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 \input{preamble} % OK, start here. % \begin{document} \title{Sheaves on Algebraic Stacks} \maketitle \phantomsection \label{section-phantom} \tableofcontents \section{Introduction} \label{section-introduction} \noindent There is a myriad of ways to think about sheaves on algebraic stacks. In this chapter we discuss one approach, which is particularly well adapted to our foundations for algebraic stacks. Whenever we introduce a type of sheaves we will indicate the precise relationship with similar notions in the literature. The goal of this chapter is to state those results that are either obviously true or straightforward to prove and leave more intricate constructions till later. \medskip\noindent In fact, it turns out that to develop a fully fledged theory of constructible \'etale sheaves and/or an adequate discussion of derived categories of complexes $\mathcal{O}$-modules whose cohomology sheaves are quasi-coherent takes a significant amount of work, see \cite{olsson_sheaves}. We will return to this in Cohomology of Stacks, Section \ref{stacks-cohomology-section-introduction}. \medskip\noindent In the literature and in research papers on sheaves on algebraic stacks the lisse-\'etale site of an algebraic stack often plays a prominent role. However, it is a problematic beast, because it turns out that a morphism of algebraic stacks does not induce a morphism of lisse-\'etale topoi. We have therefore made the design decision to avoid any mention of the lisse-\'etale site as long as possible. Arguments that traditionally use the lisse-\'etale site will be replaced by an argument using a {\v C}ech covering in the site $\mathcal{X}_{smooth}$ defined below. \medskip\noindent Some of the notation, conventions and terminology in this chapter is awkward and may seem backwards to the more experienced reader. This is intentional. Please see Quot, Section \ref{quot-section-introduction} for an explanation. \section{Conventions} \label{section-conventions} \noindent The conventions we use in this chapter are the same as those in the chapter on algebraic stacks, see Algebraic Stacks, Section \ref{algebraic-section-conventions}. For convenience we repeat them here. \medskip\noindent We work in a suitable big fppf site $\Sch_{fppf}$ as in Topologies, Definition \ref{topologies-definition-big-fppf-site}. So, if not explicitly stated otherwise all schemes will be objects of $\Sch_{fppf}$. We record what changes if you change the big fppf site elsewhere (insert future reference here). \medskip\noindent We will always work relative to a base $S$ contained in $\Sch_{fppf}$. And we will then work with the big fppf site $(\Sch/S)_{fppf}$, see Topologies, Definition \ref{topologies-definition-big-small-fppf}. The absolute case can be recovered by taking $S = \Spec(\mathbf{Z})$. \section{Presheaves} \label{section-presheaves} \noindent In this section we define presheaves on categories fibred in groupoids over $(\Sch/S)_{fppf}$, but most of the discussion works for categories over any base category. This section also serves to introduce the notation we will use later on. \begin{definition} \label{definition-presheaves} Let $p : \mathcal{X} \to (\Sch/S)_{fppf}$ be a category fibred in groupoids. \begin{enumerate} \item A {\it presheaf on $\mathcal{X}$} is a presheaf on the underlying category of $\mathcal{X}$. \item A {\it morphism of presheaves on $\mathcal{X}$} is a morphism of presheaves on the underlying category of $\mathcal{X}$. \end{enumerate} We denote $\textit{PSh}(\mathcal{X})$ the category of presheaves on $\mathcal{X}$. \end{definition} \noindent This defines presheaves of sets. Of course we can also talk about presheaves of pointed sets, abelian groups, groups, monoids, rings, modules over a fixed ring, and lie algebras over a fixed field, etc. The category of {\it abelian presheaves}, i.e., presheaves of abelian groups, is denoted $\textit{PAb}(\mathcal{X})$. \medskip\noindent Let $f : \mathcal{X} \to \mathcal{Y}$ be a $1$-morphism of categories fibred in groupoids over $(\Sch/S)_{fppf}$. Recall that this means just that $f$ is a functor over $(\Sch/S)_{fppf}$. The material in Sites, Section \ref{sites-section-more-functoriality-PSh} provides us with a pair of adjoint functors\footnote{These functors will be denoted $f^{-1}$ and $f_*$ after Lemma \ref{lemma-functoriality-sheaves} has been proved.} \begin{equation} \label{equation-pushforward-pullback} f^p : \textit{PSh}(\mathcal{Y}) \longrightarrow \textit{PSh}(\mathcal{X}) \quad\text{and}\quad {}_pf : \textit{PSh}(\mathcal{X}) \longrightarrow \textit{PSh}(\mathcal{Y}). \end{equation} The adjointness is $$\Mor_{\textit{PSh}(\mathcal{X})}(f^p\mathcal{G}, \mathcal{F}) = \Mor_{\textit{PSh}(\mathcal{Y})}(\mathcal{G}, {}_pf\mathcal{F})$$ where $\mathcal{F} \in \Ob(\textit{PSh}(\mathcal{X}))$ and $\mathcal{G} \in \Ob(\textit{PSh}(\mathcal{Y}))$. We call $f^p\mathcal{G}$ the {\it pullback} of $\mathcal{G}$. It follows from the definitions that $$f^p\mathcal{G}(x) = \mathcal{G}(f(x))$$ for any $x \in \Ob(\mathcal{X})$. The presheaf ${}_pf\mathcal{F}$ is called the {\it pushforward} of $\mathcal{F}$. It is described by the formula $$({}_pf\mathcal{F})(y) = \lim_{f(x) \to y} \mathcal{F}(x).$$ The rest of this section should probably be moved to the chapter on sites and in any case should be skipped on a first reading. \begin{lemma} \label{lemma-1-morphisms-presheaves} Let $f : \mathcal{X} \to \mathcal{Y}$ and $g : \mathcal{Y} \to \mathcal{Z}$ be $1$-morphisms of categories fibred in groupoids over $(\Sch/S)_{fppf}$. Then $(g \circ f)^p = f^p \circ g^p$ and there is a canonical isomorphism ${}_p(g \circ f) \to {}_pg \circ {}_pf$ compatible with with adjointness of $(f^p, {}_pf)$, $(g^p, {}_pg)$, and $((g \circ f)^p, {}_p(g \circ f))$. \end{lemma} \begin{proof} Let $\mathcal{H}$ be a presheaf on $\mathcal{Z}$. Then $(g \circ f)^p\mathcal{H} = f^p (g^p\mathcal{H})$ is given by the equalities $$(g \circ f)^p\mathcal{H}(x) = \mathcal{H}((g \circ f)(x)) = \mathcal{H}(g(f(x))) = f^p (g^p\mathcal{H})(x).$$ We omit the verification that this is compatible with restriction maps. \medskip\noindent Next, we define the transformation ${}_p(g \circ f) \to {}_pg \circ {}_pf$. Let $\mathcal{F}$ be a presheaf on $\mathcal{X}$. If $z$ is an object of $\mathcal{Z}$ then we get a category $\mathcal{J}$ of quadruples $(x, f(x) \to y, y, g(y) \to z)$ and a category $\mathcal{I}$ of pairs $(x, g(f(x)) \to z)$. There is a canonical functor $\mathcal{J} \to \mathcal{I}$ sending the object $(x, \alpha : f(x) \to y, y, \beta : g(y) \to z)$ to $(x, \beta \circ f(\alpha) : g(f(x)) \to z)$. This gives the arrow in \begin{align*} ({}_p(g \circ f)\mathcal{F})(z) & = \lim_{g(f(x)) \to z} \mathcal{F}(x) \\ & = \lim_\mathcal{I} \mathcal{F} \\ & \to \lim_\mathcal{J} \mathcal{F} \\ & = \lim_{g(y) \to z} \Big(\lim_{f(x) \to y} \mathcal{F}(x)\Big) \\ & = ({}_pg \circ {}_pf\mathcal{F})(x) \end{align*} by Categories, Lemma \ref{categories-lemma-functorial-limit}. We omit the verification that this is compatible with restriction maps. An alternative to this direct construction is to define ${}_p(g \circ f) \cong {}_pg \circ {}_pf$ as the unique map compatible with the adjointness properties. This also has the advantage that one does not need to prove the compatibility. \medskip\noindent Compatibility with adjointness of $(f^p, {}_pf)$, $(g^p, {}_pg)$, and $((g \circ f)^p, {}_p(g \circ f))$ means that given presheaves $\mathcal{H}$ and $\mathcal{F}$ as above we have a commutative diagram $$\xymatrix{ \Mor_{\textit{PSh}(\mathcal{X})}(f^pg^p\mathcal{H}, \mathcal{F}) \ar@{=}[r] \ar@{=}[d] & \Mor_{\textit{PSh}(\mathcal{Y})}(g^p\mathcal{H}, {}_pf\mathcal{F}) \ar@{=}[r] & \Mor_{\textit{PSh}(\mathcal{Y})}(\mathcal{H}, {}_pg{}_pf\mathcal{F}) \\ \Mor_{\textit{PSh}(\mathcal{X})}((g \circ f)^p\mathcal{G}, \mathcal{F}) \ar@{=}[rr] & & \Mor_{\textit{PSh}(\mathcal{Y})}(\mathcal{G}, {}_p(g \circ f)\mathcal{F}) \ar[u] }$$ Proof omitted. \end{proof} \begin{lemma} \label{lemma-2-morphisms-presheaves} Let $f, g : \mathcal{X} \to \mathcal{Y}$ be $1$-morphisms of categories fibred in groupoids over $(\Sch/S)_{fppf}$. Let $t : f \to g$ be a $2$-morphism of categories fibred in groupoids over $(\Sch/S)_{fppf}$. Assigned to $t$ there are canonical isomorphisms of functors $$t^p : g^p \longrightarrow f^p \quad\text{and}\quad {}_pt : {}_pf \longrightarrow {}_pg$$ which compatible with adjointness of $(f^p, {}_pf)$ and $(g^p, {}_pg)$ and with vertical and horizontal composition of $2$-morphisms. \end{lemma} \begin{proof} Let $\mathcal{G}$ be a presheaf on $\mathcal{Y}$. Then $t^p : g^p\mathcal{G} \to f^p\mathcal{G}$ is given by the family of maps $$g^p\mathcal{G}(x) = \mathcal{G}(g(x)) \xrightarrow{\mathcal{G}(t_x)} \mathcal{G}(f(x)) = f^p\mathcal{G}(x)$$ parametrized by $x \in \Ob(\mathcal{X})$. This makes sense as $t_x : f(x) \to g(x)$ and $\mathcal{G}$ is a contravariant functor. We omit the verification that this is compatible with restriction mappings. \medskip\noindent To define the transformation ${}_pt$ for $y \in \Ob(\mathcal{Y})$ define ${}_y^f\mathcal{I}$, resp.\ ${}_y^g\mathcal{I}$ to be the category of pairs $(x, \psi : f(x) \to y)$, resp.\ $(x, \psi : g(x) \to y)$, see Sites, Section \ref{sites-section-more-functoriality-PSh}. Note that $t$ defines a functor ${}_yt : {}_y^g\mathcal{I} \to {}_y^f\mathcal{I}$ given by the rule $$(x, g(x) \to y) \longmapsto (x, f(x) \xrightarrow{t_x} g(x) \to y).$$ Note that for $\mathcal{F}$ a presheaf on $\mathcal{X}$ the composition of ${}_yt$ with $\mathcal{F} : {}_y^f\mathcal{I}^{opp} \to \textit{Sets}$, $(x, f(x) \to y) \mapsto \mathcal{F}(x)$ is equal to $\mathcal{F} : {}_y^g\mathcal{I}^{opp} \to \textit{Sets}$. Hence by Categories, Lemma \ref{categories-lemma-functorial-limit} we get for every $y \in \Ob(\mathcal{Y})$ a canonical map $$({}_pf\mathcal{F})(y) = \lim_{{}_y^f\mathcal{I}} \mathcal{F} \longrightarrow \lim_{{}_y^g\mathcal{I}} \mathcal{F} = ({}_pg\mathcal{F})(y)$$ We omit the verification that this is compatible with restriction mappings. An alternative to this direct construction is to define ${}_pt$ as the unique map compatible with the adjointness properties of the pairs $(f^p, {}_pf)$ and $(g^p, {}_pg)$ (see below). This also has the advantage that one does not need to prove the compatibility. \medskip\noindent Compatibility with adjointness of $(f^p, {}_pf)$ and $(g^p, {}_pg)$ means that given presheaves $\mathcal{G}$ and $\mathcal{F}$ as above we have a commutative diagram $$\xymatrix{ \Mor_{\textit{PSh}(\mathcal{X})}(f^p\mathcal{G}, \mathcal{F}) \ar@{=}[r] \ar[d]_{- \circ t^p} & \Mor_{\textit{PSh}(\mathcal{Y})}(\mathcal{G}, {}_pf\mathcal{F}) \ar[d]^{{}_pt \circ -} \\ \Mor_{\textit{PSh}(\mathcal{X})}(g^p\mathcal{G}, \mathcal{F}) \ar@{=}[r] & \Mor_{\textit{PSh}(\mathcal{Y})}(\mathcal{G}, {}_pg\mathcal{F}) }$$ Proof omitted. Hint: Work through the proof of Sites, Lemma \ref{sites-lemma-adjoints-pu} and observe the compatibility from the explicit description of the horizontal and vertical maps in the diagram. \medskip\noindent We omit the verification that this is compatible with vertical and horizontal compositions. Hint: The proof of this for $t^p$ is straightforward and one can conclude that this holds for the ${}_pt$ maps using compatibility with adjointness. \end{proof} \section{Sheaves} \label{section-sheaves} \noindent We first make an observation that is important and trivial (especially for those readers who do not worry about set theoretical issues). \medskip\noindent Consider a big fppf site $\Sch_{fppf}$ as in Topologies, Definition \ref{topologies-definition-big-fppf-site} and denote its underlying category $\Sch_\alpha$. Besides being the underlying category of a fppf site, the category $\Sch_\alpha$ can also can serve as the underlying category for a big Zariski site, a big \'etale site, a big smooth site, and a big syntomic site, see Topologies, Remark \ref{topologies-remark-choice-sites}. We denote these sites $\Sch_{Zar}$, $\Sch_\etale$, $\Sch_{smooth}$, and $\Sch_{syntomic}$. In this situation, since we have defined the big Zariski site $(\Sch/S)_{Zar}$ of $S$, the big \'etale site $(\Sch/S)_\etale$ of $S$, the big smooth site $(\Sch/S)_{smooth}$ of $S$, the big syntomic site $(\Sch/S)_{syntomic}$ of $S$, and the big fppf site $(\Sch/S)_{fppf}$ of $S$ as the localizations (see Sites, Section \ref{sites-section-localize}) $\Sch_{Zar}/S$, $\Sch_\etale/S$, $\Sch_{smooth}/S$, $\Sch_{syntomic}/S$, and $\Sch_{fppf}/S$ of these (absolute) big sites we see that all of these have the same underlying category, namely $\Sch_\alpha/S$. \medskip\noindent It follows that if we have a category $p : \mathcal{X} \to (\Sch/S)_{fppf}$ fibred in groupoids, then $\mathcal{X}$ inherits a Zariski, \'etale, smooth, syntomic, and fppf topology, see Stacks, Definition \ref{stacks-definition-topology-inherited}. \begin{definition} \label{definition-inherited-topologies} Let $\mathcal{X}$ be a category fibred in groupoids over $(\Sch/S)_{fppf}$. \begin{enumerate} \item The {\it associated Zariski site}, denoted $\mathcal{X}_{Zar}$, is the structure of site on $\mathcal{X}$ inherited from $(\Sch/S)_{Zar}$. \item The {\it associated \'etale site}, denoted $\mathcal{X}_\etale$, is the structure of site on $\mathcal{X}$ inherited from $(\Sch/S)_\etale$. \item The {\it associated smooth site}, denoted $\mathcal{X}_{smooth}$, is the structure of site on $\mathcal{X}$ inherited from $(\Sch/S)_{smooth}$. \item The {\it associated syntomic site}, denoted $\mathcal{X}_{syntomic}$, is the structure of site on $\mathcal{X}$ inherited from $(\Sch/S)_{syntomic}$. \item The {\it associated fppf site}, denoted $\mathcal{X}_{fppf}$, is the structure of site on $\mathcal{X}$ inherited from $(\Sch/S)_{fppf}$. \end{enumerate} \end{definition} \noindent This definition makes sense by the discussion above. If $\mathcal{X}$ is an algebraic stack, the literature calls $\mathcal{X}_{fppf}$ (or a site equivalent to it) the {\it big fppf site} of $\mathcal{X}$ and similarly for the other ones. We may occasionally use this terminology to distinguish this construction from others. \begin{remark} \label{remark-ambiguity} We only use this notation when the symbol $\mathcal{X}$ refers to a category fibred in groupoids, and not a scheme, an algebraic space, etc. In this way we will avoid confusion with the small \'etale site of a scheme, or algebraic space which is denoted $X_\etale$ (in which case we use a roman capital instead of a calligraphic one). \end{remark} \noindent Now that we have these topologies defined we can say what it means to have a sheaf on $\mathcal{X}$, i.e., define the corresponding topoi. \begin{definition} \label{definition-sheaves} Let $\mathcal{X}$ be a category fibred in groupoids over $(\Sch/S)_{fppf}$. Let $\mathcal{F}$ be a presheaf on $\mathcal{X}$. \begin{enumerate} \item We say $\mathcal{F}$ is a {\it Zariski sheaf}, or a {\it sheaf for the Zariski topology} if $\mathcal{F}$ is a sheaf on the associated Zariski site $\mathcal{X}_{Zar}$. \item We say $\mathcal{F}$ is an {\it \'etale sheaf}, or a {\it sheaf for the \'etale topology} if $\mathcal{F}$ is a sheaf on the associated \'etale site $\mathcal{X}_\etale$. \item We say $\mathcal{F}$ is a {\it smooth sheaf}, or a {\it sheaf for the smooth topology} if $\mathcal{F}$ is a sheaf on the associated smooth site $\mathcal{X}_{smooth}$. \item We say $\mathcal{F}$ is a {\it syntomic sheaf}, or a {\it sheaf for the syntomic topology} if $\mathcal{F}$ is a sheaf on the associated syntomic site $\mathcal{X}_{syntomic}$. \item We say $\mathcal{F}$ is an {\it fppf sheaf}, or a {\it sheaf}, or a {\it sheaf for the fppf topology} if $\mathcal{F}$ is a sheaf on the associated fppf site $\mathcal{X}_{fppf}$. \end{enumerate} A morphism of sheaves is just a morphism of presheaves. We denote these categories of sheaves $\Sh(\mathcal{X}_{Zar})$, $\Sh(\mathcal{X}_\etale)$, $\Sh(\mathcal{X}_{smooth})$, $\Sh(\mathcal{X}_{syntomic})$, and $\Sh(\mathcal{X}_{fppf})$. \end{definition} \noindent Of course we can also talk about sheaves of pointed sets, abelian groups, groups, monoids, rings, modules over a fixed ring, and lie algebras over a fixed field, etc. The category of {\it abelian sheaves}, i.e., sheaves of abelian groups, is denoted $\textit{Ab}(\mathcal{X}_{fppf})$ and similarly for the other topologies. If $\mathcal{X}$ is an algebraic stack, then $\Sh(\mathcal{X}_{fppf})$ is equivalent (modulo set theoretical problems) to what in the literature would be termed the {\it category of sheaves on the big fppf site of $\mathcal{X}$}. Similar for other topologies. We may occasionally use this terminology to distinguish this construction from others. \medskip\noindent Since the topologies are listed in increasing order of strength we have the following strictly full inclusions $$\Sh(\mathcal{X}_{fppf}) \subset \Sh(\mathcal{X}_{syntomic}) \subset \Sh(\mathcal{X}_{smooth}) \subset \Sh(\mathcal{X}_\etale) \subset \Sh(\mathcal{X}_{Zar}) \subset \textit{PSh}(\mathcal{X})$$ We sometimes write $\Sh(\mathcal{X}_{fppf}) = \Sh(\mathcal{X})$ and $\textit{Ab}(\mathcal{X}_{fppf}) = \textit{Ab}(\mathcal{X})$ in accordance with our terminology that a sheaf on $\mathcal{X}$ is an fppf sheaf on $\mathcal{X}$. \medskip\noindent With this setup functoriality of these topoi is straightforward, and moreover, is compatible with the inclusion functors above. \begin{lemma} \label{lemma-functoriality-sheaves} Let $f : \mathcal{X} \to \mathcal{Y}$ be a $1$-morphism of categories fibred in groupoids over $(\Sch/S)_{fppf}$. Let $\tau \in \{Zar, \etale, smooth, syntomic, fppf\}$. The functors ${}_pf$ and $f^p$ of (\ref{equation-pushforward-pullback}) transform $\tau$ sheaves into $\tau$ sheaves and define a morphism of topoi $f : \Sh(\mathcal{X}_\tau) \to \Sh(\mathcal{Y}_\tau)$. \end{lemma} \begin{proof} This follows immediately from Stacks, Lemma \ref{stacks-lemma-topology-inherited-functorial}. \end{proof} \noindent In other words, pushforward and pullback of presheaves as defined in Section \ref{section-presheaves} also produces {\it pushforward} and {\it pullback} of $\tau$-sheaves. Having said all of the above we see that we can write $f^p = f^{-1}$ and ${}_pf = f_*$ without any possibility of confusion. \begin{definition} \label{definition-morphism} Let $f : \mathcal{X} \to \mathcal{Y}$ be a morphism of categories fibred in groupoids over $(\Sch/S)_{fppf}$. We denote $$f = (f^{-1}, f_*) : \Sh(\mathcal{X}_{fppf}) \longrightarrow \Sh(\mathcal{Y}_{fppf})$$ the {\it associated morphism of fppf topoi} constructed above. Similarly for the associated Zariski, \'etale, smooth, and syntomic topoi. \end{definition} \noindent As discussed in Sites, Section \ref{sites-section-sheaves-algebraic-structures} the same formula (on the underlying sheaf of sets) defines pushforward and pullback for sheaves (for one of our topologies) of pointed sets, abelian groups, groups, monoids, rings, modules over a fixed ring, and lie algebras over a fixed field, etc. \section{Computing pushforward} \label{section-pushforward} \noindent Let $f : \mathcal{X} \to \mathcal{Y}$ be a $1$-morphism of categories fibred in groupoids over $(\Sch/S)_{fppf}$. Let $\mathcal{F}$ be a presheaf on $\mathcal{X}$. Let $y \in \Ob(\mathcal{Y})$. We can compute $f_*\mathcal{F}(y)$ in the following way. Suppose that $y$ lies over the scheme $V$ and using the $2$-Yoneda lemma think of $y$ as a $1$-morphism. Consider the projection $$\text{pr} : (\Sch/V)_{fppf} \times_{y, \mathcal{Y}} \mathcal{X} \longrightarrow \mathcal{X}$$ Then we have a canonical identification \begin{equation} \label{equation-pushforward} f_*\mathcal{F}(y) = \Gamma\Big( (\Sch/V)_{fppf} \times_{y, \mathcal{Y}} \mathcal{X}, \ \text{pr}^{-1}\mathcal{F}\Big) \end{equation} Namely, objects of the $2$-fibre product are triples $(h : U \to V, x, f(x) \to h^*y)$. Dropping the $h$ from the notation we see that this is equivalent to the data of an object $x$ of $\mathcal{X}$ and a morphism $\alpha : f(x) \to y$ of $\mathcal{Y}$. Since $f_*\mathcal{F}(y) = \lim_{f(x) \to y} \mathcal{F}(x)$ by definition the equality follows. \medskip\noindent As a consequence we have the following base change'' result for pushforwards. This result is trivial and hinges on the fact that we are using big'' sites. \begin{lemma} \label{lemma-base-change} Let $S$ be a scheme. Let $$\xymatrix{ \mathcal{Y}' \times_\mathcal{Y} \mathcal{X} \ar[r]_{g'} \ar[d]_{f'} & \mathcal{X} \ar[d]^f \\ \mathcal{Y}' \ar[r]^g & \mathcal{Y} }$$ be a $2$-cartesian diagram of categories fibred in groupoids over $S$. Then we have a canonical isomorphism $$g^{-1}f_*\mathcal{F} \longrightarrow f'_*(g')^{-1}\mathcal{F}$$ functorial in the presheaf $\mathcal{F}$ on $\mathcal{X}$. \end{lemma} \begin{proof} Given an object $y'$ of $\mathcal{Y}'$ over $V$ there is an equivalence $$(\Sch/V)_{fppf} \times_{g(y'), \mathcal{Y}} \mathcal{X} = (\Sch/V)_{fppf} \times_{y', \mathcal{Y}'} (\mathcal{Y}' \times_\mathcal{Y} \mathcal{X})$$ Hence by (\ref{equation-pushforward}) a bijection $g^{-1}f_*\mathcal{F}(y') \to f'_*(g')^{-1}\mathcal{F}(y')$. We omit the verification that this is compatible with restriction mappings. \end{proof} \noindent In the case of a representable morphism of categories fibred in groupoids this formula (\ref{equation-pushforward}) simplifies. We suggest the reader skip the rest of this section. \begin{lemma} \label{lemma-representable} Let $f : \mathcal{X} \to \mathcal{Y}$ be a $1$-morphism of categories fibred in groupoids over $(\Sch/S)_{fppf}$. The following are equivalent \begin{enumerate} \item $f$ is representable, and \item for every $y \in \Ob(\mathcal{Y})$ the functor $\mathcal{X}^{opp} \to \textit{Sets}$, $x \mapsto \Mor_\mathcal{Y}(f(x), y)$ is representable. \end{enumerate} \end{lemma} \begin{proof} According to the discussion in Algebraic Stacks, Section \ref{algebraic-section-representable-morphism} we see that $f$ is representable if and only if for every $y \in \Ob(\mathcal{Y})$ lying over $U$ the $2$-fibre product $(\Sch/U)_{fppf} \times_{y, \mathcal{Y}} \mathcal{X}$ is representable, i.e., of the form $(\Sch/V_y)_{fppf}$ for some scheme $V_y$ over $U$. Objects in this $2$-fibre products are triples $(h : V \to U, x, \alpha : f(x) \to h^*y)$ where $\alpha$ lies over $\text{id}_V$. Dropping the $h$ from the notation we see that this is equivalent to the data of an object $x$ of $\mathcal{X}$ and a morphism $f(x) \to y$. Hence the $2$-fibre product is representable by $V_y$ and $f(x_y) \to y$ where $x_y$ is an object of $\mathcal{X}$ over $V_y$ if and only if the functor in (2) is representable by $x_y$ with universal object a map $f(x_y) \to y$. \end{proof} \noindent Let $$\xymatrix{ \mathcal{X} \ar[rr]_f \ar[rd]_p & & \mathcal{Y} \ar[ld]^q \\ & (\Sch/S)_{fppf} }$$ be a $1$-morphism of categories fibred in groupoids. Assume $f$ is representable. For every $y \in \Ob(\mathcal{Y})$ we choose an object $u(y) \in \Ob(\mathcal{X})$ representing the functor $x \mapsto \Mor_\mathcal{Y}(f(x), y)$ of Lemma \ref{lemma-representable} (this is possible by the axiom of choice). The objects come with canonical morphisms $f(u(y)) \to y$ by construction. For every morphism $\beta : y' \to y$ in $\mathcal{Y}$ we obtain a unique morphism $u(\beta) : u(y') \to u(y)$ in $\mathcal{X}$ such that the diagram $$\xymatrix{ f(u(y')) \ar[d] \ar[rr]_{f(u(\beta))} & & f(u(y)) \ar[d] \\ y' \ar[rr] & & y }$$ commutes. In other words, $u : \mathcal{Y} \to \mathcal{X}$ is a functor. In fact, we can say a little bit more. Namely, suppose that $V' = q(y')$, $V = q(y)$, $U' = p(u(y'))$ and $U = p(u(y))$. Then $$\xymatrix{ U' \ar[rr]_{p(u(\beta))} \ar[d] & & U \ar[d] \\ V' \ar[rr]^{q(\beta)} & & V }$$ is a fibre product square. This is true because $U' \to U$ represents the base change $(\Sch/V')_{fppf} \times_{y', \mathcal{Y}} \mathcal{X} \to (\Sch/V)_{fppf} \times_{y, \mathcal{Y}} \mathcal{X}$ of $V' \to V$. \begin{lemma} \label{lemma-representable-pushforward} Let $f : \mathcal{X} \to \mathcal{Y}$ be a representable $1$-morphism of categories fibred in groupoids over $(\Sch/S)_{fppf}$. Let $\tau \in \{Zar, \etale, smooth, syntomic, fppf\}$. Then the functor $u : \mathcal{Y}_\tau \to \mathcal{X}_\tau$ is continuous and defines a morphism of sites $\mathcal{X}_\tau \to \mathcal{Y}_\tau$ which induces the same morphism of topoi $\Sh(\mathcal{X}_\tau) \to \Sh(\mathcal{Y}_\tau)$ as the morphism $f$ constructed in Lemma \ref{lemma-functoriality-sheaves}. Moreover, $f_*\mathcal{F}(y) = \mathcal{F}(u(y))$ for any presheaf $\mathcal{F}$ on $\mathcal{X}$. \end{lemma} \begin{proof} Let $\{y_i \to y\}$ be a $\tau$-covering in $\mathcal{Y}$. By definition this simply means that $\{q(y_i) \to q(y)\}$ is a $\tau$-covering of schemes. By the final remark above the lemma we see that $\{p(u(y_i)) \to p(u(y))\}$ is the base change of the $\tau$-covering $\{q(y_i) \to q(y)\}$ by $p(u(y)) \to q(y)$, hence is itself a $\tau$-covering by the axioms of a site. Hence $\{u(y_i) \to u(y)\}$ is a $\tau$-covering of $\mathcal{X}$. This proves that $u$ is continuous. \medskip\noindent Let's use the notation $u_p, u_s, u^p, u^s$ of Sites, Sections \ref{sites-section-functoriality-PSh} and \ref{sites-section-continuous-functors}. If we can show the final assertion of the lemma, then we see that $f_* = u^p = u^s$ (by continuity of $u$ seen above) and hence by adjointness $f^{-1} = u_s$ which will prove $u_s$ is exact, hence that $u$ determines a morphism of sites, and the equality will be clear as well. To see that $f_*\mathcal{F}(y) = \mathcal{F}(u(y))$ note that by definition $$f_*\mathcal{F}(y) = ({}_pf\mathcal{F})(y) = \lim_{f(x) \to y} \mathcal{F}(x).$$ Since $u(y)$ is a final object in the category the limit is taken over we conclude. \end{proof} \section{The structure sheaf} \label{section-structure-sheaf} \noindent Let $\tau \in \{Zar, \etale, smooth, syntomic, fppf\}$. Let $p : \mathcal{X} \to (\Sch/S)_{fppf}$ be a category fibred in groupoids. The 2-category of categories fibred in groupoids over $(\Sch/S)_{fppf}$ has a final object, namely, $\text{id} : (\Sch/S)_{fppf} \to (\Sch/S)_{fppf}$ and $p$ is a $1$-morphism from $\mathcal{X}$ to this final object. Hence any presheaf $\mathcal{G}$ on $(\Sch/S)_{fppf}$ gives a presheaf $p^{-1}\mathcal{G}$ on $\mathcal{X}$ defined by the rule $p^{-1}\mathcal{G}(x) = \mathcal{G}(p(x))$. Moreover, the discussion in Section \ref{section-sheaves} shows that $p^{-1}\mathcal{G}$ is a $\tau$ sheaf whenever $\mathcal{G}$ is a $\tau$-sheaf. \medskip\noindent Recall that the site $(\Sch/S)_{fppf}$ is a ringed site with structure sheaf $\mathcal{O}$ defined by the rule $$(\Sch/S)^{opp} \longrightarrow \textit{Rings}, \quad U/S \longmapsto \Gamma(U, \mathcal{O}_U)$$ see Descent, Definition \ref{descent-definition-structure-sheaf}. \begin{definition} \label{definition-structure-sheaf} Let $p : \mathcal{X} \to (\Sch/S)_{fppf}$ be a category fibred in groupoids. The {\it structure sheaf of $\mathcal{X}$} is the sheaf of rings $\mathcal{O}_\mathcal{X} = p^{-1}\mathcal{O}$. \end{definition} \noindent For an object $x$ of $\mathcal{X}$ lying over $U$ we have $\mathcal{O}_\mathcal{X}(x) = \mathcal{O}(U) = \Gamma(U, \mathcal{O}_U)$. Needless to say $\mathcal{O}_\mathcal{X}$ is also a Zariski, \'etale, smooth, and syntomic sheaf, and hence each of the sites $\mathcal{X}_{Zar}$, $\mathcal{X}_\etale$, $\mathcal{X}_{smooth}$, $\mathcal{X}_{syntomic}$, and $\mathcal{X}_{fppf}$ is a ringed site. This construction is functorial as well. \begin{lemma} \label{lemma-functoriality-structure-sheaf} Let $f : \mathcal{X} \to \mathcal{Y}$ be a $1$-morphism of categories fibred in groupoids over $(\Sch/S)_{fppf}$. Let $\tau \in \{Zar, \etale, smooth, syntomic, fppf\}$. There is a canonical identification $f^{-1}\mathcal{O}_\mathcal{X} = \mathcal{O}_\mathcal{Y}$ which turns $f : \Sh(\mathcal{X}_\tau) \to \Sh(\mathcal{Y}_\tau)$ into a morphism of ringed topoi. \end{lemma} \begin{proof} Denote $p : \mathcal{X} \to (\Sch/S)_{fppf}$ and $q : \mathcal{Y} \to (\Sch/S)_{fppf}$ the structural functors. Then $q = p \circ f$, hence $q^{-1} = f^{-1} \circ p^{-1}$ by Lemma \ref{lemma-1-morphisms-presheaves}. The result follows. \end{proof} \begin{remark} \label{remark-flat} In the situation of Lemma \ref{lemma-functoriality-structure-sheaf} the morphism of ringed topoi $f : \Sh(\mathcal{X}_\tau) \to \Sh(\mathcal{Y}_\tau)$ is flat as is clear from the equality $f^{-1}\mathcal{O}_\mathcal{X} = \mathcal{O}_\mathcal{Y}$. This is a bit counter intuitive, for example because a closed immersion of algebraic stacks is typically not flat (as a morphism of algebraic stacks). However, exactly the same thing happens when taking a closed immersion $i : X \to Y$ of schemes: in this case the associated morphism of big $\tau$-sites $i : (\Sch/X)_\tau \to (\Sch/Y)_\tau$ also is flat. \end{remark} \section{Sheaves of modules} \label{section-modules} \noindent Since we have a structure sheaf we have modules. \begin{definition} \label{definition-modules} Let $\mathcal{X}$ be a category fibred in groupoids over $(\Sch/S)_{fppf}$. \begin{enumerate} \item A {\it presheaf of modules on $\mathcal{X}$} is a presheaf of $\mathcal{O}_\mathcal{X}$-modules. The category of presheaves of modules is denoted $\textit{PMod}(\mathcal{O}_\mathcal{X})$. \item We say a presheaf of modules $\mathcal{F}$ is an {\it $\mathcal{O}_\mathcal{X}$-module}, or more precisely a {\it sheaf of $\mathcal{O}_\mathcal{X}$-modules} if $\mathcal{F}$ is an fppf sheaf. The category of $\mathcal{O}_\mathcal{X}$-modules is denoted $\textit{Mod}(\mathcal{O}_\mathcal{X})$. \end{enumerate} \end{definition} \noindent These (pre)sheaves of modules occur in the literature as {\it (pre)sheaves of $\mathcal{O}_\mathcal{X}$-modules on the big fppf site of $\mathcal{X}$}. We will occasionally use this terminology if we want to distinguish these categories from others. We will also encounter presheaves of modules which are sheaves in the Zariski, \'etale, smooth, or syntomic topologies (without necessarily being sheaves). If need be these will be denoted $\textit{Mod}(\mathcal{X}_\etale, \mathcal{O}_\mathcal{X})$ and similarly for the other topologies. \medskip\noindent Next, we address functoriality -- first for presheaves of modules. Let $$\xymatrix{ \mathcal{X} \ar[rr]_f \ar[rd]_p & & \mathcal{Y} \ar[ld]^q \\ & (\Sch/S)_{fppf} }$$ be a $1$-morphism of categories fibred in groupoids. The functors $f^{-1}$, $f_*$ on abelian presheaves extend to functors \begin{equation} \label{equation-functoriality-presheaves-modules} f^{-1} : \textit{PMod}(\mathcal{O}_\mathcal{Y}) \longrightarrow \textit{PMod}(\mathcal{O}_\mathcal{X}) \quad\text{and}\quad f_* : \textit{PMod}(\mathcal{O}_\mathcal{Y}) \longrightarrow \textit{PMod}(\mathcal{O}_\mathcal{X}) \end{equation} This is immediate for $f^{-1}$ because $f^{-1}\mathcal{G}(x) = \mathcal{G}(f(x))$ which is a module over $\mathcal{O}_\mathcal{Y}(f(x)) = \mathcal{O}(q(f(x))) = \mathcal{O}(p(x)) = \mathcal{O}_\mathcal{X}(x)$. Alternatively it follows because $f^{-1}\mathcal{O}_\mathcal{Y} = \mathcal{O}_\mathcal{X}$ and because $f^{-1}$ commutes with limits (on presheaves). Since $f_*$ is a right adjoint it commutes with all limits (on presheaves) in particular products. Hence we can extend $f_*$ to a functor on presheaves of modules as in the proof of Modules on Sites, Lemma \ref{sites-modules-lemma-pushforward-module}. We claim that the functors (\ref{equation-functoriality-presheaves-modules}) form an adjoint pair of functors: $$\Mor_{\textit{PMod}(\mathcal{O}_\mathcal{X})}( f^{-1}\mathcal{G}, \mathcal{F}) = \Mor_{\textit{PMod}(\mathcal{O}_\mathcal{Y})}( \mathcal{G}, f_*\mathcal{F}).$$ As $f^{-1}\mathcal{O}_\mathcal{Y} = \mathcal{O}_\mathcal{X}$ this follows from Modules on Sites, Lemma \ref{sites-modules-lemma-adjoint-push-pull-modules} by endowing $\mathcal{X}$ and $\mathcal{Y}$ with the chaotic topology. \medskip\noindent Next, we discuss functoriality for modules, i.e., for sheaves of modules in the fppf topology. Denote by $f$ also the induced morphism of ringed topoi, see Lemma \ref{lemma-functoriality-structure-sheaf} (for the fppf topologies right now). Note that the functors $f^{-1}$ and $f_*$ of (\ref{equation-functoriality-presheaves-modules}) preserve the subcategories of sheaves of modules, see Lemma \ref{lemma-functoriality-sheaves}. Hence it follows immediately that \begin{equation} \label{equation-functoriality-sheaves-modules} f^{-1} : \textit{Mod}(\mathcal{O}_\mathcal{Y}) \longrightarrow \textit{Mod}(\mathcal{O}_\mathcal{X}) \quad\text{and}\quad f_* : \textit{Mod}(\mathcal{O}_\mathcal{Y}) \longrightarrow \textit{Mod}(\mathcal{O}_\mathcal{X}) \end{equation} form an adjoint pair of functors: $$\Mor_{\textit{Mod}(\mathcal{O}_\mathcal{X})}( f^{-1}\mathcal{G}, \mathcal{F}) = \Mor_{\textit{Mod}(\mathcal{O}_\mathcal{Y})}( \mathcal{G}, f_*\mathcal{F}).$$ By uniqueness of adjoints we conclude that $f^* = f^{-1}$ where $f^*$ is as defined in Modules on Sites, Section \ref{sites-modules-section-functoriality-modules} for the morphism of ringed topoi $f$ above. Of course we could have seen this directly because $f^*(-) = f^{-1}(-) \otimes_{f^{-1}\mathcal{O}_\mathcal{Y}} \mathcal{O}_\mathcal{X}$ and because $f^{-1}\mathcal{O}_\mathcal{Y} = \mathcal{O}_\mathcal{X}$. \medskip\noindent Similarly for sheaves of modules in the Zariski, \'etale, smooth, syntomic topology. \section{Representable categories} \label{section-representable} \noindent In this short section we compare our definitions with what happens in case the algebraic stacks in question are representable. \begin{lemma} \label{lemma-compare-with-scheme} Let $S$ be a scheme. Let $\mathcal{X}$ be a category fibred in groupoids over $(\Sch/S)$. Assume $\mathcal{X}$ is representable by a scheme $X$. For $\tau \in \{Zar,\linebreak[0] \etale,\linebreak[0] smooth,\linebreak[0] syntomic,\linebreak[0] fppf\}$ there is a canonical equivalence $$(\mathcal{X}_\tau, \mathcal{O}_\mathcal{X}) = ((\Sch/X)_\tau, \mathcal{O}_X)$$ of ringed sites. \end{lemma} \begin{proof} This follows by choosing an equivalence $(\Sch/X)_\tau \to \mathcal{X}$ of categories fibred in groupoids over $(\Sch/S)_{fppf}$ and using the functoriality of the construction $\mathcal{X} \leadsto \mathcal{X}_\tau$. \end{proof} \begin{lemma} \label{lemma-compare-with-morphism-of-schemes} Let $S$ be a scheme. Let $f : \mathcal{X} \to \mathcal{Y}$ be a morphism of categories fibred in groupoids over $S$. Assume $\mathcal{X}$, $\mathcal{Y}$ are representable by schemes $X$, $Y$. Let $f : X \to Y$ be the morphism of schemes corresponding to $f$. For $\tau \in \{Zar,\linebreak[0] \etale,\linebreak[0] smooth,\linebreak[0] syntomic,\linebreak[0] fppf\}$ the morphism of ringed topoi $f : (\Sh(\mathcal{X}_\tau), \mathcal{O}_\mathcal{X}) \to (\Sh(\mathcal{Y}_\tau), \mathcal{O}_\mathcal{Y})$ agrees with the morphism of ringed topoi $f : (\Sh((\Sch/X)_\tau), \mathcal{O}_X) \to (\Sh((\Sch/Y)_\tau), \mathcal{O}_Y)$ via the identifications of Lemma \ref{lemma-compare-with-scheme}. \end{lemma} \begin{proof} Follows by unwinding the definitions. \end{proof} \section{Restriction} \label{section-restriction} \noindent A trivial but useful observation is that the localization of a category fibred in groupoids at an object is equivalent to the big site of the scheme it lies over. \begin{lemma} \label{lemma-localizing} Let $p : \mathcal{X} \to (\Sch/S)_{fppf}$ be a category fibred in groupoids. Let $\tau \in \{Zar, \etale, smooth, syntomic, fppf\}$. Let $x \in \Ob(\mathcal{X})$ lying over $U = p(x)$. The functor $p$ induces an equivalence of sites $\mathcal{X}_\tau/x \to (\Sch/U)_\tau$. \end{lemma} \begin{proof} Special case of Stacks, Lemma \ref{stacks-lemma-localizing}. \end{proof} \noindent We use the lemma above to talk about the pullback and the restriction of a (pre)sheaf to a scheme. \begin{definition} \label{definition-pullback} Let $p : \mathcal{X} \to (\Sch/S)_{fppf}$ be a category fibred in groupoids. Let $x \in \Ob(\mathcal{X})$ lying over $U = p(x)$. Let $\mathcal{F}$ be a presheaf on $\mathcal{X}$. \begin{enumerate} \item The {\it pullback $x^{-1}\mathcal{F}$ of $\mathcal{F}$} is the restriction $\mathcal{F}|_{(\mathcal{X}/x)}$ viewed as a presheaf on $(\Sch/U)_{fppf}$ via the equivalence $\mathcal{X}/x \to (\Sch/U)_{fppf}$ of Lemma \ref{lemma-localizing}. \item The {\it restriction of $\mathcal{F}$ to $U_\etale$} is $x^{-1}\mathcal{F}|_{U_\etale}$, abusively written $\mathcal{F}|_{U_\etale}$. \end{enumerate} \end{definition} \noindent This notation makes sense because to the object $x$ the $2$-Yoneda lemma, see Algebraic Stacks, Section \ref{algebraic-section-2-yoneda} associates a $1$-morphism $x : (\Sch/U)_{fppf} \to \mathcal{X}/x$ which is quasi-inverse to $p : \mathcal{X}/x \to (\Sch/U)_{fppf}$. Hence $x^{-1}\mathcal{F}$ truly is the pullback of $\mathcal{F}$ via this $1$-morphism. In particular, by the material above, if $\mathcal{F}$ is a sheaf (or a Zariski, \'etale, smooth, syntomic sheaf), then $x^{-1}\mathcal{F}$ is a sheaf on $(\Sch/U)_{fppf}$ (or on $(\Sch/U)_{Zar}$, $(\Sch/U)_\etale$, $(\Sch/U)_{smooth}$, $(\Sch/U)_{syntomic}$). \medskip\noindent Let $p : \mathcal{X} \to (\Sch/S)_{fppf}$ be a category fibred in groupoids. Let $\varphi : x \to y$ be a morphism of $\mathcal{X}$ lying over the morphism of schemes $a : U \to V$. Recall that $a$ induces a morphism of small \'etale sites $a_{small} : U_\etale \to V_\etale$, see \'Etale Cohomology, Section \ref{etale-cohomology-section-functoriality}. Let $\mathcal{F}$ be a presheaf on $\mathcal{X}$. Let $\mathcal{F}|_{U_\etale}$ and $\mathcal{F}|_{V_\etale}$ be the restrictions of $\mathcal{F}$ via $x$ and $y$. There is a natural {\it comparison} map \begin{equation} \label{equation-comparison-push} c_\varphi : \mathcal{F}|_{V_\etale} \longrightarrow a_{small, *}(\mathcal{F}|_{U_\etale}) \end{equation} of presheaves on $U_\etale$. Namely, if $V' \to V$ is \'etale, set $U' = V' \times_V U$ and define $c_\varphi$ on sections over $V'$ via $$\xymatrix{ a_{small, *}(\mathcal{F}|_{U_\etale})(V') & \mathcal{F}|_{U_\etale}(U') \ar@{=}[l] & \mathcal{F}(x') \ar@{=}[l] \\ \mathcal{F}|_{V_\etale}(V') \ar@{=}[rr] \ar[u]^{c_\varphi} & & \mathcal{F}(y') \ar[u]_{\mathcal{F}(\varphi')} }$$ Here $\varphi' : x' \to y'$ is a morphism of $\mathcal{X}$ fitting into a commutative diagram $$\vcenter{ \xymatrix{ x' \ar[r] \ar[d]_{\varphi'} & x \ar[d]^\varphi \\ y' \ar[r] & y } } \quad\text{lying over}\quad \vcenter{ \xymatrix{ U' \ar[r] \ar[d] & U \ar[d]^a \\ V' \ar[r] & V } }$$ The existence and uniqueness of $\varphi'$ follow from the axioms of a category fibred in groupoids. We omit the verification that $c_\varphi$ so defined is indeed a map of presheaves (i.e., compatible with restriction mappings) and that it is functorial in $\mathcal{F}$. In case $\mathcal{F}$ is a sheaf for the \'etale topology we obtain a {\it comparison} map \begin{equation} \label{equation-comparison} c_\varphi : a_{small}^{-1}(\mathcal{F}|_{V_\etale}) \longrightarrow \mathcal{F}|_{U_\etale} \end{equation} which is also denoted $c_\varphi$ as indicated (this is the customary abuse of notation in not distinguishing between adjoint maps). \begin{lemma} \label{lemma-comparison} Let $\mathcal{F}$ be an \'etale sheaf on $\mathcal{X} \to (\Sch/S)_{fppf}$. \begin{enumerate} \item If $\varphi : x \to y$ and $\psi : y \to z$ are morphisms of $\mathcal{X}$ lying over $a : U \to V$ and $b : V \to W$, then the composition $$a_{small}^{-1}(b_{small}^{-1} (\mathcal{F}|_{W_\etale})) \xrightarrow{a_{small}^{-1}c_\psi} a_{small}^{-1}(\mathcal{F}|_{V_\etale}) \xrightarrow{c_\varphi} \mathcal{F}|_{U_\etale}$$ is equal to $c_{\psi \circ \varphi}$ via the identification $$(b \circ a)_{small}^{-1}(\mathcal{F}|_{W_\etale}) = a_{small}^{-1}(b_{small}^{-1} (\mathcal{F}|_{W_\etale})).$$ \item If $\varphi : x \to y$ lies over an \'etale morphism of schemes $a : U \to V$, then (\ref{equation-comparison}) is an isomorphism. \item Suppose $f : \mathcal{Y} \to \mathcal{X}$ is a $1$-morphism of categories fibred in groupoids over $(\Sch/S)_{fppf}$ and $y$ is an object of $\mathcal{Y}$ lying over the scheme $U$ with image $x = f(y)$. Then there is a canonical identification $f^{-1}\mathcal{F}|_{U_\etale} = \mathcal{F}|_{U_\etale}$. \item Moreover, given $\psi : y' \to y$ in $\mathcal{Y}$ lying over $a : U' \to U$ the comparison map $c_\psi : a_{small}^{-1}(F^{-1}\mathcal{F}|_{U_\etale}) \to F^{-1}\mathcal{F}|_{U'_\etale}$ is equal to the comparison map $c_{f(\psi)} : a_{small}^{-1}\mathcal{F}|_{U_\etale} \to \mathcal{F}|_{U'_\etale}$ via the identifications in (3). \end{enumerate} \end{lemma} \begin{proof} The verification of these properties is omitted. \end{proof} \noindent Next, we turn to the restriction of (pre)sheaves of modules. \begin{lemma} \label{lemma-localizing-structure-sheaf} Let $p : \mathcal{X} \to (\Sch/S)_{fppf}$ be a category fibred in groupoids. Let $\tau \in \{Zar, \etale, smooth, syntomic, fppf\}$. Let $x \in \Ob(\mathcal{X})$ lying over $U = p(x)$. The equivalence of Lemma \ref{lemma-localizing} extends to an equivalence of ringed sites $(\mathcal{X}_\tau/x, \mathcal{O}_\mathcal{X}|_x) \to ((\Sch/U)_\tau, \mathcal{O})$. \end{lemma} \begin{proof} This is immediate from the construction of the structure sheaves. \end{proof} \noindent Let $\mathcal{X}$ be a category fibred in groupoids over $(\Sch/S)_{fppf}$. Let $\mathcal{F}$ be a (pre)sheaf of modules on $\mathcal{X}$ as in Definition \ref{definition-modules}. Let $x$ be an object of $\mathcal{X}$ lying over $U$. Then Lemma \ref{lemma-localizing-structure-sheaf} guarantees that the restriction $x^{-1}\mathcal{F}$ is a (pre)sheaf of modules on $(\Sch/U)_{fppf}$. We will sometimes write $x^*\mathcal{F} = x^{-1}\mathcal{F}$ in this case. Similarly, if $\mathcal{F}$ is a sheaf for the Zariski, \'etale, smooth, or syntomic topology, then $x^{-1}\mathcal{F}$ is as well. Moreover, the restriction $\mathcal{F}|_{U_\etale} = x^{-1}\mathcal{F}|_{U_\etale}$ to $U$ is a presheaf of $\mathcal{O}_{U_\etale}$-modules. If $\mathcal{F}$ is a sheaf for the \'etale topology, then $\mathcal{F}|_{U_\etale}$ is a sheaf of modules. Moreover, if $\varphi : x \to y$ is a morphism of $\mathcal{X}$ lying over $a : U \to V$ then the comparison map (\ref{equation-comparison}) is compatible with $a_{small}^\sharp$ (see Descent, Remark \ref{descent-remark-change-topologies-ringed}) and induces a {\it comparison} map \begin{equation} \label{equation-comparison-modules} c_\varphi : a_{small}^*(\mathcal{F}|_{V_\etale}) \longrightarrow \mathcal{F}|_{U_\etale} \end{equation} of $\mathcal{O}_{U_\etale}$-modules. Note that the properties (1), (2), (3), and (4) of Lemma \ref{lemma-comparison} hold in the setting of \'etale sheaves of modules as well. We will use this in the following without further mention. \begin{lemma} \label{lemma-enough-points} Let $p : \mathcal{X} \to (\Sch/S)_{fppf}$ be a category fibred in groupoids. Let $\tau \in \{Zar, \etale, smooth, syntomic, fppf\}$. The site $\mathcal{X}_\tau$ has enough points. \end{lemma} \begin{proof} By Sites, Lemma \ref{sites-lemma-enough-points-local} we have to show that there exists a family of objects $x$ of $\mathcal{X}$ such that $\mathcal{X}_\tau/x$ has enough points and such that the sheaves $h_x^\#$ cover the final object of the category of sheaves. By Lemma \ref{lemma-localizing} and \'Etale Cohomology, Lemma \ref{etale-cohomology-lemma-points-fppf} we see that $\mathcal{X}_\tau/x$ has enough points for every object $x$ and we win. \end{proof} \section{Restriction to algebraic spaces} \label{section-restriction-algebraic-spaces} \noindent In this section we consider sheaves on categories representable by algebraic spaces. The following lemma is the analogue of Topologies, Lemma \ref{topologies-lemma-at-the-bottom-etale} for algebraic spaces. \begin{lemma} \label{lemma-compare} Let $S$ be a scheme. Let $\mathcal{X} \to (\Sch/S)_{fppf}$ be a category fibred in groupoids. Assume $\mathcal{X}$ is representable by an algebraic space $F$. Then there exists a continuous and cocontinuous functor $F_\etale \to \mathcal{X}_\etale$ which induces a morphism of ringed sites $$\pi_F : (\mathcal{X}_\etale, \mathcal{O}_\mathcal{X}) \longrightarrow (F_\etale, \mathcal{O}_F)$$ and a morphism of ringed topoi $$i_F : (\Sh(F_\etale), \mathcal{O}_F) \longrightarrow (\Sh(\mathcal{X}_\etale), \mathcal{O}_\mathcal{X})$$ such that $\pi_F \circ i_F = \text{id}$. Moreover $\pi_{F, *} = i_F^{-1}$. \end{lemma} \begin{proof} Choose an equivalence $j : \mathcal{S}_F \to \mathcal{X}$, see Algebraic Stacks, Sections \ref{algebraic-section-split} and \ref{algebraic-section-representable-by-algebraic-spaces}. An object of $F_\etale$ is a scheme $U$ together with an \'etale morphism $\varphi : U \to F$. Then $\varphi$ is an object of $\mathcal{S}_F$ over $U$. Hence $j(\varphi)$ is an object of $\mathcal{X}$ over $U$. In this way $j$ induces a functor $u : F_\etale \to \mathcal{X}$. It is clear that $u$ is continuous and cocontinuous for the \'etale topology on $\mathcal{X}$. Since $j$ is an equivalence, the functor $u$ is fully faithful. Also, fibre products and equalizers exist in $F_\etale$ and $u$ commutes with them because these are computed on the level of underlying schemes in $F_\etale$. Thus Sites, Lemmas \ref{sites-lemma-when-shriek}, \ref{sites-lemma-preserve-equalizers}, and \ref{sites-lemma-back-and-forth} apply. In particular $u$ defines a morphism of topoi $i_F : \Sh(F_\etale) \to \Sh(\mathcal{X}_\etale)$ and there exists a left adjoint $i_{F, !}$ of $i_F^{-1}$ which commutes with fibre products and equalizers. \medskip\noindent We claim that $i_{F, !}$ is exact. If this is true, then we can define $\pi_F$ by the rules $\pi_F^{-1} = i_{F, !}$ and $\pi_{F, *} = i_F^{-1}$ and everything is clear. To prove the claim, note that we already know that $i_{F, !}$ is right exact and preserves fibre products. Hence it suffices to show that $i_{F, !}* = *$ where $*$ indicates the final object in the category of sheaves of sets. Let $U$ be a scheme and let $\varphi : U \to F$ be surjective and \'etale. Set $R = U \times_F U$. Then $$\xymatrix{ h_R \ar@<1ex>[r] \ar@<-1ex>[r] & h_U \ar[r] & {*} }$$ is a coequalizer diagram in $\Sh(F_\etale)$. Using the right exactness of $i_{F, !}$, using $i_{F, !} = (u_p\ )^\#$, and using Sites, Lemma \ref{sites-lemma-pullback-representable-presheaf} we see that $$\xymatrix{ h_{u(R)} \ar@<1ex>[r] \ar@<-1ex>[r] & h_{u(U)} \ar[r] & i_{F, !}{*} }$$ is a coequalizer diagram in $\Sh(F_\etale)$. Using that $j$ is an equivalence and that $F = U/R$ it follows that the coequalizer in $\Sh(\mathcal{X}_\etale)$ of the two maps $h_{u(R)} \to h_{u(U)}$ is $*$. We omit the proof that these morphisms are compatible with structure sheaves. \end{proof} \noindent Assume $\mathcal{X}$ is an algebraic stack represented by the algebraic space $F$. Let $j : \mathcal{S}_F \to \mathcal{X}$ be an equivalence and denote $u : F_\etale \to \mathcal{X}_\etale$ the functor of the proof of Lemma \ref{lemma-compare} above. Given a sheaf $\mathcal{F}$ on $\mathcal{X}_\etale$ we have $$\pi_{F, *}\mathcal{F}(U) = i_F^{-1}\mathcal{F}(U) = \mathcal{F}(u(U)).$$ This is why we often think of $i_F^{-1}$ as a {\it restriction functor} similarly to Definition \ref{definition-pullback} and to the restriction of a sheaf on the big \'etale site of a scheme to the small \'etale site of a scheme. We often use the notation \begin{equation} \label{equation-restrict} \mathcal{F}|_{F_\etale} = i_F^{-1}\mathcal{F} = \pi_{F, *}\mathcal{F} \end{equation} in this situation. \begin{lemma} \label{lemma-compare-morphism} Let $S$ be a scheme. Let $f : \mathcal{X} \to \mathcal{Y}$ be a morphism of categories fibred in groupoids over $(\Sch/S)_{fppf}$. Assume $\mathcal{X}$, $\mathcal{Y}$ are representable by algebraic spaces $F$, $G$. Denote $f : F \to G$ the induced morphism of algebraic spaces, and $f_{small} : F_\etale \to G_\etale$ the corresponding morphism of ringed topoi. Then $$\xymatrix{ (\Sh(\mathcal{X}_\etale), \mathcal{O}_\mathcal{X}) \ar[d]_{\pi_F} \ar[rr]_f & & (\Sh(\mathcal{Y}_\etale), \mathcal{O}_\mathcal{Y}) \ar[d]^{\pi_G} \\ (\Sh(F_\etale), \mathcal{O}_F) \ar[rr]^{f_{small}} & & (\Sh(G_\etale), \mathcal{O}_G) }$$ is a commutative diagram of ringed topoi. \end{lemma} \begin{proof} This is similar to Topologies, Lemma \ref{topologies-lemma-morphism-big-small-etale} (3) but there is a small snag due to the fact that $F \to G$ may not be representable by schemes. In particular we don't get a commutative diagram of ringed sites, but only a commutative diagram of ringed topoi. \medskip\noindent Before we start the proof proper, we choose equivalences $j : \mathcal{S}_F \to \mathcal{X}$ and $j' : \mathcal{S}_G \to \mathcal{Y}$ which induce functors $u : F_\etale \to \mathcal{X}$ and $u' : G_\etale \to \mathcal{Y}$ as in the proof of Lemma \ref{lemma-compare}. Because of the 2-functoriality of sheaves on categories fibred in groupoids over $\Sch_{fppf}$ (see discussion in Section \ref{section-presheaves}) we may assume that $\mathcal{X} = \mathcal{S}_F$ and $\mathcal{Y} = \mathcal{S}_G$ and that $f : \mathcal{S}_F \to \mathcal{S}_G$ is the functor associated to the morphism $f : F \to G$. Correspondingly we will omit $u$ and $u'$ from the notation, i.e., given an object $U \to F$ of $F_\etale$ we denote $U/F$ the corresponding object of $\mathcal{X}$. Similarly for $G$. \medskip\noindent Let $\mathcal{G}$ be a sheaf on $\mathcal{X}_\etale$. To prove (2) we compute $\pi_{G, *}f_*\mathcal{G}$ and $f_{small, *}\pi_{F, *}\mathcal{G}$. To do this let $V \to G$ be an object of $G_\etale$. Then $$\pi_{G, *}f_*\mathcal{G}(V) = f_*\mathcal{G}(V/G) = \Gamma\Big( (\Sch/V)_{fppf} \times_{\mathcal{Y}} \mathcal{X}, \ \text{pr}^{-1}\mathcal{G}\Big)$$ see (\ref{equation-pushforward}). The fibre product in the formula is $$(\Sch/V)_{fppf} \times_{\mathcal{Y}} \mathcal{X} = (\Sch/V)_{fppf} \times_{\mathcal{S}_G} \mathcal{S}_F = \mathcal{S}_{V \times_G F}$$ i.e., it is the split category fibred in groupoids associated to the algebraic space $V \times_G F$. And $\text{pr}^{-1}\mathcal{G}$ is a sheaf on $\mathcal{S}_{V \times_G F}$ for the \'etale topology. \medskip\noindent In particular, if $V \times_G F$ is representable, i.e., if it is a scheme, then $\pi_{G, *}f_*\mathcal{G}(V) = \mathcal{G}(V \times_G F/F)$ and also $$f_{small, *}\pi_{F, *}\mathcal{G}(V) = \pi_{F, *}\mathcal{G}(V \times_G F) = \mathcal{G}(V \times_G F/F)$$ which proves the desired equality in this special case. \medskip\noindent In general, choose a scheme $U$ and a surjective \'etale morphism $U \to V \times_G F$. Set $R = U \times_{V \times_G F} U$. Then $U/V \times_G F$ and $R/V \times_G F$ are objects of the fibre product category above. Since $\text{pr}^{-1}\mathcal{G}$ is a sheaf for the \'etale topology on $\mathcal{S}_{V \times_G F}$ the diagram $$\xymatrix{ \Gamma\Big( (\Sch/V)_{fppf} \times_{\mathcal{Y}} \mathcal{X}, \ \text{pr}^{-1}\mathcal{G}\Big) \ar[r] & \text{pr}^{-1}\mathcal{G}(U/V \times_G F) \ar@<1ex>[r] \ar@<-1ex>[r] & \text{pr}^{-1}\mathcal{G}(R/V \times_G F) }$$ is an equalizer diagram. Note that $\text{pr}^{-1}\mathcal{G}(U/V \times_G F) = \mathcal{G}(U/F)$ and $\text{pr}^{-1}\mathcal{G}(R/V \times_G F) = \mathcal{G}(R/F)$ by the definition of pullbacks. Moreover, by the material in Properties of Spaces, Section \ref{spaces-properties-section-etale-site} (especially, Properties of Spaces, Remark \ref{spaces-properties-remark-explain-equivalence} and Lemma \ref{spaces-properties-lemma-functoriality-etale-site}) we see that there is an equalizer diagram $$\xymatrix{ f_{small, *}\pi_{F, *}\mathcal{G}(V) \ar[r] & \pi_{F, *}\mathcal{G}(U/F) \ar@<1ex>[r] \ar@<-1ex>[r] & \pi_{F, *}\mathcal{G}(R/F) }$$ Since we also have $\pi_{F, *}\mathcal{G}(U/F) = \mathcal{G}(U/F)$ and $\pi_{F, *}\mathcal{G}(U/F) = \mathcal{G}(U/F)$ we obtain a canonical identification $f_{small, *}\pi_{F, *}\mathcal{G}(V) = \pi_{G, *}f_*\mathcal{G}(V)$. We omit the proof that this is compatible with restriction mappings and that it is functorial in $\mathcal{G}$. \end{proof} \noindent Let $f : \mathcal{X} \to \mathcal{Y}$ and $f : F \to G$ be as in the second part of the lemma above. A consequence of the lemma, using (\ref{equation-restrict}), is that \begin{equation} \label{equation-compare-big-small} (f_*\mathcal{F})|_{G_\etale} = f_{small, *}(\mathcal{F}|_{F_\etale}) \end{equation} for any sheaf $\mathcal{F}$ on $\mathcal{X}_\etale$. Moreover, if $\mathcal{F}$ is a sheaf of $\mathcal{O}$-modules, then (\ref{equation-compare-big-small}) is an isomorphism of $\mathcal{O}_G$-modules on $G_\etale$. \medskip\noindent Finally, suppose that we have a $2$-commutative diagram $$\xymatrix{ \mathcal{U} \ar[r]^a \ar[dr]_f \drtwocell<\omit>{<-2>\varphi} & \mathcal{V} \ar[d]^g \\ & \mathcal{X} }$$ of $1$-morphisms of categories fibred in groupoids over $(\Sch/S)_{fppf}$, that $\mathcal{F}$ is a sheaf on $\mathcal{X}_\etale$, and that $\mathcal{U}, \mathcal{V}$ are representable by algebraic spaces $U, V$. Then we obtain a comparison map \begin{equation} \label{equation-comparison-algebraic-spaces} c_\varphi : a_{small}^{-1}(g^{-1}\mathcal{F}|_{V_\etale}) \longrightarrow f^{-1}\mathcal{F}|_{U_\etale} \end{equation} where $a : U \to V$ denotes the morphism of algebraic spaces corresponding to $a$. This is the analogue of (\ref{equation-comparison}). We define $c_\varphi$ as the adjoint to the map $$g^{-1}\mathcal{F}|_{V_\etale} \longrightarrow a_{small, *}(f^{-1}\mathcal{F}|_{U_\etale}) = (a_*f^{-1}\mathcal{F})|_{V_\etale}$$ (equality by (\ref{equation-compare-big-small})) which is the restriction to $V$ (\ref{equation-restrict}) of the map $$g^{-1}\mathcal{F} \to a_*a^{-1}g^{-1}\mathcal{F} = a_*f^{-1}\mathcal{F}$$ where the last equality uses the $2$-commutativity of the diagram above. In case $\mathcal{F}$ is a sheaf of $\mathcal{O}_\mathcal{X}$-modules $c_\varphi$ induces a {\it comparison} map \begin{equation} \label{equation-comparison-algebraic-spaces-modules} c_\varphi : a_{small}^*(g^*\mathcal{F}|_{V_\etale}) \longrightarrow f^*\mathcal{F}|_{U_\etale} \end{equation} of $\mathcal{O}_{U_\etale}$-modules. Note that the properties (1), (2), (3), and (4) of Lemma \ref{lemma-comparison} hold in this setting as well. \section{Quasi-coherent modules} \label{section-quasi-coherent} \noindent At this point we can apply the general definition of a quasi-coherent module to the situation discussed in this chapter. \begin{definition} \label{definition-quasi-coherent} Let $p : \mathcal{X} \to (\Sch/S)_{fppf}$ be a category fibred in groupoids. A {\it quasi-coherent module on $\mathcal{X}$}, or a {\it quasi-coherent $\mathcal{O}_\mathcal{X}$-module} is a quasi-coherent module on the ringed site $(\mathcal{X}_{fppf}, \mathcal{O}_\mathcal{X})$ as in Modules on Sites, Definition \ref{sites-modules-definition-site-local}. The category of quasi-coherent sheaves on $\mathcal{X}$ is denoted $\QCoh(\mathcal{O}_\mathcal{X})$. \end{definition} \noindent If $\mathcal{X}$ is an algebraic stack, then this definition agrees with all definitions in the literature in the sense that $\QCoh(\mathcal{O}_\mathcal{X})$ is equivalent (modulo set theoretic issues) to any variant of this category defined in the literature. For example, we will match our definition with the definition in \cite[Definition 6.1]{olsson_sheaves} in Cohomology on Stacks, Lemma \ref{lemma-quasi-coherent}. We will also see alternative constructions of this category later on. \medskip\noindent In general (as is the case for morphisms of schemes) the pushforward of quasi-coherent sheaf along a $1$-morphism is not quasi-coherent. Pullback does preserve quasi-coherence. \begin{lemma} \label{lemma-pullback-quasi-coherent} Let $f : \mathcal{X} \to \mathcal{Y}$ be a $1$-morphism of categories fibred in groupoids over $(\Sch/S)_{fppf}$. The pullback functor $f^* = f^{-1} : \textit{Mod}(\mathcal{O}_\mathcal{Y}) \to \textit{Mod}(\mathcal{O}_\mathcal{X})$ preserves quasi-coherent sheaves. \end{lemma} \begin{proof} This is a general fact, see Modules on Sites, Lemma \ref{sites-modules-lemma-local-pullback}. \end{proof} \noindent It turns out that quasi-coherent sheaves have a very simple characterization in terms of their pullbacks. See also Lemma \ref{lemma-quasi-coherent} for a characterization in terms of restrictions. \begin{lemma} \label{lemma-characterize-quasi-coherent} Let $p : \mathcal{X} \to (\Sch/S)_{fppf}$ be a category fibred in groupoids. Let $\mathcal{F}$ be a sheaf of $\mathcal{O}_\mathcal{X}$-modules. Then $\mathcal{F}$ is quasi-coherent if and only if $x^*\mathcal{F}$ is a quasi-coherent sheaf on $(\Sch/U)_{fppf}$ for every object $x$ of $\mathcal{X}$ with $U = p(x)$. \end{lemma} \begin{proof} By Lemma \ref{lemma-pullback-quasi-coherent} the condition is necessary. Conversely, since $x^*\mathcal{F}$ is just the restriction to $\mathcal{X}_{fppf}/x$ we see that it is sufficient directly from the definition of a quasi-coherent sheaf (and the fact that the notion of being quasi-coherent is an intrinsic property of sheaves of modules, see Modules on Sites, Section \ref{sites-modules-section-intrinsic}). \end{proof} \noindent Although there is a variant for the Zariski topology, it seems that the \'etale topology is the natural topology to use in the following definition. \begin{definition} \label{definition-locally-quasi-coherent} Let $p : \mathcal{X} \to (\Sch/S)_{fppf}$ be a category fibred in groupoids. Let $\mathcal{F}$ be a presheaf of $\mathcal{O}_\mathcal{X}$-modules. We say $\mathcal{F}$ is {\it locally quasi-coherent}\footnote{This is nonstandard notation.} if $\mathcal{F}$ is a sheaf for the \'etale topology and for every object $x$ of $\mathcal{X}$ the restriction $x^*\mathcal{F}|_{U_\etale}$ is a quasi-coherent sheaf. Here $U = p(x)$. \end{definition} \noindent We use $\textit{LQCoh}(\mathcal{O}_\mathcal{X})$ to indicate the category of locally quasi-coherent modules. We now have the following diagram of categories of modules $$\xymatrix{ \QCoh(\mathcal{O}_\mathcal{X}) \ar[r] \ar[d] & \textit{Mod}(\mathcal{O}_\mathcal{X}) \ar[d] \\ \textit{LQCoh}(\mathcal{O}_\mathcal{X}) \ar[r] & \textit{Mod}(\mathcal{X}_\etale, \mathcal{O}_\mathcal{X}) }$$ where the arrows are strictly full embeddings. It turns out that many results for quasi-coherent sheaves have a counter part for locally quasi-coherent modules. Moreover, from many points of view (as we shall see later) this is a natural category to consider. For example the quasi-coherent sheaves are exactly those locally quasi-coherent modules that are cartesian'', i.e., satisfy the second condition of the lemma below. \begin{lemma} \label{lemma-quasi-coherent} Let $p : \mathcal{X} \to (\Sch/S)_{fppf}$ be a category fibred in groupoids. Let $\mathcal{F}$ be a presheaf of $\mathcal{O}_\mathcal{X}$-modules. Then $\mathcal{F}$ is quasi-coherent if and only if the following two conditions hold \begin{enumerate} \item $\mathcal{F}$ is locally quasi-coherent, and \item for any morphism $\varphi : x \to y$ of $\mathcal{X}$ lying over $f : U \to V$ the comparison map $c_\varphi : f_{small}^*\mathcal{F}|_{V_\etale} \to \mathcal{F}|_{U_\etale}$ of (\ref{equation-comparison-modules}) is an isomorphism. \end{enumerate} \end{lemma} \begin{proof} Assume $\mathcal{F}$ is quasi-coherent. Then $\mathcal{F}$ is a sheaf for the fppf topology, hence a sheaf for the \'etale topology. Moreover, any pullback of $\mathcal{F}$ to a ringed topos is quasi-coherent, hence the restrictions $x^*\mathcal{F}|_{U_\etale}$ are quasi-coherent. This proves $\mathcal{F}$ is locally quasi-coherent. Let $y$ be an object of $\mathcal{X}$ with $V = p(y)$. We have seen that $\mathcal{X}/y = (\Sch/V)_{fppf}$. By Descent, Proposition \ref{descent-proposition-equivalence-quasi-coherent} it follows that $y^*\mathcal{F}$ is the quasi-coherent module associated to a (usual) quasi-coherent module $\mathcal{F}_V$ on the scheme $V$. Hence certainly the comparison maps (\ref{equation-comparison-modules}) are isomorphisms. \medskip\noindent Conversely, suppose that $\mathcal{F}$ satisfies (1) and (2). Let $y$ be an object of $\mathcal{X}$ with $V = p(y)$. Denote $\mathcal{F}_V$ the quasi-coherent module on the scheme $V$ corresponding to the restriction $y^*\mathcal{F}|_{V_\etale}$ which is quasi-coherent by assumption (1), see Descent, Proposition \ref{descent-proposition-equivalence-quasi-coherent}. Condition (2) now signifies that the restrictions $x^*\mathcal{F}|_{U_\etale}$ for $x$ over $y$ are each isomorphic to the (\'etale sheaf associated to the) pullback of $\mathcal{F}_V$ via the corresponding morphism of schemes $U \to V$. Hence $y^*\mathcal{F}$ is the sheaf on $(\Sch/V)_{fppf}$ associated to $\mathcal{F}_V$. Hence it is quasi-coherent (by Descent, Proposition \ref{descent-proposition-equivalence-quasi-coherent} again) and we see that $\mathcal{F}$ is quasi-coherent on $\mathcal{X}$ by Lemma \ref{lemma-characterize-quasi-coherent}. \end{proof} \begin{lemma} \label{lemma-pullback-lqc} Let $f : \mathcal{X} \to \mathcal{Y}$ be a $1$-morphism of categories fibred in groupoids over $(\Sch/S)_{fppf}$. The pullback functor $f^* = f^{-1} : \textit{Mod}(\mathcal{Y}_\etale, \mathcal{O}_\mathcal{Y}) \to \textit{Mod}(\mathcal{X}_\etale, \mathcal{O}_\mathcal{X})$ preserves locally quasi-coherent sheaves. \end{lemma} \begin{proof} Let $\mathcal{G}$ be locally quasi-coherent on $\mathcal{Y}$. Choose an object $x$ of $\mathcal{X}$ lying over the scheme $U$. The restriction $x^*f^*\mathcal{G}|_{U_\etale}$ equals $(f \circ x)^*\mathcal{G}|_{U_\etale}$ hence is a quasi-coherent sheaf by assumption on $\mathcal{G}$. \end{proof} \begin{lemma} \label{lemma-lqc-colimits} Let $p : \mathcal{X} \to (\Sch/S)_{fppf}$ be a category fibred in groupoids. \begin{enumerate} \item The category $\textit{LQCoh}(\mathcal{O}_\mathcal{X})$ has colimits and they agree with colimits in the category $\textit{Mod}(\mathcal{X}_\etale, \mathcal{O}_\mathcal{X})$. \item The category $\textit{LQCoh}(\mathcal{O}_\mathcal{X})$ is abelian with kernels and cokernels computed in $\textit{Mod}(\mathcal{X}_\etale, \mathcal{O}_\mathcal{X})$, in other words the inclusion functor is exact. \item Given a short exact sequence $0 \to \mathcal{F}_1 \to \mathcal{F}_2 \to \mathcal{F}_3 \to 0$ of $\textit{Mod}(\mathcal{X}_\etale, \mathcal{O}_\mathcal{X})$ if two out of three are locally quasi-coherent so is the third. \item Given $\mathcal{F}, \mathcal{G}$ in $\textit{LQCoh}(\mathcal{O}_\mathcal{X})$ the tensor product $\mathcal{F} \otimes_{\mathcal{O}_\mathcal{X}} \mathcal{G}$ in $\textit{Mod}(\mathcal{X}_\etale, \mathcal{O}_\mathcal{X})$ is an object of $\textit{LQCoh}(\mathcal{O}_\mathcal{X})$. \item Given $\mathcal{F}, \mathcal{G}$ in $\textit{LQCoh}(\mathcal{O}_\mathcal{X})$ with $\mathcal{F}$ locally of finite presentation on $\mathcal{X}_\etale$ the sheaf $\SheafHom_{\mathcal{O}_\mathcal{X}}(\mathcal{F}, \mathcal{G})$ in $\textit{Mod}(\mathcal{X}_\etale, \mathcal{O}_\mathcal{X})$ is an object of $\textit{LQCoh}(\mathcal{O}_\mathcal{X})$. \end{enumerate} \end{lemma} \begin{proof} Each of these statements follows from the corresponding statement of Descent, Lemma \ref{descent-lemma-equivalence-quasi-coherent-limits}. For example, suppose that $\mathcal{I} \to \textit{LQCoh}(\mathcal{O}_\mathcal{X})$, $i \mapsto \mathcal{F}_i$ is a diagram. Consider the object $\mathcal{F} = \colim_i \mathcal{F}_i$ of $\textit{Mod}(\mathcal{X}_\etale, \mathcal{O}_\mathcal{X})$. For any object $x$ of $\mathcal{X}$ with $U = p(x)$ the pullback functor $x^*$ commutes with all colimits as it is a left adjoint. Hence $x^*\mathcal{F} = \colim_i x^*\mathcal{F}_i$. Similarly we have $x^*\mathcal{F}|_{U_\etale} = \colim_i x^*\mathcal{F}_i|_{U_\etale}$. Now by assumption each $x^*\mathcal{F}_i|_{U_\etale}$ is quasi-coherent, hence the colimit is quasi-coherent by the aforementioned Descent, Lemma \ref{descent-lemma-equivalence-quasi-coherent-limits}. This proves (1). \medskip\noindent It follows from (1) that cokernels exist in $\textit{LQCoh}(\mathcal{O}_\mathcal{X})$ and agree with the cokernels computed in $\textit{Mod}(\mathcal{X}_\etale, \mathcal{O}_\mathcal{X})$. Let $\varphi : \mathcal{F} \to \mathcal{G}$ be a morphism of $\textit{LQCoh}(\mathcal{O}_\mathcal{X})$ and let $\mathcal{K} = \Ker(\varphi)$ computed in $\textit{Mod}(\mathcal{X}_\etale, \mathcal{O}_\mathcal{X})$. If we can show that $\mathcal{K}$ is a locally quasi-coherent module, then the proof of (2) is complete. To see this, note that kernels are computed in the category of presheaves (no sheafification necessary). Hence $\mathcal{K}|_{U_\etale}$ is the kernel of the map $\mathcal{F}|_{U_\etale} \to \mathcal{G}|_{U_\etale}$, i.e., is the kernel of a map of quasi-coherent sheaves on $U_\etale$ whence quasi-coherent by Descent, Lemma \ref{descent-lemma-equivalence-quasi-coherent-limits}. This proves (2). \medskip\noindent Parts (3), (4), and (5) follow in exactly the same way. Details omitted. \end{proof} \noindent In the generality discussed here the category of quasi-coherent sheaves is not abelian. See Examples, Section \ref{examples-section-nonabelian-QCoh}. Here is what we can prove without any further work. \begin{lemma} \label{lemma-qc-colimits} Let $p : \mathcal{X} \to (\Sch/S)_{fppf}$ be a category fibred in groupoids. \begin{enumerate} \item The category $\QCoh(\mathcal{O}_\mathcal{X})$ has colimits and they agree with colimits in the category $\textit{Mod}(\mathcal{O}_\mathcal{X})$ as well as with colimits in the category $\textit{LQCoh}(\mathcal{O}_\mathcal{X})$. \item Given $\mathcal{F}, \mathcal{G}$ in $\QCoh(\mathcal{O}_\mathcal{X})$ the tensor product $\mathcal{F} \otimes_{\mathcal{O}_\mathcal{X}} \mathcal{G}$ in $\textit{Mod}(\mathcal{O}_\mathcal{X})$ is an object of $\QCoh(\mathcal{O}_\mathcal{X})$. \item Given $\mathcal{F}, \mathcal{G}$ in $\QCoh(\mathcal{O}_\mathcal{X})$ with $\mathcal{F}$ locally of finite presentation on $\mathcal{X}_{fppf}$ the sheaf $\SheafHom_{\mathcal{O}_\mathcal{X}}(\mathcal{F}, \mathcal{G})$ in $\textit{Mod}(\mathcal{O}_\mathcal{X})$ is an object of $\QCoh(\mathcal{O}_\mathcal{X})$. \end{enumerate} \end{lemma} \begin{proof} Let $\mathcal{I} \to \QCoh(\mathcal{O}_\mathcal{X})$, $i \mapsto \mathcal{F}_i$ be a diagram. Consider the object $\mathcal{F} = \colim_i \mathcal{F}_i$ of $\textit{Mod}(\mathcal{O}_\mathcal{X})$. For any object $x$ of $\mathcal{X}$ with $U = p(x)$ the pullback functor $x^*$ commutes with all colimits as it is a left adjoint. Hence $x^*\mathcal{F} = \colim_i x^*\mathcal{F}_i$ in $\textit{Mod}((\Sch/U)_{fppf}, \mathcal{O})$. We conclude from Descent, Lemma \ref{descent-lemma-equivalence-quasi-coherent-limits} that $x^*\mathcal{F}$ is quasi-coherent, hence $\mathcal{F}$ is quasi-coherent, see Lemma \ref{lemma-characterize-quasi-coherent}. Thus we see that $\QCoh(\mathcal{O}_\mathcal{X})$ has colimits and they agree with colimits in the category $\textit{Mod}(\mathcal{O}_\mathcal{X})$. In particular the (fppf) sheaf $\mathcal{F}$ is also the colimit of the diagram in $\textit{Mod}(\mathcal{X}_\etale, \mathcal{O}_\mathcal{X})$, hence $\mathcal{F}$ is also the colimit in $\textit{LQCoh}(\mathcal{O}_\mathcal{X})$. This proves (1). \medskip\noindent Parts (2) and (3) are proved in the same way. Details omitted. \end{proof} \section{Stackification and sheaves} \label{section-stackification} \noindent It turns out that the category of sheaves on a category fibred in groupoids only knows about'' the stackification. \begin{lemma} \label{lemma-stackification} Let $f : \mathcal{X} \to \mathcal{Y}$ be a $1$-morphism of categories fibred in groupoids over $(\Sch/S)_{fppf}$. If $f$ induces an equivalence of stackifications, then the morphism of topoi $f : \Sh(\mathcal{X}_{fppf}) \to \Sh(\mathcal{Y}_{fppf})$ is an equivalence. \end{lemma} \begin{proof} We may assume $\mathcal{Y}$ is the stackification of $\mathcal{X}$. We claim that $f : \mathcal{X} \to \mathcal{Y}$ is a special cocontinuous functor, see Sites, Definition \ref{sites-definition-special-cocontinuous-functor} which will prove the lemma. By Stacks, Lemma \ref{stacks-lemma-topology-inherited-functorial} the functor $f$ is continuous and cocontinuous. By Stacks, Lemma \ref{stacks-lemma-stackify} we see that conditions (3), (4), and (5) of Sites, Lemma \ref{sites-lemma-equivalence} hold. \end{proof} \begin{lemma} \label{lemma-stackification-quasi-coherent} Let $f : \mathcal{X} \to \mathcal{Y}$ be a $1$-morphism of categories fibred in groupoids over $(\Sch/S)_{fppf}$. If $f$ induces an equivalence of stackifications, then $f^*$ induces equivalences $\textit{Mod}(\mathcal{O}_\mathcal{X}) \to \textit{Mod}(\mathcal{O}_\mathcal{Y})$ and $\QCoh(\mathcal{O}_\mathcal{X}) \to \QCoh(\mathcal{O}_\mathcal{Y})$. \end{lemma} \begin{proof} We may assume $\mathcal{Y}$ is the stackification of $\mathcal{X}$. The first assertion is clear from Lemma \ref{lemma-stackification} and $\mathcal{O}_\mathcal{X} = f^{-1}\mathcal{O}_\mathcal{Y}$. Pullback of quasi-coherent sheaves are quasi-coherent, see Lemma \ref{lemma-pullback-quasi-coherent}. Hence it suffices to show that if $f^*\mathcal{G}$ is quasi-coherent, then $\mathcal{G}$ is. To see this, let $y$ be an object of $\mathcal{Y}$. Translating the condition that $\mathcal{Y}$ is the stackification of $\mathcal{X}$ we see there exists an fppf covering $\{y_i \to y\}$ in $\mathcal{Y}$ such that $y_i \cong f(x_i)$ for some $x_i$ object of $\mathcal{X}$. Say $x_i$ and $y_i$ lie over the scheme $U_i$. Then $f^*\mathcal{G}$ being quasi-coherent, means that $x_i^*f^*\mathcal{G}$ is quasi-coherent. Since $x_i^*f^*\mathcal{G}$ is isomorphic to $y_i^*\mathcal{G}$ (as sheaves on $(\Sch/U_i)_{fppf}$ we see that $y_i^*\mathcal{G}$ is quasi-coherent. It follows from Modules on Sites, Lemma \ref{sites-modules-lemma-local-final-object} that the restriction of $\mathcal{G}$ to $\mathcal{Y}/y$ is quasi-coherent. Hence $\mathcal{G}$ is quasi-coherent by Lemma \ref{lemma-characterize-quasi-coherent}. \end{proof} \section{Quasi-coherent sheaves and presentations} \label{section-quasi-coherent-presentation} \noindent In Groupoids in Spaces, Definition \ref{spaces-groupoids-definition-groupoid-module} we have the defined the notion of a quasi-coherent module on an arbitrary groupoid. The following (formal) proposition tells us that we can study quasi-coherent sheaves on quotient stacks in terms of quasi-coherent modules on presentations. \begin{proposition} \label{proposition-quasi-coherent} Let $(U, R, s, t, c)$ be a groupoid in algebraic spaces over $S$. Let $\mathcal{X} = [U/R]$ be the quotient stack. The category of quasi-coherent modules on $\mathcal{X}$ is equivalent to the category of quasi-coherent modules on $(U, R, s, t, c)$. \end{proposition} \begin{proof} Denote $\QCoh(U, R, s, t, c)$ the category of quasi-coherent modules on the groupoid $(U, R, s, t, c)$. We will construct quasi-inverse functors $$\QCoh(\mathcal{O}_\mathcal{X}) \longleftrightarrow \QCoh(U, R, s, t, c).$$ According to Lemma \ref{lemma-stackification-quasi-coherent} the stackification map $[U/_{\!p}R] \to [U/R]$ (see Groupoids in Spaces, Definition \ref{spaces-groupoids-definition-quotient-stack}) induces an equivalence of categories of quasi-coherent sheaves. Thus it suffices to prove the lemma with $\mathcal{X} = [U/_{\!p}R]$. \medskip\noindent Recall that an object $x = (T, u)$ of $\mathcal{X} = [U/_{\!p}R]$ is given by a scheme $T$ and a morphism $u : T \to U$. A morphism $(T, u) \to (T', u')$ is given by a pair $(f, r)$ where $f : T \to T'$ and $r : T \to R$ with $s \circ r = u$ and $t \circ r = u' \circ f$. Let us call a {\it special morphism} any morphism of the form $(f, e \circ u' \circ f) : (T, u' \circ f) \to (T', u')$. The category of $(T, u)$ with special morphisms is just the category of schemes over $U$. \medskip\noindent Let $\mathcal{F}$ be a quasi-coherent sheaf on $\mathcal{X}$. Then we obtain for every $x = (T, u)$ a quasi-coherent sheaf $\mathcal{F}_{(T, u)} = x^*\mathcal{F}|_{T_\etale}$ on $T$. Moreover, for any morphism $(f, r) : x = (T, u) \to (T', u') = x'$ we obtain a comparison isomorphism $$c_{(f, r)} : f_{small}^*\mathcal{F}_{(T', u')} \longrightarrow \mathcal{F}_{(T, u)}$$ see Lemma \ref{lemma-quasi-coherent}. Moreover, these isomorphisms are compatible with compositions, see Lemma \ref{lemma-comparison}. If $U$, $R$ are schemes, then we can construct the quasi-coherent sheaf on the groupoid as follows: First the object $(U, \text{id})$ corresponds to a quasi-coherent sheaf $\mathcal{F}_{(U, \text{id})}$ on $U$. Next, the isomorphism $\alpha : t_{small}^*\mathcal{F}_{(U, \text{id})} \to s_{small}^*\mathcal{F}_{(U, \text{id})}$ comes from \begin{enumerate} \item the morphism $(R, \text{id}_R) : (R, s) \to (R, t)$ in the category $[U/_{\!p}R]$ which produces an isomorphism $\mathcal{F}_{(R, t)} \to \mathcal{F}_{(R, s)}$, \item the special morphism $(R, s) \to (U, \text{id})$ which produces an isomorphism $s_{small}^*\mathcal{F}_{(U, \text{id})} \to \mathcal{F}_{(R, s)}$, and \item the special morphism $(R, t) \to (U, \text{id})$ which produces an isomorphism $t_{small}^*\mathcal{F}_{(U, \text{id})} \to \mathcal{F}_{(R, t)}$. \end{enumerate} The cocycle condition for $\alpha$ follows from the condition that $(U, R, s, t, c)$ is groupoid, i.e., that composition is associative (details omitted). \medskip\noindent To do this in general, i.e., when $U$ and $R$ are algebraic spaces, it suffices to explain how to associate to an algebraic space $(W, u)$ over $U$ a quasi-coherent sheaf $\mathcal{F}_{(W, u)}$ and to construct the comparison maps for morphisms between these. We set $\mathcal{F}_{(W, u)} = x^*\mathcal{F}|_{W_\etale}$ where $x$ is the $1$-morphism $\mathcal{S}_W \to \mathcal{S}_U \to [U/_{\!p}R]$ and the comparison maps are explained in (\ref{equation-comparison-algebraic-spaces-modules}). \medskip\noindent Conversely, suppose that $(\mathcal{G}, \alpha)$ is a quasi-coherent module on $(U, R, s, t, c)$. We are going to define a presheaf of modules $\mathcal{F}$ on $\mathcal{X}$ as follows. Given an object $(T, u)$ of $[U/_{\!p}R]$ we set $$\mathcal{F}(T, u) : = \Gamma(T, u_{small}^*\mathcal{G}).$$ Given a morphism $(f, r) : (T, u) \to (T', u')$ we get a map \begin{align*} \mathcal{F}(T', u') & = \Gamma(T', (u')_{small}^*\mathcal{G}) \\ & \to \Gamma(T, f_{small}^*(u')_{small}^*\mathcal{G}) = \Gamma(T, (u' \circ f)_{small}^*\mathcal{G}) \\ & = \Gamma(T, (t \circ r)_{small}^*\mathcal{G}) = \Gamma(T, r_{small}^*t_{small}^*\mathcal{G}) \\ & \to \Gamma(T, r_{small}^*s_{small}^*\mathcal{G}) = \Gamma(T, (s \circ r)_{small}^*\mathcal{G}) \\ & = \Gamma(T, u_{small}^*\mathcal{G}) \\ & = \mathcal{F}(T, u) \end{align*} where the first arrow is pullback along $f$ and the second arrow is $\alpha$. Note that if $(T, r)$ is a special morphism, then this map is just pullback along $f$ as $e_{small}^*\alpha = \text{id}$ by the axioms of a sheaf of quasi-coherent modules on a groupoid. The cocycle condition implies that $\mathcal{F}$ is a presheaf of modules (details omitted). It is immediate from the definition that $\mathcal{F}$ is quasi-coherent when pulled back to $(\Sch/T)_{fppf}$ (by the simple description of the restriction maps of $\mathcal{F}$ in case of a special morphism). \medskip\noindent We omit the verification that the functors constructed above are quasi-inverse to each other. \end{proof} \noindent We finish this section with a technical lemma on maps out of quasi-coherent sheaves. It is an analogue of Schemes, Lemma \ref{schemes-lemma-compare-constructions}. We will see later (Criteria for Representability, Theorem \ref{criteria-theorem-flat-groupoid-gives-algebraic-stack}) that the assumptions on the groupoid imply that $\mathcal{X}$ is an algebraic stack. \begin{lemma} \label{lemma-map-from-quasi-coherent} Let $(U, R, s, t, c)$ be a groupoid in algebraic spaces over $S$. Assume $s, t$ are flat and locally of finite presentation. Let $\mathcal{X} = [U/R]$ be the quotient stack. Denote $\pi : \mathcal{S}_U \to \mathcal{X}$ the quotient map. Let $\mathcal{F}$ be a quasi-coherent $\mathcal{O}_\mathcal{X}$-module, and let $\mathcal{H}$ be any object of $\textit{Mod}(\mathcal{O}_\mathcal{X})$. The map $$\Hom_{\mathcal{O}_\mathcal{X}}(\mathcal{F}, \mathcal{H}) \longrightarrow \Hom_{\mathcal{O}_U}(x^*\mathcal{F}|_{U_\etale}, x^*\mathcal{H}|_{U_\etale}), \quad \phi \longmapsto x^*\phi|_{U_\etale}$$ is injective and its image consists of exactly those $\varphi : x^*\mathcal{F}|_{U_\etale} \to x^*\mathcal{H}|_{U_\etale}$ which give rise to a commutative diagram $$\xymatrix{ s_{small}^*(x^*\mathcal{F}|_{U_\etale}) \ar[r] \ar[d]^{s_{small}^*\varphi} & (x \circ s)^*\mathcal{F}|_{R_\etale} = (x \circ t)^*\mathcal{F}|_{R_\etale} & t_{small}^*(x^*\mathcal{F}|_{U_\etale}) \ar[l] \ar[d]_{t_{small}^*\varphi} \\ s_{small}^*(x^*\mathcal{H}|_{U_\etale}) \ar[r] & (x \circ s)^*\mathcal{H}|_{R_\etale} = (x \circ t)^*\mathcal{H}|_{R_\etale} & t_{small}^*(x^*\mathcal{H}|_{U_\etale}) \ar[l] }$$ of modules on $R_\etale$ where the horizontal arrows are the comparison maps (\ref{equation-comparison-algebraic-spaces-modules}). \end{lemma} \begin{proof} According to Lemma \ref{lemma-stackification-quasi-coherent} the stackification map $[U/_{\!p}R] \to [U/R]$ (see Groupoids in Spaces, Definition \ref{spaces-groupoids-definition-quotient-stack}) induces an equivalence of categories of quasi-coherent sheaves and of fppf $\mathcal{O}$-modules. Thus it suffices to prove the lemma with $\mathcal{X} = [U/_{\!p}R]$. By Proposition \ref{proposition-quasi-coherent} and its proof there exists a quasi-coherent module $(\mathcal{G}, \alpha)$ on $(U, R, s, t, c)$ such that $\mathcal{F}$ is given by the rule $\mathcal{F}(T, u) = \Gamma(T, u^*\mathcal{G})$. In particular $x^*\mathcal{F}|_{U_\etale} = \mathcal{G}$ and it is clear that the map of the statement of the lemma is injective. Moreover, given a map $\varphi : \mathcal{G} \to x^*\mathcal{H}|_{U_\etale}$ and given any object $y = (T, u)$ of $[U/_{\!p}R]$ we can consider the map $$\mathcal{F}(y) = \Gamma(T, u^*\mathcal{G}) \xrightarrow{u_{small}^*\varphi} \Gamma(T, u_{small}^*x^*\mathcal{H}|_{U_\etale}) \rightarrow \Gamma(T, y^*\mathcal{H}|_{T_\etale}) = \mathcal{H}(y)$$ where the second arrow is the comparison map (\ref{equation-comparison-modules}) for the sheaf $\mathcal{H}$. This assignment is compatible with the restriction mappings of the sheaves $\mathcal{F}$ and $\mathcal{G}$ for morphisms of $[U/_{\!p}R]$ if the cocycle condition of the lemma is satisfied. Proof omitted. Hint: the restriction maps of $\mathcal{F}$ are made explicit in terms of $(\mathcal{G}, \alpha)$ in the proof of Proposition \ref{proposition-quasi-coherent}. \end{proof} \section{Quasi-coherent sheaves on algebraic stacks} \label{section-quasi-coherent-algebraic-stacks} \noindent Let $\mathcal{X}$ be an algebraic stack over $S$. By Algebraic Stacks, Lemma \ref{algebraic-lemma-stack-presentation} we can find an equivalence $[U/R] \to \mathcal{X}$ where $(U, R, s, t, c)$ is a smooth groupoid in algebraic spaces. Then $$\QCoh(\mathcal{O}_\mathcal{X}) \cong \QCoh(\mathcal{O}_{[U/R]}) \cong \QCoh(U, R, s, t, c)$$ where the second equivalence is Proposition \ref{proposition-quasi-coherent}. Hence the category of quasi-coherent sheaves on an algebraic stack is equivalent to the category of quasi-coherent modules on a smooth groupoid in algebraic spaces. In particular, by Groupoids in Spaces, Lemma \ref{spaces-groupoids-lemma-abelian} we see that $\QCoh(\mathcal{O}_\mathcal{X})$ is abelian! \medskip\noindent There is something slightly disconcerting about our current setup. It is that the fully faithful embedding $$\QCoh(\mathcal{O}_\mathcal{X}) \longrightarrow \textit{Mod}(\mathcal{O}_\mathcal{X})$$ is in general {\bf not} exact. However, exactly the same thing happens for schemes: for most schemes $X$ the embedding $$\QCoh(\mathcal{O}_X) \cong \QCoh((\Sch/X)_{fppf}, \mathcal{O}_X) \longrightarrow \textit{Mod}((\Sch/X)_{fppf}, \mathcal{O}_X)$$ isn't exact, see Descent, Lemma \ref{descent-lemma-equivalence-quasi-coherent-limits}. Parenthetically, the example in the proof of Descent, Lemma \ref{descent-lemma-equivalence-quasi-coherent-limits} shows that in general the strictly full embedding $\QCoh(\mathcal{O}_\mathcal{X}) \to \textit{LQCoh}(\mathcal{O}_\mathcal{X})$ isn't exact either. \medskip\noindent We collect all the positive results obtained so far in a single statement. \begin{lemma} \label{lemma-quasi-coherent-algebraic-stack} Let $\mathcal{X}$ be an algebraic stack over $S$. \begin{enumerate} \item If $[U/R] \to \mathcal{X}$ is a presentation of $\mathcal{X}$ then there is a canonical equivalence $\QCoh(\mathcal{O}_\mathcal{X}) \cong \QCoh(U, R, s, t, c)$. \item The category $\QCoh(\mathcal{O}_\mathcal{X})$ is abelian. \item The category $\QCoh(\mathcal{O}_\mathcal{X})$ has colimits and they agree with colimits in the category $\textit{Mod}(\mathcal{O}_\mathcal{X})$. \item Given $\mathcal{F}, \mathcal{G}$ in $\QCoh(\mathcal{O}_\mathcal{X})$ the tensor product $\mathcal{F} \otimes_{\mathcal{O}_\mathcal{X}} \mathcal{G}$ in $\textit{Mod}(\mathcal{O}_\mathcal{X})$ is an object of $\QCoh(\mathcal{O}_\mathcal{X})$. \item Given $\mathcal{F}, \mathcal{G}$ in $\QCoh(\mathcal{O}_\mathcal{X})$ with $\mathcal{F}$ locally of finite presentation on $\mathcal{X}_{fppf}$ the sheaf $\SheafHom_{\mathcal{O}_\mathcal{X}}(\mathcal{F}, \mathcal{G})$ in $\textit{Mod}(\mathcal{O}_\mathcal{X})$ is an object of $\QCoh(\mathcal{O}_\mathcal{X})$. \end{enumerate} \end{lemma} \begin{proof} Properties (3), (4), and (5) were proven in Lemma \ref{lemma-qc-colimits}. Part (1) is Proposition \ref{proposition-quasi-coherent}. Part (2) follows from Groupoids in Spaces, Lemma \ref{spaces-groupoids-lemma-abelian} as discussed above. \end{proof} \begin{proposition} \label{proposition-coherator} Let $\mathcal{X}$ be an algebraic stack over $S$. \begin{enumerate} \item The category $\QCoh(\mathcal{O}_\mathcal{X})$ is a Grothendieck abelian category. Consequently, $\QCoh(\mathcal{O}_\mathcal{X})$ has enough injectives and all limits. \item The inclusion functor $\QCoh(\mathcal{O}_\mathcal{X}) \to \textit{Mod}(\mathcal{O}_\mathcal{X})$ has a right adjoint\footnote{This functor is sometimes called the {\it coherator}.} $$Q : \textit{Mod}(\mathcal{O}_\mathcal{X}) \longrightarrow \QCoh(\mathcal{O}_\mathcal{X})$$ such that for every quasi-coherent sheaf $\mathcal{F}$ the adjunction mapping $Q(\mathcal{F}) \to \mathcal{F}$ is an isomorphism. \end{enumerate} \end{proposition} \begin{proof} This proof is a repeat of the proof in the case of schemes, see Properties, Proposition \ref{properties-proposition-coherator} and the case of algebraic spaces, see Properties of Spaces, Proposition \ref{spaces-properties-proposition-coherator}. We advise the reader to read either of those proofs first. \medskip\noindent Part (1) means $\QCoh(\mathcal{O}_\mathcal{X})$ (a) has all colimits, (b) filtered colimits are exact, and (c) has a generator, see Injectives, Section \ref{injectives-section-grothendieck-conditions}. By Lemma \ref{lemma-quasi-coherent-algebraic-stack} colimits in $\QCoh(\mathcal{O}_X)$ exist and agree with colimits in $\textit{Mod}(\mathcal{O}_X)$. By Modules on Sites, Lemma \ref{sites-modules-lemma-limits-colimits} filtered colimits are exact. Hence (a) and (b) hold. \medskip\noindent Choose a presentation $\mathcal{X} = [U/R]$ so that $(U, R, s, t, c)$ is a smooth groupoid in algebraic spaces and in particular $s$ and $t$ are flat morphisms of algebraic spaces. By Lemma \ref{lemma-quasi-coherent-algebraic-stack} above we have $\QCoh(\mathcal{O}_\mathcal{X}) = \QCoh(U, R, s, t, c)$. By Groupoids in Spaces, Lemma \ref{spaces-groupoids-lemma-set-generators} there exists a set $T$ and a family $(\mathcal{F}_t)_{t \in T}$ of quasi-coherent sheaves on $\mathcal{X}$ such that every quasi-coherent sheaf on $\mathcal{X}$ is the directed colimit of its subsheaves which are isomorphic to one of the $\mathcal{F}_t$. Thus $\bigoplus_t \mathcal{F}_t$ is a generator of $\QCoh(\mathcal{O}_X)$ and we conclude that (c) holds. The assertions on limits and injectives hold in any Grothendieck abelian category, see Injectives, Theorem \ref{injectives-theorem-injective-embedding-grothendieck} and Lemma \ref{injectives-lemma-grothendieck-products}. \medskip\noindent Proof of (2). To construct $Q$ we use the following general procedure. Given an object $\mathcal{F}$ of $\textit{Mod}(\mathcal{O}_\mathcal{X})$ we consider the functor $$\QCoh(\mathcal{O}_\mathcal{X})^{opp} \longrightarrow \textit{Sets}, \quad \mathcal{G} \longmapsto \Hom_\mathcal{X}(\mathcal{G}, \mathcal{F})$$ This functor transforms colimits into limits, hence is representable, see Injectives, Lemma \ref{injectives-lemma-grothendieck-brown}. Thus there exists a quasi-coherent sheaf $Q(\mathcal{F})$ and a functorial isomorphism $\Hom_\mathcal{X}(\mathcal{G}, \mathcal{F}) = \Hom_\mathcal{X}(\mathcal{G}, Q(\mathcal{F}))$ for $\mathcal{G}$ in $\QCoh(\mathcal{O}_\mathcal{X})$. By the Yoneda lemma (Categories, Lemma \ref{categories-lemma-yoneda}) the construction $\mathcal{F} \leadsto Q(\mathcal{F})$ is functorial in $\mathcal{F}$. By construction $Q$ is a right adjoint to the inclusion functor. The fact that $Q(\mathcal{F}) \to \mathcal{F}$ is an isomorphism when $\mathcal{F}$ is quasi-coherent is a formal consequence of the fact that the inclusion functor $\QCoh(\mathcal{O}_\mathcal{X}) \to \textit{Mod}(\mathcal{O}_\mathcal{X})$ is fully faithful. \end{proof} \section{Cohomology} \label{section-cohomology-general} \noindent Let $S$ be a scheme and let $\mathcal{X}$ be a category fibred in groupoids over $(\Sch/S)_{fppf}$. For any $\tau \in \{Zariski, \etale, smooth, syntomic, fppf\}$ the categories $\textit{Ab}(\mathcal{X}_\tau)$ and $\textit{Mod}(\mathcal{X}_\tau, \mathcal{O}_\mathcal{X})$ have enough injectives, see Injectives, Theorems \ref{injectives-theorem-sheaves-injectives} and \ref{injectives-theorem-sheaves-modules-injectives}. Thus we can use the machinery of Cohomology on Sites, Section \ref{sites-cohomology-section-cohomology-sheaves} to define the cohomology groups $$H^p(\mathcal{X}_\tau, \mathcal{F}) = H^p_\tau(\mathcal{X}, \mathcal{F}) \quad\text{and}\quad H^p(x, \mathcal{F}) = H^p_\tau(x, \mathcal{F})$$ for any $x \in \Ob(\mathcal{X})$ and any object $\mathcal{F}$ of $\textit{Ab}(\mathcal{X}_\tau)$ or $\textit{Mod}(\mathcal{X}_\tau, \mathcal{O}_\mathcal{X})$. Moreover, if $f : \mathcal{X} \to \mathcal{Y}$ is a $1$-morphism of categories fibred in groupoids over $(\Sch/S)_{fppf}$, then we obtain the higher direct images $R^if_*\mathcal{F}$ in $\textit{Ab}(\mathcal{Y}_\tau)$ or $\textit{Mod}(\mathcal{Y}_\tau, \mathcal{O}_\mathcal{Y})$. Of course, as explained in Cohomology on Sites, Section \ref{sites-cohomology-section-derived-functors} there are also derived versions of $H^p(-)$ and $R^if_*$. \begin{lemma} \label{lemma-cohomology-restriction} Let $S$ be a scheme. Let $\mathcal{X}$ be a category fibred in groupoids over $(\Sch/S)_{fppf}$. Let $\tau \in \{Zariski, \etale, smooth, syntomic, fppf\}$. Let $x \in \Ob(\mathcal{X})$ be an object lying over the scheme $U$. Let $\mathcal{F}$ be an object of $\textit{Ab}(\mathcal{X}_\tau)$ or $\textit{Mod}(\mathcal{X}_\tau, \mathcal{O}_\mathcal{X})$. Then $$H^p_\tau(x, \mathcal{F}) = H^p((\Sch/U)_\tau, x^{-1}\mathcal{F})$$ and if $\tau = \etale$, then we also have $$H^p_\etale(x, \mathcal{F}) = H^p(U_\etale, \mathcal{F}|_{U_\etale}).$$ \end{lemma} \begin{proof} The first statement follows from Cohomology on Sites, Lemma \ref{sites-cohomology-lemma-cohomology-of-open} and the equivalence of Lemma \ref{lemma-localizing-structure-sheaf}. The second statement follows from the first combined with \'Etale Cohomology, Lemma \ref{etale-cohomology-lemma-compare-cohomology-big-small}. \end{proof} \section{Injective sheaves} \label{section-lower-shriek} \noindent The pushforward of an injective abelian sheaf or module is injective. \begin{lemma} \label{lemma-pushforward-injective} Let $f : \mathcal{X} \to \mathcal{Y}$ be a $1$-morphism of categories fibred in groupoids over $(\Sch/S)_{fppf}$. Let $\tau \in \{Zar, \etale, smooth, syntomic, fppf\}$. \begin{enumerate} \item $f_*\mathcal{I}$ is injective in $\textit{Ab}(\mathcal{Y}_\tau)$ for $\mathcal{I}$ injective in $\textit{Ab}(\mathcal{X}_\tau)$, and \item $f_*\mathcal{I}$ is injective in $\textit{Mod}(\mathcal{Y}_\tau, \mathcal{O}_\mathcal{Y})$ for $\mathcal{I}$ injective in $\textit{Mod}(\mathcal{X}_\tau, \mathcal{O}_\mathcal{X})$. \end{enumerate} \end{lemma} \begin{proof} This follows formally from the fact that $f^{-1}$ is an exact left adjoint of $f_*$, see Homology, Lemma \ref{homology-lemma-adjoint-preserve-injectives}. \end{proof} \noindent In the rest of this section we prove that pullback $f^{-1}$ has a left adjoint $f_!$ on abelian sheaves and modules. If $f$ is representable (by schemes or by algebraic spaces), then it will turn out that $f_!$ is exact and $f^{-1}$ will preserve injectives. We first prove a few preliminary lemmas about fibre products and equalizers in categories fibred in groupoids and their behaviour with respect to morphisms. \begin{lemma} \label{lemma-fibre-products} Let $p : \mathcal{X} \to (\Sch/S)_{fppf}$ be a category fibred in groupoids. \begin{enumerate} \item The category $\mathcal{X}$ has fibre products. \item If the $\mathit{Isom}$-presheaves of $\mathcal{X}$ are representable by algebraic spaces, then $\mathcal{X}$ has equalizers. \item If $\mathcal{X}$ is an algebraic stack (or more generally a quotient stack), then $\mathcal{X}$ has equalizers. \end{enumerate} \end{lemma} \begin{proof} Part (1) follows Categories, Lemma \ref{categories-lemma-fibred-groupoids-fibre-product-goes-up} as $(\Sch/S)_{fppf}$ has fibre products. \medskip\noindent Let $a, b : x \to y$ be morphisms of $\mathcal{X}$. Set $U = p(x)$ and $V = p(y)$. The category of schemes has equalizers hence we can let $W \to U$ be the equalizer of $p(a)$ and $p(b)$. Denote $c : z \to x$ a morphism of $\mathcal{X}$ lying over $W \to U$. The equalizer of $a$ and $b$, if it exists, is the equalizer of $a \circ c$ and $b \circ c$. Thus we may assume that $p(a) = p(b) = f : U \to V$. As $\mathcal{X}$ is fibred in groupoids, there exists a unique automorphism $i : x \to x$ in the fibre category of $\mathcal{X}$ over $U$ such that $a \circ i = b$. Again the equalizer of $a$ and $b$ is the equalizer of $\text{id}_x$ and $i$. Recall that the $\mathit{Isom}_\mathcal{X}(x)$ is the presheaf on $(\Sch/U)_{fppf}$ which to $V/U$ associates the set of automorphisms of $x|_V$ in the fibre category of $\mathcal{X}$ over $V$, see Stacks, Definition \ref{stacks-definition-mor-presheaf}. If $\mathit{Isom}_\mathcal{X}(x)$ is representable by an algebraic space $G \to U$, then we see that $\text{id}_x$ and $i$ define morphisms $e, i : U \to G$ over $U$. Set $V = U \times_{e, G, i} U$, which by Morphisms of Spaces, Lemma \ref{spaces-morphisms-lemma-section-immersion} is a scheme. Then it is clear that $x|_V \to x$ is the equalizer of the maps $\text{id}_x$ and $i$ in $\mathcal{X}$. This proves (2). \medskip\noindent If $\mathcal{X} = [U/R]$ for some groupoid in algebraic spaces $(U, R, s, t, c)$ over $S$, then the hypothesis of (2) holds by Bootstrap, Lemma \ref{bootstrap-lemma-quotient-stack-isom}. If $\mathcal{X}$ is an algebraic stack, then we can choose a presentation $[U/R] \cong \mathcal{X}$ by Algebraic Stacks, Lemma \ref{algebraic-lemma-stack-presentation}. \end{proof} \begin{lemma} \label{lemma-fibre-products-morphism} Let $f : \mathcal{X} \to \mathcal{Y}$ be a $1$-morphism of categories fibred in groupoids over $(\Sch/S)_{fppf}$. \begin{enumerate} \item The functor $f$ transforms fibre products into fibre products. \item If $f$ is faithful, then $f$ transforms equalizers into equalizers. \end{enumerate} \end{lemma} \begin{proof} By Categories, Lemma \ref{categories-lemma-fibred-groupoids-fibre-product-goes-up} we see that a fibre product in $\mathcal{X}$ is any commutative square lying over a fibre product diagram in $(\Sch/S)_{fppf}$. Similarly for $\mathcal{Y}$. Hence (1) is clear. \medskip\noindent Let $x \to x'$ be the equalizer of two morphisms $a, b : x' \to x''$ in $\mathcal{X}$. We will show that $f(x) \to f(x')$ is the equalizer of $f(a)$ and $f(b)$. Let $y \to f(x)$ be a morphism of $\mathcal{Y}$ equalizing $f(a)$ and $f(b)$. Say $x, x', x''$ lie over the schemes $U, U', U''$ and $y$ lies over $V$. Denote $h : V \to U'$ the image of $y \to f(x)$ in the category of schemes. The morphism $y \to f(x)$ is isomorphic to $f(h^*x') \to f(x')$ by the axioms of fibred categories. Hence, as $f$ is faithful, we see that $h^*x' \to x'$ equalizes $a$ and $b$. Thus we obtain a unique morphism $h^*x' \to x$ whose image $y = f(h^*x') \to f(x)$ is the desired morphism in $\mathcal{Y}$. \end{proof} \begin{lemma} \label{lemma-fibre-products-preserve-properties} Let $f : \mathcal{X} \to \mathcal{Y}$, $g : \mathcal{Z} \to \mathcal{Y}$ be faithful $1$-morphisms of categories fibred in groupoids over $(\Sch/S)_{fppf}$. \begin{enumerate} \item the functor $\mathcal{X} \times_\mathcal{Y} \mathcal{Z} \to \mathcal{Y}$ is faithful, and \item if $\mathcal{X}, \mathcal{Z}$ have equalizers, so does $\mathcal{X} \times_\mathcal{Y} \mathcal{Z}$. \end{enumerate} \end{lemma} \begin{proof} We think of objects in $\mathcal{X} \times_\mathcal{Y} \mathcal{Z}$ as quadruples $(U, x, z, \alpha)$ where $\alpha : f(x) \to g(z)$ is an isomorphism over $U$, see Categories, Lemma \ref{categories-lemma-2-product-categories-over-C}. A morphism $(U, x, z, \alpha) \to (U', x', z', \alpha')$ is a pair of morphisms $a : x \to x'$ and $b : z \to z'$ compatible with $\alpha$ and $\alpha'$. Thus it is clear that if $f$ and $g$ are faithful, so is the functor $\mathcal{X} \times_\mathcal{Y} \mathcal{Z} \to \mathcal{Y}$. Now, suppose that $(a, b), (a', b') : (U, x, z, \alpha) \to (U', x', z', \alpha')$ are two morphisms of the $2$-fibre product. Then consider the equalizer $x'' \to x$ of $a$ and $a'$ and the equalizer $z'' \to z$ of $b$ and $b'$. Since $f$ commutes with equalizers (by Lemma \ref{lemma-fibre-products-morphism}) we see that $f(x'') \to f(x)$ is the equalizer of $f(a)$ and $f(a')$. Similarly, $g(z'') \to g(z)$ is the equalizer of $g(b)$ and $g(b')$. Picture $$\xymatrix{ f(x'') \ar[r] \ar@{..>}[d]_{\alpha''}& f(x) \ar[d]_\alpha \ar@<0.5ex>[r]^{f(a)} \ar@<-0.5ex>[r]_{f(a')} & f(x') \ar[d]^{\alpha'} \\ g(z'') \ar[r] & g(z) \ar@<0.5ex>[r]^{g(b)} \ar@<-0.5ex>[r]_{g(b')} & g(z') }$$ It is clear that the dotted arrow exists and is an isomorphism. However, it is not a priori the case that the image of $\alpha''$ in the category of schemes is the identity of its source. On the other hand, the existence of $\alpha''$ means that we can assume that $x''$ and $z''$ are defined over the same scheme and that the morphisms $x'' \to x$ and $z'' \to z$ have the same image in the category of schemes. Redoing the diagram above we see that the dotted arrow now does project to an identity morphism and we win. Some details omitted. \end{proof} \noindent As we are working with big sites we have the following somewhat counter intuitive result (which also holds for morphisms of big sites of schemes). Warning: This result isn't true if we drop the hypothesis that $f$ is faithful. \begin{lemma} \label{lemma-pullback-injective} Let $f : \mathcal{X} \to \mathcal{Y}$ be a $1$-morphism of categories fibred in groupoids over $(\Sch/S)_{fppf}$. Let $\tau \in \{Zar, \etale, smooth, syntomic, fppf\}$. The functor $f^{-1} : \textit{Ab}(\mathcal{Y}_\tau) \to \textit{Ab}(\mathcal{X}_\tau)$ has a left adjoint $f_! : \textit{Ab}(\mathcal{X}_\tau) \to \textit{Ab}(\mathcal{Y}_\tau)$. If $f$ is faithful and $\mathcal{X}$ has equalizers, then \begin{enumerate} \item $f_!$ is exact, and \item $f^{-1}\mathcal{I}$ is injective in $\textit{Ab}(\mathcal{X}_\tau)$ for $\mathcal{I}$ injective in $\textit{Ab}(\mathcal{Y}_\tau)$. \end{enumerate} \end{lemma} \begin{proof} By Stacks, Lemma \ref{stacks-lemma-topology-inherited-functorial} the functor $f$ is continuous and cocontinuous. Hence by Modules on Sites, Lemma \ref{sites-modules-lemma-g-shriek-adjoint} the functor $f^{-1} : \textit{Ab}(\mathcal{Y}_\tau) \to \textit{Ab}(\mathcal{X}_\tau)$ has a left adjoint $f_! : \textit{Ab}(\mathcal{X}_\tau) \to \textit{Ab}(\mathcal{Y}_\tau)$. To see (1) we apply Modules on Sites, Lemma \ref{sites-modules-lemma-exactness-lower-shriek} and to see that the hypotheses of that lemma are satisfied use Lemmas \ref{lemma-fibre-products} and \ref{lemma-fibre-products-morphism} above. Part (2) follows from this formally, see Homology, Lemma \ref{homology-lemma-adjoint-preserve-injectives}. \end{proof} \begin{lemma} \label{lemma-pullback-injective-modules} Let $f : \mathcal{X} \to \mathcal{Y}$ be a $1$-morphism of categories fibred in groupoids over $(\Sch/S)_{fppf}$. Let $\tau \in \{Zar, \etale, smooth, syntomic, fppf\}$. The functor $f^* : \textit{Mod}(\mathcal{Y}_\tau, \mathcal{O}_\mathcal{Y}) \to \textit{Mod}(\mathcal{X}_\tau, \mathcal{O}_\mathcal{X})$ has a left adjoint $f_! : \textit{Mod}(\mathcal{X}_\tau, \mathcal{O}_\mathcal{X}) \to \textit{Mod}(\mathcal{Y}_\tau, \mathcal{O}_\mathcal{Y})$ which agrees with the functor $f_!$ of Lemma \ref{lemma-pullback-injective} on underlying abelian sheaves. If $f$ is faithful and $\mathcal{X}$ has equalizers, then \begin{enumerate} \item $f_!$ is exact, and \item $f^{-1}\mathcal{I}$ is injective in $\textit{Mod}(\mathcal{X}_\tau, \mathcal{O}_\mathcal{X})$ for $\mathcal{I}$ injective in $\textit{Mod}(\mathcal{Y}_\tau, \mathcal{O}_\mathcal{X})$. \end{enumerate} \end{lemma} \begin{proof} Recall that $f$ is a continuous and cocontinuous functor of sites and that $f^{-1}\mathcal{O}_\mathcal{Y} = \mathcal{O}_\mathcal{X}$. Hence Modules on Sites, Lemma \ref{sites-modules-lemma-lower-shriek-modules} implies $f^*$ has a left adjoint $f_!^{Mod}$. Let $x$ be an object of $\mathcal{X}$ lying over the scheme $U$. Then $f$ induces an equivalence of ringed sites $$\mathcal{X}/x \longrightarrow \mathcal{Y}/f(x)$$ as both sides are equivalent to $(\Sch/U)_\tau$, see Lemma \ref{lemma-localizing-structure-sheaf}. Modules on Sites, Remark \ref{sites-modules-remark-when-shriek-equal} shows that $f_!$ agrees with the functor on abelian sheaves. \medskip\noindent Assume now that $\mathcal{X}$ has equalizers and that $f$ is faithful. Lemma \ref{lemma-pullback-injective} tells us that $f_!$ is exact. Finally, Homology, Lemma \ref{homology-lemma-adjoint-preserve-injectives} implies the statement on pullbacks of injective modules. \end{proof} \section{The {\v C}ech complex} \label{section-cech} \noindent To compute the cohomology of a sheaf on an algebraic stack we compare it to the cohomology of the sheaf restricted to coverings of the given algebraic stack. \medskip\noindent Throughout this section the situation will be as follows. We are given a $1$-morphism of categories fibred in groupoids \begin{equation} \label{equation-covering} \vcenter{ \xymatrix{ \mathcal{U} \ar[rr]_f \ar[rd]_q & & \mathcal{X} \ar[ld]^p \\ & (\Sch/S)_{fppf} } } \end{equation} We are going to think about $\mathcal{U}$ as a covering'' of $\mathcal{X}$. Hence we want to consider the simplicial object $$\xymatrix{ \mathcal{U} \times_\mathcal{X} \mathcal{U} \times_\mathcal{X} \mathcal{U} \ar@<1ex>[r] \ar@<0ex>[r] \ar@<-1ex>[r] & \mathcal{U} \times_\mathcal{X} \mathcal{U} \ar@<0.5ex>[r] \ar@<-0.5ex>[r] & \mathcal{U} }$$ in the category of categories fibred in groupoids over $(\Sch/S)_{fppf}$. However, since this is a $(2, 1)$-category and not a category, we should say explicitly what we mean. Namely, we let $\mathcal{U}_n$ be the category with objects $(u_0, \ldots, u_n, x, \alpha_0, \ldots, \alpha_n)$ where $\alpha_i : f(u_i) \to x$ is an isomorphism in $\mathcal{X}$. We denote $f_n : \mathcal{U}_n \to \mathcal{X}$ the $1$-morphism which assigns to $(u_0, \ldots, u_n, x, \alpha_0, \ldots, \alpha_n)$ the object $x$. Note that $\mathcal{U}_0 = \mathcal{U}$ and $f_0 = f$. Given a map $\varphi : [m] \to [n]$ we consider the $1$-morphism $\mathcal{U}_\varphi : \mathcal{U}_n \longrightarrow \mathcal{U}_n$ given by $$(u_0, \ldots, u_n, x, \alpha_0, \ldots, \alpha_n) \longmapsto (u_{\varphi(0)}, \ldots, u_{\varphi(n)}, x, \alpha_{\varphi(0)}, \ldots, \alpha_{\varphi(n)})$$ on objects. All of these $1$-morphisms compose correctly on the nose (no $2$-morphisms required) and all of these $1$-morphisms are $1$-morphisms over $\mathcal{X}$. We denote $\mathcal{U}_\bullet$ this simplicial object. If $\mathcal{F}$ is a presheaf of sets on $\mathcal{X}$, then we obtain a cosimplicial set $$\xymatrix{ \Gamma(\mathcal{U}_0, f_0^{-1}\mathcal{F}) \ar@<0.5ex>[r] \ar@<-0.5ex>[r] & \Gamma(\mathcal{U}_1, f_1^{-1}\mathcal{F}) \ar@<1ex>[r] \ar@<0ex>[r] \ar@<-1ex>[r] & \Gamma(\mathcal{U}_2, f_2^{-1}\mathcal{F}) }$$ Here the arrows are the pullback maps along the given morphisms of the simplicial object. If $\mathcal{F}$ is a presheaf of abelian groups, this is a cosimplicial abelian group. \medskip\noindent Let $\mathcal{U} \to \mathcal{X}$ be as above and let $\mathcal{F}$ be an abelian presheaf on $\mathcal{X}$. The {\it {\v C}ech complex} associated to the situation is denoted $\check{\mathcal{C}}^\bullet(\mathcal{U} \to \mathcal{X}, \mathcal{F})$. It is the cochain complex associated to the cosimplicial abelian group above, see Simplicial, Section \ref{simplicial-section-dold-kan-cosimplicial}. It has terms $$\check{\mathcal{C}}^n(\mathcal{U} \to \mathcal{X}, \mathcal{F}) = \Gamma(\mathcal{U}_n, f_n^{-1}\mathcal{F}).$$ The boundary maps are the maps $$d^n = \sum\nolimits_{i = 0}^{n + 1} (-1)^i \delta^{n + 1}_i : \Gamma(\mathcal{U}_n, f_n^{-1}\mathcal{F}) \longrightarrow \Gamma(\mathcal{U}_{n + 1}, f_{n + 1}^{-1}\mathcal{F})$$ where $\delta^{n + 1}_i$ corresponds to the map $[n] \to [n + 1]$ omitting the index $i$. Note that the map $\Gamma(\mathcal{X}, \mathcal{F}) \to \Gamma(\mathcal{U}_0, f_0^{-1}\mathcal{F}_0)$ is in the kernel of the differential $d^0$. Hence we define the {\it extended {\v C}ech complex} to be the complex $$\ldots \to 0 \to \Gamma(\mathcal{X}, \mathcal{F}) \to \Gamma(\mathcal{U}_0, f_0^{-1}\mathcal{F}_0) \to \Gamma(\mathcal{U}_1, f_1^{-1}\mathcal{F}_1) \to \ldots$$ with $\Gamma(\mathcal{X}, \mathcal{F})$ placed in degree $-1$. The extended {\v C}ech complex is acyclic if and only if the canonical map $$\Gamma(\mathcal{X}, \mathcal{F})[0] \longrightarrow \check{\mathcal{C}}^\bullet(\mathcal{U} \to \mathcal{X}, \mathcal{F})$$ is a quasi-isomorphism of complexes. \begin{lemma} \label{lemma-generalities} Generalities on {\v C}ech complexes. \begin{enumerate} \item If $$\xymatrix{ \mathcal{V} \ar[d]_g \ar[r]_h & \mathcal{U} \ar[d]^f \\ \mathcal{Y} \ar[r]^e & \mathcal{X} }$$ is $2$-commutative diagram of categories fibred in groupoids over $(\Sch/S)_{fppf}$, then there is a morphism of {\v C}ech complexes $$\check{\mathcal{C}}^\bullet(\mathcal{U} \to \mathcal{X}, \mathcal{F}) \longrightarrow \check{\mathcal{C}}^\bullet(\mathcal{V} \to \mathcal{Y}, e^{-1}\mathcal{F})$$ \item if $h$ and $e$ are equivalences, then the map of (1) is an isomorphism, \item if $f, f' : \mathcal{U} \to \mathcal{X}$ are $2$-isomorphic, then the associated {\v C}ech complexes are isomorphic, \end{enumerate} \end{lemma} \begin{proof} In the situation of (1) let $t : f \circ h \to e \circ g$ be a $2$-morphism. The map on complexes is given in degree $n$ by pullback along the $1$-morphisms $\mathcal{V}_n \to \mathcal{U}_n$ given by the rule $$(v_0, \ldots, v_n, y, \beta_0, \ldots, \beta_n) \longmapsto (h(v_0), \ldots, h(v_n), e(y), e(\beta_0) \circ t_{v_0}, \ldots, e(\beta_n) \circ t_{v_n}).$$ For (2), note that pullback on global sections is an isomorphism for any presheaf of sets when the pullback is along an equivalence of categories. Part (3) follows on combining (1) and (2). \end{proof} \begin{lemma} \label{lemma-homotopy} If there exists a $1$-morphism $s : \mathcal{X} \to \mathcal{U}$ such that $f \circ s$ is $2$-isomorphic to $\text{id}_\mathcal{X}$ then the extended {\v C}ech complex is homotopic to zero. \end{lemma} \begin{proof} Set $\mathcal{U}' = \mathcal{U} \times_\mathcal{X} \mathcal{X}$ equal to the fibre product as described in Categories, Lemma \ref{categories-lemma-2-product-categories-over-C}. Set $f' : \mathcal{U}' \to \mathcal{X}$ equal to the second projection. Then $\mathcal{U} \to \mathcal{U}'$, $u \mapsto (u, f(x), 1)$ is an equivalence over $\mathcal{X}$, hence we may replace $(\mathcal{U}, f)$ by $(\mathcal{U}', f')$ by Lemma \ref{lemma-generalities}. The advantage of this is that now $f'$ has a section $s'$ such that $f' \circ s' = \text{id}_\mathcal{X}$ on the nose. Namely, if $t : s \circ f \to \text{id}_\mathcal{X}$ is a $2$-isomorphism then we can set $s'(x) = (s(x), x, t_x)$. Thus we may assume that $f \circ s = \text{id}_\mathcal{X}$. \medskip\noindent In the case that $f \circ s = \text{id}_\mathcal{X}$ the result follows from general principles. We give the homotopy explicitly. Namely, for $n \geq 0$ define $s_n : \mathcal{U}_n \to \mathcal{U}_{n + 1}$ to be the $1$-morphism defined by the rule on objects $$(u_0, \ldots, u_n, x, \alpha_0, \ldots, \alpha_n) \longmapsto (u_0, \ldots, u_n, s(x), x, \alpha_0, \ldots, \alpha_n, \text{id}_x).$$ Define $$h^{n + 1} : \Gamma(\mathcal{U}_{n + 1}, f_{n + 1}^{-1}\mathcal{F}) \longrightarrow \Gamma(\mathcal{U}_n, f_n^{-1}\mathcal{F})$$ as pullback along $s_n$. We also set $s_{-1} = s$ and $h^0 : \Gamma(\mathcal{U}_0, f_0^{-1}\mathcal{F}) \to \Gamma(\mathcal{X}, \mathcal{F})$ equal to pullback along $s_{-1}$. Then the family of maps $\{h^n\}_{n \geq 0}$ is a homotopy between $1$ and $0$ on the extended {\v C}ech complex. \end{proof} \section{The relative {\v C}ech complex} \label{section-sheaf-cech-complex} \noindent Let $f : \mathcal{U} \to \mathcal{X}$ be a $1$-morphism of categories fibred in groupoids over $(\Sch/S)_{fppf}$ as in (\ref{equation-covering}). Consider the associated simplicial object $\mathcal{U}_\bullet$ and the maps $f_n : \mathcal{U}_n \to \mathcal{X}$. Let $\tau \in \{Zar, \etale, smooth, syntomic, fppf\}$. Finally, suppose that $\mathcal{F}$ is a sheaf (of sets) on $\mathcal{X}_\tau$. Then $$\xymatrix{ f_{0, *}f_0^{-1}\mathcal{F} \ar@<0.5ex>[r] \ar@<-0.5ex>[r] & f_{1, *}f_1^{-1}\mathcal{F} \ar@<1ex>[r] \ar@<0ex>[r] \ar@<-1ex>[r] & f_{2, *}f_2^{-1}\mathcal{F} }$$ is a cosimplicial sheaf on $\mathcal{X}_\tau$ where we use the pullback maps introduced in Sites, Section \ref{sites-section-pullback}. If $\mathcal{F}$ is an abelian sheaf, then $f_{n, *}f_n^{-1}\mathcal{F}$ form a cosimplicial abelian sheaf on $\mathcal{X}_\tau$. The associated complex (see Simplicial, Section \ref{simplicial-section-dold-kan-cosimplicial}) $$\ldots \to 0 \to f_{0, *}f_0^{-1}\mathcal{F} \to f_{1, *}f_1^{-1}\mathcal{F} \to f_{2, *}f_2^{-1}\mathcal{F} \to \ldots$$ is called the {\it relative {\v C}ech complex} associated to the situation. We will denote this complex $\mathcal{K}^\bullet(f, \mathcal{F})$. The {\it extended relative {\v C}ech complex} is the complex $$\ldots \to 0 \to \mathcal{F} \to f_{0, *}f_0^{-1}\mathcal{F} \to f_{1, *}f_1^{-1}\mathcal{F} \to f_{2, *}f_2^{-1}\mathcal{F} \to \ldots$$ with $\mathcal{F}$ in degree $-1$. The extended relative {\v C}ech complex is acyclic if and only if the map $\mathcal{F}[0] \to \mathcal{K}^\bullet(f, \mathcal{F})$ is a quasi-isomorphism of complexes of sheaves. \begin{remark} \label{remark-cech-complex-presheaves} We can define the complex $\mathcal{K}^\bullet(f, \mathcal{F})$ also if $\mathcal{F}$ is a presheaf, only we cannot use the reference to Sites, Section \ref{sites-section-pullback} to define the pullback maps. To explain the pullback maps, suppose given a commutative diagram $$\xymatrix{ \mathcal{V} \ar[rd]_g \ar[rr]_h & & \mathcal{U} \ar[ld]^f \\ & \mathcal{X} }$$ of categories fibred in groupoids over $(\Sch/S)_{fppf}$ and a presheaf $\mathcal{G}$ on $\mathcal{U}$ we can define the pullback map $f_*\mathcal{G} \to g_*h^{-1}\mathcal{G}$ as the composition $$f_*\mathcal{G} \longrightarrow f_*h_*h^{-1}\mathcal{G} = g_*h^{-1}\mathcal{G}$$ where the map comes from the adjunction map $\mathcal{G} \to h_*h^{-1}\mathcal{G}$. This works because in our situation the functors $h_*$ and $h^{-1}$ are adjoint in presheaves (and agree with their counter parts on sheaves). See Sections \ref{section-presheaves} and \ref{section-sheaves}. \end{remark} \begin{lemma} \label{lemma-generalities-sheafified} Generalities on relative {\v C}ech complexes. \begin{enumerate} \item If $$\xymatrix{ \mathcal{V} \ar[d]_g \ar[r]_h & \mathcal{U} \ar[d]^f \\ \mathcal{Y} \ar[r]^e & \mathcal{X} }$$ is $2$-commutative diagram of categories fibred in groupoids over $(\Sch/S)_{fppf}$, then there is a morphism $e^{-1}\mathcal{K}^\bullet(f, \mathcal{F}) \to \mathcal{K}^\bullet(g, e^{-1}\mathcal{F})$. \item if $h$ and $e$ are equivalences, then the map of (1) is an isomorphism, \item if $f, f' : \mathcal{U} \to \mathcal{X}$ are $2$-isomorphic, then the associated relative {\v C}ech complexes are isomorphic, \end{enumerate} \end{lemma} \begin{proof} Literally the same as the proof of Lemma \ref{lemma-generalities} using the pullback maps of Remark \ref{remark-cech-complex-presheaves}. \end{proof} \begin{lemma} \label{lemma-homotopy-sheafified} If there exists a $1$-morphism $s : \mathcal{X} \to \mathcal{U}$ such that $f \circ s$ is $2$-isomorphic to $\text{id}_\mathcal{X}$ then the extended relative {\v C}ech complex is homotopic to zero. \end{lemma} \begin{proof} Literally the same as the proof of Lemma \ref{lemma-homotopy}. \end{proof} \begin{remark} \label{remark-cech-complex-sections} Let us compute'' the value of the relative {\v C}ech complex on an object $x$ of $\mathcal{X}$. Say $p(x) = U$. Consider the $2$-fibre product diagram (which serves to introduce the notation $g : \mathcal{V} \to \mathcal{Y}$) $$\xymatrix{ \mathcal{V} \ar@{=}[r] \ar[d]_g & (\Sch/U)_{fppf} \times_{x, \mathcal{X}} \mathcal{U} \ar[r] \ar[d] & \mathcal{U} \ar[d]^f \\ \mathcal{Y} \ar@{=}[r] & (\Sch/U)_{fppf} \ar[r]^-x & \mathcal{X} }$$ Note that the morphism $\mathcal{V}_n \to \mathcal{U}_n$ of the proof of Lemma \ref{lemma-generalities} induces an equivalence $\mathcal{V}_n = (\Sch/U)_{fppf} \times_{x, \mathcal{X}} \mathcal{U}_n$. Hence we see from (\ref{equation-pushforward}) that $$\Gamma(x, \mathcal{K}^\bullet(f, \mathcal{F})) = \check{\mathcal{C}}^\bullet(\mathcal{V} \to \mathcal{Y}, x^{-1}\mathcal{F})$$ In words: The value of the relative {\v C}ech complex on an object $x$ of $\mathcal{X}$ is the {\v C}ech complex of the base change of $f$ to $\mathcal{X}/x \cong (\Sch/U)_{fppf}$. This implies for example that Lemma \ref{lemma-homotopy} implies Lemma \ref{lemma-homotopy-sheafified} and more generally that results on the (usual) {\v C}ech complex imply results for the relative {\v C}ech complex. \end{remark} \begin{lemma} \label{lemma-base-change-cech-complex} Let $$\xymatrix{ \mathcal{V} \ar[d]_g \ar[r]_h & \mathcal{U} \ar[d]^f \\ \mathcal{Y} \ar[r]^e & \mathcal{X} }$$ be a $2$-fibre product of categories fibred in groupoids over $(\Sch/S)_{fppf}$ and let $\mathcal{F}$ be an abelian presheaf on $\mathcal{X}$. Then the map $e^{-1}\mathcal{K}^\bullet(f, \mathcal{F}) \to \mathcal{K}^\bullet(g, e^{-1}\mathcal{F})$ of Lemma \ref{lemma-generalities-sheafified} is an isomorphism of complexes of abelian presheaves. \end{lemma} \begin{proof} Let $y$ be an object of $\mathcal{Y}$ lying over the scheme $T$. Set $x = e(y)$. We are going to show that the map induces an isomorphism on sections over $y$. Note that $$\Gamma(y, e^{-1}\mathcal{K}^\bullet(f, \mathcal{F})) = \Gamma(x, \mathcal{K}^\bullet(f, \mathcal{F})) = \check{\mathcal{C}}^\bullet( (\Sch/T)_{fppf} \times_{x, \mathcal{X}} \mathcal{U} \to (\Sch/T)_{fppf}, x^{-1}\mathcal{F})$$ by Remark \ref{remark-cech-complex-sections}. On the other hand, $$\Gamma(y, \mathcal{K}^\bullet(g, e^{-1}\mathcal{F})) = \check{\mathcal{C}}^\bullet( (\Sch/T)_{fppf} \times_{y, \mathcal{Y}} \mathcal{V} \to (\Sch/T)_{fppf}, y^{-1}e^{-1}\mathcal{F})$$ also by Remark \ref{remark-cech-complex-sections}. Note that $y^{-1}e^{-1}\mathcal{F} = x^{-1}\mathcal{F}$ and since the diagram is $2$-cartesian the $1$-morphism $$(\Sch/T)_{fppf} \times_{y, \mathcal{Y}} \mathcal{V} \to (\Sch/T)_{fppf} \times_{x, \mathcal{X}} \mathcal{U}$$ is an equivalence. Hence the map on sections over $y$ is an isomorphism by Lemma \ref{lemma-generalities}. \end{proof} \noindent Exactness can be checked on a covering''. \begin{lemma} \label{lemma-check-exactness-covering} Let $f : \mathcal{U} \to \mathcal{X}$ be a $1$-morphism of categories fibred in groupoids over $(\Sch/S)_{fppf}$. Let $\tau \in \{Zar, \etale, smooth, syntomic, fppf\}$. Let $$\mathcal{F} \to \mathcal{G} \to \mathcal{H}$$ be a complex in $\textit{Ab}(\mathcal{X}_\tau)$. Assume that \begin{enumerate} \item for every object $x$ of $\mathcal{X}$ there exists a covering $\{x_i \to x\}$ in $\mathcal{X}_\tau$ such that each $x_i$ is isomorphic to $f(u_i)$ for some object $u_i$ of $\mathcal{U}$, and \item $f^{-1}\mathcal{F} \to f^{-1}\mathcal{G} \to f^{-1}\mathcal{H}$ is exact. \end{enumerate} Then the sequence $\mathcal{F} \to \mathcal{G} \to \mathcal{H}$ is exact. \end{lemma} \begin{proof} Let $x$ be an object of $\mathcal{X}$ lying over the scheme $T$. Consider the sequence $x^{-1}\mathcal{F} \to x^{-1}\mathcal{G} \to x^{-1}\mathcal{H}$ of abelian sheaves on $(\Sch/T)_\tau$. It suffices to show this sequence is exact. By assumption there exists a $\tau$-covering $\{T_i \to T\}$ such that $x|_{T_i}$ is isomorphic to $f(u_i)$ for some object $u_i$ of $\mathcal{U}$ over $T_i$ and moreover the sequence $u_i^{-1}f^{-1}\mathcal{F} \to u_i^{-1}f^{-1}\mathcal{G} \to u_i^{-1}f^{-1}\mathcal{H}$ of abelian sheaves on $(\Sch/T_i)_\tau$ is exact. Since $u_i^{-1}f^{-1}\mathcal{F} = x^{-1}\mathcal{F}|_{(\Sch/T_i)_\tau}$ we conclude that the sequence $x^{-1}\mathcal{F} \to x^{-1}\mathcal{G} \to x^{-1}\mathcal{H}$ become exact after localizing at each of the members of a covering, hence the sequence is exact. \end{proof} \begin{proposition} \label{proposition-exactness-cech-complex} Let $f : \mathcal{U} \to \mathcal{X}$ be a $1$-morphism of categories fibred in groupoids over $(\Sch/S)_{fppf}$. Let $\tau \in \{Zar, \etale, smooth, syntomic, fppf\}$. If \begin{enumerate} \item $\mathcal{F}$ is an abelian sheaf on $\mathcal{X}_\tau$, and \item for every object $x$ of $\mathcal{X}$ there exists a covering $\{x_i \to x\}$ in $\mathcal{X}_\tau$ such that each $x_i$ is isomorphic to $f(u_i)$ for some object $u_i$ of $\mathcal{U}$, \end{enumerate} then the extended relative {\v C}ech complex $$\ldots \to 0 \to \mathcal{F} \to f_{0, *}f_0^{-1}\mathcal{F} \to f_{1, *}f_1^{-1}\mathcal{F} \to f_{2, *}f_2^{-1}\mathcal{F} \to \ldots$$ is exact in $\textit{Ab}(\mathcal{X}_\tau)$. \end{proposition} \begin{proof} By Lemma \ref{lemma-check-exactness-covering} it suffices to check exactness after pulling back to $\mathcal{U}$. By Lemma \ref{lemma-base-change-cech-complex} the pullback of the extended relative {\v C}ech complex is isomorphic to the extend relative {\v C}ech complex for the morphism $\mathcal{U} \times_\mathcal{X} \mathcal{U} \to \mathcal{U}$ and an abelian sheaf on $\mathcal{U}_\tau$. Since there is a section $\Delta_{\mathcal{U}/\mathcal{X}} : \mathcal{U} \to \mathcal{U} \times_\mathcal{X} \mathcal{U}$ exactness follows from Lemma \ref{lemma-homotopy-sheafified}. \end{proof} \noindent Using this we can construct the {\v C}ech-to-cohomology spectral sequence as follows. We first give a technical, precise version. In the next section we give a version that applies only to algebraic stacks. \begin{lemma} \label{lemma-cech-to-cohomology} Let $f : \mathcal{U} \to \mathcal{X}$ be a $1$-morphism of categories fibred in groupoids over $(\Sch/S)_{fppf}$. Let $\tau \in \{Zar, \etale, smooth, syntomic, fppf\}$. Assume \begin{enumerate} \item $\mathcal{F}$ is an abelian sheaf on $\mathcal{X}_\tau$, \item for every object $x$ of $\mathcal{X}$ there exists a covering $\{x_i \to x\}$ in $\mathcal{X}_\tau$ such that each $x_i$ is isomorphic to $f(u_i)$ for some object $u_i$ of $\mathcal{U}$, \item the category $\mathcal{U}$ has equalizers, and \item the functor $f$ is faithful. \end{enumerate} Then there is a first quadrant spectral sequence of abelian groups $$E_1^{p, q} = H^q((\mathcal{U}_p)_\tau, f_p^{-1}\mathcal{F}) \Rightarrow H^{p + q}(\mathcal{X}_\tau, \mathcal{F})$$ converging to the cohomology of $\mathcal{F}$ in the $\tau$-topology. \end{lemma} \begin{proof} Before we start the proof we make some remarks. By Lemma \ref{lemma-fibre-products-preserve-properties} (and induction) all of the categories fibred in groupoids $\mathcal{U}_p$ have equalizers and all of the morphisms $f_p : \mathcal{U}_p \to \mathcal{X}$ are faithful. Let $\mathcal{I}$ be an injective object of $\textit{Ab}(\mathcal{X}_\tau)$. By Lemma \ref{lemma-pullback-injective} we see $f_p^{-1}\mathcal{I}$ is an injective object of $\textit{Ab}((\mathcal{U}_p)_\tau)$. Hence $f_{p, *}f_p^{-1}\mathcal{I}$ is an injective object of $\textit{Ab}(\mathcal{X}_\tau)$ by Lemma \ref{lemma-pushforward-injective}. Hence Proposition \ref{proposition-exactness-cech-complex} shows that the extended relative {\v C}ech complex $$\ldots \to 0 \to \mathcal{I} \to f_{0, *}f_0^{-1}\mathcal{I} \to f_{1, *}f_1^{-1}\mathcal{I} \to f_{2, *}f_2^{-1}\mathcal{I} \to \ldots$$ is an exact complex in $\textit{Ab}(\mathcal{X}_\tau)$ all of whose terms are injective. Taking global sections of this complex is exact and we see that the {\v C}ech complex $\check{\mathcal{C}}^\bullet(\mathcal{U} \to \mathcal{X}, \mathcal{I})$ is quasi-isomorphic to $\Gamma(\mathcal{X}_\tau, \mathcal{I})[0]$. \medskip\noindent With these preliminaries out of the way consider the two spectral sequences associated to the double complex (see Homology, Section \ref{homology-section-double-complex}) $$\check{\mathcal{C}}^\bullet(\mathcal{U} \to \mathcal{X}, \mathcal{I}^\bullet)$$ where $\mathcal{F} \to \mathcal{I}^\bullet$ is an injective resolution in $\textit{Ab}(\mathcal{X}_\tau)$. The discussion above shows that Homology, Lemma \ref{homology-lemma-double-complex-gives-resolution} applies which shows that $\Gamma(\mathcal{X}_\tau, \mathcal{I}^\bullet)$ is quasi-isomorphic to the total complex associated to the double complex. By our remarks above the complex $f_p^{-1}\mathcal{I}^\bullet$ is an injective resolution of $f_p^{-1}\mathcal{F}$. Hence the other spectral sequence is as indicated in the lemma. \end{proof} \noindent To be sure there is a version for modules as well. \begin{lemma} \label{lemma-cech-to-cohomology-modules} Let $f : \mathcal{U} \to \mathcal{X}$ be a $1$-morphism of categories fibred in groupoids over $(\Sch/S)_{fppf}$. Let $\tau \in \{Zar, \etale, smooth, syntomic, fppf\}$. Assume \begin{enumerate} \item $\mathcal{F}$ is an object of $\textit{Mod}(\mathcal{X}_\tau, \mathcal{O}_\mathcal{X})$, \item for every object $x$ of $\mathcal{X}$ there exists a covering $\{x_i \to x\}$ in $\mathcal{X}_\tau$ such that each $x_i$ is isomorphic to $f(u_i)$ for some object $u_i$ of $\mathcal{U}$, \item the category $\mathcal{U}$ has equalizers, and \item the functor $f$ is faithful. \end{enumerate} Then there is a first quadrant spectral sequence of $\Gamma(\mathcal{O}_\mathcal{X})$-modules $$E_1^{p, q} = H^q((\mathcal{U}_p)_\tau, f_p^*\mathcal{F}) \Rightarrow H^{p + q}(\mathcal{X}_\tau, \mathcal{F})$$ converging to the cohomology of $\mathcal{F}$ in the $\tau$-topology. \end{lemma} \begin{proof} The proof of this lemma is identical to the proof of Lemma \ref{lemma-cech-to-cohomology} except that it uses an injective resolution in $\textit{Mod}(\mathcal{X}_\tau, \mathcal{O}_\mathcal{X})$ and it uses Lemma \ref{lemma-pullback-injective-modules} instead of Lemma \ref{lemma-pullback-injective}. \end{proof} \noindent Here is a lemma that translates a more usual kind of covering in the kinds of coverings we have encountered above. \begin{lemma} \label{lemma-surjective-flat-locally-finite-presentation} Let $f : \mathcal{X} \to \mathcal{Y}$ be a $1$-morphism of categories fibred in groupoids over $(\Sch/S)_{fppf}$. \begin{enumerate} \item Assume that $f$ is representable by algebraic spaces, surjective, flat, and locally of finite presentation. Then for any object $y$ of $\mathcal{Y}$ there exists an fppf covering $\{y_i \to y\}$ and objects $x_i$ of $\mathcal{X}$ such that $f(x_i) \cong y_i$ in $\mathcal{Y}$. \item Assume that $f$ is representable by algebraic spaces, surjective, and smooth. Then for any object $y$ of $\mathcal{Y}$ there exists an \'etale covering $\{y_i \to y\}$ and objects $x_i$ of $\mathcal{X}$ such that $f(x_i) \cong y_i$ in $\mathcal{Y}$. \end{enumerate} \end{lemma} \begin{proof} Proof of (1). Suppose that $y$ lies over the scheme $V$. We may think of $y$ as a morphism $(\Sch/V)_{fppf} \to \mathcal{Y}$. By definition the $2$-fibre product $\mathcal{X} \times_\mathcal{Y} (\Sch/V)_{fppf}$ is representable by an algebraic space $W$ and the morphism $W \to V$ is surjective, flat, and locally of finite presentation. Choose a scheme $U$ and a surjective \'etale morphism $U \to W$. Then $U \to V$ is also surjective, flat, and locally of finite presentation (see Morphisms of Spaces, Lemmas \ref{spaces-morphisms-lemma-etale-flat}, \ref{spaces-morphisms-lemma-etale-locally-finite-presentation}, \ref{spaces-morphisms-lemma-composition-surjective}, \ref{spaces-morphisms-lemma-composition-finite-presentation}, and \ref{spaces-morphisms-lemma-composition-flat}). Hence $\{U \to V\}$ is an fppf covering. Denote $x$ the object of $\mathcal{X}$ over $U$ corresponding to the $1$-morphism $(\Sch/U)_{fppf} \to \mathcal{X}$. Then $\{f(x) \to y\}$ is the desired fppf covering of $\mathcal{Y}$. \medskip\noindent Proof of (1). Suppose that $y$ lies over the scheme $V$. We may think of $y$ as a morphism $(\Sch/V)_{fppf} \to \mathcal{Y}$. By definition the $2$-fibre product $\mathcal{X} \times_\mathcal{Y} (\Sch/V)_{fppf}$ is representable by an algebraic space $W$ and the morphism $W \to V$ is surjective and smooth. Choose a scheme $U$ and a surjective \'etale morphism $U \to W$. Then $U \to V$ is also surjective and smooth (see Morphisms of Spaces, Lemmas \ref{spaces-morphisms-lemma-etale-smooth}, \ref{spaces-morphisms-lemma-composition-surjective}, and \ref{spaces-morphisms-lemma-composition-smooth}). Hence $\{U \to V\}$ is a smooth covering. By More on Morphisms, Lemma \ref{more-morphisms-lemma-etale-dominates-smooth} there exists an \'etale covering $\{V_i \to V\}$ such that each $V_i \to V$ factors through $U$. Denote $x_i$ the object of $\mathcal{X}$ over $V_i$ corresponding to the $1$-morphism $$(\Sch/V_i)_{fppf} \to (\Sch/U)_{fppf} \to \mathcal{X}.$$ Then $\{f(x_i) \to y\}$ is the desired \'etale covering of $\mathcal{Y}$. \end{proof} \begin{lemma} \label{lemma-cech-to-cohomology-relative} Let $f : \mathcal{U} \to \mathcal{X}$ and $g : \mathcal{X} \to \mathcal{Y}$ be composable $1$-morphisms of categories fibred in groupoids over $(\Sch/S)_{fppf}$. Let $\tau \in \{Zar, \etale, smooth, syntomic, \linebreak[0] fppf\}$. Assume \begin{enumerate} \item $\mathcal{F}$ is an abelian sheaf on $\mathcal{X}_\tau$, \item for every object $x$ of $\mathcal{X}$ there exists a covering $\{x_i \to x\}$ in $\mathcal{X}_\tau$ such that each $x_i$ is isomorphic to $f(u_i)$ for some object $u_i$ of $\mathcal{U}$, \item the category $\mathcal{U}$ has equalizers, and \item the functor $f$ is faithful. \end{enumerate} Then there is a first quadrant spectral sequence of abelian sheaves on $\mathcal{Y}_\tau$ $$E_1^{p, q} = R^q(g \circ f_p)_*f_p^{-1}\mathcal{F} \Rightarrow R^{p + q}g_*\mathcal{F}$$ where all higher direct images are computed in the $\tau$-topology. \end{lemma} \begin{proof} Note that the assumptions on $f : \mathcal{U} \to \mathcal{X}$ and $\mathcal{F}$ are identical to those in Lemma \ref{lemma-cech-to-cohomology}. Hence the preliminary remarks made in the proof of that lemma hold here also. These remarks imply in particular that $$0 \to g_*\mathcal{I} \to (g \circ f_0)_*f_0^{-1}\mathcal{I} \to (g \circ f_1)_*f_1^{-1}\mathcal{I} \to \ldots$$ is exact if $\mathcal{I}$ is an injective object of $\textit{Ab}(\mathcal{X}_\tau)$. Having said this, consider the two spectral sequences of Homology, Section \ref{homology-section-double-complex} associated to the double complex $\mathcal{C}^{\bullet, \bullet}$ with terms $$\mathcal{C}^{p, q} = (g \circ f_p)_*\mathcal{I}^q$$ where $\mathcal{F} \to \mathcal{I}^\bullet$ is an injective resolution in $\textit{Ab}(\mathcal{X}_\tau)$. The first spectral sequence implies, via Homology, Lemma \ref{homology-lemma-double-complex-gives-resolution}, that $g_*\mathcal{I}^\bullet$ is quasi-isomorphic to the total complex associated to $\mathcal{C}^{\bullet, \bullet}$. Since $f_p^{-1}\mathcal{I}^\bullet$ is an injective resolution of $f_p^{-1}\mathcal{F}$ (see Lemma \ref{lemma-pullback-injective}) the second spectral sequence has terms $E_1^{p, q} = R^q(g \circ f_p)_*f_p^{-1}\mathcal{F}$ as in the statement of the lemma. \end{proof} \begin{lemma} \label{lemma-cech-to-cohomology-relative-modules} Let $f : \mathcal{U} \to \mathcal{X}$ and $g : \mathcal{X} \to \mathcal{Y}$ be composable $1$-morphisms of categories fibred in groupoids over $(\Sch/S)_{fppf}$. Let $\tau \in \{Zar, \etale, smooth, syntomic, \linebreak[0] fppf\}$. Assume \begin{enumerate} \item $\mathcal{F}$ is an object of $\textit{Mod}(\mathcal{X}_\tau, \mathcal{O}_\mathcal{X})$, \item for every object $x$ of $\mathcal{X}$ there exists a covering $\{x_i \to x\}$ in $\mathcal{X}_\tau$ such that each $x_i$ is isomorphic to $f(u_i)$ for some object $u_i$ of $\mathcal{U}$, \item the category $\mathcal{U}$ has equalizers, and \item the functor $f$ is faithful. \end{enumerate} Then there is a first quadrant spectral seque