Sheaves on Algebraic Stacks

Introduction

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.

In fact, it turns out that to develop a fully fledged theory of constructible étale 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 [olsson_sheaves]. We will return to this in Cohomology of Stacks, Section 073Q.

In the literature and in research papers on sheaves on algebraic stacks the lisse-étale 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-étale topoi. We have therefore made the design decision to avoid any mention of the lisse-étale site as long as possible. Arguments that traditionally use the lisse-étale site will be replaced by an argument using a Čech covering in the site \(\mathcal{X}_{smooth}\) defined below.

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 05X6 for an explanation.

Conventions

The conventions we use in this chapter are the same as those in the chapter on algebraic stacks, see Algebraic Stacks, Section 026M. For convenience we repeat them here.

We work in a suitable big fppf site \(\Sch_{fppf}\) as in Topologies, Definition 021R. 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).

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 021S. The absolute case can be recovered by taking \(S = \Spec(\mathbf{Z})\).

Presheaves

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.

Definition

Let \(p : \mathcal{X} \to (\Sch/S)_{fppf}\) be a category fibred in groupoids.

1. A presheaf on \(\mathcal{X}\) is a presheaf on the underlying category of \(\mathcal{X}\).

2. A morphism of presheaves on \(\mathcal{X}\) is a morphism of presheaves on the underlying category of \(\mathcal{X}\).

We denote \(\textit{PSh}(\mathcal{X})\) the category of presheaves on \(\mathcal{X}\).

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 abelian presheaves, i.e., presheaves of abelian groups, is denoted \(\textit{PAb}(\mathcal{X})\).

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 00XF provides us with a pair of adjoint functors¹ [06TK]\[\begin{equation} f^p : \textit{PSh}(\mathcal{Y}) \longrightarrow \textit{PSh}(\mathcal{X}) \quad\text{and}\quad {}_pf : \textit{PSh}(\mathcal{X}) \longrightarrow \textit{PSh}(\mathcal{Y}). \end{equation}\] 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 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 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.

Lemma

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 adjointness of \((f^p, {}_pf)\), \((g^p, {}_pg)\), and \(((g \circ f)^p, {}_p(g \circ f))\).

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.

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 002L. 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.

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.

Lemma

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.

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.

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 00XF. 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 002L 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.

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 00XH and observe the compatibility from the explicit description of the horizontal and vertical maps in the diagram.

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.

Sheaves

We first make an observation that is important and trivial (especially for those readers who do not worry about set theoretical issues).

Consider a big fppf site \(\Sch_{fppf}\) as in Topologies, Definition 021R 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 étale site, a big smooth site, and a big syntomic site, see Topologies, Remark 03FF. 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 étale 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 00XZ) \(\Sch_{Zar}/S\), \(\Sch_\etale/S\), \(\Sch_{smooth}/S\), \(\Sch_{syntomic}/S\), and \(\Sch_{fppf}/S\) of these (absolute) big sites we see that all of these have the same underlying category, namely \(\Sch_\alpha/S\).

It follows that if we have a category \(p : \mathcal{X} \to (\Sch/S)_{fppf}\) fibred in groupoids, then \(\mathcal{X}\) inherits a Zariski, étale, smooth, syntomic, and fppf topology, see Stacks, Definition 06NV.

Definition

Let \(\mathcal{X}\) be a category fibred in groupoids over \((\Sch/S)_{fppf}\).

1. The associated Zariski site, denoted \(\mathcal{X}_{Zar}\), is the structure of site on \(\mathcal{X}\) inherited from \((\Sch/S)_{Zar}\).

2. The associated étale site, denoted \(\mathcal{X}_\etale\), is the structure of site on \(\mathcal{X}\) inherited from \((\Sch/S)_\etale\).

3. The associated smooth site, denoted \(\mathcal{X}_{smooth}\), is the structure of site on \(\mathcal{X}\) inherited from \((\Sch/S)_{smooth}\).

4. The associated syntomic site, denoted \(\mathcal{X}_{syntomic}\), is the structure of site on \(\mathcal{X}\) inherited from \((\Sch/S)_{syntomic}\).

5. The associated fppf site, denoted \(\mathcal{X}_{fppf}\), is the structure of site on \(\mathcal{X}\) inherited from \((\Sch/S)_{fppf}\).

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 big fppf site of \(\mathcal{X}\) and similarly for the other ones. We may occasionally use this terminology to distinguish this construction from others.

Remark

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 étale 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).

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.

Definition

Let \(\mathcal{X}\) be a category fibred in groupoids over \((\Sch/S)_{fppf}\). Let \(\mathcal{F}\) be a presheaf on \(\mathcal{X}\).

1. We say \(\mathcal{F}\) is a Zariski sheaf, or a sheaf for the Zariski topology if \(\mathcal{F}\) is a sheaf on the associated Zariski site \(\mathcal{X}_{Zar}\).

2. We say \(\mathcal{F}\) is an étale sheaf, or a sheaf for the étale topology if \(\mathcal{F}\) is a sheaf on the associated étale site \(\mathcal{X}_\etale\).

3. We say \(\mathcal{F}\) is a smooth sheaf, or a sheaf for the smooth topology if \(\mathcal{F}\) is a sheaf on the associated smooth site \(\mathcal{X}_{smooth}\).

4. We say \(\mathcal{F}\) is a syntomic sheaf, or a sheaf for the syntomic topology if \(\mathcal{F}\) is a sheaf on the associated syntomic site \(\mathcal{X}_{syntomic}\).

5. We say \(\mathcal{F}\) is an fppf sheaf, or a sheaf, or a sheaf for the fppf topology if \(\mathcal{F}\) is a sheaf on the associated fppf site \(\mathcal{X}_{fppf}\).

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 \(\Sh(\mathcal{X}_{fppf})\).

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 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 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.

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 \(\mathcal{X}\).

With this setup functoriality of these topoi is straightforward, and moreover, is compatible with the inclusion functors above.

Lemma

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 (06TK) transform \(\tau\) sheaves into \(\tau\) sheaves and define a morphism of topoi \(f : \Sh(\mathcal{X}_\tau) \to \Sh(\mathcal{Y}_\tau)\).

Proof

This follows immediately from Stacks, Lemma 06NW.

In other words, pushforward and pullback of presheaves as defined in Section 06TI also produces pushforward and 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.

Definition

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 associated morphism of fppf topoi constructed above. Similarly for the associated Zariski, étale, smooth, and syntomic topoi.

As discussed in Sites, Section 00YR 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.

Computing pushforward

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 [06W6]\[\begin{equation} f_*\mathcal{F}(y) = \Gamma\Big( (\Sch/V)_{fppf} \times_{y, \mathcal{Y}} \mathcal{X}, \ \text{pr}^{-1}\mathcal{F}\Big) \end{equation}\] 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.

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.

Lemma

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 \(\mathcal{X}\).

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 (06W6) 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.

In the case of a representable morphism of categories fibred in groupoids this formula (06W6) simplifies. We suggest the reader skip the rest of this section.

Lemma

Let \(f : \mathcal{X} \to \mathcal{Y}\) be a \(1\)-morphism of categories fibred in groupoids over \((\Sch/S)_{fppf}\). The following are equivalent

1. \(f\) is representable, and

2. 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.

Proof

According to the discussion in Algebraic Stacks, Section 04ST 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 \(f(x_y) \to y\).

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 06W7 (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 \(V' \to V\).

Lemma

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 06TS. Moreover, \(f_*\mathcal{F}(y) = \mathcal{F}(u(y))\) for any presheaf \(\mathcal{F}\) on \(\mathcal{X}\).

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.

Let’s use the notation \(u_p, u_s, u^p, u^s\) of Sites, Sections 00VC and 00WU. 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.

The structure sheaf

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 06TN shows that \(p^{-1}\mathcal{G}\) is a \(\tau\) sheaf whenever \(\mathcal{G}\) is a \(\tau\)-sheaf.

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 03DU.

Definition

Let \(p : \mathcal{X} \to (\Sch/S)_{fppf}\) be a category fibred in groupoids. The structure sheaf of \(\mathcal{X}\) is the sheaf of rings \(\mathcal{O}_\mathcal{X} = p^{-1}\mathcal{O}\).

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, étale, 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.

Lemma

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 \(f^{-1}\mathcal{O}_\mathcal{Y} = \mathcal{O}_\mathcal{X}\) which turns \(f : \Sh(\mathcal{X}_\tau) \to \Sh(\mathcal{Y}_\tau)\) into a morphism of ringed topoi.

Proof

Denote \(p : \mathcal{X} \to (\Sch/S)_{fppf}\) and \(q : \mathcal{Y} \to (\Sch/S)_{fppf}\) the structural functors. Then \(p = q \circ f\), hence \(p^{-1} = f^{-1} \circ q^{-1}\) by Lemma 06TL. Since \(\mathcal{O}_\mathcal{X} = p^{-1}\mathcal{O}\) and \(\mathcal{O}_\mathcal{Y} = q^{-1}\mathcal{O}\) the result follows.

Remark

In the situation of Lemma 06TW 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.

Sheaves of modules

Since we have a structure sheaf we have modules.

Definition

Let \(\mathcal{X}\) be a category fibred in groupoids over \((\Sch/S)_{fppf}\).

1. A presheaf of modules on \(\mathcal{X}\) is a presheaf of \(\mathcal{O}_\mathcal{X}\)-modules. The category of presheaves of modules is denoted \(\textit{PMod}(\mathcal{O}_\mathcal{X})\).

2. We say a presheaf of modules \(\mathcal{F}\) is an \(\mathcal{O}_\mathcal{X}\)-module, or more precisely a sheaf of \(\mathcal{O}_\mathcal{X}\)-modules if \(\mathcal{F}\) is an fppf sheaf. The category of \(\mathcal{O}_\mathcal{X}\)-modules is denoted \(\textit{Mod}(\mathcal{O}_\mathcal{X})\).

These (pre)sheaves of modules occur in the literature as (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, étale, 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.

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 [06WD]\[\begin{equation} f^{-1} : \textit{PMod}(\mathcal{O}_\mathcal{Y}) \longrightarrow \textit{PMod}(\mathcal{O}_\mathcal{X}) \quad\text{and}\quad f_* : \textit{PMod}(\mathcal{O}_\mathcal{X}) \longrightarrow \textit{PMod}(\mathcal{O}_\mathcal{Y}) \end{equation}\] 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 03D1. We claim that the functors (06WD) 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 03D3 by endowing \(\mathcal{X}\) and \(\mathcal{Y}\) with the chaotic topology.

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 06TW (for the fppf topologies right now). Note that the functors \(f^{-1}\) and \(f_*\) of (06WD) preserve the subcategories of sheaves of modules, see Lemma 06TS. Hence it follows immediately that [06WE]\[\begin{equation} f^{-1} : \textit{Mod}(\mathcal{O}_\mathcal{Y}) \longrightarrow \textit{Mod}(\mathcal{O}_\mathcal{X}) \quad\text{and}\quad f_* : \textit{Mod}(\mathcal{O}_\mathcal{X}) \longrightarrow \textit{Mod}(\mathcal{O}_\mathcal{Y}) \end{equation}\] 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 03D5 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 \(f^{-1}\mathcal{O}_\mathcal{Y} = \mathcal{O}_\mathcal{X}\).

Similarly for sheaves of modules in the Zariski, étale, smooth, syntomic topology.

Representable categories

In this short section we compare our definitions with what happens in case the algebraic stacks in question are representable.

Lemma

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.

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 \(\mathcal{X} \leadsto \mathcal{X}_\tau\).

Lemma

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 075I.

Proof

Follows by unwinding the definitions.

Restriction

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.

Lemma

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 \(\mathcal{X}_\tau/x \to (\Sch/U)_\tau\).

Proof

Special case of Stacks, Lemma 0CN0.

We use the lemma above to talk about the pullback and the restriction of a (pre)sheaf to a scheme.

Definition

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 \(\mathcal{X}\).

1. The 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 06W0.

2. The restriction of \(\mathcal{F}\) to \(U_\etale\) is \(x^{-1}\mathcal{F}|_{U_\etale}\), abusively written \(\mathcal{F}|_{U_\etale}\).

This notation makes sense because to the object \(x\) the \(2\)-Yoneda lemma, see Algebraic Stacks, Section 04SS 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, étale, smooth, syntomic sheaf), then \(x^{-1}\mathcal{F}\) is a sheaf on \((\Sch/U)_{fppf}\) (or on \((\Sch/U)_{Zar}\), \((\Sch/U)_\etale\), \((\Sch/U)_{smooth}\), \((\Sch/U)_{syntomic}\)).

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 étale sites \(a_{small} : U_\etale \to V_\etale\), see Étale Cohomology, Section 04I0. 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 comparison map [06W2]\[\begin{equation} c_\varphi : \mathcal{F}|_{V_\etale} \longrightarrow a_{small, *}(\mathcal{F}|_{U_\etale}) \end{equation}\] of presheaves on \(U_\etale\). Namely, if \(V' \to V\) is étale, 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 étale topology we obtain a comparison map [06W3]\[\begin{equation} c_\varphi : a_{small}^{-1}(\mathcal{F}|_{V_\etale}) \longrightarrow \mathcal{F}|_{U_\etale} \end{equation}\] which is also denoted \(c_\varphi\) as indicated (this is the customary abuse of notation in not distinguishing between adjoint maps).

Lemma

Let \(\mathcal{F}\) be an étale sheaf on \(\mathcal{X} \to (\Sch/S)_{fppf}\).

1. 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 \[(b \circ a)_{small}^{-1}(\mathcal{F}|_{W_\etale}) = a_{small}^{-1}(b_{small}^{-1} (\mathcal{F}|_{W_\etale})).\]

2. If \(\varphi : x \to y\) lies over an étale morphism of schemes \(a : U \to V\), then (06W3) is an isomorphism.

3. 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 \(f^{-1}\mathcal{F}|_{U_\etale} = \mathcal{F}|_{U_\etale}\).

4. 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).

Proof

The verification of these properties is omitted.

Next, we turn to the restriction of (pre)sheaves of modules.

Lemma

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 06W0 extends to an equivalence of ringed sites \((\mathcal{X}_\tau/x, \mathcal{O}_\mathcal{X}|_x) \to ((\Sch/U)_\tau, \mathcal{O})\).

Proof

This is immediate from the construction of the structure sheaves.

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 06WB. Let \(x\) be an object of \(\mathcal{X}\) lying over \(U\). Then Lemma 06W9 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, étale, 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 étale 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 (06W3) is compatible with \(a_{small}^\sharp\) (see Descent, Remark 070R) and induces a comparison map [06WC]\[\begin{equation} c_\varphi : a_{small}^*(\mathcal{F}|_{V_\etale}) \longrightarrow \mathcal{F}|_{U_\etale} \end{equation}\] of \(\mathcal{O}_{U_\etale}\)-modules. Note that the properties (1), (2), (3), and (4) of Lemma 075D hold in the setting of étale sheaves of modules as well. We will use this in the following without further mention.

Lemma

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.

Proof

By Sites, Lemma 06UL 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 06W0 and Étale Cohomology, Lemma 06VX we see that \(\mathcal{X}_\tau/x\) has enough points for every object \(x\) and we win.

Restriction to algebraic spaces

In this section we consider sheaves on categories representable by algebraic spaces. The following lemma is the analogue of Topologies, Lemma 021G for algebraic spaces.

Lemma

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 \(\pi_{F, *} = i_F^{-1}\).

Proof

Choose an equivalence \(j : \mathcal{S}_F \to \mathcal{X}\), see Algebraic Stacks, Sections 04SU and 02ZV. An object of \(F_\etale\) is a scheme \(U\) together with an étale 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 étale 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 00XR, 00XS, and 00XT 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.

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 étale. 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 04D2 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 \(\Sh(\mathcal{X}_\etale)\). 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.

Remark

The constructions in Lemma 073M are compatible with étale localization. Here is a precise formulation. 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 \(F\), \(G\), and that the induced morphism \(f : F \to G\) of algebraic spaces is étale. Denote \(f_{small} : F_\etale \to G_\etale\) the corresponding morphism of ringed topoi. Then \[\xymatrix{ (\Sh(F_\etale), \mathcal{O}_F) \ar[rr]_{f_{small}} \ar[d]_{i_F} & & (\Sh(G_\etale), \mathcal{O}_G) \ar[d]^{i_G} \\ (\Sh(\mathcal{X}_\etale), \mathcal{O}_\mathcal{X}) \ar[d]_{\pi_F} \ar[rr]_f & & (\Sh(\mathcal{Y}_\etale), \mathcal{O}_\mathcal{Y}) \ar[d]^{\pi_G} \\ (\Sh(F_\etale), \mathcal{O}_F) \ar[rr]^{f_{small}} & & (\Sh(G_\etale), \mathcal{O}_G) }\] is a commutative diagram of ringed topoi. We omit the details.

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 073M 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 restriction functor similarly to Definition 06W1 and to the restriction of a sheaf on the big étale site of a scheme to the small étale site of a scheme. We often use the notation [075K]\[\begin{equation} \mathcal{F}|_{F_\etale} = i_F^{-1}\mathcal{F} = \pi_{F, *}\mathcal{F} \end{equation}\] in this situation.

Lemma

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.

Proof

This is similar to Topologies, Lemma 021I (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.

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 073M. Because of the 2-functoriality of sheaves on categories fibred in groupoids over \(\Sch_{fppf}\) (see discussion in Section 06TI) 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 \(G\).

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 (06W6). 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 étale topology.

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.

In general, choose a scheme \(U\) and a surjective étale 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 étale 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 03EB (especially, Properties of Spaces, Remark 03H7 and Lemma 03G2) 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 \(\mathcal{G}\).

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 (075K), is that [075M]\[\begin{equation} (f_*\mathcal{F})|_{G_\etale} = f_{small, *}(\mathcal{F}|_{F_\etale}) \end{equation}\] for any sheaf \(\mathcal{F}\) on \(\mathcal{X}_\etale\). Moreover, if \(\mathcal{F}\) is a sheaf of \(\mathcal{O}\)-modules, then (075M) is an isomorphism of \(\mathcal{O}_G\)-modules on \(G_\etale\).

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 [076Q]\[\begin{equation} c_\varphi : a_{small}^{-1}(g^{-1}\mathcal{F}|_{V_\etale}) \longrightarrow f^{-1}\mathcal{F}|_{U_\etale} \end{equation}\] where \(a : U \to V\) denotes the morphism of algebraic spaces corresponding to \(a\). This is the analogue of (06W3). 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 (075M)) which is the restriction to \(V\) (075K) 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 comparison map [076R]\[\begin{equation} c_\varphi : a_{small}^*(g^*\mathcal{F}|_{V_\etale}) \longrightarrow f^*\mathcal{F}|_{U_\etale} \end{equation}\] of \(\mathcal{O}_{U_\etale}\)-modules. This is the analogue of (06WC). Note that the properties (1), (2), (3), and (4) of Lemma 075D hold in this setting as well.

Quasi-coherent modules

At this point we can apply the general definition of a quasi-coherent module to the situation discussed in this chapter.

Definition

Let \(p : \mathcal{X} \to (\Sch/S)_{fppf}\) be a category fibred in groupoids. A quasi-coherent module on \(\mathcal{X}\), or a 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 03DL. The category of quasi-coherent sheaves on \(\mathcal{X}\) is denoted \(\QCoh(\mathcal{O}_\mathcal{X})\).

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 [olsson_sheaves, Definition 6.1] in Cohomology on Stacks, Lemma 06WK. We will also see alternative constructions of this category later on.

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.

Lemma

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.

Proof

This is a general fact, see Modules on Sites, Lemma 03DO.

It turns out that quasi-coherent sheaves have a very simple characterization in terms of their pullbacks. See also Lemma 06WK for a characterization in terms of restrictions.

Lemma

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 \(U = p(x)\).

Proof

By Lemma 06WH 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 03DG).

Lemma

Let \(p : \mathcal{X} \to (\Sch/S)_{fppf}\) be a category fibred in groupoids. Let \(\mathcal{F}\) be a presheaf of modules on \(\mathcal{X}\). The following are equivalent

1. \(\mathcal{F}\) is an object of \(\textit{Mod}(\mathcal{X}_{Zar}, \mathcal{O}_\mathcal{X})\) and \(\mathcal{F}\) is a quasi-coherent module on \((\mathcal{X}_{Zar}, \mathcal{O}_\mathcal{X})\) in the sense of Modules on Sites, Definition 03DL,

2. \(\mathcal{F}\) is an object of \(\textit{Mod}(\mathcal{X}_\etale, \mathcal{O}_\mathcal{X})\) and \(\mathcal{F}\) is a quasi-coherent module on \((\mathcal{X}_\etale, \mathcal{O}_\mathcal{X})\) in the sense of Modules on Sites, Definition 03DL, and

3. \(\mathcal{F}\) is a quasi-coherent module on \(\mathcal{X}\) in the sense of Definition 06WG.

Proof

Assume either (1), (2), or (3) holds. Let \(x\) be an object of \(\mathcal{X}\) lying over the scheme \(U\). Recall that \(x^*\mathcal{F} = x^{-1}\mathcal{F}\) is just the restriction to \(\mathcal{X}/x = (\Sch/U)_\tau\) where \(\tau = fppf\), \(\tau = \etale\), or \(\tau = Zar\), see Section 075C. By the definition of quasi-coherent modules on a ringed site this restriction is quasi-coherent provided \(\mathcal{F}\) is. By Descent, Proposition 03DX we see that \(x^*\mathcal{F}\) is the sheaf associated to a quasi-coherent \(\mathcal{O}_U\)-module and is therefore a quasi-coherent module in the fppf, étale, and Zariski topology; here we also use Descent, Lemma 03DT and Definition 03DU. Since this holds for every object \(x\) of \(\mathcal{X}\), we see that \(\mathcal{F}\) is a sheaf in any of the three topologies. Moreover, we find that \(\mathcal{F}\) is quasi-coherent in any of the three topologies directly from the definition of being quasi-coherent and the fact that \(x\) is an arbitrary object of \(\mathcal{X}\).

Locally quasi-coherent modules

Although there is a variant for the Zariski topology, it seems that the étale topology is the natural topology to use in the following definition.

Definition

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 locally quasi-coherent² if \(\mathcal{F}\) is a sheaf for the étale topology and for every object \(x\) of \(\mathcal{X}\) the restriction \(x^*\mathcal{F}|_{U_\etale}\) is a quasi-coherent sheaf. Here \(U = p(x)\).

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.

Lemma

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

1. \(\mathcal{F}\) is locally quasi-coherent, and

2. 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 (06WC) is an isomorphism.

Proof

Assume \(\mathcal{F}\) is quasi-coherent. Then \(\mathcal{F}\) is a sheaf for the fppf topology, hence a sheaf for the étale 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 03DX 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 (06WC) are isomorphisms.

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 03DX. Condition (2) now signifies that the restrictions \(x^*\mathcal{F}|_{U_\etale}\) for \(x\) over \(y\) are each isomorphic to the (étale 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 03DX again) and we see that \(\mathcal{F}\) is quasi-coherent on \(\mathcal{X}\) by Lemma 06WI.

Lemma

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.

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 \(\mathcal{G}\).

Lemma

Let \(p : \mathcal{X} \to (\Sch/S)_{fppf}\) be a category fibred in groupoids.

1. The category \(\textit{LQCoh}(\mathcal{O}_\mathcal{X})\) has colimits and they agree with colimits in the category \(\textit{Mod}(\mathcal{X}_\etale, \mathcal{O}_\mathcal{X})\).

2. 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.

3. 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.

4. 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 \(\textit{LQCoh}(\mathcal{O}_\mathcal{X})\).

5. Given \(\mathcal{F}, \mathcal{G}\) in \(\textit{LQCoh}(\mathcal{O}_\mathcal{X})\) with \(\mathcal{F}\) 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 \(\textit{LQCoh}(\mathcal{O}_\mathcal{X})\).

Proof

In the arguments below \(x\) denotes an arbitrary object of \(\mathcal{X}\) lying over the scheme \(U\). To show that an object \(\mathcal{H}\) of \(\textit{Mod}(\mathcal{X}_\etale, \mathcal{O}_\mathcal{X})\) is in \(\textit{LQCoh}(\mathcal{O}_\mathcal{X})\) we will show that the restriction \(x^*\mathcal{H}|_{U_\etale} = \mathcal{H}|_{U_\etale}\) is a quasi-coherent object of \(\textit{Mod}(U_\etale, \mathcal{O}_U)\).

Proof of (1). Let \(\mathcal{I} \to \textit{LQCoh}(\mathcal{O}_\mathcal{X})\), \(i \mapsto \mathcal{F}_i\) be a diagram. Consider the object \(\mathcal{F} = \colim_i \mathcal{F}_i\) of \(\textit{Mod}(\mathcal{X}_\etale, \mathcal{O}_\mathcal{X})\). 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 \(\colim_i x^*\mathcal{F}_i|_{U_\etale}\) is quasi-coherent by Descent, Lemma 0GNB. Thus \(x^*\mathcal{F}|_{U_\etale}\) is quasi-coherent as desired.

Proof of (2). 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 0GNB. This proves (2).

Proof of (3). Let \(0 \to \mathcal{F}_1 \to \mathcal{F}_2 \to \mathcal{F}_3 \to 0\) be a short exact sequence of \(\textit{Mod}(\mathcal{X}_\etale, \mathcal{O}_\mathcal{X})\). Since we are using the étale topology, the restriction \(0 \to \mathcal{F}_1|_{U_\etale} \to \mathcal{F}_2|_{U_\etale} \to \mathcal{F}_3|_{U_\etale} \to 0\) is a short exact sequence too. Hence (3) follows from the corresponding statement in Descent, Lemma 0GNB.

Proof of (4). Let \(\mathcal{F}\) and \(\mathcal{G}\) be in \(\textit{LQCoh}(\mathcal{O}_\mathcal{X})\). Since restriction to \(U_\etale\) is given by pullback along the morphism of ringed topoi \(U_\etale \to (\Sch/U)_\etale \to \mathcal{X}_\etale\) we see that the restriction of the tensor product \(\mathcal{F} \otimes_{\mathcal{O}_\mathcal{X}} \mathcal{G}\) to \(U_\etale\) is equal to \(\mathcal{F}|_{U_\etale} \otimes_{\mathcal{O}_U} \mathcal{G}|_{U_\etale}\), see Modules on Sites, Lemma 03EL. Since \(\mathcal{F}|_{U_\etale}\) and \(\mathcal{G}|_{U_\etale}\) are quasi-coherent, so is their tensor product, see Descent, Lemma 0GNB.

Proof of (5). Let \(\mathcal{F}\) and \(\mathcal{G}\) be in \(\textit{LQCoh}(\mathcal{O}_\mathcal{X})\) with \(\mathcal{F}\) of finite presentation. Since \((\Sch/U)_\etale = \mathcal{X}_\etale/x\) is a localization of \(\mathcal{X}_\etale\) at an object we see that the restriction of \(\SheafHom_{\mathcal{O}_\mathcal{X}}(\mathcal{F}, \mathcal{G})\) to \((\Sch/U)_\etale\) is equal to \[\mathcal{H} = \SheafHom_{\mathcal{O}|_{(\Sch/U)_\etale}}( \mathcal{F}|_{(\Sch/U)_\etale}, \mathcal{G}|_{(\Sch/U)_\etale})\] by Modules on Sites, Lemma 0E8H. The morphism of ringed topoi \((U_\etale, \mathcal{O}_U) \to ((\Sch/U)_\etale, \mathcal{O})\) is flat as the pullback of \(\mathcal{O}\) is \(\mathcal{O}_U\). Hence the pullback of \(\mathcal{H}\) by this morphism is equal to \(\SheafHom_{\mathcal{O}_U}(\mathcal{F}|_{U_\etale}, \mathcal{G}|_{U_\etale})\) by Modules on Sites, Lemma 0GN2. In other words, the restriction of \(\SheafHom_{\mathcal{O}_\mathcal{X}}(\mathcal{F}, \mathcal{G})\) to \(U_\etale\) is \(\SheafHom_{\mathcal{O}_U}(\mathcal{F}|_{U_\etale}, \mathcal{G}|_{U_\etale})\). Since \(\mathcal{F}|_{U_\etale}\) and \(\mathcal{G}|_{U_\etale}\) are quasi-coherent, so is \(\SheafHom_{\mathcal{O}_U}(\mathcal{F}|_{U_\etale}, \mathcal{G}|_{U_\etale})\), see Descent, Lemma 0GNB. We conclude as before.

In the generality discussed here the category of quasi-coherent sheaves is not abelian. See Examples, Section 0ALF. Here is what we can prove without any further work.

Lemma

Let \(p : \mathcal{X} \to (\Sch/S)_{fppf}\) be a category fibred in groupoids.

1. The category \(\QCoh(\mathcal{O}_\mathcal{X})\) has colimits and they agree with colimits in the categories \(\textit{Mod}(\mathcal{X}_{Zar}, \mathcal{O}_\mathcal{X})\), \(\textit{Mod}(\mathcal{X}_\etale, \mathcal{O}_\mathcal{X})\), \(\textit{Mod}(\mathcal{O}_\mathcal{X})\), and \(\textit{LQCoh}(\mathcal{O}_\mathcal{X})\).

2. Given \(\mathcal{F}, \mathcal{G}\) in \(\QCoh(\mathcal{O}_\mathcal{X})\) the tensor products \(\mathcal{F} \otimes_{\mathcal{O}_\mathcal{X}} \mathcal{G}\) computed in \(\textit{Mod}(\mathcal{X}_{Zar}, \mathcal{O}_\mathcal{X})\), \(\textit{Mod}(\mathcal{X}_\etale, \mathcal{O}_\mathcal{X})\), or \(\textit{Mod}(\mathcal{O}_\mathcal{X})\) agree and the common value is an object of \(\QCoh(\mathcal{O}_\mathcal{X})\).

3. Given \(\mathcal{F}, \mathcal{G}\) in \(\QCoh(\mathcal{O}_\mathcal{X})\) with \(\mathcal{F}\) finite locally free (in fppf, or equivalently étale, or equivalently Zariski topology) the internal homs \(\SheafHom_{\mathcal{O}_\mathcal{X}}(\mathcal{F}, \mathcal{G})\) computed in \(\textit{Mod}(\mathcal{X}_{Zar}, \mathcal{O}_\mathcal{X})\), \(\textit{Mod}(\mathcal{X}_\etale, \mathcal{O}_\mathcal{X})\), or \(\textit{Mod}(\mathcal{O}_\mathcal{X})\) agree and the common value is an object of \(\QCoh(\mathcal{O}_\mathcal{X})\).

Proof

Let \(x\) be an arbitrary object of \(\mathcal{X}\) lying over the scheme \(U\). Let \(\tau \in \{Zariski, \etale, fppf\}\). To show that an object \(\mathcal{H}\) of \(\textit{Mod}(\mathcal{X}_\tau, \mathcal{O}_\mathcal{X})\) is in \(\QCoh(\mathcal{O}_\mathcal{X})\) it suffices show that the restriction \(x^*\mathcal{H}\) (Section 075C) is a quasi-coherent object of \(\textit{Mod}((\Sch/U)_\tau, \mathcal{O})\). See Lemmas 06WI and 0EM8. Similarly for being finite locally free. Recall that \((\Sch/U)_\tau = \mathcal{X}_\tau/x\) is a localization of \(\mathcal{X}_\tau\) at an object. Hence restriction commutes with colimits, tensor products, and forming internal hom (see Modules on Sites, Lemmas 03DC, 03EL, and 0E8H). This reduces the lemma to Descent, Lemma 0GNE.

Stackification and sheaves

It turns out that the category of sheaves on a category fibred in groupoids only “knows about” the stackification.

Lemma

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.

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 03CG which will prove the lemma. By Stacks, Lemma 06NW the functor \(f\) is continuous and cocontinuous. By Stacks, Lemma 02ZN we see that conditions (3), (4), and (5) of Sites, Lemma 03A0 hold.

Lemma

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 \(\QCoh(\mathcal{O}_\mathcal{X}) \to \QCoh(\mathcal{O}_\mathcal{Y})\).

Proof

We may assume \(\mathcal{Y}\) is the stackification of \(\mathcal{X}\). The first assertion is clear from Lemma 06WQ and \(\mathcal{O}_\mathcal{X} = f^{-1}\mathcal{O}_\mathcal{Y}\). Pullback of quasi-coherent sheaves are quasi-coherent, see Lemma 06WH. 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 03DN that the restriction of \(\mathcal{G}\) to \(\mathcal{Y}/y\) is quasi-coherent. Hence \(\mathcal{G}\) is quasi-coherent by Lemma 06WI.

Quasi-coherent sheaves and presentations

Let us first match quasi-coherent sheaves with our previously defined notions for schemes and algebraic spaces.

Lemma

Let \(S\) be a scheme. Let \(\mathcal{X} \to (\Sch/S)_{fppf}\) be a category fibred in groupoids which is representable by an algebraic space \(F\). If \(\mathcal{F}\) is in \(\textit{LQCoh}(\mathcal{O}_\mathcal{X})\) then the restriction \(\mathcal{F}|_{F_\etale}\) (075K) is quasi-coherent.

Proof

Let \(U\) be a scheme étale over \(F\). Then \(\mathcal{F}|_{U_\etale} = (\mathcal{F}|_{F_\etale})|_{U_\etale}\). This is clear but see also Remark 0GQA. Thus the assertion follows from the definitions.

Lemma

Let \(S\) be a scheme. Let \(\mathcal{X} \to (\Sch/S)_{fppf}\) be a category fibred in groupoids which is representable by an algebraic space \(F\). The functor (075K) defines an equivalence \[\QCoh(\mathcal{O}_\mathcal{X}) \to \QCoh(\mathcal{O}_F),\quad \mathcal{F} \longmapsto \mathcal{F}|_{F_\etale}\] with quasi-inverse given by \(\mathcal{G} \mapsto \pi_F^*\mathcal{G}\). This equivalence is compatible with pullback for morphisms between categories fibred in groupoids representable by algebraic spaces.

Proof

By Lemma 0EM8 we may work with the étale topology. We will use the notation and results of Lemma 073M without further mention. Recall that the restriction functor \(\textit{Mod}(\mathcal{X}_\etale, \mathcal{O}_\mathcal{X}) \to \textit{Mod}(F_\etale, \mathcal{O}_F)\), \(\mathcal{F} \mapsto \mathcal{F}|_{F_\etale}\) is given by \(i_F^*\). By Lemma 0GQC or by Modules on Sites, Lemma 03DO we see that \(\mathcal{F}|_{F_\etale}\) is quasi-coherent if \(\mathcal{F}\) is quasi-coherent. Hence we get a functor as indicated in the statement of the lemma and we get a functor \(\pi_F^*\) in the opposite direction. Since \(\pi_F \circ i_F = \text{id}\) we see that \(i_F^*\pi_F^*\mathcal{G} = \mathcal{G}\).

For \(\mathcal{F}\) in \(\textit{Mod}(\mathcal{X}_\etale, \mathcal{O}_\mathcal{X})\) there is a canonical map \(\pi_F^*(\mathcal{F}|_{F_\etale}) \to \mathcal{F}\), namely the map adjoint to the identification \(\mathcal{F}|_{F_\etale} = \pi_{F, *}\mathcal{F}\). We will show that this map is an isomorphism if \(\mathcal{F}\) is a quasi-coherent module on \(\mathcal{X}\). Choose a scheme \(U\) and a surjective étale morphism \(U \to F\). Denote \(x : U \to \mathcal{X}\) the corresponding object of \(\mathcal{X}\) over \(U\). It suffices to show that \(\pi_F^*(\mathcal{F}|_{F_\etale}) \to \mathcal{F}\) is an isomorphism after restricting to \(\mathcal{X}_\etale/x = (\Sch/U)_\etale\). Since \(U \to F\) is étale, it follows from Remark 0GQA that \[\pi_F^*(\mathcal{F}|_{F_\etale})|_{\mathcal{X}_\etale/x} = \pi_U^*(\mathcal{F}|_{U_\etale})\] and that the restriction of the map \(\pi_F^*(\mathcal{F}|_{F_\etale}) \to \mathcal{F}\) to \(\mathcal{X}_\etale/x = (\Sch/U)_\etale\) is equal to the corresponding map \(\pi_U^*(\mathcal{F}|_{U_\etale}) \to \mathcal{F}|_{(\Sch/U)_\etale}\). Since we have seen the result is true for schemes in Descent, Section 03DR³ we conclude.

Compatibility with pullbacks follows from the fact that the quasi-inverse is given by \(\pi_F^*\) and the commutative diagram of ringed topoi in Lemma 073N.

In Groupoids in Spaces, Definition 0441 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.

Proposition

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 \((U, R, s, t, c)\).

Proof

We will construct quasi-inverse functors \[\QCoh(\mathcal{O}_\mathcal{X}) \longleftrightarrow \QCoh(U, R, s, t, c).\] where \(\QCoh(U, R, s, t, c)\) denotes the category of quasi-coherent modules on the groupoid \((U, R, s, t, c)\).

Let \(\mathcal{F}\) be an object of \(\QCoh(\mathcal{O}_\mathcal{X})\). Denote \(\mathcal{U}\), \(\mathcal{R}\) the categories fibred in groupoids corresponding to \(U\) and \(R\). Denote \(x\) the (defining) object of \(\mathcal{X}\) over \(U\). Recall that we have a \(2\)-commutative diagram \[\xymatrix{ \mathcal{R} \ar[r]_s \ar[d]_t & \mathcal{U} \ar[d]^x \\ \mathcal{U} \ar[r]^x & \mathcal{X} }\] See Groupoids in Spaces, Lemma 044S. By Lemma 06TM the \(2\)-arrow inherent in the diagram induces an isomorphism \(\alpha : t^*x^*\mathcal{F} \to s^*x^*\mathcal{F}\) which satisfies the cocycle condition over \(\mathcal{R} \times_{s, \mathcal{U}, t} \mathcal{R}\); this is a consequence of Groupoids in Spaces, Lemma 044T. Thus if we set \(\mathcal{G} = x^*\mathcal{F}|_{U_\etale}\) then the equivalence of categories in Lemma 0GQD (used several times compatibly with pullbacks) gives an isomorphism \(\alpha : t_{small}^*\mathcal{G} \to s_{small}^*\mathcal{G}\) satisfying the cocycle condition on \(R \times_{s, U, t} R\), i.e., \((\mathcal{G}, \alpha)\) is an object of \(\QCoh(U, R, s, t, c)\). The rule \(\mathcal{F} \mapsto (\mathcal{G}, \alpha)\) is our functor from left to right.

Construction of the functor in the other direction. Let \((\mathcal{G}, \alpha)\) be an object of \(\QCoh(U, R, s, t, c)\). According to Lemma 06WR the stackification map \([U/_{\!p}R] \to [U/R]\) (see Groupoids in Spaces, Definition 044Q) induces an equivalence of categories of quasi-coherent sheaves. Thus it suffices to construct a quasi-coherent module \(\mathcal{F}\) on \([U/_{\!p}R]\).

Recall that an object \(x = (T, u)\) of \([U/_{\!p}R]\) 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 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 \(U\).

With this notation in place, 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 \((f, r)\) 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). We see that the restriction of \(\mathcal{F}\) to \((\Sch/T)_{fppf}\) is quasi-coherent by the simple description of the restriction maps of \(\mathcal{F}\) in case of a special morphism. Hence \(\mathcal{F}\) is a sheaf on \([U/_{\!p}R]\) and quasi-coherent (Lemma 06WI).

We omit the verification that the functors constructed above are quasi-inverse to each other.

We finish this section with a technical lemma on maps out of quasi-coherent sheaves. It is an analogue of Schemes, Lemma 01I7. We will see later (Criteria for Representability, Theorem 06FI) that the assumptions on the groupoid imply that \(\mathcal{X}\) is an algebraic stack.

Lemma

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 \(x\) the object of \(\mathcal{X}\) over \(U\). 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 (076R).

Proof

According to Lemma 06WR the stackification map \([U/_{\!p}R] \to [U/R]\) (see Groupoids in Spaces, Definition 044Q) 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 06WT 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 (06WC) 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 06WT.

Quasi-coherent sheaves on algebraic stacks

Let \(\mathcal{X}\) be an algebraic stack over \(S\). By Algebraic Stacks, Lemma 04T5 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 06WT. 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 06VZ we see that \(\QCoh(\mathcal{O}_\mathcal{X})\) is abelian!

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 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 06VE. Parenthetically, the example in the proof of Descent, Lemma 06VE shows that in general the strictly full embedding \(\QCoh(\mathcal{O}_\mathcal{X}) \to \textit{LQCoh}(\mathcal{O}_\mathcal{X})\) isn’t exact either.

We collect all the results obtained so far in a single statement.

Lemma

Let \(\mathcal{X}\) be an algebraic stack over \(S\).

1. If \([U/R] \to \mathcal{X}\) is a presentation of \(\mathcal{X}\) then there is a canonical equivalence \(\QCoh(\mathcal{O}_\mathcal{X}) \cong \QCoh(U, R, s, t, c)\).

2. The category \(\QCoh(\mathcal{O}_\mathcal{X})\) is abelian.

3. The inclusion functor \(\QCoh(\mathcal{O}_\mathcal{X}) \to \textit{Mod}(\mathcal{O}_\mathcal{X})\) is right exact but not exact in general.

4. The category \(\QCoh(\mathcal{O}_\mathcal{X})\) has colimits and they agree with colimits in the category \(\textit{Mod}(\mathcal{O}_\mathcal{X})\).

5. 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 \(\QCoh(\mathcal{O}_\mathcal{X})\).

6. Given \(\mathcal{F}, \mathcal{G}\) in \(\QCoh(\mathcal{O}_\mathcal{X})\) with \(\mathcal{F}\) finite locally free the sheaf \(\SheafHom_{\mathcal{O}_\mathcal{X}}(\mathcal{F}, \mathcal{G})\) in \(\textit{Mod}(\mathcal{O}_\mathcal{X})\) is an object of \(\QCoh(\mathcal{O}_\mathcal{X})\).

7. Given a short exact sequence \(0 \to \mathcal{F}_1 \to \mathcal{F}_2 \to \mathcal{F}_3 \to 0\) in \(\textit{Mod}(\mathcal{O}_\mathcal{X})\) with \(\mathcal{F}_1\) and \(\mathcal{F}_3\) quasi-coherent, then \(\mathcal{F}_2\) is quasi-coherent.

Proof

Properties (4), (5), and (6) were proven in Lemma 06WN. Part (1) is Proposition 06WT. Part (2) follows from part (1) and Groupoids in Spaces, Lemma 06VZ as discussed above. Right exactness of the inclusion functor in (3) follows from (4); please compare with Homology, Lemma 010N. For the nonexactness of the inclusion functor in part (3) see Descent, Lemma 06VE. To see (7) observe that it suffices to check the restriction of \(\mathcal{F}_2\) to the big site of a scheme is quasi-coherent (Lemma 06WI), hence this follows from the corresponding part of Descent, Lemma 06VE.

Next we construct the coherator for modules on an algebraic stack.

Proposition

Let \(\mathcal{X}\) be an algebraic stack over \(S\).

1. The category \(\QCoh(\mathcal{O}_\mathcal{X})\) is a Grothendieck abelian category. Consequently, \(\QCoh(\mathcal{O}_\mathcal{X})\) has enough injectives and all limits.

2. The inclusion functor \(\QCoh(\mathcal{O}_\mathcal{X}) \to \textit{Mod}(\mathcal{O}_\mathcal{X})\) has a right adjoint⁴ \[Q : \textit{Mod}(\mathcal{O}_\mathcal{X}) \longrightarrow \QCoh(\mathcal{O}_\mathcal{X})\] such that for every quasi-coherent sheaf \(\mathcal{F}\) the adjunction mapping \(Q(\mathcal{F}) \to \mathcal{F}\) is an isomorphism.

Proof

This proof is a repeat of the proof in the case of schemes, see Properties, Proposition 077P and the case of algebraic spaces, see Properties of Spaces, Proposition 077V. We advise the reader to read either of those proofs first.

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 079A. By Lemma 06WV colimits in \(\QCoh(\mathcal{O}_X)\) exist and agree with colimits in \(\textit{Mod}(\mathcal{O}_X)\). By Modules on Sites, Lemma 03DB filtered colimits are exact. Hence (a) and (b) hold.

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 06WV above we have \(\QCoh(\mathcal{O}_\mathcal{X}) = \QCoh(U, R, s, t, c)\). By Groupoids in Spaces, Lemma 0780 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 079H and Lemma 07D8.

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 07D7. 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 001P) 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.

Cohomology

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 01DP and 01DU. Thus we can use the machinery of Cohomology on Sites, Section 01FT 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 071J there are also derived versions of \(H^p(-)\) and \(R^if_*\).

Lemma

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 \[H^p_\etale(x, \mathcal{F}) = H^p(U_\etale, \mathcal{F}|_{U_\etale}).\]

Proof

The first statement follows from Cohomology on Sites, Lemma 03F3 and the equivalence of Lemma 06W9. The second statement follows from the first combined with Étale Cohomology, Lemma 03YX.

Injective sheaves

The pushforward of an injective abelian sheaf or module is injective.

Lemma

Let \(f : \mathcal{X} \to \mathcal{Y}\) be a \(1\)-morphism of categories fibred in groupoids over \((\Sch/S)_{fppf}\). Let \(\tau \in \{Zar, \etale, smooth, syntomic, fppf\}\).

1. \(f_*\mathcal{I}\) is injective in \(\textit{Ab}(\mathcal{Y}_\tau)\) for \(\mathcal{I}\) injective in \(\textit{Ab}(\mathcal{X}_\tau)\), and

2. \(f_*\mathcal{I}\) is injective in \(\textit{Mod}(\mathcal{Y}_\tau, \mathcal{O}_\mathcal{Y})\) for \(\mathcal{I}\) injective in \(\textit{Mod}(\mathcal{X}_\tau, \mathcal{O}_\mathcal{X})\).

Proof

This follows formally from the fact that \(f^{-1}\) is an exact left adjoint of \(f_*\), see Homology, Lemma 015Z.

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.

Lemma

Let \(p : \mathcal{X} \to (\Sch/S)_{fppf}\) be a category fibred in groupoids.

1. The category \(\mathcal{X}\) has fibre products.

2. If the \(\mathit{Isom}\)-presheaves of \(\mathcal{X}\) are representable by algebraic spaces, then \(\mathcal{X}\) has equalizers.

3. If \(\mathcal{X}\) is an algebraic stack (or more generally a quotient stack), then \(\mathcal{X}\) has equalizers.

Proof

Part (1) follows Categories, Lemma 06N6 as \((\Sch/S)_{fppf}\) has fibre products.

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 \(T/U\) associates the set of automorphisms of \(x|_T\) in the fibre category of \(\mathcal{X}\) over \(T\), see Stacks, Definition 02ZB. 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 \(M = U \times_{e, G, i} U\), which by Morphisms of Spaces, Lemma 03KP is a scheme. Then it is clear that \(x|_M \to x\) is the equalizer of the maps \(\text{id}_x\) and \(i\) in \(\mathcal{X}\). This proves (2).

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 04TB. If \(\mathcal{X}\) is an algebraic stack, then we can choose a presentation \([U/R] \cong \mathcal{X}\) by Algebraic Stacks, Lemma 04T5.

Lemma

Let \(f : \mathcal{X} \to \mathcal{Y}\) be a \(1\)-morphism of categories fibred in groupoids over \((\Sch/S)_{fppf}\).

1. The functor \(f\) transforms fibre products into fibre products.

2. If \(f\) is faithful, then \(f\) transforms equalizers into equalizers.

Proof

By Categories, Lemma 06N6 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.

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 \(\mathcal{Y}\).

Lemma

Let \(f : \mathcal{X} \to \mathcal{Y}\), \(g : \mathcal{Z} \to \mathcal{Y}\) be faithful \(1\)-morphisms of categories fibred in groupoids over \((\Sch/S)_{fppf}\).

1. the functor \(\mathcal{X} \times_\mathcal{Y} \mathcal{Z} \to \mathcal{Y}\) is faithful, and

2. if \(\mathcal{X}, \mathcal{Z}\) have equalizers, so does \(\mathcal{X} \times_\mathcal{Y} \mathcal{Z}\).

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 0040. 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 06WZ) 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.

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.

Lemma

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

1. \(f_!\) is exact, and

2. \(f^{-1}\mathcal{I}\) is injective in \(\textit{Ab}(\mathcal{X}_\tau)\) for \(\mathcal{I}\) injective in \(\textit{Ab}(\mathcal{Y}_\tau)\).

Proof

By Stacks, Lemma 06NW the functor \(f\) is continuous and cocontinuous. Hence by Modules on Sites, Lemma 04BG 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 04BH and to see that the hypotheses of that lemma are satisfied use Lemmas 06WY and 06WZ above. Part (2) follows from this formally, see Homology, Lemma 015Z.

Lemma

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 06X1 on underlying abelian sheaves. If \(f\) is faithful and \(\mathcal{X}\) has equalizers, then

1. \(f_!\) is exact, and

2. \(f^{-1}\mathcal{I}\) is injective in \(\textit{Mod}(\mathcal{X}_\tau, \mathcal{O}_\mathcal{X})\) for \(\mathcal{I}\) injective in \(\textit{Mod}(\mathcal{Y}_\tau, \mathcal{O}_\mathcal{X})\).

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 0797 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 06W9. Modules on Sites, Remark 0798 shows that \(f_!\) agrees with the functor on abelian sheaves.

Assume now that \(\mathcal{X}\) has equalizers and that \(f\) is faithful. Lemma 06X1 tells us that \(f_!\) is exact. Finally, Homology, Lemma 015Z implies the statement on pullbacks of injective modules.

The Čech complex

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.

Throughout this section the situation will be as follows. We are given a \(1\)-morphism of categories fibred in groupoids [06X4]\[\begin{equation} \vcenter{ \xymatrix{ \mathcal{U} \ar[rr]_f \ar[rd]_q & & \mathcal{X} \ar[ld]^p \\ & (\Sch/S)_{fppf} } } \end{equation}\] 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 \[(u_0, \ldots, u_n, x, \alpha_0, \ldots, \alpha_n) \longmapsto (u_{\varphi(0)}, \ldots, u_{\varphi(m)}, x, \alpha_{\varphi(0)}, \ldots, \alpha_{\varphi(m)})\] 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.

Let \(\mathcal{U} \to \mathcal{X}\) be as above and let \(\mathcal{F}\) be an abelian presheaf on \(\mathcal{X}\). The Čech 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 019H. 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 extended Čech 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 Čech 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.

Lemma

Generalities on Čech complexes.

1. 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 Čech complexes \[\check{\mathcal{C}}^\bullet(\mathcal{U} \to \mathcal{X}, \mathcal{F}) \longrightarrow \check{\mathcal{C}}^\bullet(\mathcal{V} \to \mathcal{Y}, e^{-1}\mathcal{F})\]

2. if \(h\) and \(e\) are equivalences, then the map of (1) is an isomorphism,

3. if \(f, f' : \mathcal{U} \to \mathcal{X}\) are \(2\)-isomorphic, then the associated Čech complexes are isomorphic.

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).

Lemma

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 Čech complex is homotopic to zero.

Proof

Set \(\mathcal{U}' = \mathcal{U} \times_\mathcal{X} \mathcal{X}\) equal to the fibre product as described in Categories, Lemma 0040. 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 06X5. 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 \(f \circ s = \text{id}_\mathcal{X}\).

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 Čech complex.

The relative Čech complex

Let \(f : \mathcal{U} \to \mathcal{X}\) be a \(1\)-morphism of categories fibred in groupoids over \((\Sch/S)_{fppf}\) as in (06X4). 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 06UM. 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 019H) \[\ldots \to 0 \to f_{0, *}f_0^{-1}\mathcal{F} \to f_{1, *}f_1^{-1}\mathcal{F} \to f_{2, *}f_2^{-1}\mathcal{F} \to \ldots\] is called the relative Čech complex associated to the situation. We will denote this complex \(\mathcal{K}^\bullet(f, \mathcal{F})\). The extended relative Čech 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 Čech 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.

Remark

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 06UM 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 06TI and 06TN.

Lemma

Generalities on relative Čech complexes.

1. 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 \(e^{-1}\mathcal{K}^\bullet(f, \mathcal{F}) \to \mathcal{K}^\bullet(g, e^{-1}\mathcal{F})\).

2. if \(h\) and \(e\) are equivalences, then the map of (1) is an isomorphism,

3. if \(f, f' : \mathcal{U} \to \mathcal{X}\) are \(2\)-isomorphic, then the associated relative Čech complexes are isomorphic,

Proof

Literally the same as the proof of Lemma 06X5 using the pullback maps of Remark 06X8.

Lemma

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 Čech complex is homotopic to zero.

Proof

Literally the same as the proof of Lemma 06X6.

Remark

Let us “compute” the value of the relative Čech 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 06X5 induces an equivalence \(\mathcal{V}_n = (\Sch/U)_{fppf} \times_{x, \mathcal{X}} \mathcal{U}_n\). Hence we see from (06W6) 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 Čech complex on an object \(x\) of \(\mathcal{X}\) is the Čech complex of the base change of \(f\) to \(\mathcal{X}/x \cong (\Sch/U)_{fppf}\). This implies for example that Lemma 06X6 implies Lemma 06XA and more generally that results on the (usual) Čech complex imply results for the relative Čech complex.

Lemma

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 06X9 is an isomorphism of complexes of abelian presheaves.

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 06XB. 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 06XB. 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 06X5.

Exactness can be checked on a “covering”.

Lemma

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

1. 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

2. \(f^{-1}\mathcal{F} \to f^{-1}\mathcal{G} \to f^{-1}\mathcal{H}\) is exact.

Then the sequence \(\mathcal{F} \to \mathcal{G} \to \mathcal{H}\) is exact.

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.

Proposition

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

1. \(\mathcal{F}\) is an abelian sheaf on \(\mathcal{X}_\tau\), and

2. 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 \(\mathcal{U}\),

then the extended relative Čech 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 \(\textit{Ab}(\mathcal{X}_\tau)\).

Proof

By Lemma 06XD it suffices to check exactness after pulling back to \(\mathcal{U}\). By Lemma 06XC the pullback of the extended relative Čech complex is isomorphic to the extend relative Čech 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 06XA.

Using this we can construct the Čech-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.

Lemma

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

1. \(\mathcal{F}\) is an abelian sheaf on \(\mathcal{X}_\tau\),

2. 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 \(\mathcal{U}\),

3. the category \(\mathcal{U}\) has equalizers, and

4. the functor \(f\) is faithful.

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.

Proof

Before we start the proof we make some remarks. By Lemma 06X0 (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 06X1 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 06WX. Hence Proposition 06XE shows that the extended relative Čech 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 Čech complex \(\check{\mathcal{C}}^\bullet(\mathcal{U} \to \mathcal{X}, \mathcal{I})\) is quasi-isomorphic to \(\Gamma(\mathcal{X}_\tau, \mathcal{I})[0]\).

With these preliminaries out of the way consider the two spectral sequences associated to the double complex (see Homology, Section 012X) \[\check{\mathcal{C}}^\bullet(\mathcal{U} \to \mathcal{X}, \mathcal{I}^\bullet)\] where \(\mathcal{F} \to \mathcal{I}^\bullet\) is an injective resolution in \(\textit{Ab}(\mathcal{X}_\tau)\). The discussion above shows that Homology, Lemma 0133 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.

To be sure there is a version for modules as well.

Lemma

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

1. \(\mathcal{F}\) is an object of \(\textit{Mod}(\mathcal{X}_\tau, \mathcal{O}_\mathcal{X})\),

2. 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 \(\mathcal{U}\),

3. the category \(\mathcal{U}\) has equalizers, and

4. the functor \(f\) is faithful.

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.

Proof

The proof of this lemma is identical to the proof of Lemma 06XF except that it uses an injective resolution in \(\textit{Mod}(\mathcal{X}_\tau, \mathcal{O}_\mathcal{X})\) and it uses Lemma 06X2 instead of Lemma 06X1.

Here is a lemma that translates a more usual kind of covering in the kinds of coverings we have encountered above.

Lemma

Let \(f : \mathcal{X} \to \mathcal{Y}\) be a \(1\)-morphism of categories fibred in groupoids over \((\Sch/S)_{fppf}\).

1. 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 \(\mathcal{Y}\).

2. Assume that \(f\) is representable by algebraic spaces, surjective, and smooth. Then for any object \(y\) of \(\mathcal{Y}\) there exists an étale covering \(\{y_i \to y\}\) and objects \(x_i\) of \(\mathcal{X}\) such that \(f(x_i) \cong y_i\) in \(\mathcal{Y}\).

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 étale morphism \(U \to W\). Then \(U \to V\) is also surjective, flat, and locally of finite presentation (see Morphisms of Spaces, Lemmas 0467, 0468, 03MG, 03XQ, and 03MN). 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 \(\mathcal{Y}\).

Proof of (2). 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 étale morphism \(U \to W\). Then \(U \to V\) is also surjective and smooth (see Morphisms of Spaces, Lemmas 04XX, 03MG, and 03ZD). Hence \(\{U \to V\}\) is a smooth covering. By More on Morphisms, Lemma 055V there exists an étale 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 étale covering of \(\mathcal{Y}\).

Lemma

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

1. \(\mathcal{F}\) is an abelian sheaf on \(\mathcal{X}_\tau\),

2. 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 \(\mathcal{U}\),

3. the category \(\mathcal{U}\) has equalizers, and

4. the functor \(f\) is faithful.

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.

Proof

Note that the assumptions on \(f : \mathcal{U} \to \mathcal{X}\) and \(\mathcal{F}\) are identical to those in Lemma 06XF. 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 012X 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 0133, 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 06X1) 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.

Lemma

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

1. \(\mathcal{F}\) is an object of \(\textit{Mod}(\mathcal{X}_\tau, \mathcal{O}_\mathcal{X})\),

2. 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 \(\mathcal{U}\),

3. the category \(\mathcal{U}\) has equalizers, and

4. the functor \(f\) is faithful.

Then there is a first quadrant spectral sequence in \(\textit{Mod}(\mathcal{Y}_\tau, \mathcal{O}_\mathcal{Y})\) \[E_1^{p, q} = R^q(g \circ f_p)_*f_p^{-1}\mathcal{F} \Rightarrow R^{p + q}g_*\mathcal{F}\] where all higher direct images are computed in the \(\tau\)-topology.

Proof

The proof is identical to the proof of Lemma 072D except that it uses an injective resolution in \(\textit{Mod}(\mathcal{X}_\tau, \mathcal{O}_\mathcal{X})\) and it uses Lemma 06X2 instead of Lemma 06X1.

Cohomology on algebraic stacks

Let \(\mathcal{X}\) be an algebraic stack over \(S\). In the sections above we have seen how to define sheaves for the étale, ..., fppf topologies on \(\mathcal{X}\). In fact, we have constructed a site \(\mathcal{X}_\tau\) for each \(\tau \in \{Zar, \etale, smooth, syntomic, fppf\}\). There is a notion of an abelian sheaf \(\mathcal{F}\) on these sites. In the chapter on cohomology of sites we have explained how to define cohomology. Putting all of this together, let’s define the derived global sections or total cohomology \[R\Gamma_{Zar}(\mathcal{X}, \mathcal{F}), R\Gamma_\etale(\mathcal{X}, \mathcal{F}), \ldots, R\Gamma_{fppf}(\mathcal{X}, \mathcal{F})\] as \(\Gamma(\mathcal{X}_\tau, \mathcal{I}^\bullet)\) where \(\mathcal{F} \to \mathcal{I}^\bullet\) is an injective resolution in \(\textit{Ab}(\mathcal{X}_\tau)\). The \(i\)th cohomology group of \(\mathcal{F}\) is the \(i\)th cohomology of the total cohomology. We will denote this \[H^i_{Zar}(\mathcal{X}, \mathcal{F}), H^i_\etale(\mathcal{X}, \mathcal{F}), \ldots, H^i_{fppf}(\mathcal{X}, \mathcal{F}).\] It will turn out that \(H^i_\etale = H^i_{smooth}\) because of More on Morphisms, Lemma 055V.

If \(\mathcal{F}\) is a presheaf of \(\mathcal{O}_\mathcal{X}\)-modules which is a sheaf in the \(\tau\)-topology, then we use injective resolutions in \(\textit{Mod}(\mathcal{X}_\tau, \mathcal{O}_\mathcal{X})\) to compute its total cohomology, resp. cohomology groups; the end result is quasi-isomorphic, resp. isomorphic to the cohomology of \(\mathcal{F}\) viewed as a sheaf of abelian groups by the very general Cohomology on Sites, Lemma 03FD.

So far our only tool to compute cohomology groups is the result on Čech complexes proved above. We rephrase it here in the language of algebraic stacks for the étale and the fppf topology. Let \(f : \mathcal{U} \to \mathcal{X}\) be a \(1\)-morphism of algebraic stacks. Recall that \[f_p : \mathcal{U}_p = \mathcal{U} \times_\mathcal{X} \ldots \times_\mathcal{X} \mathcal{U} \longrightarrow \mathcal{X}\] is the structure morphism where there are \((p + 1)\)-factors. Also, recall that a sheaf on \(\mathcal{X}\) is a sheaf for the fppf topology. Note that if \(\mathcal{U}\) is an algebraic space, then \(f : \mathcal{U} \to \mathcal{X}\) is representable by algebraic spaces, see Algebraic Stacks, Lemma 045G. Thus the proposition applies in particular to a smooth cover of the algebraic stack \(\mathcal{X}\) by a scheme.

Proposition

Let \(f : \mathcal{U} \to \mathcal{X}\) be a \(1\)-morphism of algebraic stacks.

1. Let \(\mathcal{F}\) be an abelian étale sheaf on \(\mathcal{X}\). Assume that \(f\) is representable by algebraic spaces, surjective, and smooth. Then there is a spectral sequence \[E_1^{p, q} = H^q_\etale(\mathcal{U}_p, f_p^{-1}\mathcal{F}) \Rightarrow H^{p + q}_\etale(\mathcal{X}, \mathcal{F})\]

2. Let \(\mathcal{F}\) be an abelian sheaf on \(\mathcal{X}\). Assume that \(f\) is representable by algebraic spaces, surjective, flat, and locally of finite presentation. Then there is a spectral sequence \[E_1^{p, q} = H^q_{fppf}(\mathcal{U}_p, f_p^{-1}\mathcal{F}) \Rightarrow H^{p + q}_{fppf}(\mathcal{X}, \mathcal{F})\]

Proof

To see this we will check the hypotheses (1) – (4) of Lemma 06XF. The \(1\)-morphism \(f\) is faithful by Algebraic Stacks, Lemma 04Y5. This proves (4). Hypothesis (3) follows from the fact that \(\mathcal{U}\) is an algebraic stack, see Lemma 06WY. To see (2) apply Lemma 06XH. Condition (1) is satisfied by fiat.

Higher direct images and algebraic stacks

Let \(g : \mathcal{X} \to \mathcal{Y}\) be a \(1\)-morphism of algebraic stacks over \(S\). In the sections above we have constructed a morphism of ringed topoi \(g : \Sh(\mathcal{X}_\tau) \to \Sh(\mathcal{Y}_\tau)\) for each \(\tau \in \{Zar, \etale, smooth, syntomic, fppf\}\). In the chapter on cohomology of sites we have explained how to define higher direct images. Hence the total direct image \(Rg_*\mathcal{F}\) is defined as \(g_*\mathcal{I}^\bullet\) where \(\mathcal{F} \to \mathcal{I}^\bullet\) is an injective resolution in \(\textit{Ab}(\mathcal{X}_\tau)\). The \(i\)th higher direct image \(R^ig_*\mathcal{F}\) is the \(i\)th cohomology of the total direct image. Important: it matters which topology \(\tau\) is used here!

If \(\mathcal{F}\) is a presheaf of \(\mathcal{O}_\mathcal{X}\)-modules which is a sheaf in the \(\tau\)-topology, then we use injective resolutions in \(\textit{Mod}(\mathcal{X}_\tau, \mathcal{O}_\mathcal{X})\) to compute total direct image and higher direct images.

So far our only tool to compute the higher direct images of \(g_*\) is the result on Čech complexes proved above. This requires the choice of a “covering” \(f : \mathcal{U} \to \mathcal{X}\). If \(\mathcal{U}\) is an algebraic space, then \(f : \mathcal{U} \to \mathcal{X}\) is representable by algebraic spaces, see Algebraic Stacks, Lemma 045G. Thus the proposition applies in particular to a smooth cover of the algebraic stack \(\mathcal{X}\) by a scheme.

Proposition

Let \(f : \mathcal{U} \to \mathcal{X}\) and \(g : \mathcal{X} \to \mathcal{Y}\) be composable \(1\)-morphisms of algebraic stacks.

1. Assume that \(f\) is representable by algebraic spaces, surjective and smooth.

   1. If \(\mathcal{F}\) is in \(\textit{Ab}(\mathcal{X}_\etale)\) then there is a spectral sequence \[E_1^{p, q} = R^q(g \circ f_p)_*f_p^{-1}\mathcal{F} \Rightarrow R^{p + q}g_*\mathcal{F}\] in \(\textit{Ab}(\mathcal{Y}_\etale)\) with higher direct images computed in the étale topology.

   2. If \(\mathcal{F}\) is in \(\textit{Mod}(\mathcal{X}_\etale, \mathcal{O}_\mathcal{X})\) then there is a spectral sequence \[E_1^{p, q} = R^q(g \circ f_p)_*f_p^{-1}\mathcal{F} \Rightarrow R^{p + q}g_*\mathcal{F}\] in \(\textit{Mod}(\mathcal{Y}_\etale, \mathcal{O}_\mathcal{Y})\).

2. Assume that \(f\) is representable by algebraic spaces, surjective, flat, and locally of finite presentation.

   1. If \(\mathcal{F}\) is in \(\textit{Ab}(\mathcal{X})\) then there is a spectral sequence \[E_1^{p, q} = R^q(g \circ f_p)_*f_p^{-1}\mathcal{F} \Rightarrow R^{p + q}g_*\mathcal{F}\] in \(\textit{Ab}(\mathcal{Y})\) with higher direct images computed in the fppf topology.

   2. If \(\mathcal{F}\) is in \(\textit{Mod}(\mathcal{O}_\mathcal{X})\) then there is a spectral sequence \[E_1^{p, q} = R^q(g \circ f_p)_*f_p^{-1}\mathcal{F} \Rightarrow R^{p + q}g_*\mathcal{F}\] in \(\textit{Mod}(\mathcal{O}_\mathcal{Y})\).

Proof

To see this we will check the hypotheses (1) – (4) of Lemma 072D and Lemma 072E. The \(1\)-morphism \(f\) is faithful by Algebraic Stacks, Lemma 04Y5. This proves (4). Hypothesis (3) follows from the fact that \(\mathcal{U}\) is an algebraic stack, see Lemma 06WY. To see (2) apply Lemma 06XH. Condition (1) is satisfied by fiat in all four cases.

Here is a description of higher direct images for a morphism of algebraic stacks.

Lemma

Let \(S\) be a scheme. Let \(f : \mathcal{X} \to \mathcal{Y}\) be a \(1\)-morphism of algebraic stacks⁵ over \(S\). Let \(\tau \in \{Zariski,\linebreak[0] \etale,\linebreak[0] smooth,\linebreak[0] syntomic,\linebreak[0] fppf\}\). Let \(\mathcal{F}\) be an object of \(\textit{Ab}(\mathcal{X}_\tau)\) or \(\textit{Mod}(\mathcal{X}_\tau, \mathcal{O}_\mathcal{X})\). Then the sheaf \(R^if_*\mathcal{F}\) is the sheaf associated to the presheaf \[y \longmapsto H^i_\tau\Big((\Sch/V)_{fppf} \times_{y, \mathcal{Y}} \mathcal{X}, \ \text{pr}^{-1}\mathcal{F}\Big)\] Here \(y\) is an object of \(\mathcal{Y}\) lying over the scheme \(V\).

Proof

Choose an injective resolution \(\mathcal{F}[0] \to \mathcal{I}^\bullet\). By the formula for pushforward (06W6) we see that \(R^if_*\mathcal{F}\) is the sheaf associated to the presheaf which associates to \(y\) the cohomology of the complex \[\begin{matrix} \Gamma\Big((\Sch/V)_{fppf} \times_{y, \mathcal{Y}} \mathcal{X}, \ \text{pr}^{-1}\mathcal{I}^{i - 1}\Big) \\ \downarrow \\ \Gamma\Big((\Sch/V)_{fppf} \times_{y, \mathcal{Y}} \mathcal{X}, \ \text{pr}^{-1}\mathcal{I}^i\Big) \\ \downarrow \\ \Gamma\Big((\Sch/V)_{fppf} \times_{y, \mathcal{Y}} \mathcal{X}, \ \text{pr}^{-1}\mathcal{I}^{i + 1}\Big) \end{matrix}\] Since \(\text{pr}^{-1}\) is exact, it suffices to show that \(\text{pr}^{-1}\) preserves injectives. This follows from Lemmas 06X1 and 06X2 as well as the fact that \(\text{pr}\) is a representable morphism of algebraic stacks (so that \(\text{pr}\) is faithful by Algebraic Stacks, Lemma 04Y5 and that \((\Sch/V)_{fppf} \times_{y, \mathcal{Y}} \mathcal{X}\) has equalizers by Lemma 06WY).

Here is a trivial base change result.

Lemma

Let \(S\) be a scheme. Let \(\tau \in \{Zariski,\linebreak[0] \etale,\linebreak[0] smooth,\linebreak[0] syntomic,\linebreak[0] fppf\}\). 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 algebraic stacks over \(S\). Then the base change map is an isomorphism \[g^{-1}Rf_*\mathcal{F} \longrightarrow Rf'_*(g')^{-1}\mathcal{F}\] functorial for \(\mathcal{F}\) in \(\textit{Ab}(\mathcal{X}_\tau)\) or \(\mathcal{F}\) in \(\textit{Mod}(\mathcal{X}_\tau, \mathcal{O}_\mathcal{X})\).

Proof

The isomorphism \(g^{-1}f_*\mathcal{F} = f'_*(g')^{-1}\mathcal{F}\) is Lemma 075B (and it holds for arbitrary presheaves). For the total direct images, there is a base change map because the morphisms \(g\) and \(g'\) are flat, see Cohomology on Sites, Section 0735. To see that this map is a quasi-isomorphism we can use that for an object \(y'\) of \(\mathcal{Y}'\) over a scheme \(V\) there is an equivalence \[(\Sch/V)_{fppf} \times_{g(y'), \mathcal{Y}} \mathcal{X} = (\Sch/V)_{fppf} \times_{y', \mathcal{Y}'} (\mathcal{Y}' \times_\mathcal{Y} \mathcal{X})\] We conclude that the induced map \(g^{-1}R^if_*\mathcal{F} \to R^if'_*(g')^{-1}\mathcal{F}\) is an isomorphism by Lemma 075G.

Comparison

In this section we collect some results on comparing cohomology defined using stacks and using algebraic spaces.

Lemma

Let \(S\) be a scheme. Let \(\mathcal{X}\) be an algebraic stack over \(S\) representable by the algebraic space \(F\).

1. If \(\mathcal{I}\) injective in \(\textit{Ab}(\mathcal{X}_\etale)\), then \(\mathcal{I}|_{F_\etale}\) is injective in \(\textit{Ab}(F_\etale)\),

2. If \(\mathcal{I}^\bullet\) is a K-injective complex in \(\textit{Ab}(\mathcal{X}_\etale)\), then \(\mathcal{I}^\bullet|_{F_\etale}\) is a K-injective complex in \(\textit{Ab}(F_\etale)\).

The same does not hold for modules.

Proof

This follows formally from the fact that the restriction functor \(\pi_{F, *} = i_F^{-1}\) (see Lemma 073M) is right adjoint to the exact functor \(\pi_F^{-1}\), see Homology, Lemma 015Z and Derived Categories, Lemma 08BJ. To see that the lemma does not hold for modules, we refer the reader to Étale Cohomology, Lemma 0758.

Lemma

Let \(S\) be a scheme. Let \(f : \mathcal{X} \to \mathcal{Y}\) be a morphism of algebraic stacks over \(S\). Assume \(\mathcal{X}\), \(\mathcal{Y}\) are representable by algebraic spaces \(F\), \(G\). Denote \(f : F \to G\) the induced morphism of algebraic spaces.

1. For any \(\mathcal{F} \in \textit{Ab}(\mathcal{X}_\etale)\) we have \[(Rf_*\mathcal{F})|_{G_\etale} = Rf_{small, *}(\mathcal{F}|_{F_\etale})\] in \(D(G_\etale)\).

2. For any object \(\mathcal{F}\) of \(\textit{Mod}(\mathcal{X}_\etale, \mathcal{O}_\mathcal{X})\) we have \[(Rf_*\mathcal{F})|_{G_\etale} = Rf_{small, *}(\mathcal{F}|_{F_\etale})\] in \(D(\mathcal{O}_G)\).

Proof

Part (1) follows immediately from Lemma 075L and (075M) on choosing an injective resolution of \(\mathcal{F}\).

Part (2) can be proved as follows. In Lemma 073N we have seen that \(\pi_G \circ f = f_{small} \circ \pi_F\) as morphisms of ringed sites. Hence we obtain \(R\pi_{G, *} \circ Rf_* = Rf_{small, *} \circ R\pi_{F, *}\) by Cohomology on Sites, Lemma 0D6E. Since the restriction functors \(\pi_{F, *}\) and \(\pi_{G, *}\) are exact, we conclude.

Lemma

Let \(S\) be a scheme. Consider a \(2\)-fibre product square \[\xymatrix{ \mathcal{X}' \ar[r]_{g'} \ar[d]_{f'} & \mathcal{X} \ar[d]^f \\ \mathcal{Y}' \ar[r]^g & \mathcal{Y} }\] of algebraic stacks over \(S\). Assume that \(f\) is representable by algebraic spaces and that \(\mathcal{Y}'\) is representable by an algebraic space \(G'\). Then \(\mathcal{X}'\) is representable by an algebraic space \(F'\) and denoting \(f' : F' \to G'\) the induced morphism of algebraic spaces we have \[g^{-1}(Rf_*\mathcal{F})|_{G'_\etale} = Rf'_{small, *}((g')^{-1}\mathcal{F}|_{F'_\etale})\] for any \(\mathcal{F}\) in \(\textit{Ab}(\mathcal{X}_\etale)\) or in \(\textit{Mod}(\mathcal{X}_\etale, \mathcal{O}_\mathcal{X})\)

Proof

Follows formally on combining Lemmas 075H and 075N.

Change of topology

Here is a technical lemma which tells us that the fppf cohomology of a locally quasi-coherent sheaf is equal to its étale cohomology provided the comparison maps are isomorphisms for morphisms of \(\mathcal{X}\) lying over flat morphisms.

Lemma

Let \(S\) be a scheme. Let \(\mathcal{X}\) be an algebraic stack over \(S\). Let \(\mathcal{F}\) be a presheaf of \(\mathcal{O}_\mathcal{X}\)-modules. Assume

1. \(\mathcal{F}\) is locally quasi-coherent, and

2. for any morphism \(\varphi : x \to y\) of \(\mathcal{X}\) which lies over a morphism of schemes \(f : U \to V\) which is flat and locally of finite presentation the comparison map \(c_\varphi : f_{small}^*\mathcal{F}|_{V_\etale} \to \mathcal{F}|_{U_\etale}\) of (06WC) is an isomorphism.

Then \(\mathcal{F}\) is a sheaf for the fppf topology.

Proof

Let \(\{x_i \to x\}\) be an fppf covering of \(\mathcal{X}\) lying over the fppf covering \(\{f_i : U_i \to U\}\) of schemes over \(S\). By assumption the restriction \(\mathcal{G} = \mathcal{F}|_{U_\etale}\) is quasi-coherent and the comparison maps \(f_{i, small}^*\mathcal{G} \to \mathcal{F}|_{U_{i, \etale}}\) are isomorphisms. Hence the sheaf condition for \(\mathcal{F}\) and the covering \(\{x_i \to x\}\) is equivalent to the sheaf condition for \(\mathcal{G}^a\) on \((\Sch/U)_{fppf}\) and the covering \(\{U_i \to U\}\) which holds by Descent, Lemma 03DT.

Lemma

Let \(S\) be a scheme. Let \(\mathcal{X}\) be an algebraic stack over \(S\). Let \(\mathcal{F}\) be a presheaf \(\mathcal{O}_\mathcal{X}\)-module such that

1. \(\mathcal{F}\) is locally quasi-coherent, and

2. for any morphism \(\varphi : x \to y\) of \(\mathcal{X}\) which lies over a morphism of schemes \(f : U \to V\) which is flat and locally of finite presentation, the comparison map \(c_\varphi : f_{small}^*\mathcal{F}|_{V_\etale} \to \mathcal{F}|_{U_\etale}\) of (06WC) is an isomorphism.

Then \(\mathcal{F}\) is an \(\mathcal{O}_\mathcal{X}\)-module and we have the following

1. If \(\epsilon : \mathcal{X}_{fppf} \to \mathcal{X}_\etale\) is the comparison morphism, then \(R\epsilon_*\mathcal{F} = \epsilon_*\mathcal{F}\).

2. The cohomology groups \(H^p_{fppf}(\mathcal{X}, \mathcal{F})\) are equal to the cohomology groups computed in the étale topology on \(\mathcal{X}\). Similarly for the cohomology groups \(H^p_{fppf}(x, \mathcal{F})\) and the derived versions \(R\Gamma(\mathcal{X}, \mathcal{F})\) and \(R\Gamma(x, \mathcal{F})\).

3. If \(f : \mathcal{X} \to \mathcal{Y}\) is a \(1\)-morphism of categories fibred in groupoids over \((\Sch/S)_{fppf}\) then \(R^if_*\mathcal{F}\) is equal to the fppf-sheafification of the higher direct image computed in the étale cohomology. Similarly for derived pullback.

Proof

The assertion that \(\mathcal{F}\) is an \(\mathcal{O}_\mathcal{X}\)-module follows from Lemma 076T. Note that \(\epsilon\) is a morphism of sites given by the identity functor on \(\mathcal{X}\). The sheaf \(R^p\epsilon_*\mathcal{F}\) is therefore the sheaf associated to the presheaf \(x \mapsto H^p_{fppf}(x, \mathcal{F})\), see Cohomology on Sites, Lemma 072W. To prove (1) it suffices to show that \(H^p_{fppf}(x, \mathcal{F}) = 0\) for \(p > 0\) whenever \(x\) lies over an affine scheme \(U\). By Lemma 075F we have \(H^p_{fppf}(x, \mathcal{F}) = H^p((\Sch/U)_{fppf}, x^{-1}\mathcal{F})\). Combining Descent, Lemma 0756 with Cohomology of Schemes, Lemma 01XB we see that these cohomology groups are zero.

We have seen above that \(\epsilon_*\mathcal{F}\) and \(\mathcal{F}\) are the sheaves on \(\mathcal{X}_\etale\) and \(\mathcal{X}_{fppf}\) corresponding to the same presheaf on \(\mathcal{X}\) (and this is true more generally for any sheaf in the fppf topology on \(\mathcal{X}\)). We often abusively identify \(\mathcal{F}\) and \(\epsilon_*\mathcal{F}\) and this is the sense in which parts (2) and (3) of the lemma should be understood. Thus part (2) follows formally from (1) and the Leray spectral sequence, see Cohomology on Sites, Lemma 0733.

Finally we prove (3). The sheaf \(R^if_*\mathcal{F}\) (resp. \(Rf_{\etale, *}\mathcal{F}\)) is the sheaf associated to the presheaf \[y \longmapsto H^i_\tau\Big((\Sch/V)_{fppf} \times_{y, \mathcal{Y}} \mathcal{X}, \ \text{pr}^{-1}\mathcal{F}\Big)\] where \(\tau\) is \(fppf\) (resp. \(\etale\)), see Lemma 075G. Note that \(\text{pr}^{-1}\mathcal{F}\) satisfies properties (a) and (b) also (by Lemmas 06WL and 075D), hence these two presheaves are equal by (2). This immediately implies (3).

We will use the following lemma to compare étale cohomology of sheaves on algebraic stacks with cohomology on the lisse-étale topos.

Lemma

Let \(S\) be a scheme. Let \(\mathcal{X}\) be an algebraic stack over \(S\). Let \(\tau = \etale\) (resp. \(\tau = fppf\)). Let \(\mathcal{X}' \subset \mathcal{X}\) be a full subcategory with the following properties

1. if \(x \to x'\) is a morphism of \(\mathcal{X}\) which lies over a smooth (resp. flat and locally finitely presented) morphism of schemes and \(x' \in \Ob(\mathcal{X}')\), then \(x \in \Ob(\mathcal{X}')\), and

2. there exists an object \(x \in \Ob(\mathcal{X}')\) lying over a scheme \(U\) such that the associated \(1\)-morphism \(x : (\Sch/U)_{fppf} \to \mathcal{X}\) is smooth and surjective.

We get a site \(\mathcal{X}'_\tau\) by declaring a covering of \(\mathcal{X}'\) to be any family of morphisms \(\{x_i \to x\}\) in \(\mathcal{X}'\) which is a covering in \(\mathcal{X}_\tau\). Then the inclusion functor \(\mathcal{X}' \to \mathcal{X}_\tau\) is fully faithful, cocontinuous, and continuous, whence defines a morphism of topoi \[g : \Sh(\mathcal{X}'_\tau) \longrightarrow \Sh(\mathcal{X}_\tau)\] and \(H^p(\mathcal{X}'_\tau, g^{-1}\mathcal{F}) = H^p(\mathcal{X}_\tau, \mathcal{F})\) for all \(p \geq 0\) and all \(\mathcal{F} \in \textit{Ab}(\mathcal{X}_\tau)\).

Proof

Note that assumption (1) implies that if \(\{x_i \to x\}\) is a covering of \(\mathcal{X}_\tau\) and \(x \in \Ob(\mathcal{X}')\), then we have \(x_i \in \Ob(\mathcal{X}')\). Hence we see that \(\mathcal{X}' \to \mathcal{X}\) is continuous and cocontinuous as the coverings of objects of \(\mathcal{X}'_\tau\) agree with their coverings seen as objects of \(\mathcal{X}_\tau\). We obtain the morphism \(g\) and the functor \(g^{-1}\) is identified with the restriction functor, see Sites, Lemma 00XR.

In particular, if \(\{x_i \to x\}\) is a covering in \(\mathcal{X}'_\tau\), then for any abelian sheaf \(\mathcal{F}\) on \(\mathcal{X}\) then \[\check H^p(\{x_i \to x\}, g^{-1}\mathcal{F}) = \check H^p(\{x_i \to x\}, \mathcal{F})\] Thus if \(\mathcal{I}\) is an injective abelian sheaf on \(\mathcal{X}_\tau\) then we see that the higher Čech cohomology groups are zero (Cohomology on Sites, Lemma 03AW). Hence \(H^p(x, g^{-1}\mathcal{I}) = 0\) for all objects \(x\) of \(\mathcal{X}'\) (Cohomology on Sites, Lemma 03F9). In other words injective abelian sheaves on \(\mathcal{X}_\tau\) are right acyclic for the functor \(H^0(x, g^{-1}-)\). It follows that \(H^p(x, g^{-1}\mathcal{F}) = H^p(x, \mathcal{F})\) for all \(\mathcal{F} \in \textit{Ab}(\mathcal{X})\) and all \(x \in \Ob(\mathcal{X}')\).

Choose an object \(x \in \mathcal{X}'\) lying over a scheme \(U\) as in assumption (2). In particular \(\mathcal{X}/x \to \mathcal{X}\) is a morphism of algebraic stacks which representable by algebraic spaces, surjective, and smooth. (Note that \(\mathcal{X}/x\) is equivalent to \((\Sch/U)_{fppf}\), see Lemma 06W0.) The map of sheaves \[h_x \longrightarrow *\] in \(\Sh(\mathcal{X}_\tau)\) is surjective. Namely, for any object \(x'\) of \(\mathcal{X}\) there exists a \(\tau\)-covering \(\{x'_i \to x'\}\) such that there exist morphisms \(x'_i \to x\), see Lemma 06XH. Since \(g\) is exact, the map of sheaves \[g^{-1}h_x \longrightarrow * = g^{-1}*\] in \(\Sh(\mathcal{X}'_\tau)\) is surjective also. Let \(h_{x, n}\) be the \((n + 1)\)-fold product \(h_x \times \ldots \times h_x\). Then we have spectral sequences [07AL]\[\begin{equation} E_1^{p, q} = H^q(h_{x, p}, \mathcal{F}) \Rightarrow H^{p + q}(\mathcal{X}_\tau, \mathcal{F}) \end{equation}\] and [07AM]\[\begin{equation} E_1^{p, q} = H^q(g^{-1}h_{x, p}, g^{-1}\mathcal{F}) \Rightarrow H^{p + q}(\mathcal{X}'_\tau, g^{-1}\mathcal{F}) \end{equation}\] see Cohomology on Sites, Lemma 079Z.

Case I: \(\mathcal{X}\) has a final object \(x\) which is also an object of \(\mathcal{X}'\). This case follows immediately from the discussion in the second paragraph above.

Case II: \(\mathcal{X}\) is representable by an algebraic space \(F\). In this case the sheaves \(h_{x, n}\) are representable by an object \(x_n\) in \(\mathcal{X}\). (Namely, if \(\mathcal{S}_F = \mathcal{X}\) and \(x : U \to F\) is the given object, then \(h_{x, n}\) is representable by the object \(U \times_F \ldots \times_F U \to F\) of \(\mathcal{S}_F\).) It follows that \(H^q(h_{x, p}, \mathcal{F}) = H^q(x_p, \mathcal{F})\). The morphisms \(x_n \to x\) lie over smooth morphisms of schemes, hence \(x_n \in \mathcal{X}'\) for all \(n\). Hence \(H^q(g^{-1}h_{x, p}, g^{-1}\mathcal{F}) = H^q(x_p, g^{-1}\mathcal{F})\). Thus in the two spectral sequences (07AL) and (07AM) above the \(E_1^{p, q}\) terms agree by the discussion in the second paragraph. The lemma follows in Case II as well.

Case III: \(\mathcal{X}\) is an algebraic stack. We claim that in this case the cohomology groups \(H^q(h_{x, p}, \mathcal{F})\) and \(H^q(g^{-1}h_{x, n}, g^{-1}\mathcal{F})\) agree by Case II above. Once we have proved this the result will follow as before.

Namely, consider the category \(\mathcal{X}/h_{x, n}\), see Sites, Lemma 0791. Since \(h_{x, n}\) is the \((n + 1)\)-fold product of \(h_x\) an object of this category is an \((n + 2)\)-tuple \((y, s_0, \ldots, s_n)\) where \(y\) is an object of \(\mathcal{X}\) and each \(s_i : y \to x\) is a morphism of \(\mathcal{X}\). This is a category over \((\Sch/S)_{fppf}\). There is an equivalence \[\mathcal{X}/h_{x, n} \longrightarrow (\Sch/U)_{fppf} \times_\mathcal{X} \ldots \times_\mathcal{X} (\Sch/U)_{fppf} =: \mathcal{U}_n\] over \((\Sch/S)_{fppf}\). Namely, if \(x : (\Sch/U)_{fppf} \to \mathcal{X}\) also denotes the \(1\)-morphism associated with \(x\) and \(p : \mathcal{X} \to (\Sch/S)_{fppf}\) the structure functor, then we can think of \((y, s_0, \ldots, s_n)\) as \((y, f_0, \ldots, f_n, \alpha_0, \ldots, \alpha_n)\) where \(y\) is an object of \(\mathcal{X}\), \(f_i : p(y) \to p(x)\) is a morphism of schemes, and \(\alpha_i : y \to x(f_i)\) an isomorphism. The category of \(2n+3\)-tuples \((y, f_0, \ldots, f_n, \alpha_0, \ldots, \alpha_n)\) is an incarnation of the \((n + 1)\)-fold fibred product \(\mathcal{U}_n\) of algebraic stacks displayed above, as we discussed in Section 06X3. By Cohomology on Sites, Lemma 07A0 we have \[H^p(\mathcal{U}_n, \mathcal{F}|_{\mathcal{U}_n}) = H^p(\mathcal{X}/h_{x, n}, \mathcal{F}|_{\mathcal{X}/h_{x, n}}) = H^p(h_{x, n}, \mathcal{F}).\] Finally, we discuss the “primed” analogue of this. Namely, \(\mathcal{X}'/h_{x, n}\) corresponds, via the equivalence above to the full subcategory \(\mathcal{U}'_n \subset \mathcal{U}_n\) consisting of those tuples \((y, f_0, \ldots, f_n, \alpha_0, \ldots, \alpha_n)\) with \(y \in \mathcal{X}'\). Hence certainly property (1) of the statement of the lemma holds for the inclusion \(\mathcal{U}'_n \subset \mathcal{U}_n\). To see property (2) choose an object \(\xi = (y, s_0, \ldots, s_n)\) which lies over a scheme \(W\) such that \((\Sch/W)_{fppf} \to \mathcal{U}_n\) is smooth and surjective (this is possible as \(\mathcal{U}_n\) is an algebraic stack). Then \((\Sch/W)_{fppf} \to \mathcal{U}_n \to (\Sch/U)_{fppf}\) is smooth as a composition of base changes of the morphism \(x : (\Sch/U)_{fppf} \to \mathcal{X}\), see Algebraic Stacks, Lemmas 045C and 045B. Thus axiom (1) for \(\mathcal{X}\) implies that \(y\) is an object of \(\mathcal{X}'\) whence \(\xi\) is an object of \(\mathcal{U}'_n\). Using again \[H^p(\mathcal{U}'_n, \mathcal{F}|_{\mathcal{U}'_n}) = H^p(\mathcal{X}'/h_{x, n}, \mathcal{F}|_{\mathcal{X}'/h_{x, n}}) = H^p(g^{-1}h_{x, n}, g^{-1}\mathcal{F}).\] we now can use Case II for \(\mathcal{U}'_n \subset \mathcal{U}_n\) to conclude.

Restricting to affines

In this section, given a category \(\mathcal{X}\) fibred in groupoids over \((\Sch/S)_{fppf}\) we will consider the full subcategory \(\mathcal{X}_{affine}\) of \(\mathcal{X}\) consisting of objects \(x\) lying over affine schemes \(U\). We will see how, for any topology \(\tau\) finer than the Zariski topology, the category of sheaves on \(\mathcal{X}\) and \(\mathcal{X}_{affine, \tau}\) agree.

Definition

Let \(p : \mathcal{X} \to (\Sch/S)_{fppf}\) be a category fibred in groupoids. The associated affine site is the full subcategory \(\mathcal{X}_{affine}\) of \(\mathcal{X}\) whose objects are those \(x \in \Ob(\mathcal{X})\) lying over a scheme \(U\) such that \(U\) is affine. The topology on \(\mathcal{X}_{affine}\) will be the chaotic one, i.e., such that sheaves on \(\mathcal{X}_{affine}\) are the same as presheaves.

Thus the functor \(p : \mathcal{X} \to (\Sch/S)_{fppf}\) restricts to a functor \[p : \mathcal{X}_{affine} \longrightarrow (\textit{Aff}/S)_{fppf}\] where the notation on the right hand side is the one introduced in Topologies, Definition 021S. It is clear that \(\mathcal{X}_{affine}\) is fibred in groupoids over \((\textit{Aff}/S)_{fppf}\). It follows that \(\mathcal{X}_{affine}\) inherits a Zariski, étale, smooth, syntomic, and fppf topology from \((\textit{Aff}/S)_{Zar}\), \((\textit{Aff}/S)_\etale\), \((\textit{Aff}/S)_{smooth}\), \((\textit{Aff}/S)_{syntomic}\), and \((\textit{Aff}/S)_{fppf}\), see Stacks, Definition 06NV.

Definition

Let \(p : \mathcal{X} \to (\Sch/S)_{fppf}\) be a category fibred in groupoids.

1. The associated affine Zariski site \(\mathcal{X}_{affine, Zar}\) is the structure of site on \(\mathcal{X}_{affine}\) inherited from \((\textit{Aff}/S)_{Zar}\).

2. The associated affine étale site \(\mathcal{X}_{affine, \etale}\) is the structure of site on \(\mathcal{X}_{affine}\) inherited from \((\textit{Aff}/S)_\etale\).

3. The associated affine smooth site \(\mathcal{X}_{affine, smooth}\) is the structure of site on \(\mathcal{X}_{affine}\) inherited from \((\textit{Aff}/S)_{smooth}\).

4. The associated affine syntomic site \(\mathcal{X}_{affine, syntomic}\) is the structure of site on \(\mathcal{X}_{affine}\) inherited from \((\textit{Aff}/S)_{syntomic}\).

5. The associated affine fppf site \(\mathcal{X}_{affine, fppf}\) is the structure of site on \(\mathcal{X}_{affine}\) inherited from \((\textit{Aff}/S)_{fppf}\).

This definition makes sense by the discussion above. For each \(\tau \in \{Zariski, \etale, smooth, syntomic, fppf\}\) a family of morphisms \(\{x_i \to x\}_{i \in I}\) with fixed target in \(\mathcal{X}_{affine}\) is a covering in \(\mathcal{X}_{affine, \tau}\) if and only if the family of morphisms \(\{p(x_i) \to p(x)\}_{i \in I}\) of affine schemes is a standard \(\tau\)-covering as defined in Topologies, Definitions 020R, 0219, 0223, 0229, and 021Q.

Lemma

Let \(p : \mathcal{X} \to (\Sch/S)_{fppf}\) be a category fibred in groupoids. Let \(\tau \in \{Zariski, \etale, smooth, syntomic, fppf\}\). The functor \(\mathcal{X}_{affine, \tau} \to \mathcal{X}_\tau\) is a special cocontinuous functor. Hence it induces an equivalence of topoi from \(\Sh(\mathcal{X}_{affine, \tau})\) to \(\Sh(\mathcal{X}_\tau)\).

Proof

Omitted. Hint: the proof is exactly the same as the proof of Topologies, Lemmas 020W, 021E, 06VC, 06VD, and 021V.

Let \(p : \mathcal{X} \to (\Sch/S)_{fppf}\) be a category fibred in groupoids. Let us denote \(\mathcal{O}\) the restriction of \(\mathcal{O}_\mathcal{X}\) to \(\mathcal{X}_{affine}\). Then \(\mathcal{O}\) is a sheaf in the Zariski, étale, smooth, syntomic, and fppf topologies on \(\mathcal{X}_{affine}\). Furthermore, the equivalence of topoi of Lemma 0H0B extends to an equivalence [0H0C]\[\begin{equation} (\Sh(\mathcal{X}_{affine, \tau}), \mathcal{O}) \longrightarrow (\Sh(\mathcal{X}_\tau), \mathcal{O}_\mathcal{X}) \end{equation}\] of ringed topoi for \(\tau \in \{Zariski, \etale, smooth, syntomic, fppf\}\).

Quasi-coherent modules and affines

Let \(p : \mathcal{X} \to (\Sch/S)_{fppf}\) be a category fibred in groupoids. In Section 0H08 we have associated to this a ringed site \((\mathcal{X}_{affine}, \mathcal{O})\).

Lemma

Let \(p : \mathcal{X} \to (\Sch/S)_{fppf}\) be a category fibred in groupoids. Let \(\mathcal{F}\) be an \(\mathcal{O}\)-module on \(\mathcal{X}_{affine}\). The following are equivalent

1. for every morphism \(x \to x'\) of \(\mathcal{X}_{affine}\) the map \(\mathcal{F}(x') \otimes_{\mathcal{O}(x')} \mathcal{O}(x) \to \mathcal{F}(x)\) is an isomorphism,

2. \(\mathcal{F}\) is a quasi-coherent module on \((\mathcal{X}_{affine}, \mathcal{O})\) in the sense of Modules on Sites, Definition 03DL,

3. \(\mathcal{F}\) is a sheaf for the Zariski topology on \(\mathcal{X}_{affine}\) and a quasi-coherent module on \((\mathcal{X}_{affine, Zar}, \mathcal{O})\) in the sense of Modules on Sites, Definition 03DL,

4. same as in (3) for the étale topology,

5. same as in (3) for the smooth topology,

6. same as in (3) for the syntomic topology,

7. same as in (3) for the fppf topology, and

8. \(\mathcal{F}\) corresponds to a quasi-coherent module on \(\mathcal{X}\) via the equivalence (0H0C).

Proof

To make sense out of part (2), recall that \(\mathcal{X}_{affine}\) is a site gotten by endowing the category \(\mathcal{X}_{affine}\) with the chaotic topology (Definition 0H09) and hence a sheaf of \(\mathcal{O}\)-modules \(\mathcal{F}\) is the same thing as a presheaf of \(\mathcal{O}\)-modules. Conditions (1) and (2) are equivalent by Modules on Sites, Lemma 0GZN. Observe that for \(\tau \in \{Zariski, \etale, smooth, syntomic, fppf\}\) the presheaf \(\mathcal{F}\) is a \(\tau\)-sheaf if and only if for all \(x \in \Ob(\mathcal{X}_{affine})\) the restriction to \(\mathcal{X}_{affine}/x\) is a \(\tau\)-sheaf. Set \(U = p(x)\). Similarly to the discussion in Section 075C the object \(x\) of \(\mathcal{X}_{affine}\) induces an equivalence \(\mathcal{X}_{affine, \etale}/x \to (\textit{Aff}/U)_\etale\) of sites. In this way we see that the equivalence of (1) with (3) – (7) follows from Descent, Lemma 0GZV applied to each of these sites. The equivalence of (8) and (7) is immediate from the fact that “being quasi-coherent” is an intrinsic property of sheaves of modules, see Modules on Sites, Section 03DG

Lemma

Let \(p : \mathcal{X} \to (\Sch/S)_{fppf}\) be a category fibred in groupoids. Let \(\mathcal{F}\) be an \(\mathcal{O}\)-module on \(\mathcal{X}_{affine}\). The following are equivalent

1. for every morphism \(x \to x'\) of \(\mathcal{X}_{affine}\) such that \(p(x) \to p(x')\) is an étale morphism (of affine schemes), the map \(\mathcal{F}(x') \otimes_{\mathcal{O}(x')} \mathcal{O}(x) \to \mathcal{F}(x)\) is an isomorphism,

2. \(\mathcal{F}\) is a sheaf for the étale topology on \(\mathcal{X}_{affine}\) and for every object \(x\) of \(\mathcal{X}_{affine}\) the restriction \(x^*\mathcal{F}|_{U_{affine, \etale}}\) is quasi-coherent where \(U = p(x)\),

3. \(\mathcal{F}\) corresponds to a locally quasi-coherent module on \(\mathcal{X}\) via the equivalence (0H0C) for the étale topology.

Proof

To make sense out of condition (2), recall that \(U_{affine, \etale}\) is the full subcategory of \(U_\etale\) consisting of affine objects, see Topologies, Definition 021B. Similarly to the discussion in Section 075C the object \(x\) of \(\mathcal{X}_{affine}\) induces an equivalence \(\mathcal{X}_{affine, \etale}/x \to (\textit{Aff}/U)_\etale\) of sites. Then \(x^*\mathcal{F}\) is the sheaf of modules on \((\textit{Aff}/U)_\etale\) corresponding to the restriction \(\mathcal{F}|_{\mathcal{X}_{affine, \etale}/x}\). Finally, using the continuous and cocontinuous inclusion functor \(U_{affine, \etale} \to (\textit{Aff}/U)_\etale\) we can further restrict and obtain \(x^*\mathcal{F}|_{U_{affine, \etale}}\).

The equivalence of (1) and (2) follows from the remarks above and Descent, Lemma 0GZX applied to the restriction of \(\mathcal{F}\) to \(U_{affine, \etale}\) for every object \(x\) of \(\mathcal{X}\) lying over an affine scheme \(U\). The equivalence of (2) and (3) is immediate from the definitions and the fact that quasi-coherent modules on \(U_{affine, \etale}\) and \(U_\etale\) correspond (again by Descent, Lemma 0GZX for example).

Quasi-coherent objects in the derived category

Algebraic geometers have contemplated invariants for non-representable functors \(X\) (valued in sets or groupoids) on \(\Sch/S\) for decades. For instance, before the notion of a stack was invented, Mumford defined [mumford_picard] the Picard groupoid \(\Pic(X)\) for the moduli functor \(X\) of elliptic curves as the \(2\)-limit \(Pic(U)\) over the category of all schemes \(U\) equipped with a map to \(X\) (i.e., with a family of elliptic curves). Similarly, Beilinson-Drinfeld defined [BVGD] the category \(\QCoh(X)\) for an ind-scheme \(X = \colim X_i\) as the \(2\)-limit \(\lim \QCoh(X_i)\). This strategy is sufficient for defining \(1\)-categorical invariants like \(\QCoh(-)\), but inadequate for derived categorical ones (such as the quasi-coherent derived category) as \(2\)-limits of triangulated categories are poorly behaved. With the advent of higher categorical technology and derived algebraic geometry, this problem can be resolved gracefully: one can define the quasi-coherent derived \(\infty\)-category \(\mathcal{D}_{qc}(X)\) of the functor \(X\) as the limit \(\lim \mathcal{D}_{qc}(U)\), where \(U\) ranges over all derived affines over X (see [lurie-thesis]).

The goal of this section is to attach a triangulated category \(\mathit{QC}(X)\) to a functor \(X\) (valued in sets or groupoids) as above. In fact, the construction works for any category \(p : \mathcal{X} \to (\Sch/S)_{fppf}\) fibred in groupoids (not just split ones). In good cases, the category \(\mathit{QC}(\mathcal{X})\) can be shown to agree with the homotopy category of \(\mathcal{D}_{qc}(\mathcal{X})\), though it is outside the scope of this document to explain this comparison. The salient features of the construction are:

1. \(\mathit{QC}(\mathcal{X})\) is a full subcategory of \(D(\mathcal{X}_{affine}, \mathcal{O})\) by construction,

2. \(\mathit{QC}(\mathcal{X})\) agrees with \(D_\QCoh(\mathcal{O}_X)\) when \(\mathcal{X}\) is representable by the algebraic space \(X\),

3. \(\mathit{QC}(\mathcal{X})\) agrees with \(D_\QCoh(\mathcal{O}_\mathcal{X})\) when \(\mathcal{X}\) is an algebraic stack,

4. when \(X = \text{Spf}(A)\) is an affine formal algebraic space attached to a noetherian ring \(A\) equipped with the \(I\)-adic topology for an ideal \(I\), the triangulated category \(\mathit{QC}(X)\) agrees with the full subcategory \(D_{comp}(A, I) \subset D(A)\) of derived complete objects.

These results are proven in Proposition 0H0K, Derived Categories of Stacks, Proposition 0H16, and Proposition 0H0L.

As a motivation for the precise definition of \(\mathit{QC}(\mathcal{X})\) we point the reader to the characterization, in Lemma 0H0E, of quasi-coherent modules on \(\mathcal{X}\) as presheaves of \(\mathcal{O}\)-modules on \(\mathcal{X}_{affine}\) which satisfy a kind of base change property.

Definition

Let \(p : \mathcal{X} \to (\Sch/S)_{fppf}\) be a category fibred in groupoids. Let \(\mathcal{O}\) be the sheaf of rings on \(\mathcal{X}_{affine}\) introduced in Section 0H08. We define the triangulated category of quasi-coherent objects in the derived category by the formula \[\mathit{QC}(\mathcal{X}) = \mathit{QC}(\mathcal{X}_{affine}, \mathcal{O})\] where the right hand side is as defined in Cohomology on Sites, Definition 0GYV.

Note that this makes sense as \(\mathcal{X}_{affine}\) is a category and is viewed as a site by endowing it with the chaotic topology and \(\mathcal{O}\) is a sheaf of rings on this category, exactly as required in Cohomology on Sites, Definition 0GYV.

The relationship of this definition with the category of quasi-coherent modules on \(\mathcal{X}\) is not so clear in general! For example, suppose that \(M\) is an object of \(\mathit{QC}(\mathcal{X})\). Then the cohomology sheaves \(H^i(M)\) of \(M\) are (pre)sheaves of \(\mathcal{O}\)-modules on \(\mathcal{X}_{affine}\), but in general they are not quasi-coherent. The last nonvanishing cohomology sheaf is quasi-coherent however.

Lemma

In the situation of Definition 0H0H suppose that \(M\) is an object of \(\mathit{QC}(\mathcal{X})\) and \(b \in \mathbf{Z}\) such that \(H^i(M) = 0\) for all \(i > b\). Then \(H^b(M)\) is a quasi-coherent module on \((\mathcal{X}_{affine}, \mathcal{O})\), see Lemma 0H0E.

Proof

Special case of Cohomology on Sites, Lemma 0GZQ.

Lemma

Let \(S\) be a scheme. Let \(\mathcal{X} \to (\Sch/S)_{fppf}\) be a category fibred in groupoids. The comparison morphism \(\epsilon : \mathcal{X}_{affine, \etale} \to \mathcal{X}_{affine}\) satisfies the assumptions and conclusions of Cohomology on Sites, Lemma 0GZS.

Proof

Assumption (1) holds by definition of \(\mathcal{X}_{affine}\). For condition (2) we use that for \(x \in \Ob(\mathcal{X})\) lying over the affine scheme \(U = p(x)\) we have an equivalence \(\mathcal{X}_{affine, \etale}/x = (\textit{Aff}/U)_\etale\) compatible with structure sheaves; see discussion in Section 075C. Thus it suffices to show: given an affine scheme \(U = \Spec(R)\) and a complex of \(R\)-modules \(M^\bullet\) the total cohomology of the complex of modules on \((\textit{Aff}/U)_\etale\) associated to \(M^\bullet\) is quasi-isomorphic to \(M^\bullet\). This follows from a combination of: Derived Categories of Schemes, Lemma 06Z0 (total cohomology of complexes of modules over affines in the Zariski topology), Derived Categories of Spaces, Remark 08GH (agreement between total cohomology in small Zariski and étale topologies for quasi-coherent complexes of modules), and Étale Cohomology, Lemma 0DDH (to see that the étale cohomology of a complex of modules on the big étale site of a scheme may be computed after restricting to the small étale site).

If we apply the definition in case our category fibred in groupoids \(\mathcal{X}\) is representable by an algebraic space \(X\), then we recover \(D_\QCoh(\mathcal{O}_X)\). We will later state and prove the analogous result for algebraic stacks (insert future reference here).

Proposition

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 \(X\). Then \(\mathit{QC}(\mathcal{X})\) is canonically equivalent to \(D_\QCoh(\mathcal{O}_X)\).

Proof

Denote \(X_{affine}\) the category of affine schemes étale over \(X\) endowed with the chaotic topology and its structure sheaf \(\mathcal{O}_X\), see Derived Categories of Spaces, Section 0H05. The functor \(u : X_\etale \to \mathcal{X}_\etale\) of Lemma 073M gives rise to a functor \(X_{affine} \to \mathcal{X}_{affine}\). This is compatible with structure sheaves and produces a functor \[G : \mathit{QC}(\mathcal{X}) = \mathit{QC}(\mathcal{X}_{affine}, \mathcal{O}) \longrightarrow \mathit{QC}(X_{affine}, \mathcal{O}_X)\] See Cohomology on Sites, Lemma 0GZ1. By Derived Categories of Spaces, Lemma 0H06 the triangulated category \(\mathit{QC}(X_{affine}, \mathcal{O}_X)\) is equivalent to \(D_\QCoh(\mathcal{O}_X)\). Hence it suffices to prove that \(G\) is an equivalence.

Consider the flat comparison morphisms \(\epsilon_\mathcal{X} : \mathcal{X}_{affine, \etale} \to \mathcal{X}_{affine}\) and \(\epsilon_X : X_{affine, \etale} \to X_{affine}\) of ringed sites. Lemma 0H0J and (the proof of) Derived Categories of Spaces, Lemma 0H06 show that the functors \(\epsilon_\mathcal{X}^*\) and \(\epsilon_X^*\) identify \(\mathit{QC}(\mathcal{X}_{affine}, \mathcal{O})\) and \(\mathit{QC}(X_{affine}, \mathcal{O}_X)\) with subcategories \(Q_\mathcal{X} \subset D(\mathcal{X}_{affine, \etale}, \mathcal{O})\) and \(Q_X \subset D(X_{affine, \etale}, \mathcal{O}_X)\). With these identifications the functor \(G\) in the first paragraph is induced by the functor \[Li_X^* = R\pi_{X, *}: D(\mathcal{X}_{affine, \etale}, \mathcal{O}) \longrightarrow D(X_{affine, \etale}, \mathcal{O}_X)\] where \(i_X\) and \(\pi_X\) are the morphisms from Lemma 073M but with the étale sites replaced by the corresponding affine ones. The reader can show that this replacement is permissible either by reproving the lemma for the affine sites directly or by using the equivalences of topoi \(\Sh(\mathcal{X}_{affine, \etale}) = \Sh(\mathcal{X}_\etale)\) and \(\Sh(X_{affine, \etale}) = \Sh(X_\etale)\). The lemma also tells us \(Li_X^*\) has a left adjoint \[L\pi_X^*: D(X_{affine, \etale}, \mathcal{O}_X) \longrightarrow D(\mathcal{X}_{affine, \etale}, \mathcal{O})\] and moreover we have \(Li_X^* \circ L\pi_X^* = \text{id}\) since \(\pi_X \circ i_X\) is the identity. Thus it suffices to show that (a) \(L\pi_X^*\) sends \(Q_X\) into \(Q_\mathcal{X}\) and (b) the kernel of \(Li_X^*\) is \(0\). See Derived Categories, Lemma 09J1.

Proof of (a). By Derived Categories of Spaces, Lemma 0H06 we have \(Q_X = D_\QCoh(X_{affine, \etale}, \mathcal{O}_X)\). Let \(K\) be an object of \(Q_X\). Let \(x\) be an object of \(\mathcal{X}_{affine, \etale}\) lying over the affine scheme \(U = p(x)\). Denote \(f : U \to X\) the morphism corresponding to \(x\). Then we see that \[R\Gamma(x, L\pi_X^*K) = R\Gamma(U, Lf^*K)\] This follows from transitivity of pullbacks; see discussion in Section 076P. Next, suppose that \(x \to x'\) is a morphism of \(\mathcal{X}_{affine, \etale}\) lying over the morphism \(h : U \to U'\) of affine schemes. As before denote \(f : U \to X\) and \(f' : U' \to X\) the morphisms corresponding to \(x\) and \(x'\) so that we have \(f = f' \circ h\). Then \[\begin{align*} R\Gamma(x, L\pi_X^*K) & = R\Gamma(U, Lf^*K) \\ & = R\Gamma(U, Lh^*L(f')^*K) \\ & = R\Gamma(U', L(f')^*K) \otimes_{\mathcal{O}(U')}^\mathbf{L} \mathcal{O}(U) \\ & = R\Gamma(x', L\pi_X^*K) \otimes_{\mathcal{O}(x')}^\mathbf{L} \mathcal{O}(x) \end{align*}\] and hence we have (a) by the footnote in the statement of Cohomology on Sites, Lemma 0GZS. The third equality is Derived Categories of Schemes, Lemma 08DW.

Proof of (b). Let \(M\) be an object of \(Q_\mathcal{X}\) such that \(Li_X^*M = 0\). Let \(x'\) be an object of \(\mathcal{X}_{affine, \etale}\) lying over the affine scheme \(U' = p(x')\) and assume that the corresponding morphism \(f' : U' \to X\) is étale. Then \(f' : U' \to X\) is an object of \(X_{affine, \etale}\) and the condition \(Li_X^*M = 0\) implies that \(M|_{U'_\etale} = 0\). In particular, we see that \(R\Gamma(x', M) = 0\). However, for an arbitrary object \(x\) of the site \(\mathcal{X}_{affine, \etale}\) there exists a covering \(\{x_i \to x\}\) such that for each \(i\) there is a morphism \(x_i \to x'_i\) with \(x'_i\) corresponding to an object of \(X_{affine, \etale}\). Now since \(M\) is in \(Q_\mathcal{X}\) we have \[R\Gamma(x_i, M) = R\Gamma(x_i', M) \otimes_{\mathcal{O}(x_i')}^\mathbf{L} \mathcal{O}(x_i) = 0\] and we conclude that \(M\) is zero as desired.

To show that the construction produces an interesting category in another case, let us state and prove a characterization of \(\mathit{QC}(\text{Spf}(A))\) for the formal spectrum of a Noetherian adic ring \(A\).

Proposition

Let \(S\) be a scheme. Let \(X = \text{Spf}(A)\) where \(A\) is an an adic Noetherian topological \(S\)-algebra with ideal of definition \(I\), see More on Algebra, Definition 07E8 and Formal Spaces, Definition 0AIF. Let \(p : \mathcal{X} \to (\Sch/S)_{fppf}\) the be category fibred in sets associated to the functor \(X\), see Categories, Example 04TM. Then \(\mathit{QC}(\mathcal{X})\) is canonically equivalent to the category \(D_{comp}(A, I)\) of objects of \(D(A)\) which are derived complete with respect to \(I\).

Proof

Recall that \(X = \colim \Spec(A/I^n)\) as an fppf sheaf. An object of \(\mathcal{X}_{affine}\) is the same thing as an affine scheme \(U = \Spec(R)\) with a given morphism \(f : U \to X\). By Formal Spaces, Lemma 0AIA there exists an \(n \geq 1\) such that \(f\) factors through the monomorphism \(\Spec(A/I^n) \to X\). Consider the full subcategory \(\mathcal{C} \subset \mathcal{X}_{affine}\) consisting of the objects \(\Spec(A/I^n) \to X\). By the remarks just made and Differential Graded Sheaves, Lemma 0GZE restriction to \(\mathcal{C}\) is an exact equivalence \(\mathit{QC}(\mathcal{X}) \to \mathit{QC}(\mathcal{C}, \mathcal{O}|_\mathcal{C})\). For simplicity, let us assume that \(I^n \not = I^{n + 1}\) for all \(n \geq 1\). Then \((\mathcal{C}, \mathcal{O}|_\mathcal{C})\) is isomorphic as a ringed site to the ringed site \((\mathbf{N}, (A/I^n))\), see Differential Graded Sheaves, Section 0GZG. Hence we conclude by Differential Graded Sheaves, Proposition 0GZK.

The following lemma will be used in comparing \(\mathit{QC}(\mathcal{X})\) to \(D_\QCoh(\mathcal{O}_\mathcal{X})\) when \(\mathcal{X}\) is an algebraic stack.

Lemma

Let \(S\) be a scheme. Let \(\mathcal{X} \to (\Sch/S)_{fppf}\) be a category fibred in groupoids. The comparison morphism \(\epsilon : \mathcal{X}_{affine, fppf} \to \mathcal{X}_{affine}\) satisfies the assumptions and conclusions of Cohomology on Sites, Lemma 0GZS.

Proof

The proof is exactly the same as the proof of Lemma 0H0J. Assumption (1) holds by definition of \(\mathcal{X}_{affine}\). For condition (2) we use that for \(x \in \Ob(\mathcal{X})\) lying over the affine scheme \(U = p(x)\) we have an equivalence \(\mathcal{X}_{affine, \etale}/x = (\textit{Aff}/U)_\etale\) compatible with structure sheaves; see discussion in Section 075C. Thus it suffices to show: given an affine scheme \(U = \Spec(R)\) and a complex of \(R\)-modules \(M^\bullet\) the total cohomology of the complex of modules on \((\textit{Aff}/U)_{fppf}\) associated to \(M^\bullet\) is quasi-isomorphic to \(M^\bullet\). This is Étale Cohomology, Lemma 0H0U.