 	
in 	
$ 	
D^+(X_\etale)$ 	
with 	
torsion 	
cohomology 	
sheaves 	
. 	

 	
\end{enumerate 	
} 	

 	
\end{lemma 	
} 	


 	
\begin{proof 	
} 	

 	
Proof 	
of 	
( 	
1 	
) 	
. 	
You 	
can 	
prove 	
this 	
by 	
repeating 	
the 	
proof 	
of 	

 	
Lemma 	
\ref{lemma 	
- 	
compare 	
- 	
higher 	
- 	
direct 	
- 	
image 	
- 	
proper 	
} 	
part 	
( 	
1 	
) 	
; 	

 	
we 	
will 	
instead 	
deduce 	
the 	
result 	
from 	
this 	
. 	

 	
As 	
$ 	
\epsilon_{Y 	
, 	
* 	
} 	
$ 	
is 	
the 	
identity 	
functor 	
on 	
underlying 	
presheaves 	
, 	

 	
it 	
reflects 	
isomorphisms 	
. 	
Lemma 	
\ref{lemma 	
- 	
comparison 	
- 	
fppf 	
- 	
etale 	
} 	

 	
shows 	
that 	
$ 	
\epsilon_{Y 	
, 	
* 	
} 	
\circ 	
a_Y^{-1 	
} 	
= 	
\pi_Y^{-1}$ 	

 	
and 	
similarly 	
for 	
$ 	
X$. 	
To 	
show 	
that 	
the 	
canonical 	
map 	

 	
$ 	
a_Y^{-1}f_{small 	
, 	
* 	
} 	
\mathcal{F 	
} 	
\to 	
f_{big 	
, 	
fppf 	
, 	
* 	
} 	
a_X^{-1}\mathcal{F}$ 	

 	
is 	
an 	
isomorphism 	
, 	
it 	
suffices 	
to 	
show 	
that 	

 	
\begin{align 	
* 	
} 	

 	
\pi_Y^{-1}f_{small 	
, 	
* 	
} 	
\mathcal{F 	
} 	

 	
& 	
= 	

 	
\epsilon_{Y 	
, 	
* 	
} 	
a_Y^{-1}f_{small 	
, 	
* 	
} 	
\mathcal{F 	
} 	
\\ 	

 	
& 	
\to 	

 	
\epsilon_{Y 	
, 	
* 	
} 	
f_{big 	
, 	
fppf 	
, 	
* 	
} 	
a_X^{-1}\mathcal{F 	
} 	
\\ 	

 	
& 	
= 	

 	
f_{big 	
, 	
\etale 	
, 	
* 	
} 	
\epsilon_{X 	
, 	
* 	
} 	
a_X^{-1}\mathcal{F 	
} 	
\\ 	

 	
& 	
= 	

 	
f_{big 	
, 	
\etale 	
, 	
* 	
} 	
\pi_X^{-1}\mathcal{F 	
} 	

 	
\end{align 	
* 	
} 	

 	
is 	
an 	
isomorphism 	
. 	
This 	
is 	
part 	

 	
( 	
1 	
) 	
of 	
Lemma 	
\ref{lemma 	
- 	
compare 	
- 	
higher 	
- 	
direct 	
- 	
image 	
- 	
proper}. 	


 	
\medskip\noindent 	

 	
To 	
see 	
( 	
2 	
) 	
we 	
use 	
that 	

 	
\begin{align 	
* 	
} 	

 	
R\epsilon_{Y 	
, 	
* 	
} 	
Rf_{big 	
, 	
fppf 	
, 	
* 	
} 	
a_X^{-1}K 	

 	
& 	
= 	

 	
Rf_{big 	
, 	
\etale 	
, 	
* 	
} 	
R\epsilon_{X 	
, 	
* 	
} 	
a_X^{-1}K 	
\\ 	

 	
& 	
= 	

 	
Rf_{big 	
, 	
\etale 	
, 	
* 	
} 	
\pi_X^{-1}K 	
\\ 	

 	
& 	
= 	

 	
\pi_Y^{-1}Rf_{small 	
, 	
* 	
} 	
K 	
\\ 	

 	
& 	
= 	

 	
R\epsilon_{Y 	
, 	
* 	
} 	
a_Y^{-1}Rf_{small 	
, 	
* 	
} 	
K 	

 	
\end{align 	
* 	
} 	

 	
The 	
first 	
equality 	
by 	
the 	
commutative 	
diagram 	
in 	

 	
Lemma 	
\ref{lemma 	
- 	
push 	
- 	
pull 	
- 	
fppf 	
- 	
etale 	
} 	

 	
and 	
Cohomology 	
on 	
Sites 	
, 	
Lemma 	

 	
\ref{sites 	
- 	
cohomology 	
- 	
lemma 	
- 	
derived 	
- 	
pushforward 	
- 	
composition}. 	

 	
Then 	
second 	
equality 	
is 	
Lemma 	
\ref{lemma 	
- 	
cohomological 	
- 	
descent 	
- 	
etale 	
- 	
fppf}. 	

 	
The 	
third 	
is 	

 	
Lemma 	
\ref{lemma 	
- 	
compare 	
- 	
higher 	
- 	
direct 	
- 	
image 	
- 	
proper 	
} 	
part 	
( 	
2 	
) 	
. 	

 	
The 	
fourth 	
is 	
Lemma 	
\ref{lemma 	
- 	
cohomological 	
- 	
descent 	
- 	
etale 	
- 	
fppf 	
} 	
again 	
. 	

 	
Thus 	
the 	
base 	
change 	
map 	

 	
$ 	
a_Y^{-1}(Rf_{small 	
, 	
* 	
} 	
K 	
) 	
\to 	
Rf_{big 	
, 	
fppf 	
, 	
* 	
} 	
( 	
a_X^{-1}K)$ 	

 	
induces 	
an 	
isomorphism 	

 	
$ 	
$ 	

 	
R\epsilon_{Y 	
, 	
* 	
} 	
a_Y^{-1}Rf_{small 	
, 	
* 	
} 	
K 	
\to 	

 	
R\epsilon_{Y 	
, 	
* 	
} 	
Rf_{big 	
, 	
fppf 	
, 	
* 	
} 	
a_X^{-1}K 	

 	
$ 	
$ 	

 	
The 	
proof 	
is 	
finished 	
by 	
the 	
following 	
remark 	
: 	
a 	
map 	

 	
$ 	
\alpha 	
: 	
a_Y^{-1}L 	
\to 	
M$ 	
with 	
$ 	
L$ 	
in 	
$ 	
D^+(Y_\etale)$ 	

 	
and 	
$ 	
M$ 	
in 	
$ 	
D^+((\textit{Spaces}/Y)_{fppf})$ 	
such 	
that 	

 	
$ 	
R\epsilon_{Y 	
, 	
* 	
} 	
\alpha$ 	
is 	
an 	
isomorphism 	
, 	
is 	
an 	
isomorphism 	
. 	
Namely 	
, 	

 	
we 	
show 	
by 	
induction 	
on 	
$ 	
i$ 	
that 	
$ 	
H^i(\alpha)$ 	
is 	
an 	
isomorphism 	
. 	

 	
This 	
is 	
true 	
for 	
all 	
sufficiently 	
small 	
$ 	
i$. 	

 	
If 	
it 	
holds 	
for 	
$ 	
i 	
\leq 	
i_0 	
$ 	
, 	
then 	
we 	
see 	
that 	

 	
$ 	
R^j\epsilon_{Y 	
, 	
* 	
} 	
H^i(M 	
) 	
= 	
0 	
$ 	
for 	
$ 	
j 	
> 	
0 	
$ 	
and 	
$ 	
i 	
\leq 	
i_0 	
$ 	

 	
by 	
Lemma 	
\ref{lemma 	
- 	
comparison 	
- 	
fppf 	
- 	
etale 	
} 	

 	
because 	
$ 	
H^i(M 	
) 	
= 	
a_Y^{-1}H^i(L)$ 	
in 	
this 	
range 	
. 	

 	
Hence 	
$ 	
\epsilon_{Y 	
, 	
* 	
} 	
H^{i_0 	
+ 	
1}(M 	
) 	
= 	
H^{i_0 	
+ 	
1}(R\epsilon_{Y 	
, 	
* 	
} 	
M)$ 	

 	
by 	
a 	
spectral 	
sequence 	
argument 	
. 	

 	
Thus 	
$ 	
\epsilon_{Y 	
, 	
* 	
} 	
H^{i_0 	
+ 	
1}(M 	
) 	
= 	
\pi_Y^{-1}H^{i_0 	
+ 	
1}(L 	
) 	
= 	

 	
\epsilon_{Y 	
, 	
* 	
} 	
a_Y^{-1}H^{i_0 	
+ 	
1}(L)$. 	

 	
This 	
implies 	
$ 	
H^{i_0 	
+ 	
1}(\alpha)$ 	
is 	
an 	
isomorphism 	

 	
( 	
because 	
$ 	
\epsilon_{Y 	
, 	
* 	
} 	
$ 	
reflects 	
isomorphisms 	
as 	
it 	
is 	
the 	

 	
identity 	
on 	
underlying 	
presheaves 	
) 	
as 	
desired 	
. 	

 	
\end{proof 	
} 	


 	
\begin{lemma 	
} 	

 	
\label{lemma 	
- 	
finite 	
- 	
push 	
- 	
pull 	
- 	
fppf 	
- 	
etale 	
} 	

 	
In 	
Lemma 	
\ref{lemma 	
- 	
push 	
- 	
pull 	
- 	
fppf 	
- 	
etale 	
} 	
if 	
$ 	
f$ 	
is 	
finite 	
, 	
then 	

 	
$ 	
a_Y^{-1}(Rf_{small 	
, 	
* 	
} 	
K 	
) 	
= 	
Rf_{big 	
, 	
fppf 	
, 	
* 	
} 	
( 	
a_X^{-1}K)$ 	

 	
for 	
$ 	
K$ 	
in 	
$ 	
D^+(X_\etale)$. 	

 	
\end{lemma 	
} 	


 	
\begin{proof 	
} 	

 	
Let 	
$ 	
V 	
\to 	
Y$ 	
be 	
a 	
surjective 	TYPE
\'etale 	
morphism 	
where 	
$ 	
V$ 	
is 	
a 	
scheme 	
. 	

 	
It 	
suffices 	
to 	
prove 	
the 	
base 	
change 	
map 	
is 	
an 	
isomorphism 	
after 	

 	
restricting 	
to 	
$ 	
V$. 	
Hence 	
we 	
may 	
assume 	
that 	
$ 	
Y$ 	
is 	
a 	
scheme 	
. 	

 	
As 	
the 	
morphism 	
is 	
finite 	
, 	
hence 	
representable 	
, 	
we 	
conclude 	

 	
that 	
we 	
may 	
assume 	
both 	
$ 	
X$ 	
and 	
$ 	
Y$ 	
are 	
schemes 	
. 	
In 	
this 	
case 	

 	
the 	
result 	
follows 	
from 	
the 	
case 	
of 	
schemes 	

 	
( 	
\'Etale 	
Cohomology 	
, 	
Lemma 	
\ref{etale 	
- 	
cohomology 	
- 	
lemma 	
- 	
V 	
- 	
C 	
- 	
all 	
- 	
n 	
- 	
etale 	
- 	
fppf 	
} 	

 	
part 	
( 	
2 	
) 	
) 	
using 	
the 	
comparison 	
of 	
topoi 	
discussed 	
in 	

 	
Section 	
\ref{section 	
- 	
api 	
} 	

 	
and 	
in 	
particular 	
given 	
in 	

 	
Lemma 	
\ref{lemma 	
- 	
compare 	
- 	
cohomology 	
- 	
other 	
- 	
topologies}. 	

 	
Some 	
details 	
omitted 	
. 	

 	
\end{proof 	
} 	


 	
\begin{lemma 	
} 	

 	
\label{lemma 	
- 	
descent 	
- 	
sheaf 	
- 	
fppf 	
- 	
etale 	
} 	

 	
In 	
Lemma 	
\ref{lemma 	
- 	
push 	
- 	
pull 	
- 	
fppf 	
- 	
etale 	
} 	
assume 	

 	
$ 	
f$ 	
is 	
flat 	
, 	
locally 	
of 	
finite 	
presentation 	
, 	
and 	
surjective 	
. 	

 	
Then 	
the 	
functor 	

 	
$ 	
$ 	

 	
\Sh(Y_\etale 	
) 	
\longrightarrow 	

 	
\left\ 	
{ 	

 	
( 	
\mathcal{G 	
} 	
, 	
\mathcal{H 	
} 	
, 	
\alpha 	
) 	

 	
\middle| 	

 	
\begin{matrix 	
} 	

 	
\mathcal{G 	
} 	
\in 	
\Sh(X_\etale),\ 	
\mathcal{H 	
} 	
\in 	
\Sh((\Sch 	
/ 	
Y)_{fppf 	
} 	
) 	
, 	
\\ 	

 	
\alpha 	
: 	
a_X^{-1}\mathcal{G 	
} 	
\to 	
f_{big 	
, 	
fppf}^{-1}\mathcal{H 	
} 	

 	
\text 	
{ 	
an 	
isomorphism 	
} 	

 	
\end{matrix 	
} 	

 	
\right\ 	
} 	

 	
$ 	
$ 	

 	
sending 	
$ 	
\mathcal{F}$ 	
to 	

 	
$ 	
( 	
f_{small}^{-1}\mathcal{F 	
} 	
, 	
a_Y^{-1}\mathcal{F 	
} 	
, 	
can)$ 	
is 	
an 	
equivalence 	
. 	

 	
\end{lemma 	
} 	


 	
\begin{proof 	
} 	

 	
The 	
functor 	
$ 	
a_X^{-1}$ 	
is 	
fully 	
faithful 	
( 	
as 	
$ 	
a_{X 	
, 	
* 	
} 	
a_X^{-1 	
} 	
= 	
\text{id}$ 	
by 	

 	
Lemma 	
\ref{lemma 	
- 	
comparison 	
- 	
fppf 	
- 	
etale 	
} 	
) 	
. 	
Hence 	
the 	
forgetful 	
functor 	

 	
$ 	
( 	
\mathcal{G 	
} 	
, 	
\mathcal{H 	
} 	
, 	
\alpha 	
) 	
\mapsto 	
\mathcal{H}$ 	
identifies 	
the 	

 	
category 	
of 	
triples 	
with 	
a 	
full 	
subcategory 	
of 	
$ 	
\Sh((\Sch 	
/ 	
Y)_{fppf})$. 	

 	
Moreover 	
, 	
the 	
functor 	
$ 	
a_Y^{-1}$ 	
is 	
fully 	
faithful 	
, 	
hence 	
the 	
functor 	

 	
in 	
the 	
lemma 	
is 	
fully 	
faithful 	
as 	
well 	
. 	


 	
\medskip\noindent 	

 	
Suppose 	
that 	
we 	
have 	
an 	
\'etale 	
covering 	
$ 	
\{Y_i 	
\to 	
Y\}$. 	

 	
Let 	
$ 	
f_i 	
: 	
X_i 	
\to 	
Y_i$ 	
be 	
the 	
base 	
change 	
of 	
$ 	
f$. 	

 	
Denote 	
$ 	
f_{ij 	
} 	
= 	
f_i 	
\times 	
f_j 	
: 	
X_i 	
\times_X 	
X_j 	
  	
\to 	
Y_i 	
\times_Y 	
Y_j$. 	

 	
Claim 	
: 	
if 	
the 	
lemma 	
is 	
true 	
for 	
$ 	
f_i$ 	
and 	
$ 	
f_{ij}$ 	
for 	
all 	
$ 	
i 	
, 	
j$ 	
, 	
then 	

 	
the 	
lemma 	
is 	
true 	
for 	
$ 	
f$. 	
To 	
see 	
this 	
, 	
note 	
that 	
the 	
given 	
\'etale 	
covering 	

 	
determines 	
an 	
\'etale 	
covering 	
of 	
the 	
final 	
object 	
in 	
each 	
of 	

 	
the 	
four 	
sites 	
$ 	
Y_\etale 	
, 	
X_\etale 	
, 	
( 	
\Sch 	
/ 	
Y)_{fppf 	
} 	
, 	
( 	
\Sch 	
/ 	
X)_{fppf}$. 	

 	
Thus 	
the 	
category 	
of 	
sheaves 	
is 	
equivalent 	
to 	
the 	
category 	
of 	

 	
glueing 	
data 	
for 	
this 	
covering 	

 	
( 	
Sites 	
, 	
Lemma 	
\ref{sites 	
- 	
lemma 	
- 	
mapping 	
- 	
property 	
- 	
glue 	
} 	
) 	

 	
in 	
each 	
of 	
the 	
four 	
cases 	
. 	
A 	
huge 	
commutative 	
diagram 	
of 	

 	
categories 	
then 	
finishes 	
the 	
proof 	
of 	
the 	
claim 	
. 	
We 	
omit 	
the 	
details 	
. 	

 	
The 	
claim 	
shows 	
that 	
we 	
may 	
work 	
\'etale 	
locally 	
on 	
$ 	
Y$. 	

 	
In 	
particular 	
, 	
we 	
may 	
assume 	
$ 	
Y$ 	
is 	
a 	
scheme 	
. 	


 	
\medskip\noindent 	

 	
Assume 	
$ 	
Y$ 	
is 	
a 	
scheme 	
. 	
Choose 	
a 	
scheme 	
$ 	
X'$ 	
and 	
a 	
surjective 	
\'etale 	

 	
morphism 	
$ 	
s 	
: 	
X 	
' 	
\to 	
X$. 	
Set 	
$ 	
f 	
' 	
= 	
f 	
\circ 	
s 	
: 	
X 	
' 	
\to 	
Y$ 	
and 	
observe 	

 	
that 	
$ 	
f'$ 	
is 	
surjective 	
, 	
locally 	
of 	
finite 	
presentation 	
, 	
and 	
flat 	
. 	

 	
Claim 	
: 	
if 	
the 	
lemma 	
is 	
true 	
for 	
$ 	
f'$ 	
, 	
then 	
it 	
is 	
true 	
for 	
$ 	
f$. 	

 	
Namely 	
, 	
given 	
a 	
triple 	
$ 	
( 	
\mathcal{G 	
} 	
, 	
\mathcal{H 	
} 	
, 	
\alpha)$ 	

 	
for 	
$ 	
f$ 	
, 	
we 	
can 	
pullback 	
by 	
$ 	
s$ 	
to 	
get 	
a 	
triple 	

 	
$ 	
( 	
s_{small}^{-1}\mathcal{G 	
} 	
, 	
\mathcal{H 	
} 	
, 	
s_{big 	
, 	
fppf}^{-1}\alpha)$ 	

 	
for 	
$ 	
f'$. 	
A 	
solution 	
for 	
this 	
triple 	
gives 	
a 	
sheaf 	

 	
$ 	
\mathcal{F}$ 	
on 	
$ 	
Y_\etale$ 	
with 	
$ 	
a_Y^{-1}\mathcal{F 	
} 	
= 	
\mathcal{H}$. 	

 	
By 	
the 	
first 	
paragraph 	
of 	
the 	
proof 	
this 	
means 	
the 	
triple 	
is 	

 	
in 	
the 	
essential 	
image 	
. 	
This 	
reduces 	
us 	
to 	
the 	
case 	

 	
where 	
both 	
$ 	
X$ 	
and 	
$ 	
Y$ 	
are 	
schemes 	
. 	
This 	
case 	
follows 	
from 	

 	
\'Etale 	
Cohomology 	
, 	
Lemma 	

 	
\ref{etale 	
- 	
cohomology 	
- 	
lemma 	
- 	
descent 	
- 	
sheaf 	
- 	
fppf 	
- 	
etale 	
} 	

 	
via 	
the 	
discussion 	
in 	
Section 	
\ref{section 	
- 	
api 	
} 	

 	
and 	
in 	
particular 	
Lemma 	
\ref{lemma 	
- 	
compare 	
- 	
cohomology 	
- 	
other 	
- 	
topologies}. 	

 	
\end{proof 	
} 	




 	
\section{Comparing 	
fppf 	
and 	
\'etale 	
topologies 	
: 	
modules 	
} 	

 	
\label{section 	
- 	
fppf 	
- 	
etale 	
- 	
modules 	
} 	


 	
\noindent 	

 	
We 	
continue 	
the 	
discussion 	
in 	
Section 	
\ref{section 	
- 	
fppf 	
- 	
etale 	
} 	
but 	
in 	
this 	

 	
section 	
we 	
briefly 	
discuss 	
what 	
happens 	
for 	
sheaves 	
of 	
modules 	
. 	


 	
\medskip\noindent 	

 	
Let 	
$ 	
S$ 	
be 	
a 	
scheme 	TYPE
. 	TYPE
Let 	
$ 	
X$ 	
be 	
an 	
algebraic 	TYPE
space 	
over 	
$ 	
S$. 	

 	
The 	
morphisms 	
of 	
sites 	
$ 	
\epsilon_X$ 	
, 	
$ 	
\pi_X$ 	
, 	
and 	

 	
their 	
composition 	
$ 	
a_X$ 	
introduced 	
in 	
Section 	
\ref{section 	
- 	
fppf 	
- 	
etale 	
} 	

 	
have 	
natural 	
enhancements 	
to 	
morphisms 	
of 	
ringed 	
sites 	
. 	
The 	
first 	

 	
is 	
written 	
as 	

 	
$ 	
$ 	

 	
\epsilon_X 	
: 	

 	
( 	
( 	
\textit{Spaces}/X)_{fppf 	
} 	
, 	
\mathcal{O 	
} 	
) 	

 	
\longrightarrow 	

 	
( 	
( 	
\textit{Spaces}/X)_\etale 	
, 	
\mathcal{O 	
} 	
) 	

 	
$ 	
$ 	

 	
Note 	
that 	
we 	
can 	
use 	
the 	
same 	
symbol 	
for 	
the 	
structure 	
sheaf 	
as 	
indeed 	

 	
the 	
sheaves 	
have 	
the 	
same 	
underlying 	
presheaf 	
. 	
The 	
second 	
is 	

 	
$ 	
$ 	

 	
\pi_X 	
: 	

 	
( 	
( 	
\textit{Spaces}/X)_\etale 	
, 	
\mathcal{O 	
} 	
) 	

 	
\longrightarrow 	

 	
( 	
X_\etale 	
, 	
\mathcal{O}_X 	
) 	

 	
$ 	
$ 	

 	
The 	
third 	
is 	
the 	
morphism 	

 	
$ 	
$ 	

 	
a_X 	
: 	

 	
( 	
( 	
\textit{Spaces}/X)_{fppf 	
} 	
, 	
\mathcal{O 	
} 	
) 	

 	
\longrightarrow 	

 	
( 	
X_\etale 	
, 	
\mathcal{O}_X 	
) 	

 	
$ 	
$ 	

 	
Let 	
us 	
review 	
what 	
we 	
already 	
know 	
about 	
quasi 	
- 	
coherent 	

 	
modules 	
on 	
these 	
sites 	
. 	


 	
\begin{lemma 	
} 	

 	
\label{lemma 	
- 	
review 	
- 	
quasi 	
- 	
coherent 	
} 	

 	
Let 	
$ 	
S$ 	
be 	
a 	
scheme 	TYPE
. 	TYPE
Let 	
$ 	
X$ 	
be 	
an 	
algebraic 	TYPE
space 	
over 	
$ 	
S$. 	

 	
Let 	
$ 	
\mathcal{F}$ 	
be 	
a 	
quasi 	TYPE
- 	TYPE
coherent 	TYPE
$ 	
\mathcal{O}_X$-module 	
. 	

 	
\begin{enumerate 	
} 	

 	
\item 	
The 	
rule 	

 	
$ 	
$ 	

 	
\mathcal{F}^a 	
: 	
( 	
\textit{Spaces}/X)_\etale 	
\longrightarrow 	
\textit{Ab},\quad 	

 	
( 	
f 	
: 	
Y 	
\to 	
X 	
) 	
\longmapsto 	
\Gamma(Y 	
, 	
f^*\mathcal{F 	
} 	
) 	

 	
$ 	
$ 	

 	
satisfies 	
the 	
sheaf 	
condition 	
for 	
fpqc 	
and 	
a 	
fortiori 	

 	
fppf 	
and 	
\'etale 	
coverings 	
, 	

 	
\item 	
$ 	
\mathcal{F}^a 	
= 	
\pi_X^*\mathcal{F}$ 	
on 	
$ 	
( 	
\textit{Spaces}/X)_\etale$ 	
, 	

 	
\item 	
$ 	
\mathcal{F}^a 	
= 	
a_X^*\mathcal{F}$ 	
on 	
$ 	
( 	
\textit{Spaces}/X)_{fppf}$ 	
, 	

 	
\item 	
the 	
rule 	
$ 	
\mathcal{F 	
} 	
\mapsto 	
\mathcal{F}^a$ 	
defines 	

 	
an 	
equivalence 	
between 	
quasi 	
- 	
coherent 	
$ 	
\mathcal{O}_X$-modules 	

 	
and 	
quasi 	
- 	
coherent 	
modules 	
on 	

 	
$ 	
( 	
( 	
\textit{Spaces}/X)_\etale 	
, 	
\mathcal{O})$ 	
, 	

 	
\item 	
the 	
rule 	
$ 	
\mathcal{F 	
} 	
\mapsto 	
\mathcal{F}^a$ 	
defines 	

 	
an 	
equivalence 	
between 	
quasi 	
- 	
coherent 	
$ 	
\mathcal{O}_X$-modules 	

 	
and 	
quasi 	
- 	
coherent 	
modules 	
on 	

 	
$ 	
( 	
( 	
\textit{Spaces}/X)_{fppf 	
} 	
, 	
\mathcal{O})$ 	
, 	

 	
\item 	
we 	
have 	
$ 	
\epsilon_{X 	
, 	
* 	
} 	
a_X^*\mathcal{F 	
} 	
= 	
\pi_X^*\mathcal{F}$ 	

 	
and 	
$ 	
a_{X 	
, 	
* 	
} 	
a_X^*\mathcal{F 	
} 	
= 	
\mathcal{F}$ 	
, 	

 	
\item 	
we 	
have 	
$ 	
R^i\epsilon_{X 	
, 	
* 	
} 	
( 	
a_X^*\mathcal{F 	
} 	
) 	
= 	
0 	
$ 	

 	
and 	
$ 	
R^ia_{X 	
, 	
* 	
} 	
( 	
a_X^*\mathcal{F 	
} 	
) 	
= 	
0 	
$ 	
for 	
$ 	
i 	
> 	
0$. 	

 	
\end{enumerate 	
} 	

 	
\end{lemma 	
} 	


 	
\begin{proof 	
} 	

 	
Part 	
( 	
1 	
) 	
is 	
a 	
consequence 	
of 	
fppf 	
descent 	
of 	
quasi 	
- 	
coherent 	
modules 	
. 	

 	
Namely 	
, 	
suppose 	
that 	
$ 	
\{f_i 	
: 	
U_i 	
\to 	
U\}$ 	
is 	
an 	
fpqc 	
covering 	

 	
in 	
$ 	
( 	
\textit{Spaces}/X)_\etale$. 	
Denote 	
$ 	
g 	
: 	
U 	
\to 	
X$ 	
the 	
structure 	

 	
morphism 	
. 	
Suppose 	
that 	

 	
we 	
have 	
a 	
family 	
of 	
sections 	
$ 	
s_i 	
\in 	
\Gamma(U_i 	
, 	
f_i^*g^*\mathcal{F})$ 	

 	
such 	
that 	
$ 	
s_i|_{U_i 	
\times_U 	
U_j 	
} 	
= 	
s_j|_{U_i 	
\times_U 	
U_j}$. 	

 	
We 	
have 	
to 	
find 	
the 	
correspond 	
section 	
$ 	
s 	
\in 	
\Gamma(U 	
, 	
g^*\mathcal{F})$. 	

 	
We 	
can 	
reinterpret 	
the 	
$ 	
s_i$ 	
as 	
a 	
family 	
of 	
maps 	

 	
$ 	
\varphi_i 	
: 	
f_i^*\mathcal{O}_U 	
= 	
\mathcal{O}_{U_i 	
} 	
\to 	
f_i^*g^*\mathcal{F}$ 	

 	
compatible 	
with 	
the 	
canonical 	
descent 	
data 	
associated 	
to 	
the 	

 	
quasi 	
- 	
coherent 	
sheaves 	
$ 	
\mathcal{O}_U$ 	
and 	
$ 	
g^*\mathcal{F}$ 	
on 	
$ 	
U$. 	

 	
Hence 	
by 	
Descent 	
on 	
Spaces 	
, 	
Proposition 	

 	
\ref{spaces 	
- 	
descent 	
- 	
proposition 	
- 	
fpqc 	
- 	
descent 	
- 	
quasi 	
- 	
coherent 	
} 	

 	
we 	
see 	
that 	
we 	
may 	
( 	
uniquely 	
) 	
descend 	

 	
these 	
to 	
a 	
map 	
$ 	
\mathcal{O}_U 	
\to 	
g^*\mathcal{F}$ 	
which 	
gives 	

 	
us 	
our 	
section 	
$ 	
s$. 	


 	
\medskip\noindent 	

 	
We 	
will 	
deduce 	
( 	
2 	
) 	
-- 	
( 	
7 	
) 	
from 	
the 	
corresponding 	
statement 	
for 	
schemes 	
. 	

 	
Choose 	
an 	
\'etale 	
covering 	
$ 	
\{X_i 	
\to 	
X\}_{i 	
\in 	
I}$ 	

 	
where 	
each 	
$ 	
X_i$ 	
is 	
a 	
scheme 	
. 	
Observe 	
that 	
$ 	
X_i 	
\times_X 	
X_j$ 	

 	
is 	
a 	
scheme 	
too 	
. 	
This 	
covering 	
induces 	
a 	
covering 	
of 	

 	
the 	
final 	
object 	
in 	
each 	
of 	
the 	
three 	
sites 	

 	
$ 	
( 	
\textit{Spaces}/X)_{fppf}$ 	
, 	
$ 	
( 	
\textit{Spaces}/X)_\etale$ 	
, 	
and 	
$ 	
X_\etale$. 	

 	
Hence 	
we 	
see 	
that 	
the 	
category 	
of 	
sheaves 	
on 	
these 	
sites 	

 	
are 	
equivalent 	
to 	
descent 	
data 	
for 	
these 	
coverings 	
, 	
see 	

 	
Sites 	
, 	
Lemma 	
\ref{sites 	
- 	
lemma 	
- 	
mapping 	
- 	
property 	
- 	
glue}. 	

 	
Parts 	
( 	
2 	
) 	
, 	
( 	
3 	
) 	
are 	
local 	
( 	
because 	
we 	
have 	
the 	
glueing 	
statement 	
) 	
. 	

 	
Being 	
quasi 	
- 	
coherent 	
is 	
a 	
local 	
property 	
, 	
hence 	
parts 	

 	
( 	
4 	
) 	
, 	
( 	
5 	
) 	
are 	
local 	
. 	
Clearly 	
( 	
6 	
) 	
and 	
( 	
7 	
) 	
are 	
local 	
. 	

 	
It 	
follows 	
that 	
it 	
suffices 	
to 	
prove 	
parts 	
( 	
2 	
) 	
-- 	
( 	
7 	
) 	

 	
of 	
the 	
lemma 	
when 	
$ 	
X$ 	
is 	
a 	
scheme 	
. 	


 	
\medskip\noindent 	

 	
Assume 	
$ 	
X$ 	
is 	
a 	
scheme 	
. 	
The 	
embeddings 	

 	
$ 	
( 	
\Sch 	
/ 	
X)_\etale 	
\subset 	
( 	
\textit{Spaces}/X)_\etale$ 	
and 	

 	
$ 	
( 	
\Sch 	
/ 	
X)_{fppf 	
} 	
\subset 	
( 	
\textit{Spaces}/X)_{fppf}$ 	

 	
determine 	
equivalences 	
of 	
ringed 	
topoi 	
by 	

 	
Lemma 	
\ref{lemma 	
- 	
compare 	
- 	
cohomology 	
- 	
other 	
- 	
topologies}. 	

 	
We 	
conclude 	
that 	
( 	
2 	
) 	
-- 	
( 	
7 	
) 	
follows 	
from 	
the 	
case 	
of 	
schemes 	
. 	

 	
\'Etale 	
Cohomology 	
, 	
Lemma 	

 	
\ref{etale 	
- 	
cohomology 	
- 	
lemma 	
- 	
review 	
- 	
quasi 	
- 	
coherent}. 	

 	
To 	
transport 	
the 	
property 	
of 	
being 	
quasi 	
- 	
coherent 	
via 	

 	
this 	
equivalence 	
use 	
that 	
being 	
quasi 	
- 	
coherent 	
is 	
an 	

 	
intrinsic 	
property 	
of 	
modules 	
as 	
explained 	
in 	

 	
Modules 	
on 	
Sites 	
, 	
Section 	
\ref{sites 	
- 	
modules 	
- 	
section 	
- 	
local}. 	

 	
Some 	
minor 	
details 	
omitted 	
. 	

 	
\end{proof 	
} 	


 	
\begin{lemma 	
} 	

 	
\label{lemma 	
- 	
cohomological 	
- 	
descent 	
- 	
etale 	
- 	
fppf 	
- 	
modules 	
} 	

 	
Let 	
$ 	
S$ 	
be 	
a 	
scheme 	TYPE
. 	TYPE
Let 	
$ 	
X$ 	
be 	
an 	
algebraic 	TYPE
space 	
over 	
$ 	
S$. 	

 	
For 	
$ 	
\mathcal{F}$ 	
a 	
quasi 	
- 	
coherent 	
$ 	
\mathcal{O}_X$-module 	
the 	
maps 	

 	
$ 	
$ 	

 	
\pi_X^*\mathcal{F 	
} 	
\longrightarrow 	
R\epsilon_{X 	
, 	
* 	
} 	
( 	
a_X^*\mathcal{F 	
} 	
) 	

 	
\quad\text{and}\quad 	

 	
\mathcal{F 	
} 	
\longrightarrow 	
Ra_{X 	
, 	
* 	
} 	
( 	
a_X^*\mathcal{F 	
} 	
) 	

 	
$ 	
$ 	

 	
are 	
isomorphisms 	
. 	

 	
\end{lemma 	
} 	


 	
\begin{proof 	
} 	

 	
This 	
is 	
an 	
immediate 	
consequence 	
of 	
parts 	
( 	
6 	
) 	
and 	
( 	
7 	
) 	
of 	

 	
Lemma 	
\ref{lemma 	
- 	
review 	
- 	
quasi 	
- 	
coherent}. 	

 	
\end{proof 	
} 	


 	
\begin{lemma 	
} 	

 	
\label{lemma 	
- 	
vanishing 	
- 	
adequate 	
} 	

 	
Let 	
$ 	
S$ 	
be 	
a 	
scheme 	TYPE
. 	TYPE
Let 	
$ 	
X$ 	
be 	
an 	
algebraic 	TYPE
space 	
over 	
$ 	
S$. 	

 	
Let 	
$ 	
\mathcal{F}_1 	
\to 	
\mathcal{F}_2 	
\to 	
\mathcal{F}_3 	
$ 	

 	
be 	
a 	
complex 	TYPE
of 	
quasi 	
- 	
coherent 	
$ 	
\mathcal{O}_X$-modules 	
. 	

 	
Set 	

 	
$ 	
$ 	

 	
\mathcal{H}_\etale 	
= 	

 	
\Ker(\pi_X^*\mathcal{F}_2 	
\to 	
\pi_X^*\mathcal{F}_3)/ 	

 	
\Im(\pi_X^*\mathcal{F}_1 	
\to 	
\pi_X^*\mathcal{F}_2 	
) 	

 	
$ 	
$ 	

 	
on 	
$ 	
( 	
\textit{Spaces}/X)_\etale$ 	
and 	
set 	

 	
$ 	
$ 	

 	
\mathcal{H}_{fppf 	
} 	
= 	

 	
\Ker(a_X^*\mathcal{F}_2 	
\to 	
a_X^*\mathcal{F}_3)/ 	

 	
\Im(a_X^*\mathcal{F}_1 	
\to 	
a_X^*\mathcal{F}_2 	
) 	

 	
$ 	
$ 	

 	
on 	
$ 	
( 	
\textit{Spaces}/X)_{fppf}$. 	

 	
Then 	
$ 	
\mathcal{H}_\etale 	
= 	
\epsilon_{X 	
, 	
* 	
} 	
\mathcal{H}_{fppf}$ 	

 	
and 	

 	
$ 	
$ 	

 	
H^p_\etale(U 	
, 	
\mathcal{H}_\etale 	
) 	
= 	
H^p_{fppf}(U 	
, 	
\mathcal{H}_{fppf 	
} 	
) 	
= 	
0 	

 	
$ 	
$ 	

 	
for 	
$ 	
p 	
> 	
0 	
$ 	
and 	
any 	
affine 	
object 	
$ 	
U$ 	
of 	
$ 	
( 	
\textit{Spaces}/X)_\etale$. 	

 	
\end{lemma 	
} 	


 	
\noindent 	

 	
More 	
is 	
true 	
, 	
namely 	
the 	
collection 	
of 	
modules 	
on 	

 	
$ 	
( 	
\textit{Spaces}/X)_{fppf}$ 	
which 	
fppf 	
locally 	
look 	
like 	
those 	

 	
in 	
the 	
lemma 	
are 	
called 	
adquate 	
modules 	
. 	
They 	
form 	
a 	

 	
weak 	
Serre 	
subcategory 	
of 	
the 	
category 	
of 	
all 	
$ 	
\mathcal{O}$-modules 	

 	
and 	
their 	
cohomology 	
is 	
studied 	
in 	

 	
Adequate 	
Modules 	
, 	
Section 	
\ref{adequate 	
- 	
section 	
- 	
adequate}. 	


 	
\begin{proof 	
} 	

 	
For 	
any 	
object 	
$ 	
f 	
: 	
U 	
\to 	
X$ 	
of 	
$ 	
( 	
\textit{Spaces}/X)_\etale$ 	

 	
consider 	
the 	
restriction 	

 	
$ 	
\mathcal{H}_\etale|_{U_\etale}$ 	
of 	
$ 	
\mathcal{H}_\etale$ 	
to 	
$ 	
U_\etale$ 	
via 	

 	
the 	
functor 	
$ 	
i_f^ 	
* 	
= 	
i_f^{-1}$ 	
discussed 	
in 	
Section 	
\ref{section 	
- 	
compare}. 	

 	
The 	
sheaf 	
$ 	
\mathcal{H}_\etale|_{U_\etale}$ 	

 	
is 	
equal 	
to 	
the 	
homology 	
of 	
complex 	
$ 	
f^*\mathcal{F}_\bullet$ 	
in 	
degree 	
$ 	
1$. 	

 	
This 	
is 	
true 	
because 	
$ 	
i_f 	
\circ 	
\pi_X 	
= 	
f$ 	
as 	
morphisms 	
of 	
ringed 	
sites 	

 	
$ 	
U_\etale 	
\to 	
X_\etale$. 	
In 	
particular 	
we 	
see 	
that 	

 	
$ 	
\mathcal{H}_\etale|_{U_\etale}$ 	
is 	
a 	
quasi 	
- 	
coherent 	
$ 	
\mathcal{O}_U$-module 	
. 	

 	
Next 	
, 	
let 	
$ 	
g 	
: 	
V 	
\to 	
U$ 	
be 	
a 	
flat 	TYPE

 	
morphism 	
in 	
$ 	
( 	
\textit{Spaces}/X)_\etale$. 	
Since 	

 	
$ 	
$ 	

 	
i_{f 	
\circ 	
g}^ 	
* 	
\circ 	
\pi_X^ 	
* 	
= 	
( 	
f 	
\circ 	
g)^ 	
* 	
= 	
g^ 	
* 	
\circ 	
f^ 	
* 	

 	
$ 	
$ 	

 	
as 	
morphisms 	
of 	
sites 	
$ 	
V_\etale 	
\to 	
X_\etale$ 	
and 	
since 	
$ 	
g$ 	
is 	
flat 	

 	
hence 	
$ 	
g^*$ 	
is 	
exact 	
, 	
we 	
obtain 	

 	
$ 	
$ 	

 	
\mathcal{H}_\etale|_{V_\etale 	
} 	
= 	

 	
g^*\left(\mathcal{H}_\etale|_{U_\etale}\right 	
) 	

 	
$ 	
$ 	

 	
With 	
these 	
preparations 	
we 	
are 	
ready 	
to 	
prove 	
the 	
lemma 	
. 	


 	
\medskip\noindent 	

 	
Let 	
$ 	
\mathcal{U 	
} 	
= 	
\{g_i 	
: 	
U_i 	
\to 	
U\}_{i 	
\in 	
I}$ 	
be 	
an 	
fppf 	
covering 	

 	
with 	
$ 	
f 	
: 	
U 	
\to 	
X$ 	
as 	
above 	
. 	
The 	
sheaf 	
propery 	
holds 	
for 	

 	
$ 	
\mathcal{H}_\etale$ 	
and 	
the 	
covering 	
$ 	
\mathcal{U}$ 	

 	
by 	
( 	
1 	
) 	
of 	
Lemma 	
\ref{lemma 	
- 	
review 	
- 	
quasi 	
- 	
coherent 	
} 	

 	
applied 	
to 	
$ 	
\mathcal{H}_\etale|_{U_\etale}$ 	
and 	
the 	
above 	
. 	

 	
Therefore 	
we 	
see 	
that 	
$ 	
\mathcal{H}_\etale$ 	
is 	
already 	
an 	
fppf 	

 	
sheaf 	
and 	
this 	
means 	
that 	
$ 	
\mathcal{H}_{fppf}$ 	
is 	

 	
equal 	
to 	
$ 	
\mathcal{H}_\etale$ 	

 	
as 	
a 	
presheaf 	
. 	
In 	
particular 	

 	
$ 	
\mathcal{H}_\etale 	
= 	
\epsilon_{X 	
, 	
* 	
} 	
\mathcal{H}_{fppf}$. 	


 	
\medskip\noindent 	

 	
Finally 	
, 	
to 	
prove 	
the 	
vanishing 	
, 	
we 	
use 	

 	
Cohomology 	
on 	
Sites 	
, 	
Lemma 	
\ref{sites 	
- 	
cohomology 	
- 	
lemma 	
- 	
cech 	
- 	
vanish 	
- 	
collection}. 	

 	
We 	
let 	
$ 	
\mathcal{B}$ 	
be 	
the 	
affine 	
objects 	
of 	

 	
$ 	
( 	
\textit{Spaces}/X)_{fppf}$ 	
and 	
we 	
let 	

 	
$ 	
\text{Cov}$ 	
be 	
the 	
set 	
of 	
finite 	
fppf 	
coverings 	

 	
$ 	
\mathcal{U 	
} 	
= 	
\{U_i 	
\to 	
U\}_{i 	
= 	
1 	
, 	
\ldots 	
, 	
n}$ 	
with 	
$ 	
U$ 	
, 	
$ 	
U_i$ 	
affine 	
. 	

 	
We 	
have 	

 	
$ 	
$ 	

 	
{ 	
\check 	
H}^p(\mathcal{U 	
} 	
, 	
\mathcal{H}_\etale 	
) 	
= 	

 	
{ 	
\check 	
H}^p(\mathcal{U 	
} 	
, 	
\left(\mathcal{H}_\etale|_{U_\etale}\right)^a 	
) 	

 	
$ 	
$ 	

 	
because 	
the 	
values 	
of 	
$ 	
\mathcal{H}_\etale$ 	
on 	
the 	

 	
affine 	
schemes 	
$ 	
U_{i_0 	
} 	
\times_U 	
\ldots 	
\times_U 	
U_{i_p}$ 	

 	
flat 	
over 	
$ 	
U$ 	
agree 	
with 	
the 	
values 	
of 	
the 	
pullback 	

 	
of 	
the 	
quasi 	
- 	
coherent 	
module 	
$ 	
\mathcal{H}_\etale|_{U_\etale}$ 	
by 	

 	
the 	
first 	
paragraph 	
. 	
Hence 	
we 	
obtain 	
vanishing 	
by 	

 	
Descent 	
, 	
Lemma 	
\ref{descent 	
- 	
lemma 	
- 	
standard 	
- 	
covering 	
- 	
Cech 	
- 	
quasi 	
- 	
coherent}. 	

 	
This 	
finishes 	
the 	
proof 	
. 	

 	
\end{proof 	
} 	


 	
\begin{lemma 	
} 	

 	
\label{lemma 	
- 	
cohomological 	
- 	
descent 	
- 	
etale 	
- 	
fppf 	
- 	
modules 	
- 	
unbounded 	
} 	

 	
Let 	
$ 	
S$ 	
be 	
a 	
scheme 	TYPE
. 	TYPE
Let 	
$ 	
X$ 	
be 	
an 	
algebraic 	TYPE
space 	
over 	
$ 	
S$. 	

 	
For 	
$ 	
K 	
\in 	
D_\QCoh(\mathcal{O}_X)$ 	
the 	
maps 	

 	
$ 	
$ 	

 	
L\pi_X^*K 	
\longrightarrow 	
R\epsilon_{X 	
, 	
* 	
} 	
( 	
La_X^*\mathcal{F 	
} 	
) 	

 	
\quad\text{and}\quad 	

 	
K 	
\longrightarrow 	
Ra_{X 	
, 	
* 	
} 	
( 	
La_X^*K 	
) 	

 	
$ 	
$ 	

 	
are 	
isomorphisms 	
. 	
Here 	

 	
$ 	
a_X 	
: 	
\Sh((\textit{Spaces}/X)_{fppf 	
} 	
) 	
\to 	
\Sh(X_\etale)$ 	
is 	
as 	
above 	
. 	

 	
\end{lemma 	
} 	


 	
\begin{proof 	
} 	

 	
The 	
question 	
is 	
\'etale 	
local 	
on 	
$ 	
X$ 	
hence 	
we 	
may 	
assume 	
$ 	
X$ 	
is 	
affine 	
. 	

 	
Say 	
$ 	
X 	
= 	
\Spec(A)$. 	
Then 	
we 	
have 	
$ 	
D_\QCoh(\mathcal{O}_X 	
) 	
= 	
D(A)$ 	
by 	

 	
Derived 	
Categories 	
of 	
Spaces 	
, 	
Lemma 	

 	
\ref{spaces 	
- 	
perfect 	
- 	
lemma 	
- 	
derived 	
- 	
quasi 	
- 	
coherent 	
- 	
small 	
- 	
etale 	
- 	
site 	
} 	

 	
and 	

 	
Derived 	
Categories 	
of 	
Schemes 	
, 	
Lemma 	

 	
\ref{perfect 	
- 	
lemma 	
- 	
affine 	
- 	
compare 	
- 	
bounded}. 	

 	
Hence 	
we 	
can 	
choose 	
an 	
K 	
- 	
flat 	
complex 	
of 	
$ 	
A$ 	
-modules 	

 	
$ 	
K^\bullet$ 	
whose 	
corresponding 	
complex 	

 	
$ 	
\mathcal{K}^\bullet$ 	
of 	
quasi 	
- 	
coherent 	
$ 	
\mathcal{O}_X$-modules 	

 	
represents 	
$ 	
K$. 	

 	
We 	
claim 	
that 	
$ 	
\mathcal{K}^\bullet$ 	
is 	
a 	
K 	
- 	
flat 	
complex 	

 	
of 	
$ 	
\mathcal{O}_X$-modules 	
. 	


 	
\medskip\noindent 	

 	
Proof 	
of 	
the 	
claim 	
. 	
By 	

 	
Derived 	
Categories 	
of 	
Schemes 	
, 	
Lemma 	

 	
\ref{perfect 	
- 	
lemma 	
- 	
affine 	
- 	
K 	
- 	
flat 	
} 	

 	
we 	
see 	
that 	
$ 	
\widetilde{K}^\bullet$ 	
is 	
K 	
- 	
flat 	
on 	
the 	
scheme 	

 	
$ 	
( 	
\Spec(A 	
) 	
, 	
\mathcal{O}_{\Spec(A)})$. 	

 	
Next 	
, 	
note 	
that 	
$ 	
\mathcal{K}^\bullet 	
= 	
\epsilon^*\widetilde{K}^\bullet$ 	

 	
where 	
$ 	
\epsilon$ 	
is 	
as 	
in 	
Derived 	
Categories 	
of 	
Spaces 	
, 	
Lemma 	

 	
\ref{spaces 	
- 	
perfect 	
- 	
lemma 	
- 	
derived 	
- 	
quasi 	
- 	
coherent 	
- 	
small 	
- 	
etale 	
- 	
site 	
} 	

 	
whence 	
$ 	
\mathcal{K}^\bullet$ 	
is 	
K 	
- 	
flat 	
by 	

 	
Cohomology 	
on 	
Sites 	
, 	
Lemma 	
\ref{sites 	
- 	
cohomology 	
- 	
lemma 	
- 	
pullback 	
- 	
K 	
- 	
flat 	
- 	
points 	
} 	

 	
and 	
the 	
fact 	
that 	
the 	
\'etale 	
site 	
of 	
a 	
scheme 	
has 	
enough 	
points 	

 	
( 	
\'Etale 	
Cohomology 	
, 	
Remarks 	
\ref{etale 	
- 	
cohomology 	
- 	
remarks 	
- 	
enough 	
- 	
points 	
} 	
) 	
. 	


 	
\medskip\noindent 	

 	
By 	
the 	
claim 	
we 	
see 	
that 	

 	
$ 	
La_X^*K 	
= 	
a_X^*\mathcal{K}^\bullet$ 	
and 	

 	
$ 	
L\pi_X^*K 	
= 	
\pi_X^*\mathcal{K}^\bullet$. 	

 	
Since 	
the 	
first 	
part 	
of 	
the 	
proof 	
shows 	
that 	
the 	
pullback 	

 	
$ 	
a_X^*\mathcal{K}^n$ 	
of 	
the 	
quasi 	
- 	
coherent 	
module 	

 	
is 	
acyclic 	
for 	
$ 	
\epsilon_{X 	
, 	
* 	
} 	
$ 	
, 	
resp.\ 	
$ 	
a_{X 	
, 	
* 	
} 	
$ 	
, 	
surely 	
the 	
proof 	
is 	
done 	

 	
by 	
Leray 	
's 	
acyclicity 	
lemma 	
? 	
Actually 	
... 	
, 	
no 	
because 	
Leray 	
's 	

 	
acyclicity 	
lemma 	
only 	
applies 	
to 	
bounded 	
below 	
complexes 	
. 	

 	
However 	
, 	
in 	
the 	
next 	
paragraph 	
we 	
will 	
show 	
the 	
result 	
does 	
follow 	

 	
from 	
the 	
bounded 	
below 	
case 	
because 	
our 	
complex 	
is 	
the 	
derived 	
limit 	

 	
of 	
bounded 	
below 	
complexes 	
of 	
quasi 	
- 	
coherent 	
modules 	
. 	


 	
\medskip\noindent 	

 	
The 	
cohomology 	
sheaves 	
of 	

 	
$ 	
\pi_X^*\mathcal{K}^\bullet$ 	
and 	
$ 	
a_X^*\mathcal{K}^\bullet$ 	

 	
have 	
vanishing 	
higher 	
cohomology 	

 	
groups 	
over 	
affine 	
objects 	
of 	
$ 	
( 	
\textit{Spaces}/X)_\etale$ 	
by 	

 	
Lemma 	
\ref{lemma 	
- 	
vanishing 	
- 	
adequate}. 	

 	
Therefore 	
we 	
have 	

 	
$ 	
$ 	

 	
L\pi_X^*K 	
= 	
R\lim 	
\tau_{\geq 	
-n}(L\pi_X^*K 	
) 	

 	
\quad\text{and}\quad 	

 	
La_X^*K 	
= 	
R\lim 	
\tau_{\geq 	
-n}(La_X^*K 	
) 	

 	
$ 	
$ 	

 	
by 	
Cohomology 	
on 	
Sites 	
, 	
Lemma 	
\ref{sites 	
- 	
cohomology 	
- 	
lemma 	
- 	
is 	
- 	
limit 	
- 	
dimension}. 	


 	
\medskip\noindent 	

 	
Proof 	
of 	
$ 	
L\pi_X^*K 	
= 	
R\epsilon_{X 	
, 	
* 	
} 	
( 	
La_X^*\mathcal{F})$. 	

 	
By 	
the 	
above 	
we 	
have 	

 	
$ 	
$ 	

 	
R\epsilon_{X 	
, 	
* 	
} 	
La_X^*K 	
= 	

 	
R\lim 	
R\epsilon_{X 	
, 	
* 	
} 	
( 	
\tau_{\geq 	
-n}(La_X^*K 	
) 	
) 	

 	
$ 	
$ 	

 	
by 	
Cohomology 	
on 	
Sites 	
, 	
Lemma 	

 	
\ref{sites 	
- 	
cohomology 	
- 	
lemma 	
- 	
Rf 	
- 	
commutes 	
- 	
with 	
- 	
Rlim}. 	

 	
Note 	
that 	
$ 	
\tau_{\geq 	
-n}(La_X^*K)$ 	
is 	
represented 	
by 	

 	
$ 	
\tau_{\geq 	
-n}(a_X^*\mathcal{K}^\bullet)$ 	
which 	
may 	
not 	
be 	
the 	

 	
same 	
as 	
$ 	
a_X^*(\tau_{\geq 	
-n}\mathcal{K}^\bullet)$. 	

 	
But 	
clearly 	
the 	
systems 	

 	
$ 	
$ 	

 	
\{\tau_{\geq 	
-n}(a_X^*\mathcal{K}^\bullet)\}_{n 	
\geq 	
1 	
} 	

 	
\quad\text{and}\quad 	

 	
\{a_X^*(\tau_{\geq 	
-n}\mathcal{K}^\bullet)\}_{n 	
\geq 	
1 	
} 	

 	
$ 	
$ 	

 	
are 	
isomorphic 	
as 	
pro 	
- 	
systems 	
. 	

 	
By 	
Leray 	
's 	
acyclicity 	
lemma 	

 	
( 	
Derived 	
Categories 	
, 	
Lemma 	
\ref{derived 	
- 	
lemma 	
- 	
leray 	
- 	
acyclicity 	
} 	
) 	

 	
and 	
the 	
first 	
part 	
of 	
the 	
lemma 	
we 	
see 	
that 	

 	
$ 	
$ 	

 	
R\epsilon_{X 	
, 	
* 	
} 	
( 	
a_X^*(\tau_{\geq 	
-n}\mathcal{K}^\bullet 	
) 	
) 	
= 	

 	
\pi_X^*(\tau_{\geq 	
-n}\mathcal{K}^\bullet 	
) 	

 	
$ 	
$ 	

 	
Then 	
we 	
can 	
use 	
that 	
the 	
systems 	

 	
$ 	
$ 	

 	
\{\tau_{\geq 	
-n}(\pi_X^*\mathcal{K}^\bullet)\}_{n 	
\geq 	
1 	
} 	

 	
\quad\text{and}\quad 	

 	
\{\pi_X^*(\tau_{\geq 	
-n}\mathcal{K}^\bullet)\}_{n 	
\geq 	
1 	
} 	

 	
$ 	
$ 	

 	
are 	
isomorphic 	
as 	
pro 	
- 	
systems 	
. 	
Finally 	
, 	
we 	
put 	
everything 	
together 	
as 	
follows 	

 	
\begin{align 	
* 	
} 	

 	
R\epsilon_{X 	
, 	
* 	
} 	
La_X^*K 	

 	
& 	
= 	

 	
R\epsilon_{X 	
, 	
* 	
} 	
( 	
R\lim 	
\tau_{\geq 	
-n}(La_X^*K 	
) 	
) 	
\\ 	

 	
& 	
= 	

 	
R\lim 	
R\epsilon_{X 	
, 	
* 	
} 	
( 	
\tau_{\geq 	
-n}(La_X^*K 	
) 	
) 	
\\ 	

 	
& 	
= 	

 	
R\lim 	
R\epsilon_{X 	
, 	
* 	
} 	
( 	
\tau_{\geq 	
-n}(a_X^*\mathcal{K}^\bullet 	
) 	
) 	
\\ 	

 	
& 	
= 	

 	
R\lim 	
R\epsilon_{X 	
, 	
* 	
} 	
( 	
a_X^*(\tau_{\geq 	
-n}\mathcal{K}^\bullet 	
) 	
) 	
\\ 	

 	
& 	
= 	

 	
R\lim 	
\pi_X^*(\tau_{\geq 	
-n}\mathcal{K}^\bullet 	
) 	
\\ 	

 	
& 	
= 	

 	
R\lim 	
\tau_{\geq 	
-n}(\pi_X^*\mathcal{K}^\bullet 	
) 	
\\ 	

 	
& 	
= 	

 	
R\lim 	
\tau_{\geq 	
-n}(L\pi_X^*K 	
) 	
\\ 	

 	
& 	
= 	

 	
L\pi_X^*K 	

 	
\end{align 	
* 	
} 	

 	
Here 	
in 	
equalities 	
four 	
and 	
six 	
we 	
have 	
used 	
that 	
isomorphic 	

 	
pro 	
- 	
systems 	
have 	
the 	
same 	
$ 	
R\lim$ 	
( 	
small 	
detail 	
omitted 	
) 	
. 	

 	
You 	
can 	
avoid 	
this 	
step 	
by 	
using 	
more 	
about 	
cohomology 	
of 	
the 	
terms 	

 	
of 	
the 	
complex 	
$ 	
\tau_{\geq 	
-n}a_X^*\mathcal{K}^\bullet$ 	
proved 	

 	
in 	
Lemma 	
\ref{lemma 	
- 	
vanishing 	
- 	
adequate 	
} 	
as 	
this 	
will 	
prove 	

 	
directly 	
that 	
$ 	
R\epsilon_{X 	
, 	
* 	
} 	
( 	
\tau_{\geq 	
-n}(a_X^*\mathcal{K}^\bullet 	
) 	
) 	
= 	

 	
\tau_{\geq 	
-n}(\pi_X^*\mathcal{K}^\bullet)$. 	


 	
\medskip\noindent 	

 	
The 	
equality 	
$ 	
K 	
= 	
Ra_{X 	
, 	
* 	
} 	
( 	
La_X^*\mathcal{F})$ 	
is 	

 	
proved 	
in 	
exactly 	
the 	
same 	
way 	
using 	
in 	
the 	
final 	
step 	
that 	

 	
$ 	
K 	
= 	
R\lim 	
\tau_{\geq 	
-n}K$ 	
by 	

 	
Derived 	
Categories 	
of 	
Spaces 	
, 	
Lemma 	
\ref{spaces 	
- 	
perfect 	
- 	
lemma 	
- 	
nice 	
- 	
K 	
- 	
injective}. 	

 	
\end{proof 	
} 	









 	
\section{Comparing 	
ph 	
and 	
\'etale 	
topologies 	
} 	

 	
\label{section 	
- 	
ph 	
- 	
etale 	
} 	


 	
\noindent 	

 	
This 	
section 	
is 	
the 	
analogue 	
of 	

 	
\'Etale 	
Cohomology 	
, 	
Section 	
\ref{etale 	
- 	
cohomology 	
- 	
section 	
- 	
ph 	
- 	
etale}. 	


 	
\medskip\noindent 	

 	
Let 	
$ 	
S$ 	
be 	
a 	
scheme 	TYPE
. 	TYPE
Let 	
$ 	
X$ 	
be 	
an 	
algebraic 	TYPE
space 	
over 	
$ 	
S$. 	

 	
On 	
the 	
category 	
$ 	
\textit{Spaces}/X$ 	
we 	
consider 	
the 	
ph 	

 	
and 	
\'etale 	
topologies 	
. 	
The 	
identity 	
functor 	

 	
$ 	
( 	
\textit{Spaces}/X)_\etale 	
\to 	
( 	
\textit{Spaces}/X)_{ph}$ 	

 	
is 	
continuous 	
as 	
every 	
\'etale 	
covering 	
is 	
a 	
ph 	
covering 	
by 	

 	
Topologies 	
on 	
Spaces 	
, 	
Lemma 	

 	
\ref{spaces 	
- 	
topologies 	
- 	
lemma 	
- 	
zariski 	
- 	
etale 	
- 	
smooth 	
- 	
syntomic 	
- 	
fppf 	
- 	
ph}. 	

 	
Hence 	
it 	
defines 	
a 	
morphism 	
of 	
sites 	

 	
$ 	
$ 	

 	
\epsilon_X 	
: 	

 	
( 	
\textit{Spaces}/X)_{ph 	
} 	
\longrightarrow 	
( 	
\textit{Spaces}/X)_\etale 	

 	
$ 	
$ 	

 	
by 	
an 	
application 	
of 	
Sites 	
, 	
Proposition 	
\ref{sites 	
- 	
proposition 	
- 	
get 	
- 	
morphism}. 	

 	
Please 	
note 	
that 	
$ 	
\epsilon_{X 	
, 	
* 	
} 	
$ 	
is 	
the 	
identity 	
functor 	
on 	
underlying 	

 	
presheaves 	
and 	
that 	
$ 	
\epsilon_X^{-1}$ 	
associates 	
to 	
an 	
\'etale 	
sheaf 	
the 	

 	
ph 	
sheafification 	
. 	

 	
Consider 	
the 	
morphism 	
of 	
sites 	

 	
$ 	
$ 	

 	
\pi_X 	
: 	
( 	
\textit{Spaces}/X)_\etale 	
\longrightarrow 	
X_{spaces 	
, 	
\etale 	
} 	

 	
$ 	
$ 	

 	
comparing 	
big 	
and 	
small 	
\'etale 	
sites 	
, 	
see 	
Section 	
\ref{section 	
- 	
compare}. 	

 	
The 	
composition 	
determines 	
a 	
morphism 	
of 	
sites 	

 	
$ 	
$ 	

 	
a_X 	
= 	
\pi_X 	
\circ 	
\epsilon_X 	
: 	

 	
( 	
\textit{Spaces}/X)_{ph 	
} 	

 	
\longrightarrow 	

 	
X_{spaces 	
, 	
\etale 	
} 	

 	
$ 	
$ 	

 	
If 	
$ 	
\mathcal{H}$ 	
is 	
an 	
abelian 	
sheaf 	
on 	
$ 	
( 	
\textit{Spaces}/X)_{ph}$ 	
, 	

 	
then 	
we 	
will 	
write 	
$ 	
H^n_{ph}(U 	
, 	
\mathcal{H})$ 	
for 	
the 	
cohomology 	

 	
of 	
$ 	
\mathcal{H}$ 	
over 	
an 	
object 	
$ 	
U$ 	
of 	
$ 	
( 	
\textit{Spaces}/X)_{ph}$. 	


 	
\begin{lemma 	
} 	

 	
\label{lemma 	
- 	
comparison 	
- 	
ph 	
- 	
etale 	
} 	

 	
Let 	
$ 	
S$ 	
be 	
a 	
scheme 	TYPE
. 	TYPE
Let 	
$ 	
X$ 	
be 	
an 	
algebraic 	TYPE
space 	
over 	
$ 	
S$. 	

 	
\begin{enumerate 	
} 	

 	
\item 	
For 	
$ 	
\mathcal{F 	
} 	
\in 	
\Sh(X_\etale)$ 	
we 	
have 	

 	
$ 	
\epsilon_{X 	
, 	
* 	
} 	
a_X^{-1}\mathcal{F 	
} 	
= 	
\pi_X^{-1}\mathcal{F}$ 	

 	
and 	
$ 	
a_{X 	
, 	
* 	
} 	
a_X^{-1}\mathcal{F 	
} 	
= 	
\mathcal{F}$. 	

 	
\item 	
For 	
$ 	
\mathcal{F 	
} 	
\in 	
\textit{Ab}(X_\etale)$ 	
torsion 	
we 	
have 	

 	
$ 	
R^i\epsilon_{X 	
, 	
* 	
} 	
( 	
a_X^{-1}\mathcal{F 	
} 	
) 	
= 	
0 	
$ 	
for 	
$ 	
i 	
> 	
0$. 	

 	
\end{enumerate 	
} 	

 	
\end{lemma 	
} 	


 	
\begin{proof 	
} 	

 	
We 	
have 	
$ 	
a_X^{-1}\mathcal{F 	
} 	
= 	
\epsilon_X^{-1 	
} 	
\pi_X^{-1}\mathcal{F}$. 	

 	
By 	
Lemma 	
\ref{lemma 	
- 	
describe 	
- 	
pullback 	
} 	
the 	
\'etale 	
sheaf 	

 	
$ 	
\pi_X^{-1}\mathcal{F}$ 	
is 	
a 	
sheaf 	
for 	
the 	
ph 	
topology 	

 	
and 	
therefore 	
is 	
equal 	
to 	
$ 	
a_X^{-1}\mathcal{F}$ 	
( 	
as 	
pulling 	

 	
back 	
by 	
$ 	
\epsilon_X$ 	
is 	
given 	
by 	
ph 	
sheafification 	
) 	
. 	

 	
Recall 	
moreover 	
that 	
$ 	
\epsilon_{X 	
, 	
* 	
} 	
$ 	
is 	
the 	
identity 	

 	
on 	
underlying 	
presheaves 	
. 	

 	
Now 	
part 	
( 	
1 	
) 	
is 	
immediate 	
from 	
the 	
explicit 	
description 	
of 	
$ 	
\pi_X^{-1}$ 	

 	
in 	
Lemma 	
\ref{lemma 	
- 	
describe 	
- 	
pullback}. 	


 	
\medskip\noindent 	

 	
We 	
will 	
prove 	
part 	
( 	
2 	
) 	
by 	
reducing 	
it 	
to 	
the 	
case 	
of 	
schemes 	
-- 	

 	
see 	
part 	
( 	
1 	
) 	
of 	

 	
\'Etale 	
Cohomology 	
, 	
Lemma 	
\ref{etale 	
- 	
cohomology 	
- 	
lemma 	
- 	
V 	
- 	
C 	
- 	
all 	
- 	
n 	
- 	
etale 	
- 	
ph}. 	

 	
This 	
will 	
`` 	
clearly 	
work 	
'' 	
as 	
every 	
algebraic 	
space 	
is 	

 	
\'etale 	
locally 	
a 	
scheme 	
. 	
The 	
details 	
are 	
given 	
below 	
but 	
we 	
urge 	

 	
the 	
reader 	
to 	
skip 	
the 	
proof 	
. 	


 	
\medskip\noindent 	

 	
For 	
an 	
abelian 	
sheaf 	
$ 	
\mathcal{H}$ 	
on 	
$ 	
( 	
\textit{Spaces}/X)_{ph}$ 	
the 	

 	
higher 	
direct 	
image 	
$ 	
R^p\epsilon_{X 	
, 	
* 	
} 	
\mathcal{H}$ 	
is 	
the 	
sheaf 	

 	
associated 	
to 	
the 	
presheaf 	
$ 	
U 	
\mapsto 	
H^p_{ph}(U 	
, 	
\mathcal{H})$ 	

 	
on 	
$ 	
( 	
\textit{Spaces}/X)_\etale$. 	
See 	

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

 	
Since 	
every 	
object 	
of 	
$ 	
( 	
\textit{Spaces}/X)_\etale$ 	
has 	
a 	
covering 	

 	
by 	
schemes 	
, 	
it 	
suffices 	
to 	
prove 	
that 	
given 	
$ 	
U 	
/ 	
X$ 	
a 	
scheme 	
and 	

 	
$ 	
\xi 	
\in 	
H^p_{ph}(U 	
, 	
a_X^{-1}\mathcal{F})$ 	
we 	
can 	
find 	

 	
an 	
\'etale 	
covering 	
$ 	
\{U_i 	
\to 	
U\}$ 	
such 	
that 	
$ 	
\xi$ 	

 	
restricts 	
to 	
zero 	
on 	
$ 	
U_i$. 	
We 	
have 	

 	
\begin{align 	
* 	
} 	

 	
H^p_{ph}(U 	
, 	
a_X^{-1}\mathcal{F 	
} 	
) 	

 	
& 	
= 	

 	
H^p((\textit{Spaces}/U)_{ph 	
} 	
, 	
( 	
a_X^{-1}\mathcal{F})|_{\textit{Spaces}/U 	
} 	
) 	
\\ 	

 	
& 	
= 	

 	
H^p((\Sch 	
/ 	
U)_{ph 	
} 	
, 	
( 	
a_X^{-1}\mathcal{F})|_{\Sch 	
/ 	
U 	
} 	
) 	

 	
\end{align 	
* 	
} 	

 	
where 	
the 	
second 	
identification 	
is 	

 	
Lemma 	
\ref{lemma 	
- 	
compare 	
- 	
cohomology 	
- 	
other 	
- 	
topologies 	
} 	

 	
and 	
the 	
first 	
is 	
a 	
general 	
fact 	
about 	
restriction 	

 	
( 	
Cohomology 	
on 	
Sites 	
, 	
Lemma 	
\ref{sites 	
- 	
cohomology 	
- 	
lemma 	
- 	
cohomology 	
- 	
of 	
- 	
open 	
} 	
) 	
. 	

 	
Looking 	
at 	
the 	
first 	
paragraph 	
and 	
the 	
corresponding 	
result 	
in 	
the 	

 	
case 	
of 	
schemes 	
( 	
\'Etale 	
Cohomology 	
, 	
Lemma 	

 	
\ref{etale 	
- 	
cohomology 	
- 	
lemma 	
- 	
describe 	
- 	
pullback 	
- 	
pi 	
- 	
ph 	
} 	
) 	

 	
we 	
conclude 	
that 	
the 	
sheaf 	
$ 	
( 	
a_X^{-1}\mathcal{F})|_{\Sch 	
/ 	
U}$ 	

 	
matches 	
the 	
pullback 	
by 	
the 	
`` 	
schemes 	
version 	
of 	
$ 	
a_U$ 	
'' 	
. 	

 	
Therefore 	
we 	
can 	
find 	
an 	
\'etale 	
covering 	

 	
$ 	
\{U_i 	
\to 	
U\}$ 	
such 	
that 	
our 	
class 	
dies 	
in 	

 	
$ 	
H^p((\Sch 	
/ 	
U_i)_{ph 	
} 	
, 	
( 	
a_X^{-1}\mathcal{F})|_{\Sch 	
/ 	
U_i})$ 	

 	
for 	
each 	
$ 	
i$ 	
, 	
see 	

 	
\'Etale 	
Cohomology 	
, 	
Lemma 	
\ref{etale 	
- 	
cohomology 	
- 	
lemma 	
- 	
V 	
- 	
C 	
- 	
all 	
- 	
n 	
- 	
etale 	
- 	
ph 	
} 	

 	
( 	
the 	
precise 	
statement 	
one 	
should 	
use 	
here 	
is 	
that 	
$ 	
V_n$ 	
holds 	
for 	
all 	
$ 	
n$ 	

 	
which 	
is 	
the 	
statement 	
of 	
part 	
( 	
2 	
) 	
for 	
the 	
case 	
of 	
schemes 	
) 	
. 	

 	
Transporting 	
back 	
( 	
using 	
the 	
same 	
formulas 	
as 	
above 	
but 	
now 	
for 	

 	
$ 	
U_i$ 	
) 	
we 	
conclude 	
$ 	
\xi$ 	
restricts 	
to 	
zero 	
over 	
$ 	
U_i$ 	
as 	
desired 	
. 	

 	
\end{proof 	
} 	


 	
\noindent 	

 	
The 	
hard 	
work 	
done 	
in 	
the 	
case 	
of 	
schemes 	
now 	
tells 	
us 	
that 	

 	
\'etale 	
and 	
ph 	
cohomology 	
agree 	
for 	
torsion 	
abelian 	
sheaves 	
coming 	
from 	
the 	

 	
small 	
\'etale 	
site 	
. 	


 	
\begin{lemma 	
} 	

 	
\label{lemma 	
- 	
cohomological 	
- 	
descent 	
- 	
etale 	
- 	
ph 	
} 	

 	
Let 	
$ 	
S$ 	
be 	
a 	
scheme 	TYPE
. 	TYPE
Let 	
$ 	
X$ 	
be 	
an 	
algebraic 	TYPE
space 	
over 	
$ 	
S$. 	

 	
For 	
$ 	
K 	
\in 	
D^+(X_\etale)$ 	
with 	
torsion 	
cohomology 	
sheaves 	
the 	
maps 	

 	
$ 	
$ 	

 	
\pi_X^{-1}K 	
\longrightarrow 	
R\epsilon_{X 	
, 	
* 	
} 	
a_X^{-1}K 	

 	
\quad\text{and}\quad 	

 	
K 	
\longrightarrow 	
Ra_{X 	
, 	
* 	
} 	
a_X^{-1}K 	

 	
$ 	
$ 	

 	
are 	
isomorphisms 	
with 	

 	
$ 	
a_X 	
: 	
\Sh((\textit{Spaces}/X)_{ph 	
} 	
) 	
\to 	
\Sh(X_\etale)$ 	
as 	
above 	
. 	

 	
\end{lemma 	
} 	


 	
\begin{proof 	
} 	

 	
We 	
only 	
prove 	
the 	
second 	
statement 	
; 	
the 	
first 	
is 	
easier 	
and 	
proved 	
in 	
exactly 	

 	
the 	
same 	
manner 	
. 	
There 	
is 	
a 	
reduction 	
to 	
the 	
case 	
where 	

 	
$ 	
K$ 	
is 	
given 	
by 	
a 	
single 	
torsion 	
abelian 	
sheaf 	
. 	
Namely 	
, 	
represent 	
$ 	
K$ 	

 	
by 	
a 	
bounded 	
below 	
complex 	
$ 	
\mathcal{F}^\bullet$ 	
of 	
torsion 	

 	
abelian 	
sheaves 	
. 	
This 	
is 	
possible 	
by 	
Cohomology 	
on 	
Sites 	
, 	
Lemma 	

 	
\ref{sites 	
- 	
cohomology 	
- 	
lemma 	
- 	
torsion}. 	

 	
By 	
the 	
case 	
of 	
a 	
sheaf 	
we 	
see 	
that 	

 	
$ 	
\mathcal{F}^n 	
= 	
a_{X 	
, 	
* 	
} 	
a_X^{-1 	
} 	
\mathcal{F}^n$ 	

 	
and 	
that 	
the 	
sheaves 	
$ 	
R^qa_{X 	
, 	
* 	
} 	
a_X^{-1}\mathcal{F}^n$ 	

 	
are 	
zero 	
for 	
$ 	
q 	
> 	
0$. 	
By 	
Leray 	
's 	
acyclicity 	
lemma 	

 	
( 	
Derived 	
Categories 	
, 	
Lemma 	
\ref{derived 	
- 	
lemma 	
- 	
leray 	
- 	
acyclicity 	
} 	
) 	

 	
applied 	
to 	
$ 	
a_X^{-1}\mathcal{F}^\bullet$ 	

 	
and 	
the 	
functor 	
$ 	
a_{X 	
, 	
* 	
} 	
$ 	
we 	
conclude 	
. 	
From 	
now 	
on 	
assume 	

 	
$ 	
K 	
= 	
\mathcal{F}$ 	
where 	
$ 	
\mathcal{F}$ 	
is 	
a 	
torsion 	
abelian 	
sheaf 	
. 	


 	
\medskip\noindent 	

 	
By 	
Lemma 	
\ref{lemma 	
- 	
comparison 	
- 	
ph 	
- 	
etale 	
} 	
we 	
have 	

 	
$ 	
a_{X 	
, 	
* 	
} 	
a_X^{-1}\mathcal{F 	
} 	
= 	
\mathcal{F}$. 	
Thus 	
it 	
suffices 	
to 	
show 	
that 	

 	
$ 	
R^qa_{X 	
, 	
* 	
} 	
a_X^{-1}\mathcal{F 	
} 	
= 	
0 	
$ 	
for 	
$ 	
q 	
> 	
0$. 	

 	
For 	
this 	
we 	
can 	
use 	
$ 	
a_X 	
= 	
\epsilon_X 	
\circ 	
\pi_X$ 	
and 	

 	
the 	
Leray 	
spectral 	
sequence 	

 	
( 	
Cohomology 	
on 	
Sites 	
, 	
Lemma 	
\ref{sites 	
- 	
cohomology 	
- 	
lemma 	
- 	
relative 	
- 	
Leray 	
} 	
) 	
. 	

 	
By 	
Lemma 	
\ref{lemma 	
- 	
comparison 	
- 	
ph 	
- 	
etale 	
} 	

 	
we 	
have 	
$ 	
R^i\epsilon_{X 	
, 	
* 	
} 	
( 	
a_X^{-1}\mathcal{F 	
} 	
) 	
= 	
0 	
$ 	
for 	
$ 	
i 	
> 	
0$. 	

 	
We 	
have 	

 	
$ 	
\epsilon_{X 	
, 	
* 	
} 	
a_X^{-1}\mathcal{F 	
} 	
= 	
\pi_X^{-1}\mathcal{F}$ 	

 	
and 	
by 	
Lemma 	
\ref{lemma 	
- 	
cohomological 	
- 	
descent 	
- 	
etale 	
} 	
we 	
have 	

 	
$ 	
R^j\pi_{X 	
, 	
* 	
} 	
( 	
\pi_X^{-1}\mathcal{F 	
} 	
) 	
= 	
0 	
$ 	
for 	
$ 	
j 	
> 	
0$. 	

 	
This 	
concludes 	
the 	
proof 	
. 	

 	
\end{proof 	
} 	


 	
\begin{lemma 	
} 	

 	
\label{lemma 	
- 	
compare 	
- 	
cohomology 	
- 	
etale 	
- 	
ph 	
} 	

 	
Let 	
$ 	
S$ 	
be 	
a 	
scheme 	TYPE
and 	
let 	
$ 	
X$ 	
be 	
an 	
algebraic 	TYPE
space 	
over 	
$ 	
S$. 	

 	
With 	
$ 	
a_X 	
: 	
\Sh((\textit{Spaces}/X)_{ph 	
} 	
) 	
\to 	
\Sh(X_\etale)$ 	

 	
as 	
above 	
: 	

 	
\begin{enumerate 	
} 	

 	
\item 	
$ 	
H^q(X_\etale 	
, 	
\mathcal{F 	
} 	
) 	
= 	
H^q_{ph}(X 	
, 	
a_X^{-1}\mathcal{F})$ 	

 	
for 	
a 	
torsion 	
abelian 	
sheaf 	
$ 	
\mathcal{F}$ 	
on 	
$ 	
X_\etale$ 	
, 	

 	
\item 	
$ 	
H^q(X_\etale 	
, 	
K 	
) 	
= 	
H^q_{ph}(X 	
, 	
a_X^{-1}K)$ 	
for 	
$ 	
K 	
\in 	
D^+(X_\etale)$ 	

 	
with 	
torsion 	
cohomology 	
sheaves 	

 	
\end{enumerate 	
} 	

 	
Example 	
: 	
if 	
$ 	
A$ 	
is 	
a 	
torsion 	
abelian 	
group 	
, 	
then 	

 	
$ 	
H^q_\etale(X 	
, 	
\underline{A 	
} 	
) 	
= 	
H^q_{ph}(X 	
, 	
\underline{A})$. 	

 	
\end{lemma 	
} 	


 	
\begin{proof 	
} 	

 	
This 	
follows 	
from 	
Lemma 	
\ref{lemma 	
- 	
cohomological 	
- 	
descent 	
- 	
etale 	
- 	
ph 	
} 	

 	
by 	
Cohomology 	
on 	
Sites 	
, 	
Remark 	
\ref{sites 	
- 	
cohomology 	
- 	
remark 	
- 	
before 	
- 	
Leray}. 	

 	
\end{proof 	
} 	


 	
\begin{lemma 	
} 	

 	
\label{lemma 	
- 	
push 	
- 	
pull 	
- 	
ph 	
- 	
etale 	
} 	

 	
Let 	
$ 	
S$ 	
be 	
a 	
scheme 	TYPE
. 	TYPE

 	
Let 	
$ 	
f 	
: 	
X 	
\to 	
Y$ 	
be 	
a 	
morphism 	
of 	
algebraic 	
spaces 	
over 	
$ 	
S$. 	

 	
Then 	
there 	
are 	
commutative 	
diagrams 	
of 	
topoi 	

 	
$ 	
$ 	

 	
\xymatrix 	
{ 	

 	
\Sh((\textit{Spaces}/X)_{ph 	
} 	
) 	
\ar[rr]_{f_{big 	
, 	
ph 	
} 	
} 	
\ar[d]_{\epsilon_X 	
} 	
& 	
& 	

 	
\Sh((\textit{Spaces}/Y)_{ph 	
} 	
) 	
\ar[d]^{\epsilon_Y 	
} 	
\\ 	

 	
\Sh((\textit{Spaces}/X)_\etale 	
) 	
\ar[rr]^{f_{big 	
, 	
\etale 	
} 	
} 	
& 	
& 	

 	
\Sh((\textit{Spaces}/Y)_\etale 	
) 	

 	
} 	

 	
$ 	
$ 	

 	
and 	

 	
$ 	
$ 	

 	
\xymatrix 	
{ 	

 	
\Sh((\textit{Spaces}/X)_{ph 	
} 	
) 	
\ar[rr]_{f_{big 	
, 	
ph 	
} 	
} 	
\ar[d]_{a_X 	
} 	
& 	
& 	

 	
\Sh((\textit{Spaces}/Y)_{ph 	
} 	
) 	
\ar[d]^{a_Y 	
} 	
\\ 	

 	
\Sh(X_\etale 	
) 	
\ar[rr]^{f_{small 	
} 	
} 	
& 	
& 	

 	
\Sh(Y_\etale 	
) 	

 	
} 	

 	
$ 	
$ 	

 	
with 	
$ 	
a_X 	
= 	
\pi_X 	
\circ 	
\epsilon_X$ 	
and 	
$ 	
a_Y 	
= 	
\pi_X 	
\circ 	
\epsilon_X$. 	

 	
\end{lemma 	
} 	


 	
\begin{proof 	
} 	

 	
This 	
follows 	
immediately 	
from 	
working 	
out 	
the 	
definitions 	
of 	
the 	

 	
morphisms 	
involved 	
, 	
see 	

 	
Topologies 	
on 	
Spaces 	
, 	
Section 	
\ref{spaces 	
- 	
topologies 	
- 	
section 	
- 	
ph 	
} 	

 	
and 	
Section 	
\ref{section 	
- 	
compare}. 	

 	
\end{proof 	
} 	


 	
\begin{lemma 	
} 	

 	
\label{lemma 	
- 	
proper 	
- 	
push 	
- 	
pull 	
- 	
ph 	
- 	
etale 	
} 	

 	
In 	
Lemma 	
\ref{lemma 	
- 	
push 	
- 	
pull 	
- 	
ph 	
- 	
etale 	
} 	
if 	
$ 	
f$ 	
is 	
proper 	
, 	
then 	
we 	
have 	

 	
\begin{enumerate 	
} 	

 	
\item 	
$ 	
a_Y^{-1 	
} 	
\circ 	
f_{small 	
, 	
* 	
} 	
= 	
f_{big 	
, 	
ph 	
, 	
* 	
} 	
\circ 	
a_X^{-1}$ 	
, 	
and 	

 	
\item 	

 	
$ 	
a_Y^{-1}(Rf_{small 	
, 	
* 	
} 	
K 	
) 	
= 	
Rf_{big 	
, 	
ph 	
, 	
* 	
} 	
( 	
a_X^{-1}K)$ 	

 	
for 	
$ 	
K$ 	
in 	
$ 	
D^+(X_\etale)$ 	
with 	
torsion 	
cohomology 	
sheaves 	
. 	

 	
\end{enumerate 	
} 	

 	
\end{lemma 	
} 	


 	
\begin{proof 	
} 	

 	
Proof 	
of 	
( 	
1 	
) 	
. 	
You 	
can 	
prove 	
this 	
by 	
repeating 	
the 	
proof 	
of 	

 	
Lemma 	
\ref{lemma 	
- 	
compare 	
- 	
higher 	
- 	
direct 	
- 	
image 	
- 	
proper 	
} 	
part 	
( 	
1 	
) 	
; 	

 	
we 	
will 	
instead 	
deduce 	
the 	
result 	
from 	
this 	
. 	

 	
As 	
$ 	
\epsilon_{Y 	
, 	
* 	
} 	
$ 	
is 	
the 	
identity 	
functor 	
on 	
underlying 	
presheaves 	
, 	

 	
it 	
reflects 	
isomorphisms 	
. 	
Lemma 	
\ref{lemma 	
- 	
comparison 	
- 	
ph 	
- 	
etale 	
} 	

 	
shows 	
that 	
$ 	
\epsilon_{Y 	
, 	
* 	
} 	
\circ 	
a_Y^{-1 	
} 	
= 	
\pi_Y^{-1}$ 	

 	
and 	
similarly 	
for 	
$ 	
X$. 	
To 	
show 	
that 	
the 	
canonical 	
map 	

 	
$ 	
a_Y^{-1}f_{small 	
, 	
* 	
} 	
\mathcal{F 	
} 	
\to 	
f_{big 	
, 	
ph 	
, 	
* 	
} 	
a_X^{-1}\mathcal{F}$ 	

 	
is 	
an 	
isomorphism 	
, 	
it 	
suffices 	
to 	
show 	
that 	

 	
\begin{align 	
* 	
} 	

 	
\pi_Y^{-1}f_{small 	
, 	
* 	
} 	
\mathcal{F 	
} 	

 	
& 	
= 	

 	
\epsilon_{Y 	
, 	
* 	
} 	
a_Y^{-1}f_{small 	
, 	
* 	
} 	
\mathcal{F 	
} 	
\\ 	

 	
& 	
\to 	

 	
\epsilon_{Y 	
, 	
* 	
} 	
f_{big 	
, 	
ph 	
, 	
* 	
} 	
a_X^{-1}\mathcal{F 	
} 	
\\ 	

 	
& 	
= 	

 	
f_{big 	
, 	
\etale 	
, 	
* 	
} 	
\epsilon_{X 	
, 	
* 	
} 	
a_X^{-1}\mathcal{F 	
} 	
\\ 	

 	
& 	
= 	

 	
f_{big 	
, 	
\etale 	
, 	
* 	
} 	
\pi_X^{-1}\mathcal{F 	
} 	

 	
\end{align 	
* 	
} 	

 	
is 	
an 	
isomorphism 	
. 	
This 	
is 	
part 	

 	
( 	
1 	
) 	
of 	
Lemma 	
\ref{lemma 	
- 	
compare 	
- 	
higher 	
- 	
direct 	
- 	
image 	
- 	
proper}. 	


 	
\medskip\noindent 	

 	
To 	
see 	
( 	
2 	
) 	
we 	
use 	
that 	

 	
\begin{align 	
* 	
} 	

 	
R\epsilon_{Y 	
, 	
* 	
} 	
Rf_{big 	
, 	
ph 	
, 	
* 	
} 	
a_X^{-1}K 	

 	
& 	
= 	

 	
Rf_{big 	
, 	
\etale 	
, 	
* 	
} 	
R\epsilon_{X 	
, 	
* 	
} 	
a_X^{-1}K 	
\\ 	

 	
& 	
= 	

 	
Rf_{big 	
, 	
\etale 	
, 	
* 	
} 	
\pi_X^{-1}K 	
\\ 	

 	
& 	
= 	

 	
\pi_Y^{-1}Rf_{small 	
, 	
* 	
} 	
K 	
\\ 	

 	
& 	
= 	

 	
R\epsilon_{Y 	
, 	
* 	
} 	
a_Y^{-1}Rf_{small 	
, 	
* 	
} 	
K 	

 	
\end{align 	
* 	
} 	

 	
The 	
first 	
equality 	
by 	
the 	
commutative 	
diagram 	
in 	

 	
Lemma 	
\ref{lemma 	
- 	
push 	
- 	
pull 	
- 	
ph 	
- 	
etale 	
} 	

 	
and 	
Cohomology 	
on 	
Sites 	
, 	
Lemma 	

 	
\ref{sites 	
- 	
cohomology 	
- 	
lemma 	
- 	
derived 	
- 	
pushforward 	
- 	
composition}. 	

 	
Then 	
second 	
equality 	
is 	
Lemma 	
\ref{lemma 	
- 	
cohomological 	
- 	
descent 	
- 	
etale 	
- 	
ph}. 	

 	
The 	
third 	
is 	

 	
Lemma 	
\ref{lemma 	
- 	
compare 	
- 	
higher 	
- 	
direct 	
- 	
image 	
- 	
proper 	
} 	
part 	
( 	
2 	
) 	
. 	

 	
The 	
fourth 	
is 	
Lemma 	
\ref{lemma 	
- 	
cohomological 	
- 	
descent 	
- 	
etale 	
- 	
ph 	
} 	
again 	
. 	

 	
Thus 	
the 	
base 	
change 	
map 	

 	
$ 	
a_Y^{-1}(Rf_{small 	
, 	
* 	
} 	
K 	
) 	
\to 	
Rf_{big 	
, 	
ph 	
, 	
* 	
} 	
( 	
a_X^{-1}K)$ 	

 	
induces 	
an 	
isomorphism 	

 	
$ 	
$ 	

 	
R\epsilon_{Y 	
, 	
* 	
} 	
a_Y^{-1}Rf_{small 	
, 	
* 	
} 	
K 	
\to 	

 	
R\epsilon_{Y 	
, 	
* 	
} 	
Rf_{big 	
, 	
ph 	
, 	
* 	
} 	
a_X^{-1}K 	

 	
$ 	
$ 	

 	
The 	
proof 	
is 	
finished 	
by 	
the 	
following 	
remark 	
: 	
consider 	
a 	
map 	

 	
$ 	
\alpha 	
: 	
a_Y^{-1}L 	
\to 	
M$ 	
with 	
$ 	
L$ 	
in 	
$ 	
D^+(Y_\etale)$ 	

 	
having 	
torsion 	
cohomology 	
sheaves 	

 	
and 	
$ 	
M$ 	
in 	
$ 	
D^+((\textit{Spaces}/Y)_{ph})$. 	
If 	

 	
$ 	
R\epsilon_{Y 	
, 	
* 	
} 	
\alpha$ 	
is 	
an 	
isomorphism 	
, 	
then 	
$ 	
\alpha$ 	
is 	
an 	
isomorphism 	
. 	

 	
Namely 	
, 	
we 	
show 	
by 	
induction 	
on 	
$ 	
i$ 	
that 	
$ 	
H^i(\alpha)$ 	
is 	
an 	
isomorphism 	
. 	

 	
This 	
is 	
true 	
for 	
all 	
sufficiently 	
small 	
$ 	
i$. 	

 	
If 	
it 	
holds 	
for 	
$ 	
i 	
\leq 	
i_0 	
$ 	
, 	
then 	
we 	
see 	
that 	

 	
$ 	
R^j\epsilon_{Y 	
, 	
* 	
} 	
H^i(M 	
) 	
= 	
0 	
$ 	
for 	
$ 	
j 	
> 	
0 	
$ 	
and 	
$ 	
i 	
\leq 	
i_0 	
$ 	

 	
by 	
Lemma 	
\ref{lemma 	
- 	
comparison 	
- 	
ph 	
- 	
etale 	
} 	

 	
because 	
$ 	
H^i(M 	
) 	
= 	
a_Y^{-1}H^i(L)$ 	
in 	
this 	
range 	
. 	

 	
Hence 	
$ 	
\epsilon_{Y 	
, 	
* 	
} 	
H^{i_0 	
+ 	
1}(M 	
) 	
= 	
H^{i_0 	
+ 	
1}(R\epsilon_{Y 	
, 	
* 	
} 	
M)$ 	

 	
by 	
a 	
spectral 	
sequence 	
argument 	
. 	

 	
Thus 	
$ 	
\epsilon_{Y 	
, 	
* 	
} 	
H^{i_0 	
+ 	
1}(M 	
) 	
= 	
\pi_Y^{-1}H^{i_0 	
+ 	
1}(L 	
) 	
= 	

 	
\epsilon_{Y 	
, 	
* 	
} 	
a_Y^{-1}H^{i_0 	
+ 	
1}(L)$. 	

 	
This 	
implies 	
$ 	
H^{i_0 	
+ 	
1}(\alpha)$ 	
is 	
an 	
isomorphism 	

 	
( 	
because 	
$ 	
\epsilon_{Y 	
, 	
* 	
} 	
$ 	
reflects 	
isomorphisms 	
as 	
it 	
is 	
the 	

 	
identity 	
on 	
underlying 	
presheaves 	
) 	
as 	
desired 	
. 	

 	
\end{proof 	
} 	














 	
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