Cohomology of Algebraic Stacks

Introduction

In this chapter we write about cohomology of algebraic stacks. This means in particular cohomology of quasi-coherent sheaves, i.e., we prove analogues of the results in the chapters entitled “Cohomology of Schemes” and “Cohomology of Algebraic Spaces”. The results in this chapter are different from those in [LM-B] mainly because we consistently use the “big sites”. Before reading this chapter please take a quick look at the chapter “Sheaves on Algebraic Stacks” in order to become familiar with the terminology introduced there, see Sheaves on Stacks, Section 06TG.

Conventions and abuse of language

We continue to use the conventions and the abuse of language introduced in Properties of Stacks, Section 04XA.

Notation

Different topologies. If we indicate an algebraic stack by a calligraphic letter, such as \(\mathcal{X}, \mathcal{Y}, \mathcal{Z}\), then the notation \(\mathcal{X}_{Zar}, \mathcal{X}_\etale, \mathcal{X}_{smooth}, \mathcal{X}_{syntomic}, \mathcal{X}_{fppf}\) indicates the site introduced in Sheaves on Stacks, Definition 06TP. (Think “big site”.) Correspondingly the structure sheaf of \(\mathcal{X}\) is a sheaf on \(\mathcal{X}_{fppf}\). On the other hand, algebraic spaces and schemes are usually indicated by roman capitals, such as \(X, Y, Z\), and in this case \(X_\etale\) indicates the small étale site of \(X\) (as defined in Topologies, Definition 021B or Properties of Spaces, Definition 03ED). It seems that the distinction should be clear enough.

The default topology is the fppf topology. Hence we will sometimes say “sheaf on \(\mathcal{X}\)” or “sheaf of \(\mathcal{O}_\mathcal{X}\)-modules” when we mean sheaf on \(\mathcal{X}_{fppf}\) or object of \(\textit{Mod}(\mathcal{X}_{fppf}, \mathcal{O}_\mathcal{X})\).

If \(f : \mathcal{X} \to \mathcal{Y}\) is a morphism of algebraic stacks, then the functors \(f_*\) and \(f^{-1}\) defined on presheaves preserves sheaves for any of the topologies mentioned above. In particular when we discuss the pushforward or pullback of a sheaf we don’t have to mention which topology we are working with. The same isn’t true when we compute cohomology groups and/or higher direct images. In this case we will always mention which topology we are working with.

Suppose that \(f : X \to \mathcal{Y}\) is a morphism from an algebraic space \(X\) to an algebraic stack \(\mathcal{Y}\). Let \(\mathcal{G}\) be a sheaf on \(\mathcal{Y}_\tau\) for some topology \(\tau\). In this case \(f^{-1}\mathcal{G}\) is a sheaf for the \(\tau\) topology on \(\mathcal{S}_X\) (the algebraic stack associated to \(X\)) because (by our conventions) \(f\) really is a \(1\)-morphism \(f : \mathcal{S}_X \to \mathcal{Y}\). If \(\tau = \etale\) or stronger, then we write \(f^{-1}\mathcal{G}|_{X_\etale}\) to denote the restriction to the étale site of \(X\), see Sheaves on Stacks, Section 073L. If \(\mathcal{G}\) is an \(\mathcal{O}_\mathcal{X}\)-module we sometimes write \(f^*\mathcal{G}\) and \(f^*\mathcal{G}|_{X_\etale}\) instead.

Pullback of quasi-coherent modules

Let \(f : \mathcal{X} \to \mathcal{Y}\) be a morphism of algebraic stacks. It is a very general fact that quasi-coherent modules on ringed topoi are compatible with pullbacks. In particular the pullback \(f^*\) preserves quasi-coherent modules and we obtain a functor \[f^* : \QCoh(\mathcal{O}_\mathcal{Y}) \longrightarrow \QCoh(\mathcal{O}_\mathcal{X}),\] see Sheaves on Stacks, Lemma 06WH. In general this functor isn’t exact, but if \(f\) is flat then it is.

Lemma

If \(f : \mathcal{X} \to \mathcal{Y}\) is a flat morphism of algebraic stacks then \(f^* : \QCoh(\mathcal{O}_\mathcal{Y}) \to \QCoh(\mathcal{O}_\mathcal{X})\) is an exact functor.

Proof

Choose a scheme \(V\) and a surjective smooth morphism \(V \to \mathcal{Y}\). Choose a scheme \(U\) and a surjective smooth morphism \(U \to V \times_\mathcal{Y} \mathcal{X}\). Then \(U \to \mathcal{X}\) is still smooth and surjective as a composition of two such morphisms. From the commutative diagram \[\xymatrix{ U \ar[d] \ar[r]_{f'} & V \ar[d] \\ \mathcal{X} \ar[r]^f & \mathcal{Y} }\] we obtain a commutative diagram \[\xymatrix{ \QCoh(\mathcal{O}_U) & \QCoh(\mathcal{O}_V) \ar[l] \\ \QCoh(\mathcal{O}_\mathcal{X}) \ar[u] & \QCoh(\mathcal{O}_\mathcal{Y}) \ar[l] \ar[u] }\] of abelian categories. Our proof that the bottom two categories in this diagram are abelian showed that the vertical functors are faithful exact functors (see proof of Sheaves on Stacks, Lemma 06WV). Since \(f'\) is a flat morphism of schemes (by our definition of flat morphisms of algebraic stacks) we see that \((f')^*\) is an exact functor on quasi-coherent sheaves on \(V\). Thus we win.

Lemma

Let \(\mathcal{X}\) be an algebraic stack. Let \(I\) be a set and for \(i \in I\) let \(x_i : U_i \to \mathcal{X}\) be an object of \(\mathcal{X}\). Assume that \(x_i\) is flat and \(\coprod x_i : \coprod U_i \to \mathcal{X}\) is surjective. Let \(\varphi : \mathcal{F} \to \mathcal{G}\) be an arrow of \(\QCoh(\mathcal{O}_\mathcal{X})\). Denote \(\varphi_i\) the restriction of \(\varphi\) to \((U_i)_\etale\). Then \(\varphi\) is injective, resp. surjective, resp. an isomorphism if and only if each \(\varphi_i\) is so.

Proof

Choose a scheme \(U\) and a surjective smooth morphism \(x : U \to \mathcal{X}\). We may and do think of \(x\) as an object of \(\mathcal{X}\). This produces a presentation \(\mathcal{X} = [U/R]\) for some groupoid in spaces \((U, R, s, t, c)\) and correspondingly an equivalence \[\QCoh(\mathcal{O}_\mathcal{X}) = \QCoh(U, R, s, t, c)\] See discussion in Sheaves on Stacks, Section 06WU. The structure of abelian category on the right hand is such that \(\varphi\) is injective, resp. surjective, resp. an isomorphism if and only if the restriction \(\varphi|_{U_\etale}\) is so, see Groupoids in Spaces, Lemma 06VZ.

For each \(i\) we choose an étale covering \(\{W_{i, j} \to V \times_\mathcal{X} U_i\}_{j \in J_i}\) by schemes. Denote \(g_{i, j} : W_{i, j} \to V\) and \(h_{i, j} : W_{i, j} \to U_i\) the obvious arrows. Each of the morphisms of schemes \(g_{i, j} : W_{i, j} \to U\) is flat and they are jointly surjective. Similarly, for each fixed \(i\) the morphisms of schemes \(h_{i, j} : W_{i, j} \to U_i\) are flat and jointly surjective. By Sheaves on Stacks, Lemma 06WK the pullback by \((g_{i, j})_{small}\) of the restriction \(\varphi|_{U_\etale}\) is the restriction \(\varphi|_{(W_{i, j})_\etale}\) and the pullback by \((h_{i, j})_{small}\) of the restriction \(\varphi|_{(U_i)_\etale}\) is the restriction \(\varphi|_{(W_{i, j})_\etale}\). Pullback of quasi-coherent modules by a flat morphism of schemes is exact and pullback by a jointly surjective family of flat morphisms of schemes reflects injective, resp. surjective, resp. bijective maps of quasi-coherent modules (in fact this holds for all modules as we can check exactness at stalks). Thus we see \[\varphi|_{U_\etale} \text{ injective} \Leftrightarrow \varphi|_{(W_{i, j})_\etale} \text{ injective for all }i, j \Leftrightarrow \varphi|_{(U_i)_\etale} \text{ injective for all }i\] This finishes the proof.

Higher direct images of types of modules

The following lemma is the basis for our understanding of higher direct images of certain types of sheaves of modules. There are two versions: one for the étale topology and one for the fppf topology.

Lemma

Let \(\mathcal{M}\) be a rule which associates to every algebraic stack \(\mathcal{X}\) a subcategory \(\mathcal{M}_\mathcal{X}\) of \(\textit{Mod}(\mathcal{X}_\etale, \mathcal{O}_\mathcal{X})\) such that

1. \(\mathcal{M}_\mathcal{X}\) is a weak Serre subcategory of \(\textit{Mod}(\mathcal{X}_\etale, \mathcal{O}_\mathcal{X})\) (see Homology, Definition 02MO) for all algebraic stacks \(\mathcal{X}\),

2. for a smooth morphism of algebraic stacks \(f : \mathcal{Y} \to \mathcal{X}\) the functor \(f^*\) maps \(\mathcal{M}_\mathcal{X}\) into \(\mathcal{M}_\mathcal{Y}\),

3. if \(f_i : \mathcal{X}_i \to \mathcal{X}\) is a family of smooth morphisms of algebraic stacks with \(|\mathcal{X}| = \bigcup |f_i|(|\mathcal{X}_i|)\), then an object \(\mathcal{F}\) of \(\textit{Mod}(\mathcal{X}_\etale, \mathcal{O}_\mathcal{X})\) is in \(\mathcal{M}_\mathcal{X}\) if and only if \(f_i^*\mathcal{F}\) is in \(\mathcal{M}_{\mathcal{X}_i}\) for all \(i\), and

4. if \(f : \mathcal{Y} \to \mathcal{X}\) is a morphism of algebraic stacks such that \(\mathcal{X}\) and \(\mathcal{Y}\) are representable by affine schemes, then \(R^if_*\) maps \(\mathcal{M}_\mathcal{Y}\) into \(\mathcal{M}_\mathcal{X}\).

Then for any quasi-compact and quasi-separated morphism \(f : \mathcal{Y} \to \mathcal{X}\) of algebraic stacks \(R^if_*\) maps \(\mathcal{M}_\mathcal{Y}\) into \(\mathcal{M}_\mathcal{X}\). (Higher direct images computed in étale topology.)

Proof

Let \(f : \mathcal{Y} \to \mathcal{X}\) be a quasi-compact and quasi-separated morphism of algebraic stacks and let \(\mathcal{F}\) be an object of \(\mathcal{M}_\mathcal{Y}\). Choose a surjective smooth morphism \(\mathcal{U} \to \mathcal{X}\) where \(\mathcal{U}\) is representable by a scheme. By Sheaves on Stacks, Lemma 075H taking higher direct images commutes with base change. Assumption (2) shows that the pullback of \(\mathcal{F}\) to \(\mathcal{U} \times_\mathcal{X} \mathcal{Y}\) is in \(\mathcal{M}_{\mathcal{U} \times_\mathcal{X} \mathcal{Y}}\) because the projection \(\mathcal{U} \times_\mathcal{X} \mathcal{Y} \to \mathcal{Y}\) is smooth as a base change of a smooth morphism. Hence (3) shows we may replace \(\mathcal{Y} \to \mathcal{X}\) by the projection \(\mathcal{U} \times_\mathcal{X} \mathcal{Y} \to \mathcal{U}\). In other words, we may assume that \(\mathcal{X}\) is representable by a scheme. Using (3) once more, we see that the question is Zariski local on \(\mathcal{X}\), hence we may assume that \(\mathcal{X}\) is representable by an affine scheme. Since \(f\) is quasi-compact this implies that also \(\mathcal{Y}\) is quasi-compact. Thus we may choose a surjective smooth morphism \(g : \mathcal{V} \to \mathcal{Y}\) where \(\mathcal{V}\) is representable by an affine scheme.

In this situation we have the spectral sequence \[E_2^{p, q} = R^q(f \circ g_p)_*g_p^*\mathcal{F} \Rightarrow R^{p + q}f_*\mathcal{F}\] of Sheaves on Stacks, Proposition 072G. Recall that this is a first quadrant spectral sequence hence we may use the last part of Homology, Lemma 0132. Note that the morphisms \[g_p : \mathcal{V}_p = \mathcal{V} \times_\mathcal{Y} \ldots \times_\mathcal{Y} \mathcal{V} \longrightarrow \mathcal{Y}\] are smooth as compositions of base changes of the smooth morphism \(g\). Thus the sheaves \(g_p^*\mathcal{F}\) are in \(\mathcal{M}_{\mathcal{V}_p}\) by (2). Hence it suffices to prove that the higher direct images of objects of \(\mathcal{M}_{\mathcal{V}_p}\) under the morphisms \[\mathcal{V}_p = \mathcal{V} \times_\mathcal{Y} \ldots \times_\mathcal{Y} \mathcal{V} \longrightarrow \mathcal{X}\] are in \(\mathcal{M}_\mathcal{X}\). The algebraic stacks \(\mathcal{V}_p\) are quasi-compact and quasi-separated by Morphisms of Stacks, Lemma 075S. Of course each \(\mathcal{V}_p\) is representable by an algebraic space (the diagonal of the algebraic stack \(\mathcal{Y}\) is representable by algebraic spaces). This reduces us to the case where \(\mathcal{Y}\) is representable by an algebraic space and \(\mathcal{X}\) is representable by an affine scheme.

In the situation where \(\mathcal{Y}\) is representable by an algebraic space and \(\mathcal{X}\) is representable by an affine scheme, we choose anew a surjective smooth morphism \(\mathcal{V} \to \mathcal{Y}\) where \(\mathcal{V}\) is representable by an affine scheme. Going through the argument above once again we once again reduce to the morphisms \(\mathcal{V}_p \to \mathcal{X}\). But in the current situation the algebraic stacks \(\mathcal{V}_p\) are representable by quasi-compact and quasi-separated schemes (because the diagonal of an algebraic space is representable by schemes).

Thus we may assume \(\mathcal{Y}\) is representable by a scheme and \(\mathcal{X}\) is representable by an affine scheme. Choose (again) a surjective smooth morphism \(\mathcal{V} \to \mathcal{Y}\) where \(\mathcal{V}\) is representable by an affine scheme. In this case all the algebraic stacks \(\mathcal{V}_p\) are representable by separated schemes (because the diagonal of a scheme is separated).

Thus we may assume \(\mathcal{Y}\) is representable by a separated scheme and \(\mathcal{X}\) is representable by an affine scheme. Choose (yet again) a surjective smooth morphism \(\mathcal{V} \to \mathcal{Y}\) where \(\mathcal{V}\) is representable by an affine scheme. In this case all the algebraic stacks \(\mathcal{V}_p\) are representable by affine schemes (because the diagonal of a separated scheme is a closed immersion hence affine) and this case is handled by assumption (4). This finishes the proof.

Here is the version for the fppf topology.

Lemma

Let \(\mathcal{M}\) be a rule which associates to every algebraic stack \(\mathcal{X}\) a subcategory \(\mathcal{M}_\mathcal{X}\) of \(\textit{Mod}(\mathcal{O}_\mathcal{X})\) such that

1. \(\mathcal{O}_\mathcal{X}\) is a weak Serre subcategory of \(\textit{Mod}(\mathcal{O}_\mathcal{X})\) for all algebraic stacks \(\mathcal{X}\),

2. for a smooth morphism of algebraic stacks \(f : \mathcal{Y} \to \mathcal{X}\) the functor \(f^*\) maps \(\mathcal{M}_\mathcal{X}\) into \(\mathcal{M}_\mathcal{Y}\),

3. if \(f_i : \mathcal{X}_i \to \mathcal{X}\) is a family of smooth morphisms of algebraic stacks with \(|\mathcal{X}| = \bigcup |f_i|(|\mathcal{X}_i|)\), then an object \(\mathcal{F}\) of \(\textit{Mod}(\mathcal{O}_\mathcal{X})\) is in \(\mathcal{M}_\mathcal{X}\) if and only if \(f_i^*\mathcal{F}\) is in \(\mathcal{M}_{\mathcal{X}_i}\) for all \(i\), and

4. if \(f : \mathcal{Y} \to \mathcal{X}\) is a morphism of algebraic stacks and \(\mathcal{X}\) and \(\mathcal{Y}\) are representable by affine schemes, then \(R^if_*\) maps \(\mathcal{M}_\mathcal{Y}\) into \(\mathcal{M}_\mathcal{X}\).

Then for any quasi-compact and quasi-separated morphism \(f : \mathcal{Y} \to \mathcal{X}\) of algebraic stacks \(R^if_*\) maps \(\mathcal{M}_\mathcal{Y}\) into \(\mathcal{M}_\mathcal{X}\). (Higher direct images computed in fppf topology.)

Proof

Identical to the proof of Lemma 076Z.

Locally quasi-coherent modules

Let \(\mathcal{X}\) be an algebraic stack. Let \(\mathcal{F}\) be a presheaf of \(\mathcal{O}_\mathcal{X}\)-modules. We can ask whether \(\mathcal{F}\) is locally quasi-coherent, see Sheaves on Stacks, Definition 06WJ. Briefly, this means \(\mathcal{F}\) is an \(\mathcal{O}_\mathcal{X}\)-module for the étale topology such that for any morphism \(f : U \to \mathcal{X}\) the restriction \(f^*\mathcal{F}|_{U_\etale}\) is quasi-coherent on \(U_\etale\). (The actual definition is slightly different, but equivalent.) A useful fact is that \[\textit{LQCoh}(\mathcal{O}_\mathcal{X}) \subset \textit{Mod}(\mathcal{X}_\etale, \mathcal{O}_\mathcal{X})\] is a weak Serre subcategory, see Sheaves on Stacks, Lemma 06WM.

Lemma

Let \(\mathcal{X}\) be an algebraic stack. Let \(f_j : \mathcal{X}_j \to \mathcal{X}\) be a family of smooth morphisms of algebraic stacks with \(|\mathcal{X}| =\bigcup |f_j|(|\mathcal{X}_j|)\). Let \(\mathcal{F}\) be a sheaf of \(\mathcal{O}_\mathcal{X}\)-modules on \(\mathcal{X}_\etale\). If each \(f_j^{-1}\mathcal{F}\) is locally quasi-coherent, then so is \(\mathcal{F}\).

Proof

We may replace each of the algebraic stacks \(\mathcal{X}_j\) by a scheme \(U_j\) (using that any algebraic stack has a smooth covering by a scheme and that compositions of smooth morphisms are smooth, see Morphisms of Stacks, Lemma 075V). The pullback of \(\mathcal{F}\) to \((\Sch/U_j)_\etale\) is still locally quasi-coherent, see Sheaves on Stacks, Lemma 06WL. Then \(f = \coprod f_j : U = \coprod U_j \to \mathcal{X}\) is a surjective smooth morphism. Let \(x\) be an object of \(\mathcal{X}\). By Sheaves on Stacks, Lemma 06XH there exists an étale covering \(\{x_i \to x\}_{i \in I}\) such that each \(x_i\) lifts to an object \(u_i\) of \((\Sch/U)_\etale\). This just means that \(x\), \(x_i\) live over schemes \(V\), \(V_i\), that \(\{V_i \to V\}\) is an étale covering, and that \(x_i\) comes from a morphism \(u_i : V_i \to U\). The restriction \(x_i^*\mathcal{F}|_{V_{i, \etale}}\) is equal to the restriction of \(f^*\mathcal{F}\) to \(V_{i, \etale}\), see Sheaves on Stacks, Lemma 075D. Hence \(x^*\mathcal{F}|_{V_\etale}\) is a sheaf on the small étale site of \(V\) which is quasi-coherent when restricted to \(V_{i, \etale}\) for each \(i\). This implies that it is quasi-coherent (as desired), for example by Properties of Spaces, Lemma 03M0.

Lemma

Let \(f : \mathcal{X} \to \mathcal{Y}\) be a quasi-compact and quasi-separated morphism of algebraic stacks. Let \(\mathcal{F}\) be a locally quasi-coherent \(\mathcal{O}_\mathcal{X}\)-module on \(\mathcal{X}_\etale\). Then \(R^if_*\mathcal{F}\) (computed in the étale topology) is locally quasi-coherent on \(\mathcal{Y}_\etale\).

Proof

We will use Lemma 076Z to prove this. We will check its assumptions (1) – (4). Parts (1) and (2) follows from Sheaves on Stacks, Lemma 06WM. Part (3) follows from Lemma 075Y. Thus it suffices to show (4).

Suppose \(f : \mathcal{X} \to \mathcal{Y}\) is a morphism of algebraic stacks such that \(\mathcal{X}\) and \(\mathcal{Y}\) are representable by affine schemes \(X\) and \(Y\). Choose any object \(y\) of \(\mathcal{Y}\) lying over a scheme \(V\). For clarity, denote \(\mathcal{V} = (\Sch/V)_{fppf}\) the algebraic stack corresponding to \(V\). Consider the cartesian diagram \[\xymatrix{ \mathcal{Z} \ar[d] \ar[r]_g \ar[d]_{f'} & \mathcal{X} \ar[d]^f \\ \mathcal{V} \ar[r]^y & \mathcal{Y} }\] Thus \(\mathcal{Z}\) is representable by the scheme \(Z = V \times_Y X\) and \(f'\) is quasi-compact and separated (even affine). By Sheaves on Stacks, Lemma 075P we have \[R^if_*\mathcal{F}|_{V_\etale} = R^if'_{small, *}\big(g^*\mathcal{F}|_{Z_\etale}\big)\] The right hand side is a quasi-coherent sheaf on \(V_\etale\) by Cohomology of Spaces, Lemma 0720. This implies the left hand side is quasi-coherent which is what we had to prove.

Lemma

Let \(\mathcal{X}\) be an algebraic stack. Let \(f_j : \mathcal{X}_j \to \mathcal{X}\) be a family of flat and locally finitely presented morphisms of algebraic stacks with \(|\mathcal{X}| =\bigcup |f_j|(|\mathcal{X}_j|)\). Let \(\mathcal{F}\) be a sheaf of \(\mathcal{O}_\mathcal{X}\)-modules on \(\mathcal{X}_{fppf}\). If each \(f_j^{-1}\mathcal{F}\) is locally quasi-coherent, then so is \(\mathcal{F}\).

Proof

First, suppose there is a morphism \(a : \mathcal{U} \to \mathcal{X}\) which is surjective, flat, locally of finite presentation, quasi-compact, and quasi-separated such that \(a^*\mathcal{F}\) is locally quasi-coherent. Then there is an exact sequence \[0 \to \mathcal{F} \to a_*a^*\mathcal{F} \to b_*b^*\mathcal{F}\] where \(b\) is the morphism \(b : \mathcal{U} \times_\mathcal{X} \mathcal{U} \to \mathcal{X}\), see Sheaves on Stacks, Proposition 06XE and Lemma 06XH. Moreover, the pullback \(b^*\mathcal{F}\) is the pullback of \(a^*\mathcal{F}\) via one of the projection morphisms, hence is locally quasi-coherent (Sheaves on Stacks, Lemma 06WL). The modules \(a_*a^*\mathcal{F}\) and \(b_*b^*\mathcal{F}\) are locally quasi-coherent by Lemma 075Z. (Note that \(a_*\) and \(b_*\) don’t care about which topology is used to calculate them.) We conclude that \(\mathcal{F}\) is locally quasi-coherent, see Sheaves on Stacks, Lemma 06WM.

We are going to reduce the proof of the general case the situation in the first paragraph. Let \(x\) be an object of \(\mathcal{X}\) lying over the scheme \(U\). We have to show that \(\mathcal{F}|_{U_\etale}\) is a quasi-coherent \(\mathcal{O}_U\)-module. It suffices to do this (Zariski) locally on \(U\), hence we may assume that \(U\) is affine. By Morphisms of Stacks, Lemma 07AN there exists an fppf covering \(\{a_i : U_i \to U\}\) such that each \(x \circ a_i\) factors through some \(f_j\). Hence \(a_i^*\mathcal{F}\) is locally quasi-coherent on \((\Sch/U_i)_{fppf}\). After refining the covering we may assume \(\{U_i \to U\}_{i = 1, \ldots, n}\) is a standard fppf covering. Then \(x^*\mathcal{F}\) is an fppf module on \((\Sch/U)_{fppf}\) whose pullback by the morphism \(a : U_1 \amalg \ldots \amalg U_n \to U\) is locally quasi-coherent. Hence by the first paragraph we see that \(x^*\mathcal{F}\) is locally quasi-coherent, which certainly implies that \(\mathcal{F}|_{U_\etale}\) is quasi-coherent.

Flat comparison maps

Let \(\mathcal{X}\) be an algebraic stack and let \(\mathcal{F}\) be an object of \(\textit{Mod}(\mathcal{X}_\etale, \mathcal{O}_\mathcal{X})\). Given an object \(x\) of \(\mathcal{X}\) lying over the scheme \(U\) the restriction \(\mathcal{F}|_{U_\etale}\) is the restriction of \(x^{-1}\mathcal{F}\) to the small étale site of \(U\), see Sheaves on Stacks, Definition 06W1. Next, let \(\varphi : x \to x'\) be a morphism of \(\mathcal{X}\) lying over a morphism of schemes \(f : U \to U'\). Thus a \(2\)-commutative diagram \[\xymatrix{ U \ar[rd]_x \ar[rr]_f & & U' \ar[ld]^{x'} \\ & \mathcal{X} }\] Associated to \(\varphi\) we obtain a comparison map between restrictions [0761]\[\begin{equation} c_\varphi : f_{small}^*(\mathcal{F}|_{U'_\etale}) \longrightarrow \mathcal{F}|_{U_\etale} \end{equation}\] see Sheaves on Stacks, Equation (06WC). In this situation we can consider the following property of \(\mathcal{F}\).

Definition

Let \(\mathcal{X}\) be an algebraic stack and let \(\mathcal{F}\) in \(\textit{Mod}(\mathcal{X}_\etale, \mathcal{O}_\mathcal{X})\). We say \(\mathcal{F}\) has the flat base change property¹ if and only if \(c_\varphi\) is an isomorphism whenever \(f\) is flat.

Here is a lemma with some properties of this notion.

Lemma

Let \(\mathcal{X}\) be an algebraic stack. Let \(\mathcal{F}\) be an \(\mathcal{O}_\mathcal{X}\)-module on \(\mathcal{X}_\etale\).

1. If \(\mathcal{F}\) has the flat base change property then for any morphism \(g : \mathcal{Y} \to \mathcal{X}\) of algebraic stacks, the pullback \(g^*\mathcal{F}\) does too.

2. The full subcategory of \(\textit{Mod}(\mathcal{X}_\etale, \mathcal{O}_\mathcal{X})\) consisting of modules with the flat base change property is a weak Serre subcategory.

3. Let \(f_i : \mathcal{X}_i \to \mathcal{X}\) be a family of smooth morphisms of algebraic stacks such that \(|\mathcal{X}| = \bigcup_i |f_i|(|\mathcal{X}_i|)\). If each \(f_i^*\mathcal{F}\) has the flat base change property then so does \(\mathcal{F}\).

4. The category of \(\mathcal{O}_\mathcal{X}\)-modules on \(\mathcal{X}_\etale\) with the flat base change property has colimits and they agree with colimits in \(\textit{Mod}(\mathcal{X}_\etale, \mathcal{O}_\mathcal{X})\).

5. Given \(\mathcal{F}\) and \(\mathcal{G}\) in \(\textit{Mod}(\mathcal{X}_\etale, \mathcal{O}_\mathcal{X})\) with the flat base change property then the tensor product \(\mathcal{F} \otimes_{\mathcal{O}_\mathcal{X}} \mathcal{G}\) has the flat base change property.

6. Given \(\mathcal{F}\) and \(\mathcal{G}\) in \(\textit{Mod}(\mathcal{X}_\etale, \mathcal{O}_\mathcal{X})\) with \(\mathcal{F}\) of finite presentation and \(\mathcal{G}\) having the flat base change property then the sheaf \(\SheafHom_{\mathcal{O}_\mathcal{X}}(\mathcal{F}, \mathcal{G})\) has the flat base change property.

Proof

Let \(g : \mathcal{Y} \to \mathcal{X}\) be as in (1). Let \(y\) be an object of \(\mathcal{Y}\) lying over a scheme \(V\). By Sheaves on Stacks, Lemma 075D we have \((g^*\mathcal{F})|_{V_\etale} = \mathcal{F}|_{V_\etale}\). Moreover a comparison mapping for the sheaf \(g^*\mathcal{F}\) on \(\mathcal{Y}\) is a special case of a comparison map for the sheaf \(\mathcal{F}\) on \(\mathcal{X}\), see Sheaves on Stacks, Lemma 075D. In this way (1) is clear.

Proof of (2). We use the characterization of weak Serre subcategories of Homology, Lemma 0754. Kernels and cokernels of maps between sheaves having the flat base change property also have the flat base change property. This is clear because \(f_{small}^*\) is exact for a flat morphism of schemes and since the restriction functors \((-)|_{U_\etale}\) are exact (because we are working in the étale topology). Finally, if \(0 \to \mathcal{F}_1 \to \mathcal{F}_2 \to \mathcal{F}_3 \to 0\) is a short exact sequence of \(\textit{Mod}(\mathcal{X}_\etale, \mathcal{O}_\mathcal{X})\) and the outer two sheaves have the flat base change property then the middle one does as well, again because of the exactness of \(f_{small}^*\) and the restriction functors (and the 5 lemma).

Proof of (3). Let \(f_i : \mathcal{X}_i \to \mathcal{X}\) be a jointly surjective family of smooth morphisms of algebraic stacks and assume each \(f_i^*\mathcal{F}\) has the flat base change property. By part (1), the definition of an algebraic stack, and the fact that compositions of smooth morphisms are smooth (see Morphisms of Stacks, Lemma 075V) we may assume that each \(\mathcal{X}_i\) is representable by a scheme. Let \(\varphi : x \to x'\) be a morphism of \(\mathcal{X}\) lying over a flat morphism \(a : U \to U'\) of schemes. By Sheaves on Stacks, Lemma 06XH there exists a jointly surjective family of étale morphisms \(U'_i \to U'\) such that \(U'_i \to U' \to \mathcal{X}\) factors through \(\mathcal{X}_i\). Thus we obtain commutative diagrams \[\xymatrix{ U_i = U \times_{U'} U_i' \ar[r]_-{a_i} \ar[d] & U_i' \ar[r]_{x_i'} \ar[d] & \mathcal{X}_i \ar[d]^{f_i} \\ U \ar[r]^a & U' \ar[r]^{x'} & \mathcal{X} }\] Note that each \(a_i\) is a flat morphism of schemes as a base change of \(a\). Denote \(\psi_i : x_i \to x'_i\) the morphism of \(\mathcal{X}_i\) lying over \(a_i\) with target \(x_i'\). By assumption the comparison maps \(c_{\psi_i} : (a_i)_{small}^*\big(f_i^*\mathcal{F}|_{(U'_i)_\etale}\big) \to f_i^*\mathcal{F}|_{(U_i)_\etale}\) is an isomorphism. Because the vertical arrows \(U_i' \to U'\) and \(U_i \to U\) are étale, the sheaves \(f_i^*\mathcal{F}|_{(U_i')_\etale}\) and \(f_i^*\mathcal{F}|_{(U_i)_\etale}\) are the restrictions of \(\mathcal{F}|_{U'_\etale}\) and \(\mathcal{F}|_{U_\etale}\) and the map \(c_{\psi_i}\) is the restriction of \(c_\varphi\) to \((U_i)_\etale\), see Sheaves on Stacks, Lemma 075D. Since \(\{U_i \to U\}\) is an étale covering, this implies that the comparison map \(c_\varphi\) is an isomorphism which is what we wanted to prove.

Proof of (4). Let \(\mathcal{I} \to \textit{Mod}(\mathcal{X}_\etale, \mathcal{O}_\mathcal{X})\), \(i \mapsto \mathcal{F}_i\) be a diagram and assume each \(\mathcal{F}_i\) has the flat base change property. Let \(\varphi : x \to x'\) be a morphism of \(\mathcal{X}\) lying over the flat morphism of schemes \(f : U \to U'\). Recall that \(\colim_i \mathcal{F}_i\) is the sheafification of the presheaf colimit. As we are using the étale topology, it is clear that \[(\colim_i \mathcal{F}_i)|_{U_\etale} = \colim_i {\mathcal{F}_i}|_{U_\etale}\] and similarly for the restriction to \(U'_\etale\). Hence \[\begin{align*} f_{small}^*((\colim_i \mathcal{F}_i)|_{U'_\etale}) & = f_{small}^*(\colim_i {\mathcal{F}_i}|_{U'_\etale}) \\ & = \colim_i f_{small}^*({\mathcal{F}_i}|_{U'_\etale}) \\ & \xrightarrow{\colim c_\varphi} \colim_i \mathcal{F}_i|_{U_\etale} \\ & = (\colim_i \mathcal{F}_i)|_{U_\etale} \end{align*}\] For the second equality we used that \(f_{small}^*\) commutes with colimits (as a left adjoint). The arrow is an isomorphism as each \(\mathcal{F}_i\) has the flat base change property. Thus the colimit has the flat base change property and (4) is true.

Part (5) holds because tensor products commute with pullbacks, see Modules on Sites, Lemma 03EL. Details omitted.

Let \(\mathcal{F}\) and \(\mathcal{G}\) be as in (6). Since \(\mathcal{F}\) is quasi-coherent it has the flat base change property by Sheaves on Stacks, Lemma 06WK. Let \(\varphi : x \to x'\) be a morphism of \(\mathcal{X}\) lying over the flat morphism of schemes \(f : U \to U'\). As we are using the étale topology, we have \[\SheafHom_{\mathcal{O}_\mathcal{X}}(\mathcal{F}, \mathcal{G})|_{U_\etale} = \SheafHom_{\mathcal{O}_U}(\mathcal{F}|_{U_\etale}, \mathcal{G}|_{U_\etale})\] and similarly for the restriction to \(U'_\etale\) (details omitted). Hence \[\begin{align*} f_{small}^*( \SheafHom_{\mathcal{O}_\mathcal{X}}(\mathcal{F}, \mathcal{G})|_{U'_\etale}) & = f_{small}^*( \SheafHom_{\mathcal{O}_{U'}}( \mathcal{F}|_{U'_\etale}, \mathcal{G}|_{U'_\etale})) \\ & = \SheafHom_{\mathcal{O}_{U'}}( f_{small}^*(\mathcal{F}|_{U'_\etale}), f_{small}^*(\mathcal{G}|_{U'_\etale})) \\ & \xrightarrow{c_\varphi} \SheafHom_{\mathcal{O}_U}(\mathcal{F}|_{U_\etale}, \mathcal{G}|_{U_\etale}) \\ & = \SheafHom_{\mathcal{O}_\mathcal{X}}(\mathcal{F}, \mathcal{G})|_{U_\etale} \end{align*}\] Here the second equality is Modules on Sites, Lemma 0GN2 which uses that \(f : U \to U'\) is flat and hence the morphism of ringed sites \(f_{small}\) is flat too. The arrow is an isomorphism as both \(\mathcal{F}\) and \(\mathcal{G}\) have the flat base change property. Thus our \(\SheafHom\) has the flat base change property too as desired.

Lemma

Let \(f : \mathcal{X} \to \mathcal{Y}\) be a quasi-compact and quasi-separated morphism of algebraic stacks. Let \(\mathcal{F}\) be an object of \(\textit{Mod}(\mathcal{X}_\etale, \mathcal{O}_\mathcal{X})\) which is locally quasi-coherent and has the flat base change property. Then each \(R^if_*\mathcal{F}\) (computed in the étale topology) has the flat base change property.

Proof

We will use Lemma 076Z to prove this. For every algebraic stack \(\mathcal{X}\) let \(\textit{LQCoh}^{fbc}(\mathcal{O}_\mathcal{X})\) denote the full subcategory of \(\textit{Mod}(\mathcal{X}_\etale, \mathcal{O}_\mathcal{X})\) consisting of locally quasi-coherent sheaves with the flat base change property. Once we verify conditions (1) – (4) of Lemma 076Z the lemma will follow. Properties (1), (2), and (3) follow from Sheaves on Stacks, Lemmas 06WL and 06WM and Lemmas 075Y and 0764. Thus it suffices to show part (4).

Suppose \(f : \mathcal{X} \to \mathcal{Y}\) is a morphism of algebraic stacks such that \(\mathcal{X}\) and \(\mathcal{Y}\) are representable by affine schemes \(X\) and \(Y\). In this case, suppose that \(\psi : y \to y'\) is a morphism of \(\mathcal{Y}\) lying over a flat morphism \(b : V \to V'\) of schemes. For clarity denote \(\mathcal{V} = (\Sch/V)_{fppf}\) and \(\mathcal{V}' = (\Sch/V')_{fppf}\) the corresponding algebraic stacks. Consider the diagram of algebraic stacks \[\xymatrix{ \mathcal{Z} \ar[d]_{f''} \ar[r]_a & \mathcal{Z}' \ar[r]_{x'} \ar[d]_{f'} & \mathcal{X} \ar[d]^f \\ \mathcal{V} \ar[r]^b & \mathcal{V}' \ar[r]^{y'} & \mathcal{Y} }\] with both squares cartesian. As \(f\) is representable by schemes (and quasi-compact and separated – even affine) we see that \(\mathcal{Z}\) and \(\mathcal{Z}'\) are representable by schemes \(Z\) and \(Z'\) and in fact \(Z = V \times_{V'} Z'\). Since \(\mathcal{F}\) has the flat base change property we see that \[a_{small}^*\big(\mathcal{F}|_{Z'_\etale}\big) \longrightarrow \mathcal{F}|_{Z_\etale}\] is an isomorphism. Moreover, \[R^if_*\mathcal{F}|_{V'_\etale} = R^i(f')_{small, *}\big(\mathcal{F}|_{Z'_\etale}\big)\] and \[R^if_*\mathcal{F}|_{V_\etale} = R^i(f'')_{small, *}\big(\mathcal{F}|_{Z_\etale}\big)\] by Sheaves on Stacks, Lemma 075P. Hence we see that the comparison map \[c_\psi : b_{small}^*(R^if_*\mathcal{F}|_{V'_\etale}) \longrightarrow R^if_*\mathcal{F}|_{V_\etale}\] is an isomorphism by Cohomology of Spaces, Lemma 073K. Thus \(R^if_*\mathcal{F}\) has the flat base change property. Since \(R^if_*\mathcal{F}\) is locally quasi-coherent by Lemma 075Z we win.

Locally quasi-coherent modules with the flat base change property

Let \(\mathcal{X}\) be an algebraic stack. We² will denote \[\textit{LQCoh}^{fbc}(\mathcal{O}_\mathcal{X}) \subset \textit{Mod}(\mathcal{X}_\etale, \mathcal{O}_\mathcal{X})\] the full subcategory whose objects are étale \(\mathcal{O}_\mathcal{X}\)-modules \(\mathcal{F}\) which are both locally quasi-coherent (Section 075X) and have the flat base change property (Section 0760). We have \[\QCoh(\mathcal{O}_\mathcal{X}) \subset \textit{LQCoh}^{fbc}(\mathcal{O}_\mathcal{X})\] by Sheaves on Stacks, Lemma 06WK.

Proposition

Summary of results on locally quasi-coherent modules having the flat base change property.

1. Let \(\mathcal{X}\) be an algebraic stack. If \(\mathcal{F}\) is in \(\textit{LQCoh}^{fbc}(\mathcal{O}_\mathcal{X})\), then \(\mathcal{F}\) is a sheaf for the fppf topology, i.e., it is an object of \(\textit{Mod}(\mathcal{O}_\mathcal{X})\).

2. The category \(\textit{LQCoh}^{fbc}(\mathcal{O}_\mathcal{X})\) is a weak Serre subcategory of both \(\textit{Mod}(\mathcal{O}_\mathcal{X})\) and \(\textit{Mod}(\mathcal{X}_\etale, \mathcal{O}_\mathcal{X})\).

3. Pullback \(f^*\) along any morphism of algebraic stacks \(f : \mathcal{X} \to \mathcal{Y}\) induces a functor \(f^* : \textit{LQCoh}^{fbc}(\mathcal{O}_\mathcal{Y}) \to \textit{LQCoh}^{fbc}(\mathcal{O}_\mathcal{X})\).

4. If \(f : \mathcal{X} \to \mathcal{Y}\) is a quasi-compact and quasi-separated morphism of algebraic stacks and \(\mathcal{F}\) is an object of \(\textit{LQCoh}^{fbc}(\mathcal{O}_\mathcal{X})\), then

   1. the total direct image \(Rf_*\mathcal{F}\) and the higher direct images \(R^if_*\mathcal{F}\) can be computed in either the étale or the fppf topology with the same result, and

   2. each \(R^if_*\mathcal{F}\) is an object of \(\textit{LQCoh}^{fbc}(\mathcal{O}_\mathcal{Y})\).

5. The category \(\textit{LQCoh}^{fbc}(\mathcal{O}_\mathcal{X})\) has colimits and they agree with colimits in \(\textit{Mod}(\mathcal{X}_\etale, \mathcal{O}_\mathcal{X})\) as well as in \(\textit{Mod}(\mathcal{O}_\mathcal{X})\).

6. Given \(\mathcal{F}\) and \(\mathcal{G}\) in \(\textit{LQCoh}^{fbc}(\mathcal{O}_\mathcal{X})\) then the tensor product \(\mathcal{F} \otimes_{\mathcal{O}_\mathcal{X}} \mathcal{G}\) is in \(\textit{LQCoh}^{fbc}(\mathcal{O}_\mathcal{X})\).

7. Given \(\mathcal{F}\) of finite presentation and \(\mathcal{G}\) in \(\textit{LQCoh}^{fbc}(\mathcal{O}_\mathcal{X})\) then \(\SheafHom_{\mathcal{O}_\mathcal{X}}(\mathcal{F}, \mathcal{G})\) is in \(\textit{LQCoh}^{fbc}(\mathcal{O}_\mathcal{X})\).

Proof

Part (1) is Sheaves on Stacks, Lemma 076T.

Part (2) for the embedding \(\textit{LQCoh}^{fbc}(\mathcal{O}_\mathcal{X}) \subset \textit{Mod}(\mathcal{X}_\etale, \mathcal{O}_\mathcal{X})\) we have seen in the proof of Lemma 0765. Let us prove (2) for the embedding \(\textit{LQCoh}^{fbc}(\mathcal{O}_\mathcal{X}) \subset \textit{Mod}(\mathcal{O}_\mathcal{X})\). Let \(\varphi : \mathcal{F} \to \mathcal{G}\) be a morphism between objects of \(\textit{LQCoh}^{fbc}(\mathcal{O}_\mathcal{X})\). Since \(\Ker(\varphi)\) is the same whether computed in the étale or the fppf topology, we see that \(\Ker(\varphi)\) is in \(\textit{LQCoh}^{fbc}(\mathcal{O}_\mathcal{X})\) by the étale case. On the other hand, the cokernel computed in the fppf topology is the fppf sheafification of the cokernel computed in the étale topology. However, this étale cokernel is in \(\textit{LQCoh}^{fbc}(\mathcal{O}_\mathcal{X})\) hence an fppf sheaf by (1) and we see that the cokernel is in \(\textit{LQCoh}^{fbc}(\mathcal{O}_\mathcal{X})\). Finally, suppose that \[0 \to \mathcal{F}_1 \to \mathcal{F}_2 \to \mathcal{F}_3 \to 0\] is an exact sequence in \(\textit{Mod}(\mathcal{O}_\mathcal{X})\) (i.e., using the fppf topology) with \(\mathcal{F}_1\), \(\mathcal{F}_2\) in \(\textit{LQCoh}^{fbc}(\mathcal{O}_\mathcal{X})\). In order to show that \(\mathcal{F}_2\) is an object of \(\textit{LQCoh}^{fbc}(\mathcal{O}_\mathcal{X})\) it suffices to show that the sequence is also exact in the étale topology. To do this it suffices to show that any element of \(H^1_{fppf}(x, \mathcal{F}_1)\) becomes zero on the members of an étale covering of \(x\) (for any object \(x\) of \(\mathcal{X}\)). This is true because \(H^1_{fppf}(x, \mathcal{F}_1) = H^1_\etale(x, \mathcal{F}_1)\) by Sheaves on Stacks, Lemma 075R and because of locality of cohomology, see Cohomology on Sites, Lemma 01FW. This proves (2).

Part (3) follows from Lemma 0764 and Sheaves on Stacks, Lemma 06WL.

Part (4)(b) for \(R^if_*\mathcal{F}\) computed in the étale cohomology follows from Lemma 0765. Whereupon part (4)(a) follows from Sheaves on Stacks, Lemma 075R combined with (1) above.

Part (5) for the étale topology follows from Sheaves on Stacks, Lemma 06WM and Lemma 0764. The fppf version then follows as the colimit in the étale topology is already an fppf sheaf by part (1).

Parts (6) and (7) follow from the corresponding parts of Lemma 0764 and Sheaves on Stacks, Lemma 06WM.

Lemma

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

1. Let \(f_j : \mathcal{X}_j \to \mathcal{X}\) be a family of smooth morphisms of algebraic stacks with \(|\mathcal{X}| =\bigcup |f_j|(|\mathcal{X}_j|)\). Let \(\mathcal{F}\) be a sheaf of \(\mathcal{O}_\mathcal{X}\)-modules on \(\mathcal{X}_\etale\). If each \(f_j^{-1}\mathcal{F}\) is in \(\textit{LQCoh}^{fpc}(\mathcal{O}_{\mathcal{X}_i})\), then \(\mathcal{F}\) is in \(\textit{LQCoh}^{fbc}(\mathcal{O}_\mathcal{X})\).

2. Let \(f_j : \mathcal{X}_j \to \mathcal{X}\) be a family of flat and locally finitely presented morphisms of algebraic stacks with \(|\mathcal{X}| =\bigcup |f_j|(|\mathcal{X}_j|)\). Let \(\mathcal{F}\) be a sheaf of \(\mathcal{O}_\mathcal{X}\)-modules on \(\mathcal{X}_{fppf}\). If each \(f_j^{-1}\mathcal{F}\) is in \(\textit{LQCoh}^{fbc}(\mathcal{O}_{\mathcal{X}_i})\), then \(\mathcal{F}\) is in \(\textit{LQCoh}^{fbc}(\mathcal{O}_\mathcal{X})\).

Proof

Part (1) follows from a combination of Lemmas 075Y and 0764. The proof of (2) is analogous to the proof of Lemma 07AP. Let \(\mathcal{F}\) of a sheaf of \(\mathcal{O}_\mathcal{X}\)-modules on \(\mathcal{X}_{fppf}\).

First, suppose there is a morphism \(a : \mathcal{U} \to \mathcal{X}\) which is surjective, flat, locally of finite presentation, quasi-compact, and quasi-separated such that \(a^*\mathcal{F}\) is locally quasi-coherent and has the flat base change property. Then there is an exact sequence \[0 \to \mathcal{F} \to a_*a^*\mathcal{F} \to b_*b^*\mathcal{F}\] where \(b\) is the morphism \(b : \mathcal{U} \times_\mathcal{X} \mathcal{U} \to \mathcal{X}\), see Sheaves on Stacks, Proposition 06XE and Lemma 06XH. Moreover, the pullback \(b^*\mathcal{F}\) is the pullback of \(a^*\mathcal{F}\) via one of the projection morphisms, hence is locally quasi-coherent and has the flat base change property, see Proposition 0771. The modules \(a_*a^*\mathcal{F}\) and \(b_*b^*\mathcal{F}\) are locally quasi-coherent and have the flat base change property by Proposition 0771. We conclude that \(\mathcal{F}\) is locally quasi-coherent and has the flat base change property by Proposition 0771.

Choose a scheme \(U\) and a surjective smooth morphism \(x : U \to \mathcal{X}\). By part (1) it suffices to show that \(x^*\mathcal{F}\) is locally quasi-coherent and has the flat base change property. Again by part (1) it suffices to do this (Zariski) locally on \(U\), hence we may assume that \(U\) is affine. By Morphisms of Stacks, Lemma 07AN there exists an fppf covering \(\{a_i : U_i \to U\}\) such that each \(x \circ a_i\) factors through some \(f_j\). Hence the module \(a_i^*\mathcal{F}\) on \((\Sch/U_i)_{fppf}\) is locally quasi-coherent and has the flat base change property. After refining the covering we may assume \(\{U_i \to U\}_{i = 1, \ldots, n}\) is a standard fppf covering. Then \(x^*\mathcal{F}\) is an fppf module on \((\Sch/U)_{fppf}\) whose pullback by the morphism \(a : U_1 \amalg \ldots \amalg U_n \to U\) is locally quasi-coherent and has the flat base change property. Hence by the previous paragraph we see that \(x^*\mathcal{F}\) is locally quasi-coherent and has the flat base change property as desired.

Lemma

Let \(f : \mathcal{X} \to \mathcal{Y}\) be a morphism of algebraic stacks which is quasi-compact, quasi-separated, and representable by algebraic spaces. Let \(\mathcal{F}\) be in \(\textit{LQCoh}^{fbc}(\mathcal{O}_\mathcal{X})\). Then for an object \(y : V \to \mathcal{Y}\) of \(\mathcal{Y}\) we have \[(R^if_*\mathcal{F})|_{V_\etale} = R^if'_{small, *}(\mathcal{F}|_{U_\etale})\] where \(f' : U = V \times_\mathcal{Y} \mathcal{X} \to V\) is the base change of \(f\).

Proof

By Sheaves on Stacks, Lemma 075H we can reduce to the case where \(\mathcal{X}\) is represented by \(U\) and \(\mathcal{Y}\) is represented by \(V\). Of course this also uses that the pullback of \(\mathcal{F}\) to \(U\) is in \(\textit{LQCoh}^{fbc}(\mathcal{O}_U)\) by Proposition 0771. Then the result follows from Sheaves on Stacks, Lemma 075N and the fact that \(R^if_*\) may be computed in the étale topology by Proposition 0771.

Lemma

Let \(f : \mathcal{X} \to \mathcal{Y}\) be an affine morphism of algebraic stacks. The functor \(f_* : \textit{LQCoh}^{fbc}(\mathcal{O}_\mathcal{X}) \to \textit{LQCoh}^{fbc}(\mathcal{O}_\mathcal{Y})\) is exact and commutes with direct sums. The functors \(R^if_*\) for \(i > 0\) vanish on \(\textit{LQCoh}^{fbc}(\mathcal{O}_\mathcal{X})\).

Proof

The functors exist by Proposition 0771. By Lemma 0GQH this reduces to the case of an affine morphism of algebraic spaces taking higher direct images in the setting of quasi-coherent modules on algebraic spaces. By the discussion in Cohomology of Spaces, Section 071Y we reduce to the case of an affine morphism of schemes. For affine morphisms of schemes we have the vanishing of higher direct images on quasi-coherent modules by Cohomology of Schemes, Lemma 01XC. The vanishing for \(R^1f_*\) implies exactness of \(f_*\). Commuting with direct sums follows from Morphisms, Lemma 01SB for example.

Parasitic modules

The following definition is compatible with Descent, Definition 06ZL.

Definition

Let \(\mathcal{X}\) be an algebraic stack. A presheaf of \(\mathcal{O}_\mathcal{X}\)-modules \(\mathcal{F}\) is parasitic if we have \(\mathcal{F}(x) = 0\) for any object \(x\) of \(\mathcal{X}\) which lies over a scheme \(U\) such that the corresponding morphism \(x : U \to \mathcal{X}\) is flat.

Here is a lemma with some properties of this notion.

Lemma

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

1. If \(\mathcal{F}\) is parasitic and \(g : \mathcal{Y} \to \mathcal{X}\) is a flat morphism of algebraic stacks, then \(g^*\mathcal{F}\) is parasitic.

2. For \(\tau \in \{Zariski, \etale, smooth, syntomic, fppf\}\) we have

   1. the \(\tau\) sheafification of a parasitic presheaf of modules is parasitic, and

   2. the full subcategory of \(\textit{Mod}(\mathcal{X}_\tau, \mathcal{O}_\mathcal{X})\) consisting of parasitic modules is a Serre subcategory.

3. Suppose \(\mathcal{F}\) is a sheaf for the étale topology. Let \(f_i : \mathcal{X}_i \to \mathcal{X}\) be a family of smooth morphisms of algebraic stacks such that \(|\mathcal{X}| = \bigcup_i |f_i|(|\mathcal{X}_i|)\). If each \(f_i^*\mathcal{F}\) is parasitic then so is \(\mathcal{F}\).

4. Suppose \(\mathcal{F}\) is a sheaf for the fppf topology. Let \(f_i : \mathcal{X}_i \to \mathcal{X}\) be a family of flat and locally finitely presented morphisms of algebraic stacks such that \(|\mathcal{X}| = \bigcup_i |f_i|(|\mathcal{X}_i|)\). If each \(f_i^*\mathcal{F}\) is parasitic then so is \(\mathcal{F}\).

Proof

To see part (1) let \(y\) be an object of \(\mathcal{Y}\) which lies over a scheme \(V\) such that the corresponding morphism \(y : V \to \mathcal{Y}\) is flat. Then \(g(y) : V \to \mathcal{Y} \to \mathcal{X}\) is flat as a composition of flat morphisms (see Morphisms of Stacks, Lemma 06PX) hence \(\mathcal{F}(g(y))\) is zero by assumption. Since \(g^*\mathcal{F} = g^{-1}\mathcal{F}(y) = \mathcal{F}(g(y))\) we conclude \(g^*\mathcal{F}\) is parasitic.

To see part (2)(a) note that if \(\{x_i \to x\}\) is a \(\tau\)-covering of \(\mathcal{X}\), then each of the morphisms \(x_i \to x\) lies over a flat morphism of schemes. Hence if \(x\) lies over a scheme \(U\) such that \(x : U \to \mathcal{X}\) is flat, so do all of the objects \(x_i\). Hence the presheaf \(\mathcal{F}^+\) (see Sites, Section 00W1) is parasitic if the presheaf \(\mathcal{F}\) is parasitic. This proves (2)(a) as the sheafification of \(\mathcal{F}\) is \((\mathcal{F}^+)^+\).

Let \(\mathcal{F}\) be a parasitic \(\tau\)-module. It is immediate from the definitions that any submodule of \(\mathcal{F}\) is parasitic. On the other hand, if \(\mathcal{F}' \subset \mathcal{F}\) is a submodule, then it is equally clear that the presheaf \(x \mapsto \mathcal{F}(x)/\mathcal{F}'(x)\) is parasitic. Hence the quotient \(\mathcal{F}/\mathcal{F}'\) is a parasitic module by (2)(a). Finally, we have to show that given a short exact sequence \(0 \to \mathcal{F}_1 \to \mathcal{F}_2 \to \mathcal{F}_3 \to 0\) with \(\mathcal{F}_1\) and \(\mathcal{F}_3\) parasitic, then \(\mathcal{F}_2\) is parasitic. This follows immediately on evaluating on \(x\) lying over a scheme flat over \(\mathcal{X}\). This proves (2)(b), see Homology, Lemma 02MP.

Let \(f_i : \mathcal{X}_i \to \mathcal{X}\) be a jointly surjective family of smooth morphisms of algebraic stacks and assume each \(f_i^*\mathcal{F}\) is parasitic. Let \(x\) be an object of \(\mathcal{X}\) which lies over a scheme \(U\) such that \(x : U \to \mathcal{X}\) is flat. Consider a surjective smooth covering \(W_i \to U \times_{x, \mathcal{X}} \mathcal{X}_i\). Denote \(y_i : W_i \to \mathcal{X}_i\) the projection. It follows that \(\{f_i(y_i) \to x\}\) is a covering for the smooth topology on \(\mathcal{X}\). Since a composition of flat morphisms is flat we see that \(f_i^*\mathcal{F}(y_i) = 0\). On the other hand, as we saw in the proof of (1), we have \(f_i^*\mathcal{F}(y_i) = \mathcal{F}(f_i(y_i))\). Hence we see that for some smooth covering \(\{x_i \to x\}_{i \in I}\) in \(\mathcal{X}\) we have \(\mathcal{F}(x_i) = 0\). This implies \(\mathcal{F}(x) = 0\) because the smooth topology is the same as the étale topology, see More on Morphisms, Lemma 055V. Namely, \(\{x_i \to x\}_{i \in I}\) lies over a smooth covering \(\{U_i \to U\}_{i \in I}\) of schemes. By the lemma just referenced there exists an étale covering \(\{V_j \to U\}_{j \in J}\) which refines \(\{U_i \to U\}_{i \in I}\). Denote \(x'_j = x|_{V_j}\). Then \(\{x'_j \to x\}\) is an étale covering in \(\mathcal{X}\) refining \(\{x_i \to x\}_{i \in I}\). This means the map \(\mathcal{F}(x) \to \prod_{j \in J} \mathcal{F}(x'_j)\), which is injective as \(\mathcal{F}\) is a sheaf in the étale topology, factors through \(\mathcal{F}(x) \to \prod_{i \in I} \mathcal{F}(x_i)\) which is zero. Hence \(\mathcal{F}(x) = 0\) as desired.

Proof of (4): omitted. Hint: similar, but simpler, than the proof of (3).

Parasitic modules are preserved under absolutely any pushforward.

Lemma

Let \(\tau \in \{\etale, fppf\}\). Let \(\mathcal{X}\) be an algebraic stack. Let \(\mathcal{F}\) be a parasitic object of \(\textit{Mod}(\mathcal{X}_\tau, \mathcal{O}_\mathcal{X})\).

1. \(H^i_\tau(\mathcal{X}, \mathcal{F}) = 0\) for all \(i\).

2. Let \(f : \mathcal{X} \to \mathcal{Y}\) be a morphism of algebraic stacks. Then \(R^if_*\mathcal{F}\) (computed in \(\tau\)-topology) is a parasitic object of \(\textit{Mod}(\mathcal{Y}_\tau, \mathcal{O}_\mathcal{Y})\).

Proof

We first reduce (2) to (1). By Sheaves on Stacks, Lemma 075G we see that \(R^if_*\mathcal{F}\) is the sheaf associated to the presheaf \[y \longmapsto H^i_\tau\Big(V \times_{y, \mathcal{Y}} \mathcal{X}, \ \text{pr}^{-1}\mathcal{F}\Big)\] Here \(y\) is a typical object of \(\mathcal{Y}\) lying over the scheme \(V\). By Lemma 0774 it suffices to show that these cohomology groups are zero when \(y : V \to \mathcal{Y}\) is flat. Note that \(\text{pr} : V \times_{y, \mathcal{Y}} \mathcal{X} \to \mathcal{X}\) is flat as a base change of \(y\). Hence by Lemma 0774 we see that \(\text{pr}^{-1}\mathcal{F}\) is parasitic. Thus it suffices to prove (1).

To see (1) we can use the spectral sequence of Sheaves on Stacks, Proposition 06XJ to reduce this to the case where \(\mathcal{X}\) is an algebraic stack representable by an algebraic space. Note that in the spectral sequence each \(f_p^{-1}\mathcal{F} = f_p^*\mathcal{F}\) is a parasitic module by Lemma 0774 because the morphisms \(f_p : \mathcal{U}_p = \mathcal{U} \times_\mathcal{X} \ldots \times_\mathcal{X} \mathcal{U} \to \mathcal{X}\) are flat. Reusing this spectral sequence one more time (as in the proof of Lemma 076Z) we reduce to the case where the algebraic stack \(\mathcal{X}\) is representable by a scheme \(X\). Then \(H^i_\tau(\mathcal{X}, \mathcal{F}) = H^i((\Sch/X)_\tau, \mathcal{F})\). In this case the vanishing follows easily from an argument with Čech coverings, see Descent, Lemma 0755.

The following lemma is one of the major reasons we care about parasitic modules. To understand the statement, recall that the functors \(\QCoh(\mathcal{O}_\mathcal{X}) \to \textit{Mod}(\mathcal{X}_\etale, \mathcal{O}_\mathcal{X})\) and \(\QCoh(\mathcal{O}_\mathcal{X}) \to \textit{Mod}(\mathcal{O}_\mathcal{X})\) aren’t exact in general.

Lemma

Let \(\mathcal{X}\) be an algebraic stack. Let \(\alpha : \mathcal{F} \to \mathcal{G}\) and \(\beta : \mathcal{G} \to \mathcal{H}\) be maps in \(\QCoh(\mathcal{O}_\mathcal{X})\) with \(\beta \circ \alpha = 0\). The following are equivalent:

1. in the abelian category \(\QCoh(\mathcal{O}_\mathcal{X})\) the complex \(\mathcal{F} \to \mathcal{G} \to \mathcal{H}\) is exact at \(\mathcal{G}\),

2. \(\Ker(\beta)/\Im(\alpha)\) computed in either \(\textit{Mod}(\mathcal{X}_\etale, \mathcal{O}_\mathcal{X})\) or \(\textit{Mod}(\mathcal{X}_{fppf}, \mathcal{O}_\mathcal{X})\) is parasitic.

Proof

We have \(\QCoh(\mathcal{O}_\mathcal{X}) \subset \textit{LQCoh}^{fbc}(\mathcal{O}_\mathcal{X})\), see Section 0GQG. Hence \(\Ker(\beta)/\Im(\alpha)\) computed in \(\textit{Mod}(\mathcal{X}_\etale, \mathcal{O}_\mathcal{X})\) or \(\textit{Mod}(\mathcal{X}_{fppf}, \mathcal{O}_\mathcal{X})\) agree, see Proposition 0771. From now on we will use the étale topology on \(\mathcal{X}\).

Let \(\mathcal{E}\) be the cohomology of \(\mathcal{F} \to \mathcal{G} \to \mathcal{H}\) computed in the abelian category \(\QCoh(\mathcal{O}_\mathcal{X})\). Let \(x : U \to \mathcal{X}\) be a flat morphism where \(U\) is a scheme. As we are using the étale topology, the restriction functor \(\textit{Mod}(\mathcal{X}_\etale, \mathcal{O}_\mathcal{X}) \to \textit{Mod}(U_\etale, \mathcal{O}_U)\) is exact. On the other hand, by Lemma 076X and Sheaves on Stacks, Lemma 0GQD the restriction functor \[\QCoh(\mathcal{O}_\mathcal{X}) \xrightarrow{x^*} \QCoh((\Sch/U)_\etale, \mathcal{O}) \xrightarrow{{-}|_{U_\etale}} \QCoh(U_\etale, \mathcal{O}_U)\] is exact too. We conclude that \(\mathcal{E}|_{U_\etale} = (\Ker(\beta)/\Im(\alpha))|_{U_\etale}\).

If (1) holds, then \(\mathcal{E} = 0\) hence \(\Ker(\beta)/\Im(\alpha)\) restricts to zero on \(U_\etale\) for all \(U\) flat over \(\mathcal{X}\) and this is the definition of a parasitic module. If (2) holds, then \(\Ker(\beta)/\Im(\alpha)\) restricts to zero on \(U_\etale\) for all \(U\) flat over \(\mathcal{X}\) hence \(\mathcal{E}\) restricts to zero on \(U_\etale\) for all \(U\) flat over \(\mathcal{X}\). This certainly implies that the quasi-coherent module \(\mathcal{E}\) is zero, for example apply Lemma 0GQF to the map \(0 \to \mathcal{E}\).

Quasi-coherent modules

We have seen that the category of quasi-coherent modules on an algebraic stack is equivalent to the category of quasi-coherent modules on a presentation, see Sheaves on Stacks, Section 06WU. This fact is the basis for the following.

Lemma

Let \(\mathcal{X}\) be an algebraic stack. Let \(\textit{LQCoh}^{fbc}(\mathcal{O}_\mathcal{X})\) be the category of locally quasi-coherent modules with the flat base change property, see Section 0GQG. The inclusion functor \(i : \QCoh(\mathcal{O}_\mathcal{X}) \to \textit{LQCoh}^{fbc}(\mathcal{O}_\mathcal{X})\) has a right adjoint \[Q : \textit{LQCoh}^{fbc}(\mathcal{O}_\mathcal{X}) \to \QCoh(\mathcal{O}_\mathcal{X})\] such that \(Q \circ i\) is the identity functor.

Proof

Choose a scheme \(U\) and a surjective smooth morphism \(f : U \to \mathcal{X}\). Set \(R = U \times_\mathcal{X} U\) so that we obtain a smooth groupoid \((U, R, s, t, c)\) in algebraic spaces with the property that \(\mathcal{X} = [U/R]\), see Algebraic Stacks, Lemma 04T5. We may and do replace \(\mathcal{X}\) by \([U/R]\). By Sheaves on Stacks, Proposition 06WT there is an equivalence \[q_1 : \QCoh(U, R, s, t, c) \longrightarrow \QCoh(\mathcal{O}_\mathcal{X})\] Let us construct a functor \[q_2 : \textit{LQCoh}^{fbc}(\mathcal{O}_\mathcal{X}) \longrightarrow \QCoh(U, R, s, t, c)\] by the following rule: if \(\mathcal{F}\) is an object of \(\textit{LQCoh}^{fbc}(\mathcal{O}_\mathcal{X})\) then we set \[q_2(\mathcal{F}) = (f^*\mathcal{F}|_{U_\etale}, \alpha)\] where \(\alpha\) is the isomorphism \[t_{small}^*(f^*\mathcal{F}|_{U_\etale}) \to t^*f^*\mathcal{F}|_{R_\etale} \to s^*f^*\mathcal{F}|_{R_\etale} \to s_{small}^*(f^*\mathcal{F}|_{U_\etale})\] where the outer two morphisms are the comparison maps. Note that \(q_2(\mathcal{F})\) is quasi-coherent precisely because \(\mathcal{F}\) is locally quasi-coherent and that we used (and needed) the flat base change property in the construction of the descent datum \(\alpha\). We omit the verification that the cocycle condition (see Groupoids in Spaces, Definition 0441) holds. Looking at the proof of Sheaves on Stacks, Proposition 06WT we see that \(q_2 \circ i\) is the quasi-inverse to \(q_1\). We define \(Q = q_1 \circ q_2\). Let \(\mathcal{F}\) be an object of \(\textit{LQCoh}^{fbc}(\mathcal{O}_\mathcal{X})\) and let \(\mathcal{G}\) be an object of \(\QCoh(\mathcal{O}_\mathcal{X})\). We have \[\begin{align*} \Mor_{\textit{LQCoh}^{fbc}(\mathcal{O}_\mathcal{X})} (i(\mathcal{G}), \mathcal{F}) & = \Mor_{\QCoh(U, R, s, t, c)}(q_2(i(\mathcal{G})), q_2(\mathcal{F})) \\ & = \Mor_{\QCoh(\mathcal{O}_\mathcal{X})}(\mathcal{G}, Q(\mathcal{F})) \end{align*}\] where the first equality is Sheaves on Stacks, Lemma 076S and the second equality holds because \(q_1 \circ i\) and \(q_2\) are quasi-inverse equivalences of categories. The assertion \(Q \circ i \cong \text{id}\) is a formal consequence of the fact that \(i\) is fully faithful.

Lemma

Let \(\mathcal{X}\) be an algebraic stack. Let \(Q : \textit{LQCoh}^{fbc}(\mathcal{O}_\mathcal{X}) \to \QCoh(\mathcal{O}_\mathcal{X})\) be the functor constructed in Lemma 0778.

1. The kernel of \(Q\) is exactly the collection of parasitic objects of \(\textit{LQCoh}^{fbc}(\mathcal{O}_\mathcal{X})\).

2. For any object \(\mathcal{F}\) of \(\textit{LQCoh}^{fbc}(\mathcal{O}_\mathcal{X})\) both the kernel and the cokernel of the adjunction map \(Q(\mathcal{F}) \to \mathcal{F}\) are parasitic.

3. The functor \(Q\) is exact and commutes with all limits and colimits.

Proof

Write \(\mathcal{X} = [U/R]\) as in the proof of Lemma 0778. Let \(\mathcal{F}\) be an object of \(\textit{LQCoh}^{fbc}(\mathcal{O}_\mathcal{X})\). It is clear from the proof of Lemma 0778 that \(\mathcal{F}\) is in the kernel of \(Q\) if and only if \(\mathcal{F}|_{U_\etale} = 0\). In particular, if \(\mathcal{F}\) is parasitic then \(\mathcal{F}\) is in the kernel. Next, let \(x : V \to \mathcal{X}\) be a flat morphism, where \(V\) is a scheme. Set \(W = V \times_\mathcal{X} U\) and consider the diagram \[\xymatrix{ W \ar[d]_p \ar[r]_q & V \ar[d] \\ U \ar[r] & \mathcal{X} }\] Note that the projection \(p : W \to U\) is flat and the projection \(q : W \to V\) is smooth and surjective. This implies that \(q_{small}^*\) is a faithful functor on quasi-coherent modules. By assumption \(\mathcal{F}\) has the flat base change property so that we obtain \(p_{small}^*\mathcal{F}|_{U_\etale} \cong q_{small}^*\mathcal{F}|_{V_\etale}\). Thus if \(\mathcal{F}\) is in the kernel of \(Q\), then \(\mathcal{F}|_{V_\etale} = 0\) which completes the proof of (1).

Part (2) follows from the discussion above and the fact that the map \(Q(\mathcal{F}) \to \mathcal{F}\) becomes an isomorphism after restricting to \(U_\etale\).

To see part (3) note that \(Q\) is left exact as a right adjoint. Let \(0 \to \mathcal{F} \to \mathcal{G} \to \mathcal{H} \to 0\) be a short exact sequence in \(\textit{LQCoh}^{fbc}(\mathcal{O}_\mathcal{X})\). Consider the following commutative diagram \[\xymatrix{ 0 \ar[r] & Q(\mathcal{F}) \ar[r] \ar[d]_a & Q(\mathcal{G}) \ar[r] \ar[d]_b & Q(\mathcal{H}) \ar[r] \ar[d]_c & 0 \\ 0 \ar[r] & \mathcal{F} \ar[r] & \mathcal{G} \ar[r] & \mathcal{H} \ar[r] & 0 }\] Since the kernels and cokernels of \(a\), \(b\), and \(c\) are parasitic by part (2) and since the bottom row is a short exact sequence, we see that the top row as a complex of \(\mathcal{O}_\mathcal{X}\)-modules has parasitic cohomology sheaves (details omitted; this uses that the category of parasitic modules is a Serre subcategory of the category of all modules). By left exactness of \(Q\) we see that only exactness at \(Q(\mathcal{H})\) is at issue. However, the cokernel \(\mathcal{Q}\) of \(Q(\mathcal{G}) \to Q(\mathcal{H}))\) may be computed either in \(\textit{Mod}(\mathcal{O}_\mathcal{X})\) or in \(\QCoh(\mathcal{O}_\mathcal{X})\) with the same result because the inclusion functor \(\QCoh(\mathcal{O}_\mathcal{X}) \to \textit{LQCoh}^{fbc}(\mathcal{O}_\mathcal{X})\) is a left adjoint and hence right exact. Hence \(\mathcal{Q} = Q(\mathcal{Q})\) is both quasi-coherent and parasitic, whence \(0\) by part (1) as desired.

As a right adjoint \(Q\) commutes with all limits. Since \(Q\) is exact, to show that \(Q\) commutes with all colimits it suffices to show that \(Q\) commutes with direct sums, see Categories, Lemma 002P. Let \(\mathcal{F}_i\), \(i \in I\) be a family of objects of \(\textit{LQCoh}^{fbc}(\mathcal{O}_\mathcal{X})\). To see that \(Q(\bigoplus \mathcal{F}_i)\) is equal to \(\bigoplus Q(\mathcal{F}_i)\) we look at the construction of \(Q\) in the proof of Lemma 0778. This uses a presentation \(\mathcal{X} = [U/R]\) where \(U\) is a scheme. Then \(Q(\mathcal{F})\) is computed by first taking the pair \((\mathcal{F}|_{U_\etale}, \alpha)\) in \(\QCoh(U, R, s, t, c)\) and then using the equivalence \(\QCoh(U, R, s, t, c) \cong \QCoh(\mathcal{O}_\mathcal{X})\). Since the restriction functor \(\textit{Mod}(\mathcal{O}_\mathcal{X}) \to \textit{Mod}(\mathcal{O}_{U_\etale})\), \(\mathcal{F} \mapsto \mathcal{F}|_{U_\etale}\) commutes with direct sums, the desired equality is clear.

Lemma

Let \(f : \mathcal{X} \to \mathcal{Y}\) be a flat morphism of algebraic stacks. Then \(Q_\mathcal{X} \circ f^* = f^* \circ Q_\mathcal{Y}\) where \(Q_\mathcal{X}\) and \(Q_\mathcal{Y}\) are as in Lemma 0778.

Proof

Observe that \(f^*\) preserves both \(\QCoh\) and \(\textit{LQCoh}^{fbc}\), see Sheaves on Stacks, Lemma 06WH and Proposition 0771. If \(\mathcal{F}\) is in \(\textit{LQCoh}^{fbc}(\mathcal{O}_\mathcal{Y})\) then \(Q_\mathcal{Y}(\mathcal{F}) \to \mathcal{F}\) has parasitic kernel and cokernel by Lemma 0779. As \(f\) is flat we get that \(f^*Q_\mathcal{Y}(\mathcal{F}) \to f^*\mathcal{F}\) has parasitic kernel and cokernel by Lemma 0774. Thus the induced map \(f^*Q_\mathcal{Y}(\mathcal{F}) \to Q_\mathcal{X}(f^*\mathcal{F})\) has parasitic kernel and cokernel and hence is an isomorphism for example by Lemma 0776.

Lemma

Let \(\mathcal{X}\) be an algebraic stack. Let \(x\) be an object of \(\mathcal{X}\) lying over the scheme \(U\) such that \(x : U \to \mathcal{X}\) is flat. Then for \(\mathcal{F}\) in \(\QCoh^{fbc}(\mathcal{O}_\mathcal{X})\) we have \(Q(\mathcal{F})|_{U_\etale} = \mathcal{F}|_{U_\etale}\).

Proof

True because the kernel and cokernel of \(Q(\mathcal{F}) \to \mathcal{F}\) are parasitic, see Lemma 0779.

Remark

Let \(\mathcal{X}\) be an algebraic stack. The category \(\QCoh(\mathcal{O}_\mathcal{X})\) is abelian, the inclusion functor \(\QCoh(\mathcal{O}_\mathcal{X}) \to \textit{Mod}(\mathcal{O}_\mathcal{X})\) is right exact, but not exact in general, see Sheaves on Stacks, Lemma 06WV. We can use the functor \(Q\) from Lemmas 0778 and 0779 to understand this. Namely, let \(\varphi : \mathcal{F} \to \mathcal{G}\) be a map of quasi-coherent \(\mathcal{O}_\mathcal{X}\)-modules. Then

1. the cokernel \(\Coker(\varphi)\) computed in \(\textit{Mod}(\mathcal{O}_\mathcal{X})\) is quasi-coherent and is the cokernel of \(\varphi\) in \(\QCoh(\mathcal{O}_\mathcal{X})\),

2. the image \(\Im(\varphi)\) computed in \(\textit{Mod}(\mathcal{O}_\mathcal{X})\) is quasi-coherent and is the image of \(\varphi\) in \(\QCoh(\mathcal{O}_\mathcal{X})\), and

3. the kernel \(\Ker(\varphi)\) computed in \(\textit{Mod}(\mathcal{O}_\mathcal{X})\) is in \(\textit{LQCoh}^{fbc}(\mathcal{O}_\mathcal{X})\) by Proposition 0771 and \(Q(\Ker(\varphi))\) is the kernel in \(\QCoh(\mathcal{O}_\mathcal{X})\).

This follows from the references given.

Remark

Let \(\mathcal{X}\) be an algebraic stack. Given two quasi-coherent \(\mathcal{O}_\mathcal{X}\)-modules \(\mathcal{F}\) and \(\mathcal{G}\) the tensor product module \(\mathcal{F} \otimes_{\mathcal{O}_\mathcal{X}} \mathcal{G}\) is quasi-coherent, see Sheaves on Stacks, Lemma 06WV part (5). Similarly, given two locally quasi-coherent modules with the flat base change property, their tensor product has the same property, see Proposition 0771. Thus the inclusion functors \[\QCoh(\mathcal{O}_\mathcal{X}) \to \textit{LQCoh}^{fbc}(\mathcal{O}_\mathcal{X}) \to \textit{Mod}(\mathcal{O}_\mathcal{X})\] are functors of symmetric monoidal categories. What is more interesting is that the functor \[Q : \textit{LQCoh}^{fbc}(\mathcal{O}_\mathcal{X}) \longrightarrow \QCoh(\mathcal{O}_\mathcal{X})\] is a functor of symmetric monoidal categories as well. Namely, given \(\mathcal{F}\) and \(\mathcal{G}\) in \(\textit{LQCoh}^{fbc}(\mathcal{O}_\mathcal{X})\) we obtain \[\xymatrix{ Q(\mathcal{F}) \otimes_{\mathcal{O}_\mathcal{X}} Q(\mathcal{G}) \ar[rr] \ar[rd] & & \mathcal{F} \otimes_{\mathcal{O}_\mathcal{X}} \mathcal{G} \\ & Q(\mathcal{F} \otimes_{\mathcal{O}_\mathcal{X}} \mathcal{G}) \ar[ru] }\] where the south-west arrow comes from the universal property of the north-west arrow (and the fact already mentioned that the object in the upper left corner is quasi-coherent). If we restrict this diagram to \(U_\etale\) for \(U \to \mathcal{X}\) flat, then all three arrows become isomorphisms (see Lemmas 0778 and 0779 and Definition 0773). Hence \(Q(\mathcal{F}) \otimes_{\mathcal{O}_\mathcal{X}} Q(\mathcal{G}) \to Q(\mathcal{F} \otimes_{\mathcal{O}_\mathcal{X}} \mathcal{G})\) is an isomorphism, see for example Lemma 0GQF.

Remark

Let \(\mathcal{X}\) be an algebraic stack. Let \(\textit{Parasitic}(\mathcal{O}_\mathcal{X}) \subset \textit{Mod}(\mathcal{O}_\mathcal{X})\) denote the full subcategory consisting of parasitic modules. The results of Lemmas 0778 and 0779 imply that \[\QCoh(\mathcal{O}_\mathcal{X}) = \textit{LQCoh}^{fbc}(\mathcal{O}_\mathcal{X}) / \textit{Parasitic}(\mathcal{O}_\mathcal{X}) \cap \textit{LQCoh}^{fbc}(\mathcal{O}_\mathcal{X})\] in words: the category of quasi-coherent modules is the category of locally quasi-coherent modules with the flat base change property divided out by the Serre subcategory consisting of parasitic objects. See Homology, Lemma 02MS. The existence of the inclusion functor \(i : \QCoh(\mathcal{O}_\mathcal{X}) \to \textit{LQCoh}^{fbc}(\mathcal{O}_\mathcal{X})\) which is left adjoint to the quotient functor is a key feature of the situation. In Derived Categories of Stacks, Section 07B5 and especially Lemma 07BA we prove that a similar result holds on the level of derived categories.

Lemma

Let \(\mathcal{X}\) be an algebraic stack. Let \(\mathcal{F}\) be an \(\mathcal{O}_\mathcal{X}\)-module of finite presentation and let \(\mathcal{G}\) be a quasi-coherent \(\mathcal{O}_\mathcal{X}\)-module. The internal homs \(\SheafHom_{\mathcal{O}_\mathcal{X}}(\mathcal{F}, \mathcal{G})\) computed in \(\textit{Mod}(\mathcal{X}_\etale, \mathcal{O}_\mathcal{X})\) or \(\textit{Mod}(\mathcal{O}_\mathcal{X})\) agree and the common value is an object of \(\textit{LQCoh}^{fbc}(\mathcal{O}_\mathcal{X})\). The quasi-coherent module \(hom(\mathcal{F}, \mathcal{G}) = Q(\SheafHom_{\mathcal{O}_\mathcal{X}}(\mathcal{F}, \mathcal{G}))\) has the following universal property \[\Hom_\mathcal{X}(\mathcal{H}, hom(\mathcal{F}, \mathcal{G})) = \Hom_\mathcal{X}(\mathcal{H} \otimes_{\mathcal{O}_\mathcal{X}} \mathcal{F}, \mathcal{G})\] for \(\mathcal{H}\) in \(\QCoh(\mathcal{O}_\mathcal{X})\).

Proof

The construction of \(\SheafHom_{\mathcal{O}_\mathcal{X}}(\mathcal{F}, \mathcal{G})\) in Modules on Sites, Section 04TT depends only on \(\mathcal{F}\) and \(\mathcal{G}\) as presheaves of modules; the output \(\SheafHom\) is a sheaf for the fppf topology because \(\mathcal{F}\) and \(\mathcal{G}\) are assumed sheaves in the fppf topology, see Modules on Sites, Lemma 03EM. By Sheaves on Stacks, Lemma 06WM we see that \(\SheafHom_{\mathcal{O}_\mathcal{X}}(\mathcal{F}, \mathcal{G})\) is locally quasi-coherent. By Lemma 0764 we see that \(\SheafHom_{\mathcal{O}_\mathcal{X}}(\mathcal{F}, \mathcal{G})\) has the flat base change property. Hence \(\SheafHom_{\mathcal{O}_\mathcal{X}}(\mathcal{F}, \mathcal{G})\) is an object of \(\textit{LQCoh}^{fbc}(\mathcal{O}_\mathcal{X})\) and it makes sense to apply the functor \(Q\) of Lemma 0778 to it. By the universal property of \(Q\) we have \[\Hom_\mathcal{X}(\mathcal{H}, Q(\SheafHom_{\mathcal{O}_\mathcal{X}}(\mathcal{F}, \mathcal{G}))) = \Hom_\mathcal{X}(\mathcal{H}, \SheafHom_{\mathcal{O}_\mathcal{X}}(\mathcal{F}, \mathcal{G}))\] for \(\mathcal{H}\) quasi-coherent, hence the displayed formula of the lemma follows from Modules on Sites, Lemma 03EO.

Lemma

Let \(f : \mathcal{X} \to \mathcal{Y}\) be a flat morphism of algebraic stacks. Let \(\mathcal{F}\) be an \(\mathcal{O}_\mathcal{Y}\)-module of finite presentation and let \(\mathcal{G}\) be a quasi-coherent \(\mathcal{O}_\mathcal{Y}\)-module. Then \(f^*hom(\mathcal{F}, \mathcal{G}) = hom(f^*\mathcal{F}, f^*\mathcal{G})\) with notation as in Lemma 0GQN.

Proof

We have \(f^*\SheafHom_{\mathcal{O}_\mathcal{Y}}(\mathcal{F}, \mathcal{G}) = \SheafHom_{\mathcal{O}_\mathcal{X}}(f^*\mathcal{F}, f^*\mathcal{G})\) by Modules on Sites, Lemma 0GN2. (Observe that this step is not where the flatness of \(f\) is used as the morphism of ringed topoi associated to \(f\) is always flat, see Sheaves on Stacks, Remark 06TX.) Then apply Lemma 0GQJ (and here we do use flatness of \(f\)).

Pushforward of quasi-coherent modules

Let \(f : \mathcal{X} \to \mathcal{Y}\) be a morphism of algebraic stacks. Consider the pushforward \[f_* : \textit{Mod}(\mathcal{O}_\mathcal{X}) \longrightarrow \textit{Mod}(\mathcal{O}_\mathcal{Y})\] It turns out that this functor almost never preserves the subcategories of quasi-coherent sheaves. For example, consider the morphism of schemes \[j : X = \mathbf{A}^2_k \setminus \{0\} \longrightarrow \mathbf{A}^2_k = Y.\] Associated to this we have the corresponding morphism of algebraic stacks \[f = j_{big} : \mathcal{X} = (\Sch/X)_{fppf} \to (\Sch/Y)_{fppf} = \mathcal{Y}\] The pushforward \(f_*\mathcal{O}_\mathcal{X}\) of the structure sheaf has global sections \(k[x, y]\). Hence if \(f_*\mathcal{O}_\mathcal{X}\) is quasi-coherent on \(\mathcal{Y}\) then we would have \(f_*\mathcal{O}_\mathcal{X} = \mathcal{O}_\mathcal{Y}\). However, consider \(T = \Spec(k) \to \mathbf{A}^2_k = Y\) mapping to \(0\). Then \(\Gamma(T, f_*\mathcal{O}_\mathcal{X}) = 0\) because \(X \times_Y T = \emptyset\) whereas \(\Gamma(T, \mathcal{O}_\mathcal{Y}) = k\). On the positive side, for any flat morphism \(T \to Y\) we have the equality \(\Gamma(T, f_*\mathcal{O}_\mathcal{X}) = \Gamma(T, \mathcal{O}_\mathcal{Y})\) as follows from Cohomology of Schemes, Lemma 02KH using that \(j\) is quasi-compact and quasi-separated.

Let \(f : \mathcal{X} \to \mathcal{Y}\) be a quasi-compact and quasi-separated morphism of algebraic stacks. We work around the problem mentioned above using the following three observations:

1. \(f_*\) does preserve locally quasi-coherent modules (Lemma 075Z),

2. \(f_*\) transforms a quasi-coherent sheaf into a locally quasi-coherent sheaf whose flat comparison maps are isomorphisms (Lemma 0765), and

3. locally quasi-coherent \(\mathcal{O}_\mathcal{Y}\)-modules with the flat base change property give rise to quasi-coherent modules on a presentation of \(\mathcal{Y}\) and hence quasi-coherent modules on \(\mathcal{Y}\), see Sheaves on Stacks, Section 06WU.

Thus we obtain a functor \[f_{\QCoh, *} : \QCoh(\mathcal{O}_\mathcal{X}) \longrightarrow \QCoh(\mathcal{O}_\mathcal{Y})\] which is a right adjoint to \(f^* : \QCoh(\mathcal{O}_\mathcal{Y}) \to \QCoh(\mathcal{O}_\mathcal{X})\) such that moreover \[\Gamma(y, f_*\mathcal{F}) = \Gamma(y, f_{\QCoh, *}\mathcal{F})\] for any \(y \in \Ob(\mathcal{Y})\) such that the associated \(1\)-morphism \(y : V \to \mathcal{Y}\) is flat, see Lemma 0GQQ. Moreover, a similar construction will produce functors \(R^if_{\QCoh, *}\). However, these results will not be sufficient to produce a total direct image functor (of complexes with quasi-coherent cohomology sheaves).

Proposition

Let \(f : \mathcal{X} \to \mathcal{Y}\) be a quasi-compact and quasi-separated morphism of algebraic stacks. The functor \(f^* : \QCoh(\mathcal{O}_\mathcal{Y}) \to \QCoh(\mathcal{O}_\mathcal{X})\) has a right adjoint \[f_{\QCoh, *} : \QCoh(\mathcal{O}_\mathcal{X}) \longrightarrow \QCoh(\mathcal{O}_\mathcal{Y})\] which can be defined as the composition \[\QCoh(\mathcal{O}_\mathcal{X}) \to \textit{LQCoh}^{fbc}(\mathcal{O}_\mathcal{X}) \xrightarrow{f_*} \textit{LQCoh}^{fbc}(\mathcal{O}_\mathcal{Y}) \xrightarrow{Q} \QCoh(\mathcal{O}_\mathcal{Y})\] where the functors \(f_*\) and \(Q\) are as in Proposition 0771 and Lemma 0778. Moreover, if we define \(R^if_{\QCoh, *}\) as the composition \[\QCoh(\mathcal{O}_\mathcal{X}) \to \textit{LQCoh}^{fbc}(\mathcal{O}_\mathcal{X}) \xrightarrow{R^if_*} \textit{LQCoh}^{fbc}(\mathcal{O}_\mathcal{Y}) \xrightarrow{Q} \QCoh(\mathcal{O}_\mathcal{Y})\] then the sequence of functors \(\{R^if_{\QCoh, *}\}_{i \geq 0}\) forms a cohomological \(\delta\)-functor.

Proof

This is a combination of the results mentioned in the statement. The adjointness can be shown as follows: Let \(\mathcal{F}\) be a quasi-coherent \(\mathcal{O}_\mathcal{X}\)-module and let \(\mathcal{G}\) be a quasi-coherent \(\mathcal{O}_\mathcal{Y}\)-module. Then we have \[\begin{align*} \Mor_{\QCoh(\mathcal{O}_\mathcal{X})}(f^*\mathcal{G}, \mathcal{F}) & = \Mor_{\textit{LQCoh}^{fbc}(\mathcal{O}_\mathcal{Y})} (\mathcal{G}, f_*\mathcal{F}) \\ & = \Mor_{\QCoh(\mathcal{O}_\mathcal{Y})}(\mathcal{G}, Q(f_*\mathcal{F})) \\ & = \Mor_{\QCoh(\mathcal{O}_\mathcal{Y})}(\mathcal{G}, f_{\QCoh, *}\mathcal{F}) \end{align*}\] the first equality by adjointness of \(f_*\) and \(f^*\) (for arbitrary sheaves of modules). By Proposition 0771 we see that \(f_*\mathcal{F}\) is an object of \(\textit{LQCoh}^{fbc}(\mathcal{O}_\mathcal{Y})\) (and can be computed in either the fppf or étale topology) and we obtain the second equality by Lemma 0778. The third equality is the definition of \(f_{\QCoh, *}\).

To see that \(\{R^if_{\QCoh, *}\}_{i \geq 0}\) is a cohomological \(\delta\)-functor as defined in Homology, Definition 010Q let \[0 \to \mathcal{F}_1 \to \mathcal{F}_2 \to \mathcal{F}_3 \to 0\] be a short exact sequence of \(\QCoh(\mathcal{O}_\mathcal{X})\). This sequence may not be an exact sequence in \(\textit{Mod}(\mathcal{O}_\mathcal{X})\) but we know that it is up to parasitic modules, see Lemma 0776. Thus we may break up the sequence into short exact sequences \[\begin{matrix} 0 \to \mathcal{P}_1 \to \mathcal{F}_1 \to \mathcal{I}_2 \to 0 \\ 0 \to \mathcal{I}_2 \to \mathcal{F}_2 \to \mathcal{Q}_2 \to 0 \\ 0 \to \mathcal{P}_2 \to \mathcal{Q}_2 \to \mathcal{I}_3 \to 0 \\ 0 \to \mathcal{I}_3 \to \mathcal{F}_3 \to \mathcal{P}_3 \to 0 \end{matrix}\] of \(\textit{Mod}(\mathcal{O}_\mathcal{X})\) with \(\mathcal{P}_i\) parasitic. Note that each of the sheaves \(\mathcal{P}_j\), \(\mathcal{I}_j\), \(\mathcal{Q}_j\) is an object of \(\textit{LQCoh}^{fbc}(\mathcal{O}_\mathcal{X})\), see Proposition 0771. Applying \(R^if_*\) we obtain long exact sequences \[\begin{matrix} 0 \to f_*\mathcal{P}_1 \to f_*\mathcal{F}_1 \to f_*\mathcal{I}_2 \to R^1f_*\mathcal{P}_1 \to \ldots \\ 0 \to f_*\mathcal{I}_2 \to f_*\mathcal{F}_2 \to f_*\mathcal{Q}_2 \to R^1f_*\mathcal{I}_2 \to \ldots \\ 0 \to f_*\mathcal{P}_2 \to f_*\mathcal{Q}_2 \to f_*\mathcal{I}_3 \to R^1f_*\mathcal{P}_2 \to \ldots \\ 0 \to f_*\mathcal{I}_3 \to f_*\mathcal{F}_3 \to f_*\mathcal{P}_3 \to R^1f_*\mathcal{I}_3 \to \ldots \end{matrix}\] where are the terms are objects of \(\textit{LQCoh}^{fbc}(\mathcal{O}_\mathcal{Y})\) by Proposition 0771. By Lemma 0775 the sheaves \(R^if_*\mathcal{P}_j\) are parasitic, hence vanish on applying the functor \(Q\), see Lemma 0779. Since \(Q\) is exact the maps \[Q(R^if_*\mathcal{F}_3) \cong Q(R^if_*\mathcal{I}_3) \cong Q(R^if_*\mathcal{Q}_2) \rightarrow Q(R^{i + 1}f_*\mathcal{I}_2) \cong Q(R^{i + 1}f_*\mathcal{F}_1)\] can serve as the connecting map which turns the family of functors \(\{R^if_{\QCoh, *}\}_{i \geq 0}\) into a cohomological \(\delta\)-functor.

Lemma

Let \(f : \mathcal{X} \to \mathcal{Y}\) be a quasi-compact and quasi-separated morphism of algebraic stacks. Let \(y : V \to \mathcal{Y}\) in \(\Ob(\mathcal{Y})\) with \(y\) a flat morphism. Let \(\mathcal{F}\) be in \(\QCoh(\mathcal{O}_\mathcal{X})\). Then \((f_*\mathcal{F})(y) = (f_{\QCoh, *}\mathcal{F})(y)\) and \((R^if_*\mathcal{F})(y) = (R^if_{\QCoh, *}\mathcal{F})(y)\) for all \(i \in \mathbf{Z}\).

Proof

This follows from the construction of the functors \(R^if_{\QCoh, *}\) in Proposition 077A, the definition of parasitic modules in Definition 0773, and Lemma 0779 part (2).

Remark

Let \(f : \mathcal{X} \to \mathcal{Y}\) be a quasi-compact and quasi-separated morphism of algebraic stacks. Let \(\mathcal{F}\) and \(\mathcal{G}\) be in \(\QCoh(\mathcal{O}_\mathcal{X})\). Then there is a canonical commutative diagram \[\xymatrix{ f_{\QCoh, *}\mathcal{F} \otimes_{\mathcal{O}_\mathcal{Y}} f_{\QCoh, *}\mathcal{G} \ar[r] \ar[d] & f_*\mathcal{F} \otimes_{\mathcal{O}_\mathcal{Y}} f_*\mathcal{G} \ar[d]^c \\ f_{\QCoh, *}(\mathcal{F} \otimes_{\mathcal{O}_\mathcal{X}} \mathcal{G}) \ar[r] & f_*(\mathcal{F} \otimes_{\mathcal{O}_\mathcal{X}} \mathcal{G}) }\] The vertical arrow \(c\) on the right is the naive relative cup product (in degree \(0\)), see Cohomology on Sites, Section 0FPJ. The source and target of \(c\) are in \(\textit{LQCoh}^{fbc}(\mathcal{O}_\mathcal{X})\), see Proposition 0771. Applying \(Q\) to \(c\) we obtain the left vertical arrow as \(Q\) commutes with tensor products, see Remark 0GQM. This construction is functorial in \(\mathcal{F}\) and \(\mathcal{G}\).

Lemma

Let \(f : \mathcal{X} \to \mathcal{Y}\) be a quasi-compact and quasi-separated morphism of algebraic stacks. Let \(\mathcal{F}\) be a quasi-coherent sheaf on \(\mathcal{X}\). Then there exists a spectral sequence with \(E_2\)-page \[E_2^{p, q} = H^p(\mathcal{Y}, R^qf_{\QCoh, *}\mathcal{F})\] converging to \(H^{p + q}(\mathcal{X}, \mathcal{F})\).

Proof

By Cohomology on Sites, Lemma 0732 the Leray spectral sequence with \[E_2^{p, q} = H^p(\mathcal{Y}, R^qf_*\mathcal{F})\] converges to \(H^{p + q}(\mathcal{X}, \mathcal{F})\). The kernel and cokernel of the adjunction map \[R^qf_{\QCoh, *}\mathcal{F} \longrightarrow R^qf_*\mathcal{F}\] are parasitic modules on \(\mathcal{Y}\) (Lemma 0779) hence have vanishing cohomology (Lemma 0775). It follows formally that \(H^p(\mathcal{Y}, R^qf_{\QCoh, *}\mathcal{F}) = H^p(\mathcal{Y}, R^qf_*\mathcal{F})\) and we win.

Lemma

Let \(f : \mathcal{X} \to \mathcal{Y}\) and \(g : \mathcal{Y} \to \mathcal{Z}\) be quasi-compact and quasi-separated morphisms of algebraic stacks. Let \(\mathcal{F}\) be a quasi-coherent sheaf on \(\mathcal{X}\). Then there exists a spectral sequence with \(E_2\)-page \[E_2^{p, q} = R^pg_{\QCoh, *}(R^qf_{\QCoh, *}\mathcal{F})\] converging to \(R^{p + q}(g \circ f)_{\QCoh, *}\mathcal{F}\).

Proof

By Cohomology on Sites, Lemma 0734 the Leray spectral sequence with \[E_2^{p, q} = R^pg_*(R^qf_*\mathcal{F})\] converges to \(R^{p + q}(g \circ f)_*\mathcal{F}\). By the results of Proposition 0771 all the terms of this spectral sequence are objects of \(\textit{LQCoh}^{fbc}(\mathcal{O}_\mathcal{Z})\). Applying the exact functor \(Q_\mathcal{Z} : \textit{LQCoh}^{fbc}(\mathcal{O}_\mathcal{Z}) \to \QCoh(\mathcal{O}_\mathcal{Z})\) we obtain a spectral sequence in \(\QCoh(\mathcal{O}_\mathcal{Z})\) covering to \(R^{p + q}(g \circ f)_{\QCoh, *}\mathcal{F}\). Hence the result follows if we can show that \[Q_\mathcal{Z}(R^pg_*(R^qf_*\mathcal{F})) = Q_\mathcal{Z}(R^pg_*(Q_\mathcal{X}(R^qf_*\mathcal{F}))\] This follows from the fact that the kernel and cokernel of the map \[Q_\mathcal{X}(R^qf_*\mathcal{F}) \longrightarrow R^qf_*\mathcal{F}\] are parasitic (Lemma 0779) and that \(R^pg_*\) transforms parasitic modules into parasitic modules (Lemma 0775).

To end this section we make explicit the spectral sequences associated to a smooth covering by a scheme. Please compare with Sheaves on Stacks, Sections 06XI and 072F.

Proposition

Let \(f : \mathcal{U} \to \mathcal{X}\) be a morphism of algebraic stacks. Assume \(f\) is representable by algebraic spaces, surjective, flat, and locally of finite presentation. Let \(\mathcal{F}\) be a quasi-coherent \(\mathcal{O}_\mathcal{X}\)-module. Then there is a spectral sequence \[E_2^{p, q} = H^q(\mathcal{U}_p, f_p^*\mathcal{F}) \Rightarrow H^{p + q}(\mathcal{X}, \mathcal{F})\] where \(f_p\) is the morphism \(\mathcal{U} \times_\mathcal{X} \ldots \times_\mathcal{X} \mathcal{U} \to \mathcal{X}\) (\(p + 1\) factors).

Proof

This is a special case of Sheaves on Stacks, Proposition 06XJ.

Proposition

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

1. \(f\) is representable by algebraic spaces, surjective, flat, locally of finite presentation, quasi-compact, and quasi-separated, and

2. \(g\) is quasi-compact and quasi-separated.

If \(\mathcal{F}\) is in \(\QCoh(\mathcal{O}_\mathcal{X})\) then there is a spectral sequence \[E_2^{p, q} = R^q(g \circ f_p)_{\QCoh, *}f_p^*\mathcal{F} \Rightarrow R^{p + q}g_{\QCoh, *}\mathcal{F}\] in \(\QCoh(\mathcal{O}_\mathcal{Y})\).

Proof

Note that each of the morphisms \(f_p : \mathcal{U} \times_\mathcal{X} \ldots \times_\mathcal{X} \mathcal{U} \to \mathcal{X}\) is quasi-compact and quasi-separated, hence \(g \circ f_p\) is quasi-compact and quasi-separated, hence the assertion makes sense (i.e., the functors \(R^q(g \circ f_p)_{\QCoh, *}\) are defined). There is a spectral sequence \[E_2^{p, q} = R^q(g \circ f_p)_*f_p^{-1}\mathcal{F} \Rightarrow R^{p + q}g_*\mathcal{F}\] by Sheaves on Stacks, Proposition 072G. Applying the exact functor \(Q_\mathcal{Y} : \textit{LQCoh}^{fbc}(\mathcal{O}_\mathcal{Y}) \to \QCoh(\mathcal{O}_\mathcal{Y})\) gives the desired spectral sequence in \(\QCoh(\mathcal{O}_\mathcal{Y})\).

Further remarks on quasi-coherent modules

In this section we collect some results that to help understand how to use quasi-coherent modules on algebraic stacks.

Let \(f : \mathcal{U} \to \mathcal{X}\) be a morphism of algebraic stacks. Assume \(\mathcal{U}\) is represented by the algebraic space \(U\). Consider the functor \[a : \textit{Mod}(\mathcal{X}_\etale, \mathcal{O}_\mathcal{X}) \longrightarrow \textit{Mod}(U_\etale, \mathcal{O}_U),\quad \mathcal{F} \longmapsto f^*\mathcal{F}|_{U_\etale}\] given by pullback (Sheaves on Stacks, Section 06WA) followed by restriction (Sheaves on Stacks, Section 076P). Applying this functor to locally quasi-coherent modules we obtain a functor \[b : \textit{LQCoh}(\mathcal{O}_\mathcal{X}) \longrightarrow \QCoh(U_\etale, \mathcal{O}_U)\] See Sheaves on Stacks, Lemmas 06WL and 0GQC. We can further limit our functor to even smaller subcategories to obtain \[c : \textit{LQCoh}^{fbc}(\mathcal{O}_\mathcal{X}) \longrightarrow \QCoh(U_\etale, \mathcal{O}_U)\] and \[d : \QCoh(\mathcal{O}_\mathcal{X}) \longrightarrow \QCoh(U_\etale, \mathcal{O}_U)\] About these functors we can say the following:³

1. The functor \(a\) is exact. Namely, pullback \(f^* = f^{-1}\) is exact (Sheaves on Stacks, Section 06WA) and restriction to \(U_\etale\) is exact, see Sheaves on Stacks, Equation (075K).

2. The functor \(b\) is exact. Namely, by Sheaves on Stacks, Lemma 06WM the inclusion \(\textit{LQCoh}(\mathcal{O}_\mathcal{X}) \to \textit{Mod}(\mathcal{X}_\etale, \mathcal{O}_\mathcal{X})\) is exact.

3. The functor \(c\) is exact. Namely, by Proposition 0771 the inclusion functor \(\textit{LQCoh}^{fbc}(\mathcal{O}_\mathcal{X}) \to \textit{Mod}(\mathcal{X}_\etale, \mathcal{O}_\mathcal{X})\) is exact.

4. The functor \(d\) is right exact but not exact in general. Namely, by Sheaves on Stacks, Lemma 06WN the inclusion functor \(\QCoh(\mathcal{O}_\mathcal{X}) \to \textit{Mod}(\mathcal{X}_\etale, \mathcal{O}_\mathcal{X})\) is right exact. We omit giving an example showing non-exactness.

5. If \(f\) is flat, then \(d\) is exact. This follows on combining Lemma 076X and Sheaves on Stacks, Lemma 0GQD.

6. If \(f\) is flat, then \(c\) kills parasitic objects. Namely, \(f^*\) preserves parasitic object by Lemma 0774. Then for any scheme \(V\) étale over \(U\) and hence flat over \(\mathcal{X}\) we see that \(0 = f^*\mathcal{F}|_{V_\etale} = c(\mathcal{F})|_{V_\etale}\) by the compatibility of restriction with étale localization Sheaves on Stacks, Remark 0GQA. Hence clearly \(c(\mathcal{F}) = 0\).

7. If \(f\) is flat, then \(c = d \circ Q\). Namely, the kernel and cokernel of \(Q(\mathcal{F}) \to \mathcal{F}\) are parasitic by Lemma 0779. Thus, since \(c\) is exact (3) and kills parasitic objects (6), we see that \(c\) applied to \(Q(\mathcal{F}) \to \mathcal{F}\) is an isomorphism.

8. The functors \(a, b, c, d\) commute with colimits and arbitrary direct sums. This is true for \(f^*\) and restriction as left adjoints and hence it holds for \(a\). Then it follows for \(b\), \(c\), \(d\) by the references given above.

9. The functors \(a, b, c, d\) commute with tensor products.

10. If \(f\) is flat and surjective, \(\mathcal{F}\) is in \(\textit{LQCoh}^{fbc}(\mathcal{O}_\mathcal{X})\), and \(c(\mathcal{F}) = 0\), then \(\mathcal{F}\) is parasitic. Namely, by (7) we get \(d(Q(\mathcal{F})) = 0\). We may assume \(U\) is a scheme by the compatibility of restriction with étale localization (see reference above). Then Lemma 0GQF applied to \(0 \to Q(\mathcal{F})\) and the morphism \(f : U \to \mathcal{X}\) shows that \(Q(\mathcal{F}) = 0\). Thus \(\mathcal{F}\) is parasitic by Lemma 0779.

11. If \(f\) is flat and surjective, then the functor \(d\) reflects exactness. More precisely, let \(\mathcal{F}^\bullet\) be a complex in \(\QCoh(\mathcal{O}_\mathcal{X})\). Then \(\mathcal{F}^\bullet\) is exact in \(\QCoh(\mathcal{O}_\mathcal{X})\) if and only if \(d(\mathcal{F}^\bullet)\) is exact. Namely, we have seen one implication in (5). For the other, suppose that \(H^i(d(\mathcal{F}^\bullet)) = 0\). Then \(\mathcal{G} = H^i(\mathcal{F}^\bullet)\) is an object of \(\QCoh(\mathcal{O}_\mathcal{X})\) with \(d(\mathcal{G}) = 0\). Hence \(\mathcal{G}\) is both quasi-coherent and parasitic by (10), whence \(0\) for example by Remark 07B2.

12. If \(f\) is flat, \(\mathcal{F}, \mathcal{G} \in \Ob(\QCoh(\mathcal{O}_\mathcal{X}))\), and \(\mathcal{F}\) of finite presentation and let then we have \[d(hom(\mathcal{F}, \mathcal{G})) = \SheafHom_{\mathcal{O}_U}(d(\mathcal{F}), d(\mathcal{G}))\] with notation as in Lemma 0GQN. Perhaps the easiest way to see this is as follows \[\begin{align*} d(hom(\mathcal{F}, \mathcal{G})) & = d(Q(\SheafHom_{\mathcal{O}_\mathcal{X}}(\mathcal{F}, \mathcal{G}))) \\ & = c(\SheafHom_{\mathcal{O}_\mathcal{X}}(\mathcal{F}, \mathcal{G})) \\ & = f^*\SheafHom_{\mathcal{O}_\mathcal{X}}(\mathcal{F}, \mathcal{G})|_{U_\etale} \\ & = \SheafHom_{\mathcal{O}_\mathcal{U}}(f^*\mathcal{F}, f^*\mathcal{G})|_{U_\etale} \\ & = \SheafHom_{\mathcal{O}_U}(f^*\mathcal{F}|_{U_\etale}, f^*\mathcal{G}|_{U_\etale}) \end{align*}\] The first equality by construction of \(hom\). The second equality by (7). The third equality by definition of \(c\). The fourth equality by Modules on Sites, Lemma 0GN2. The final equality by the same reference applied to the flat morphism of ringed topoi \(i_U (U_\etale, \mathcal{O}_U) \to (\mathcal{U}_\etale, \mathcal{O}_\mathcal{U})\) of Sheaves on Stacks, Lemma 073M.

13. add more here.

Colimits and cohomology

The following lemma in particular applies to diagrams of quasi-coherent sheaves.

Lemma

Let \(\mathcal{X}\) be a quasi-compact and quasi-separated algebraic stack. Then \[\colim_i H^p(\mathcal{X}, \mathcal{F}_i) \longrightarrow H^p(\mathcal{X}, \colim_i \mathcal{F}_i)\] is an isomorphism for every filtered diagram of abelian sheaves on \(\mathcal{X}\). The same is true for abelian sheaves on \(\mathcal{X}_\etale\) taking cohomology in the étale topology.

Proof

Let \(\tau = fppf\), resp. \(\tau = \etale\). The lemma follows from Cohomology on Sites, Lemma 0GN3 applied to the site \(\mathcal{X}_\tau\). In order to check the assumptions we use Cohomology on Sites, Remark 0GN4. Namely, let \(\mathcal{B} \subset \Ob(\mathcal{X}_\tau)\) be the set of objects lying over affine schemes. In other words, an element of \(\mathcal{B}\) is a morphism \(x : U \to \mathcal{X}\) with \(U\) affine. We check each of the conditions (1) – (4) of the remark in turn:

1. Since \(\mathcal{X}\) is quasi-compact, there exists a surjective and smooth morphism \(x : U \to \mathcal{X}\) with \(U\) affine (Properties of Stacks, Lemma 04YC). Then \(h_x^\# \to *\) is a surjective map of sheaves on \(\mathcal{X}_\tau\).

2. Since coverings in \(\mathcal{X}_\tau\) are fppf, resp. étale coverings, we see that every covering of \(U \in \mathcal{B}\) is refined by a finite affine fppf covering, see Topologies, Lemma 021P, resp. Lemma 0218.

3. Let \(x : U \to \mathcal{X}\) and \(x' : U' \to \mathcal{X}\) be in \(\mathcal{B}\). The product \(h_x^\# \times h_{x'}^\#\) in \(\Sh(\mathcal{X}_\tau)\) is equal to the sheaf on \(\mathcal{X}_\tau\) determined by the algebraic space \(W = U \times_{x, \mathcal{X}, x'} U'\) over \(\mathcal{X}\): for an object \(y : V \to \mathcal{X}\) of \(\mathcal{X}_\tau\) we have \((h_x^\# \times h_{x'}^\#)(y) = \{f : V \to W \mid y = x \circ \text{pr}_1 \circ f = x' \circ \text{pr}_2 \circ f\}\). The algebraic space \(W\) is quasi-compact because \(\mathcal{X}\) is quasi-separated, see Morphisms of Stacks, Lemma 075S for example. Hence we can choose an affine scheme \(U''\) and a surjective étale morphism \(U'' \to W\). Denote \(x'' : U'' \to \mathcal{X}\) the composition of \(U'' \to W\) and \(W \to \mathcal{X}\). Then \(h_{x''}^\# \to h_x^\# \times h_{x'}^\#\) is surjective as desired.

4. Let \(x : U \to \mathcal{X}\) and \(x' : U' \to \mathcal{X}\) be in \(\mathcal{B}\). Let \(a, b : U \to U'\) be a morphism over \(\mathcal{X}\), i.e., \(a, b : x \to x'\) is a morphism in \(\mathcal{X}_\tau\). Then the equalizer of \(h_a\) and \(h_b\) is represented by the equalizer of \(a, b : U \to U'\) which is affine scheme over \(\mathcal{X}\) and hence in \(\mathcal{B}\).

This finished the proof.

Lemma

Let \(f : \mathcal{X} \to \mathcal{Y}\) be a quasi-compact and quasi-separated morphism of algebraic stacks. Let \(\mathcal{F} = \colim \mathcal{F}_i\) be a filtered colimit of abelian sheaves on \(\mathcal{X}\). Then for any \(p \geq 0\) we have \[R^pf_*\mathcal{F} = \colim R^pf_*\mathcal{F}_i.\] The same is true for abelian sheaves on \(\mathcal{X}_\etale\) taking higher direct images in the étale topology.

Proof

We will prove this for the fppf topology; the proof for the étale topology is the same. Recall that \(R^if_*\mathcal{F}\) is the sheaf on \(\mathcal{Y}_{fppf}\) associated to the presheaf \[(y : V \to \mathcal{Y}) \longmapsto H^i(V \times_{y, \mathcal{Y}} \mathcal{X}, \text{pr}^{-1}\mathcal{F})\] See Sheaves on Stacks, Lemma 075G. Recall that the colimit is the sheaf associated to the presheaf colimit. When \(V\) is affine, the fibre product \(V \times_\mathcal{Y} \mathcal{X}\) is quasi-compact and quasi-separated. Hence we can apply Lemma 0GQV to \(H^p(V \times_\mathcal{Y} \mathcal{X}, -)\) where \(V\) is affine. Since every \(V\) has an fppf covering by affine objects this proves the lemma. Some details omitted.

Lemma

Let \(f : \mathcal{X} \to \mathcal{Y}\) be a quasi-compact and quasi-separated morphism of algebraic stacks. The functor \(f_{\QCoh, *}\) and the functors \(R^if_{\QCoh, *}\) commute with direct sums and filtered colimits.

Proof

The functors \(f_*\) and \(R^if_*\) commute with direct sums and filtered colimits on all modules by Lemma 0GQW. The lemma follows as \(f_{\QCoh, *} = Q \circ f_*\) and \(R^if_{\QCoh, *} = Q \circ R^if_*\) and \(Q\) commutes with all colimits, see Lemma 0779.

Lemma

Let \(f : \mathcal{X} \to \mathcal{Y}\) be an affine morphism of algebraic stacks. The functors \(R^if_{\QCoh, *}\), \(i > 0\) vanish and the functor \(f_{\QCoh, *}\) is exact and commutes with direct sums and all colimits.

Proof

Since we have \(R^if_{\QCoh, *} = Q \circ R^if_*\) we obtain the vanishing from Lemma 0GQI. The vanishing implies that \(f_{\QCoh, *}\) is exact as \(\{R^if_{\QCoh, *}\}_{i \geq 0}\) form a \(\delta\)-functor, see Proposition 077A. Then \(f_{\QCoh, *}\) commutes with direct sums for example by Lemma 0GQX. An exact functor which commutes with direct sums commutes with all colimits.

The following lemma tells us that finitely presented modules behave as expected in quasi-compact and quasi-separated algebraic stacks.

Lemma

Let \(\mathcal{X}\) be a quasi-compact and quasi-separated algebraic stack. Let \(I\) be a directed set and let \((\mathcal{F}_i, \varphi_{ii'})\) be a system over \(I\) of \(\mathcal{O}_\mathcal{X}\)-modules. Let \(\mathcal{G}\) be an \(\mathcal{O}_\mathcal{X}\)-module of finite presentation. Then we have \[\colim_i \Hom_\mathcal{X}(\mathcal{G}, \mathcal{F}_i) = \Hom_\mathcal{X}(\mathcal{G}, \colim_i \mathcal{F}_i).\] In particular, \(\Hom_\mathcal{X}(\mathcal{G}, -)\) commutes with filtered colimits in \(\QCoh(\mathcal{O}_\mathcal{X})\).

Proof

The displayed equality is a special case of Modules on Sites, Lemma 0GN0. In order to apply it, we need to check the hypotheses of Sites, Lemma 0GMR part (4) for the site \(\mathcal{X}_{fppf}\). In order to do this, we will check hypotheses (2)(a), (2)(b), (2)(c) of Sites, Remark 0GMS. Namely, let \(\mathcal{B} \subset \Ob(\mathcal{X}_{fppf})\) be the set of objects lying over affine schemes. In other words, an element of \(\mathcal{B}\) is a morphism \(x : U \to \mathcal{X}\) with \(U\) affine. We check each of the conditions (2)(a), (2)(b), and (2)(c) of the remark in turn:

1. Since \(\mathcal{X}\) is quasi-compact, there exists a surjective and smooth morphism \(x : U \to \mathcal{X}\) with \(U\) affine (Properties of Stacks, Lemma 04YC). Then \(h_x^\# \to *\) is a surjective map of sheaves on \(\mathcal{X}_{fppf}\).

2. Since coverings in \(\mathcal{X}_{fppf}\) are fppf coverings, we see that every covering of \(U \in \mathcal{B}\) is refined by a finite affine fppf covering, see Topologies, Lemma 021P.

3. Let \(x : U \to \mathcal{X}\) and \(x' : U' \to \mathcal{X}\) be in \(\mathcal{B}\). The product \(h_x^\# \times h_{x'}^\#\) in \(\Sh(\mathcal{X}_{fppf})\) is equal to the sheaf on \(\mathcal{X}_{fppf}\) determined by the algebraic space \(W = U \times_{x, \mathcal{X}, x'} U'\) over \(\mathcal{X}\): for an object \(y : V \to \mathcal{X}\) of \(\mathcal{X}_{fppf}\) we have \((h_x^\# \times h_{x'}^\#)(y) = \{f : V \to W \mid y = x \circ \text{pr}_1 \circ f = x' \circ \text{pr}_2 \circ f\}\). The algebraic space \(W\) is quasi-compact because \(\mathcal{X}\) is quasi-separated, see Morphisms of Stacks, Lemma 075S for example. Hence we can choose an affine scheme \(U''\) and a surjective étale morphism \(U'' \to W\). Denote \(x'' : U'' \to \mathcal{X}\) the composition of \(U'' \to W\) and \(W \to \mathcal{X}\). Then \(h_{x''}^\# \to h_x^\# \times h_{x'}^\#\) is surjective as desired.

For the final statement, observe that the inclusion functor \(\QCoh(\mathcal{O}_X) \to \textit{Mod}(\mathcal{O}_X)\) commutes with colimits and that finitely presented modules are quasi-coherent. See Sheaves on Stacks, Lemma 06WV.

The lisse-étale and the flat-fppf sites

In the book [LM-B] many of the results above are proved using the lisse-étale site of an algebraic stack. We define this site here. In Examples, Section 07BF we show that the lisse-étale site isn’t functorial. We also define its analogue, the flat-fppf site, which is better suited to the development of algebraic stacks as given in the Stacks project (because we use the fppf topology as our base topology). Of course the flat-fppf site isn’t functorial either.

Definition

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

1. The lisse-étale site of \(\mathcal{X}\) is the full subcategory \(\mathcal{X}_{lisse,\etale}\)⁴ of \(\mathcal{X}\) whose objects are those \(x \in \Ob(\mathcal{X})\) lying over a scheme \(U\) such that \(x : U \to \mathcal{X}\) is smooth. A covering of \(\mathcal{X}_{lisse,\etale}\) is a family of morphisms \(\{x_i \to x\}_{i \in I}\) of \(\mathcal{X}_{lisse,\etale}\) which forms a covering of \(\mathcal{X}_\etale\).

2. The flat-fppf site of \(\mathcal{X}\) is the full subcategory \(\mathcal{X}_{flat,fppf}\) of \(\mathcal{X}\) whose objects are those \(x \in \Ob(\mathcal{X})\) lying over a scheme \(U\) such that \(x : U \to \mathcal{X}\) is flat. A covering of \(\mathcal{X}_{flat,fppf}\) is a family of morphisms \(\{x_i \to x\}_{i \in I}\) of \(\mathcal{X}_{flat,fppf}\) which forms a covering of \(\mathcal{X}_{fppf}\).

We denote \(\mathcal{O}_{\mathcal{X}_{lisse,\etale}}\) the restriction of \(\mathcal{O}_\mathcal{X}\) to the lisse-étale site and similarly for \(\mathcal{O}_{\mathcal{X}_{flat,fppf}}\). The relationship between the lisse-étale site and the étale site is as follows (we mainly stick to “topological” properties in this lemma).

Lemma

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

1. The inclusion functor \(\mathcal{X}_{lisse,\etale} \to \mathcal{X}_\etale\) is fully faithful, continuous and cocontinuous. It follows that

   1. there is a morphism of topoi \[g : \Sh(\mathcal{X}_{lisse,\etale}) \longrightarrow \Sh(\mathcal{X}_\etale)\] with \(g^{-1}\) given by restriction,

   2. the functor \(g^{-1}\) has a left adjoint \(g_!^{Sh}\) on sheaves of sets,

   3. the adjunction maps \(g^{-1}g_* \to \text{id}\) and \(\text{id} \to g^{-1}g_!^{Sh}\) are isomorphisms,

   4. the functor \(g^{-1}\) has a left adjoint \(g_!\) on abelian sheaves,

   5. the adjunction map \(\text{id} \to g^{-1}g_!\) is an isomorphism, and

   6. we have \(g^{-1}\mathcal{O}_\mathcal{X} = \mathcal{O}_{\mathcal{X}_{lisse,\etale}}\) hence \(g\) induces a flat morphism of ringed topoi such that \(g^{-1} = g^*\).

2. The inclusion functor \(\mathcal{X}_{flat,fppf} \to \mathcal{X}_{fppf}\) is fully faithful, continuous and cocontinuous. It follows that

   1. there is a morphism of topoi \[g : \Sh(\mathcal{X}_{flat,fppf}) \longrightarrow \Sh(\mathcal{X}_{fppf})\] with \(g^{-1}\) given by restriction,

   2. the functor \(g^{-1}\) has a left adjoint \(g_!^{Sh}\) on sheaves of sets,

   3. the adjunction maps \(g^{-1}g_* \to \text{id}\) and \(\text{id} \to g^{-1}g_!^{Sh}\) are isomorphisms,

   4. the functor \(g^{-1}\) has a left adjoint \(g_!\) on abelian sheaves,

   5. the adjunction map \(\text{id} \to g^{-1}g_!\) is an isomorphism, and

   6. we have \(g^{-1}\mathcal{O}_\mathcal{X} = \mathcal{O}_{\mathcal{X}_{flat,fppf}}\) hence \(g\) induces a flat morphism of ringed topoi such that \(g^{-1} = g^*\).

Proof

In both cases it is immediate that the functor is fully faithful, continuous, and cocontinuous (see Sites, Definitions 00WV and 00XJ). Hence properties (a), (b), (c) follow from Sites, Lemmas 00XR and 00XT. Parts (d), (e) follow from Modules on Sites, Lemmas 04BG and 077I. Part (f) is immediate.

Lemma

Let \(\mathcal{X}\) be an algebraic stack. Notation as in Lemma 0788.

1. For an abelian sheaf \(\mathcal{F}\) on \(\mathcal{X}_\etale\) we have

   1. \(H^p(\mathcal{X}_\etale, \mathcal{F}) = H^p(\mathcal{X}_{lisse,\etale}, g^{-1}\mathcal{F})\), and

   2. \(H^p(x, \mathcal{F}) = H^p(\mathcal{X}_{lisse,\etale}/x, g^{-1}\mathcal{F})\) for any object \(x\) of \(\mathcal{X}_{lisse,\etale}\).

   The same holds for sheaves of modules.

2. For an abelian sheaf \(\mathcal{F}\) on \(\mathcal{X}_{fppf}\) we have

   1. \(H^p(\mathcal{X}_{fppf}, \mathcal{F}) = H^p(\mathcal{X}_{flat,fppf}, g^{-1}\mathcal{F})\), and

   2. \(H^p(x, \mathcal{F}) = H^p(\mathcal{X}_{flat,fppf}/x, g^{-1}\mathcal{F})\) for any object \(x\) of \(\mathcal{X}_{flat,fppf}\).

   The same holds for sheaves of modules.

Proof

Part (1)(a) follows from Sheaves on Stacks, Lemma 07AK applied to the inclusion functor \(\mathcal{X}_{lisse,\etale} \to \mathcal{X}_\etale\). Part (1)(b) follows from part (1)(a). Namely, if \(x\) lies over the scheme \(U\), then the site \(\mathcal{X}_\etale/x\) is equivalent to \((\Sch/U)_\etale\) and \(\mathcal{X}_{lisse,\etale}\) is equivalent to \(U_{lisse,\etale}\). Part (2) is proved in the same manner.

Lemma

Let \(\mathcal{X}\) be an algebraic stack. Notation as in Lemma 0788.

1. There exists a functor \[g_! : \textit{Mod}(\mathcal{X}_{lisse,\etale}, \mathcal{O}_{\mathcal{X}_{lisse,\etale}}) \longrightarrow \textit{Mod}(\mathcal{X}_\etale, \mathcal{O}_{\mathcal{X}})\] which is left adjoint to \(g^*\). Moreover it agrees with the functor \(g_!\) on abelian sheaves and \(g^*g_! = \text{id}\).

2. There exists a functor \[g_! : \textit{Mod}(\mathcal{X}_{flat,fppf}, \mathcal{O}_{\mathcal{X}_{flat,fppf}}) \longrightarrow \textit{Mod}(\mathcal{X}_{fppf}, \mathcal{O}_{\mathcal{X}})\] which is left adjoint to \(g^*\). Moreover it agrees with the functor \(g_!\) on abelian sheaves and \(g^*g_! = \text{id}\).

Proof

In both cases, the existence of the functor \(g_!\) follows from Modules on Sites, Lemma 0797. To see that \(g_!\) agrees with the functor on abelian sheaves we will show the maps Modules on Sites, Equation (0799) are isomorphisms.

Lisse-étale case. Let \(x \in \Ob(\mathcal{X}_{lisse,\etale})\) lying over a scheme \(U\) with \(x : U \to \mathcal{X}\) smooth. Consider the induced fully faithful functor \[g' : \mathcal{X}_{lisse,\etale}/x \longrightarrow \mathcal{X}_\etale/x\] The right hand side is identified with \((\Sch/U)_\etale\) and the left hand side with the full subcategory of schemes \(U'/U\) such that the composition \(U' \to U \to \mathcal{X}\) is smooth. Thus Étale Cohomology, Lemma 07AJ applies.

Flat-fppf case. Let \(x \in \Ob(\mathcal{X}_{flat,fppf})\) lying over a scheme \(U\) with \(x : U \to \mathcal{X}\) flat. Consider the induced fully faithful functor \[g' : \mathcal{X}_{flat,fppf}/x \longrightarrow \mathcal{X}_{fppf}/x\] The right hand side is identified with \((\Sch/U)_{fppf}\) and the left hand side with the full subcategory of schemes \(U'/U\) such that the composition \(U' \to U \to \mathcal{X}\) is flat. Thus Étale Cohomology, Lemma 07AJ applies.

In both cases the equality \(g^*g_! = \text{id}\) follows from \(g^* = g^{-1}\) and the equality for abelian sheaves in Lemma 0788.

Lemma

Let \(\mathcal{X}\) be an algebraic stack. Notation as in Lemmas 0788 and 0789.

1. We have \(g_!\mathcal{O}_{\mathcal{X}_{lisse,\etale}} = \mathcal{O}_\mathcal{X}\).

2. We have \(g_!\mathcal{O}_{\mathcal{X}_{flat, fppf}} = \mathcal{O}_\mathcal{X}\).

Proof

In this proof we write \(\mathcal{C} = \mathcal{X}_\etale\) (resp. \(\mathcal{C} = \mathcal{X}_{fppf}\)) and we denote \(\mathcal{C}' = \mathcal{X}_{lisse,\etale}\) (resp. \(\mathcal{C}' = \mathcal{X}_{flat, fppf}\)). Then \(\mathcal{C}'\) is a full subcategory of \(\mathcal{C}\). In this proof we will think of objects \(V\) of \(\mathcal{C}\) as schemes over \(\mathcal{X}\) and objects \(U\) of \(\mathcal{C}'\) as schemes smooth (resp. flat) over \(\mathcal{X}\). Finally, we write \(\mathcal{O} = \mathcal{O}_\mathcal{X}\) and \(\mathcal{O}' = \mathcal{O}_{\mathcal{X}_{lisse,\etale}}\) (resp. \(\mathcal{O}' = \mathcal{O}_{\mathcal{X}_{flat,fppf}}\)). In the notation above we have \(\mathcal{O}(V) = \Gamma(V, \mathcal{O}_V)\) and \(\mathcal{O}'(U) = \Gamma(U, \mathcal{O}_U)\). Consider the \(\mathcal{O}\)-module homomorphism \(g_!\mathcal{O}' \to \mathcal{O}\) adjoint to the identification \(\mathcal{O}' = g^{-1}\mathcal{O}\).

Recall that \(g_!\mathcal{O}'\) is the sheaf associated to the presheaf \(g_{p!}\mathcal{O}'\) given by the rule \[V \longmapsto \colim_{V \to U} \mathcal{O}'(U)\] where the colimit is taken in the category of abelian groups (Modules on Sites, Definition 04BF). Below we will use frequently that if \[V \to U \to U'\] are morphisms and if \(f' \in \mathcal{O}'(U')\) restricts to \(f \in \mathcal{O}'(U)\), then \((V \to U, f)\) and \((V \to U', f')\) define the same element of the colimit. Also, \(g_!\mathcal{O}' \to \mathcal{O}\) maps the element \((V \to U, f)\) simply to the pullback of \(f\) to \(V\).

Let us prove that \(g_!\mathcal{O}' \to \mathcal{O}\) is surjective. Let \(h \in \mathcal{O}(V)\) for some object \(V\) of \(\mathcal{C}\). It suffices to show that \(h\) is locally in the image. Choose an object \(U\) of \(\mathcal{C}'\) corresponding to a surjective smooth morphism \(U \to \mathcal{X}\). Since \(U \times_\mathcal{X} V \to V\) is surjective smooth, after replacing \(V\) by the members of an étale covering of \(V\) we may assume there exists a morphism \(V \to U\), see Topologies on Spaces, Lemma 0CFV. Using \(h\) we obtain a morphism \(V \to U \times \mathbf{A}^1\) such that writing \(\mathbf{A}^1 = \Spec(\mathbf{Z}[t])\) the element \(t \in \mathcal{O}(U \times \mathbf{A}^1)\) pulls back to \(h\). Since \(U \times \mathbf{A}^1\) is an object of \(\mathcal{C}'\) we see that \((V \to U \times \mathbf{A}^1, t)\) is an element of the colimit above which maps to \(h \in \mathcal{O}(V)\) as desired.

Suppose that \(s \in g_!\mathcal{O}'(V)\) is a section mapping to zero in \(\mathcal{O}(V)\). To finish the proof we have to show that \(s\) is zero. After replacing \(V\) by the members of a covering we may assume \(s\) is an element of the colimit \[\colim_{V \to U} \mathcal{O}'(U)\] Say \(s = \sum (\varphi_i, s_i)\) is a finite sum with \(\varphi_i : V \to U_i\), \(U_i\) smooth (resp. flat) over \(\mathcal{X}\), and \(s_i \in \Gamma(U_i, \mathcal{O}_{U_i})\). Choose a scheme \(W\) surjective étale over the algebraic space \(U = U_1 \times_\mathcal{X} \ldots \times_\mathcal{X} U_n\). Note that \(W\) is still smooth (resp. flat) over \(\mathcal{X}\), i.e., defines an object of \(\mathcal{C}'\). The fibre product \[V' = V \times_{(\varphi_1, \ldots, \varphi_n), U} W\] is surjective étale over \(V\), hence it suffices to show that \(s\) maps to zero in \(g_!\mathcal{O}'(V')\). Note that the restriction \(\sum (\varphi_i, s_i)|_{V'}\) corresponds to the sum of the pullbacks of the functions \(s_i\) to \(W\). In other words, we have reduced to the case of \((\varphi, s)\) where \(\varphi : V \to U\) is a morphism with \(U\) in \(\mathcal{C}'\) and \(s \in \mathcal{O}'(U)\) restricts to zero in \(\mathcal{O}(V)\). By the commutative diagram \[\xymatrix{ V \ar[rr]_-{(\varphi, 0)} \ar[rrd]_\varphi & & U \times \mathbf{A}^1 \\ & & U \ar[u]_{(\text{id}, 0)} }\] we see that \(((\varphi, 0) : V \to U \times \mathbf{A}^1, \text{pr}_2^*x)\) represents zero in the colimit above. Hence we may replace \(U\) by \(U \times \mathbf{A}^1\), \(\varphi\) by \((\varphi, 0)\) and \(s\) by \(\text{pr}_1^*s + \text{pr}_2^*x\). Thus we may assume that the vanishing locus \(Z : s = 0\) in \(U\) of \(s\) is smooth (resp. flat) over \(\mathcal{X}\). Then we see that \((V \to Z, 0)\) and \((\varphi, s)\) have the same value in the colimit, i.e., we see that the element \(s\) is zero as desired.

The lisse-étale and the flat-fppf sites can be used to characterize parasitic modules as follows.

Lemma

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

1. Let \(\mathcal{F}\) be an \(\mathcal{O}_\mathcal{X}\)-module with the flat base change property on \(\mathcal{X}_\etale\). The following are equivalent

   1. \(\mathcal{F}\) is parasitic, and

   2. \(g^*\mathcal{F} = 0\) where \(g : \Sh(\mathcal{X}_{lisse,\etale}) \to \Sh(\mathcal{X}_\etale)\) is as in Lemma 0788.

2. Let \(\mathcal{F}\) be an \(\mathcal{O}_\mathcal{X}\)-module on \(\mathcal{X}_{fppf}\). The following are equivalent

   1. \(\mathcal{F}\) is parasitic, and

   2. \(g^*\mathcal{F} = 0\) where \(g : \Sh(\mathcal{X}_{flat,fppf}) \to \Sh(\mathcal{X}_{fppf})\) is as in Lemma 0788.

Proof

Part (2) is immediate from the definitions (this is one of the advantages of the flat-fppf site over the lisse-étale site). The implication (1)(a) \(\Rightarrow\) (1)(b) is immediate as well. To see (1)(b) \(\Rightarrow\) (1)(a) let \(U\) be a scheme and let \(x : U \to \mathcal{X}\) be a surjective smooth morphism. Then \(x\) is an object of the lisse-étale site of \(\mathcal{X}\). Hence we see that (1)(b) implies that \(\mathcal{F}|_{U_\etale} = 0\). Let \(V \to \mathcal{X}\) be an flat morphism where \(V\) is a scheme. Set \(W = U \times_\mathcal{X} V\) and consider the diagram \[\xymatrix{ W \ar[d]_p \ar[r]_q & V \ar[d] \\ U \ar[r] & \mathcal{X} }\] Note that the projection \(p : W \to U\) is flat and the projection \(q : W \to V\) is smooth and surjective. This implies that \(q_{small}^*\) is a faithful functor on quasi-coherent modules. By assumption \(\mathcal{F}\) has the flat base change property so that we obtain \(p_{small}^*\mathcal{F}|_{U_\etale} \cong q_{small}^*\mathcal{F}|_{V_\etale}\). Thus if \(\mathcal{F}\) is in the kernel of \(g^*\), then \(\mathcal{F}|_{V_\etale} = 0\) as desired.

Functoriality of the lisse-étale and flat-fppf sites

The lisse-étale site is functorial for smooth morphisms of algebraic stacks and the flat-fppf site is functorial for flat morphisms of algebraic stacks. We warn the reader that the lisse-étale and flat-fppf topoi are not functorial with respect to all morphisms of algebraic stacks, see Examples, Section 07BF.

Lemma

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

1. If \(f\) is smooth, then \(f\) restricts to a continuous and cocontinuous functor \(\mathcal{X}_{lisse,\etale} \to \mathcal{Y}_{lisse,\etale}\) which gives a morphism of ringed topoi fitting into the following commutative diagram \[\xymatrix{ \Sh(\mathcal{X}_{lisse,\etale}) \ar[r]_{g'} \ar[d]_{f'} & \Sh(\mathcal{X}_\etale) \ar[d]^f \\ \Sh(\mathcal{Y}_{lisse,\etale}) \ar[r]^g & \Sh(\mathcal{Y}_\etale) }\] We have \(f'_*(g')^{-1} = g^{-1}f_*\) and \(g'_!(f')^{-1} = f^{-1}g_!\).

2. If \(f\) is flat, then \(f\) restricts to a continuous and cocontinuous functor \(\mathcal{X}_{flat,fppf} \to \mathcal{Y}_{flat,fppf}\) which gives a morphism of ringed topoi fitting into the following commutative diagram \[\xymatrix{ \Sh(\mathcal{X}_{flat,fppf}) \ar[r]_{g'} \ar[d]_{f'} & \Sh(\mathcal{X}_{fppf}) \ar[d]^f \\ \Sh(\mathcal{Y}_{flat,fppf}) \ar[r]^g & \Sh(\mathcal{Y}_{fppf}) }\] We have \(f'_*(g')^{-1} = g^{-1}f_*\) and \(g'_!(f')^{-1} = f^{-1}g_!\).

Proof

The initial statement comes from the fact that if \(x \in \Ob(\mathcal{X})\) lies over a scheme \(U\) such that \(x : U \to \mathcal{X}\) is smooth (resp. flat) and if \(f\) is smooth (resp. flat) then \(f(x) : U \to \mathcal{Y}\) is smooth (resp. flat), see Morphisms of Stacks, Lemmas 075V and 06PX. The induced functor \(\mathcal{X}_{lisse,\etale} \to \mathcal{Y}_{lisse,\etale}\) (resp. \(\mathcal{X}_{flat,fppf} \to \mathcal{Y}_{flat,fppf}\)) is continuous and cocontinuous by our definition of coverings in these categories. Finally, the commutativity of the diagram is a consequence of the fact that the horizontal morphisms are given by the inclusion functors (see Lemma 0788) and Sites, Lemma 03L5.

To show that \(f'_*(g')^{-1} = g^{-1}f_*\) let \(\mathcal{F}\) be a sheaf on \(\mathcal{X}_\etale\) (resp. \(\mathcal{X}_{fppf}\)). There is a canonical pullback map \[g^{-1}f_*\mathcal{F} \longrightarrow f'_*(g')^{-1}\mathcal{F}\] see Sites, Section 06UM. We claim this map is an isomorphism. To prove this pick an object \(y\) of \(\mathcal{Y}_{lisse,\etale}\) (resp. \(\mathcal{Y}_{flat,fppf}\)). Say \(y\) lies over the scheme \(V\) such that \(y : V \to \mathcal{Y}\) is smooth (resp. flat). Since \(g^{-1}\) is the restriction we find that \[\left(g^{-1}f_*\mathcal{F}\right)(y) = \Gamma(V \times_{y, \mathcal{Y}} \mathcal{X},\ \text{pr}^{-1}\mathcal{F})\] by Sheaves on Stacks, Equation (06W6). Let \((V \times_{y, \mathcal{Y}} \mathcal{X})' \subset V \times_{y, \mathcal{Y}} \mathcal{X}\) be the full subcategory consisting of objects \(z : W \to V \times_{y, \mathcal{Y}} \mathcal{X}\) such that the induced morphism \(W \to \mathcal{X}\) is smooth (resp. flat). Denote \[\text{pr}' : (V \times_{y, \mathcal{Y}} \mathcal{X})' \longrightarrow \mathcal{X}_{lisse,\etale} \ (\text{resp. }\mathcal{X}_{flat,fppf})\] the restriction of the functor \(\text{pr}\) used in the formula above. Exactly the same argument that proves Sheaves on Stacks, Equation (06W6) shows that for any sheaf \(\mathcal{H}\) on \(\mathcal{X}_{lisse,\etale}\) (resp. \(\mathcal{X}_{flat,fppf}\)) we have [07AU]\[\begin{equation} f'_*\mathcal{H}(y) = \Gamma((V \times_{y, \mathcal{Y}} \mathcal{X})', \ (\text{pr}')^{-1}\mathcal{H}) \end{equation}\] Since \((g')^{-1}\) is restriction we see that \[\left(f'_*(g')^{-1}\mathcal{F}\right)(y) = \Gamma((V \times_{y, \mathcal{Y}} \mathcal{X})', \ \text{pr}^{-1}\mathcal{F}|_{(V \times_{y, \mathcal{Y}} \mathcal{X})'})\] By Sheaves on Stacks, Lemma 07AK we see that \[\Gamma((V \times_{y, \mathcal{Y}} \mathcal{X})', \ \text{pr}^{-1}\mathcal{F}|_{(V \times_{y, \mathcal{Y}} \mathcal{X})'}) = \Gamma(V \times_{y, \mathcal{Y}} \mathcal{X},\ \text{pr}^{-1}\mathcal{F})\] are equal as desired; although we omit the verification of the assumptions of the lemma we note that the fact that \(V \to \mathcal{Y}\) is smooth (resp. flat) is used to verify the second condition.

Finally, the equality \(g'_!(f')^{-1} = f^{-1}g_!\) follows formally from the equality \(f'_*(g')^{-1} = g^{-1}f_*\) by the adjointness of \(f^{-1}\) and \(f_*\), the adjointness of \(g_!\) and \(g^{-1}\), and their “primed” versions.

Lemma

With assumptions and notation as in Lemma 07AT. Let \(\mathcal{H}\) be an abelian sheaf on \(\mathcal{X}_{lisse,\etale}\) (resp. \(\mathcal{X}_{flat,fppf}\)). Then [07AW]\[\begin{equation} R^pf'_*\mathcal{H} = \text{sheaf associated to }y \longmapsto H^p((V \times_{y, \mathcal{Y}} \mathcal{X})', (\text{pr}')^{-1}\mathcal{H}) \end{equation}\] Here \(y\) is an object of \(\mathcal{Y}_{lisse,\etale}\) (resp. \(\mathcal{Y}_{flat,fppf}\)) lying over the scheme \(V\) and the notation \((V \times_{y, \mathcal{Y}} \mathcal{X})'\) and \(\text{pr}'\) are explained in the proof.

Proof

As in the proof of Lemma 07AT let \((V \times_{y, \mathcal{Y}} \mathcal{X})' \subset V \times_{y, \mathcal{Y}} \mathcal{X}\) be the full subcategory consisting of objects \((x, \varphi)\) where \(x\) is an object of \(\mathcal{X}_{lisse,\etale}\) (resp. \(\mathcal{X}_{flat,fppf}\)) and \(\varphi : f(x) \to y\) is a morphism in \(\mathcal{Y}\). By Equation (07AU) we have \[f'_*\mathcal{H}(y) = \Gamma((V \times_{y, \mathcal{Y}} \mathcal{X})', \ (\text{pr}')^{-1}\mathcal{H})\] where \(\text{pr}'\) is the projection. For an object \((x, \varphi)\) of \((V \times_{y, \mathcal{Y}} \mathcal{X})'\) we can think of \(\varphi\) as a section of \((f')^{-1}h_y\) over \(x\). Thus \((V \times_\mathcal{Y} \mathcal{X})'\) is the localization of the site \(\mathcal{X}_{lisse,\etale}\) (resp. \(\mathcal{X}_{flat,fppf}\)) at the sheaf of sets \((f')^{-1}h_y\), see Sites, Lemma 0791. The morphism \[\text{pr}' : (V \times_{y, \mathcal{Y}} \mathcal{X})' \to \mathcal{X}_{lisse,\etale} \ (\text{resp. } \text{pr}' : (V \times_{y, \mathcal{Y}} \mathcal{X})' \to \mathcal{X}_{flat,fppf})\] is the localization morphism. In particular, the pullback \((\text{pr}')^{-1}\) preserves injective abelian sheaves, see Cohomology on Sites, Lemma 07A0.

Choose an injective resolution \(\mathcal{H} \to \mathcal{I}^\bullet\) on \(\mathcal{X}_{lisse,\etale}\) (resp. \(\mathcal{X}_{flat,fppf}\)). By the formula for pushforward we see that \(R^if'_*\mathcal{H}\) is the sheaf associated to the presheaf which associates to \(y\) the cohomology of the complex \[\begin{matrix} \Gamma\Big((V \times_{y, \mathcal{Y}} \mathcal{X})', (\text{pr}')^{-1}\mathcal{I}^{i - 1}\Big) \\ \downarrow \\ \Gamma\Big((V \times_{y, \mathcal{Y}} \mathcal{X})', (\text{pr}')^{-1}\mathcal{I}^i\Big) \\ \downarrow \\ \Gamma\Big((V \times_{y, \mathcal{Y}} \mathcal{X})', (\text{pr}')^{-1}\mathcal{I}^{i + 1}\Big) \end{matrix}\] Since \((\text{pr}')^{-1}\) is exact and preserves injectives the complex \((\text{pr}')^{-1}\mathcal{I}^\bullet\) is an injective resolution of \((\text{pr}')^{-1}\mathcal{H}\) and the proof is complete.

Lemma

With assumptions and notation as in Lemma 07AT the canonical (base change) map \[g^{-1}Rf_*\mathcal{F} \longrightarrow Rf'_*(g')^{-1}\mathcal{F}\] is an isomorphism for any abelian sheaf \(\mathcal{F}\) on \(\mathcal{X}_\etale\) (resp. \(\mathcal{X}_{fppf}\)).

Proof

Comparing the formula for \(g^{-1}R^pf_*\mathcal{F}\) and \(R^pf'_*(g')^{-1}\mathcal{F}\) given in Sheaves on Stacks, Lemma 075G and Lemma 0GR2 we see that it suffices to show \[H^p((V \times_{y, \mathcal{Y}} \mathcal{X})', \ \text{pr}^{-1}\mathcal{F}|_{(V \times_{y, \mathcal{Y}} \mathcal{X})'}) = H^p_\tau(V \times_{y, \mathcal{Y}} \mathcal{X},\ \text{pr}^{-1}\mathcal{F})\] where \(\tau = \etale\) (resp. \(\tau = fppf\)). Here \(y\) is an object of \(\mathcal{Y}\) lying over a scheme \(V\) such that the morphism \(y : V \to \mathcal{Y}\) is smooth (resp. flat). This equality follows from Sheaves on Stacks, Lemma 07AK. Although we omit the verification of the assumptions of the lemma, we note that the fact that \(V \to \mathcal{Y}\) is smooth (resp. flat) is used to verify the second condition.

Quasi-coherent modules and the lisse-étale and flat-fppf sites

In this section we explain how to think of quasi-coherent modules on an algebraic stack in terms of its lisse-étale or flat-fppf site.

Lemma

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

1. Let \(f_j : \mathcal{X}_j \to \mathcal{X}\) be a family of smooth morphisms of algebraic stacks with \(|\mathcal{X}| =\bigcup |f_j|(|\mathcal{X}_j|)\). Let \(\mathcal{F}\) be a sheaf of \(\mathcal{O}_\mathcal{X}\)-modules on \(\mathcal{X}_\etale\). If each \(f_j^{-1}\mathcal{F}\) is quasi-coherent, then so is \(\mathcal{F}\).

2. Let \(f_j : \mathcal{X}_j \to \mathcal{X}\) be a family of flat and locally finitely presented morphisms of algebraic stacks with \(|\mathcal{X}| =\bigcup |f_j|(|\mathcal{X}_j|)\). Let \(\mathcal{F}\) be a sheaf of \(\mathcal{O}_\mathcal{X}\)-modules on \(\mathcal{X}_{fppf}\). If each \(f_j^{-1}\mathcal{F}\) is quasi-coherent, then so is \(\mathcal{F}\).

Proof

Proof of (1). We may replace each of the algebraic stacks \(\mathcal{X}_j\) by a scheme \(U_j\) (using that any algebraic stack has a smooth covering by a scheme and that compositions of smooth morphisms are smooth, see Morphisms of Stacks, Lemma 075V). The pullback of \(\mathcal{F}\) to \((\Sch/U_j)_\etale\) is still quasi-coherent, see Modules on Sites, Lemma 03DO. Then \(f = \coprod f_j : U = \coprod U_j \to \mathcal{X}\) is a smooth surjective morphism. Let \(x : V \to \mathcal{X}\) be an object of \(\mathcal{X}\). By Sheaves on Stacks, Lemma 06XH there exists an étale covering \(\{x_i \to x\}_{i \in I}\) such that each \(x_i\) lifts to an object \(u_i\) of \((\Sch/U)_\etale\). This just means that \(x_i\) lives over a scheme \(V_i\), that \(\{V_i \to V\}\) is an étale covering, and that \(x_i\) comes from a morphism \(u_i : V_i \to U\). Then \(x_i^*\mathcal{F} = u_i^*f^*\mathcal{F}\) is quasi-coherent. This implies that \(x^*\mathcal{F}\) on \((\Sch/V)_\etale\) is quasi-coherent, for example by Modules on Sites, Lemma 03DN. By Sheaves on Stacks, Lemma 0EM8 we see that \(x^*\mathcal{F}\) is an fppf sheaf and since \(x\) was arbitrary we see that \(\mathcal{F}\) is a sheaf in the fppf topology. Applying Sheaves on Stacks, Lemma 06WI we see that \(\mathcal{F}\) is quasi-coherent.

Proof of (2). This is proved using exactly the same argument, which we fully write out here. We may replace each of the algebraic stacks \(\mathcal{X}_j\) by a scheme \(U_j\) (using that any algebraic stack has a smooth covering by a scheme and that flat and locally finite presented morphisms are preserved by composition, see Morphisms of Stacks, Lemmas 06PX and 06Q3). The pullback of \(\mathcal{F}\) to \((\Sch/U_j)_\etale\) is still locally quasi-coherent, see Sheaves on Stacks, Lemma 06WH. Then \(f = \coprod f_j : U = \coprod U_j \to \mathcal{X}\) is a surjective, flat, and locally finitely presented morphism. Let \(x : V \to \mathcal{X}\) be an object of \(\mathcal{X}\). By Sheaves on Stacks, Lemma 06XH there exists an fppf covering \(\{x_i \to x\}_{i \in I}\) such that each \(x_i\) lifts to an object \(u_i\) of \((\Sch/U)_\etale\). This just means that \(x_i\) lives over a scheme \(V_i\), that \(\{V_i \to V\}\) is an fppf covering, and that \(x_i\) comes from a morphism \(u_i : V_i \to U\). Then \(x_i^*\mathcal{F} = u_i^*f^*\mathcal{F}\) is quasi-coherent. This implies that \(x^*\mathcal{F}\) on \((\Sch/V)_\etale\) is quasi-coherent, for example by Modules on Sites, Lemma 03DN. By Sheaves on Stacks, Lemma 06WI we see that \(\mathcal{F}\) is quasi-coherent.

We recall that we have defined the notion of a quasi-coherent module on any ringed topos in Modules on Sites, Section 03DK.

Lemma

Let \(\mathcal{X}\) be an algebraic stack. Notation as in Lemma 0788.

1. Let \(\mathcal{H}\) be a quasi-coherent \(\mathcal{O}_{\mathcal{X}_{lisse,\etale}}\)-module on the lisse-étale site of \(\mathcal{X}\). Then \(g_!\mathcal{H}\) is a quasi-coherent module on \(\mathcal{X}\).

2. Let \(\mathcal{H}\) be a quasi-coherent \(\mathcal{O}_{\mathcal{X}_{flat,fppf}}\)-module on the flat-fppf site of \(\mathcal{X}\). Then \(g_!\mathcal{H}\) is a quasi-coherent module on \(\mathcal{X}\).

Proof

Pick a scheme \(U\) and a surjective smooth morphism \(x : U \to \mathcal{X}\). By Modules on Sites, Definition 03DL there exists an étale (resp. fppf) covering \(\{U_i \to U\}_{i \in I}\) such that each pullback \(f_i^{-1}\mathcal{H}\) has a global presentation (see Modules on Sites, Definition 03DE). Here \(f_i : U_i \to \mathcal{X}\) is the composition \(U_i \to U \to \mathcal{X}\) which is a morphism of algebraic stacks. (Recall that the pullback “is” the restriction to \(\mathcal{X}/f_i\), see Sheaves on Stacks, Definition 06W1 and the discussion following.) Since each \(f_i\) is smooth (resp. flat) by Lemma 07AT we see that \(f_i^{-1}g_!\mathcal{H} = g_{i, !}(f'_i)^{-1}\mathcal{H}\). Using Lemma 07AZ we reduce the statement of the lemma to the case where \(\mathcal{H}\) has a global presentation. Say we have \[\bigoplus\nolimits_{j \in J} \mathcal{O} \longrightarrow \bigoplus\nolimits_{i \in I} \mathcal{O} \longrightarrow \mathcal{H} \longrightarrow 0\] of \(\mathcal{O}\)-modules where \(\mathcal{O} = \mathcal{O}_{\mathcal{X}_{lisse,\etale}}\) (resp. \(\mathcal{O} = \mathcal{O}_{\mathcal{X}_{flat,fppf}}\)). Since \(g_!\) commutes with arbitrary colimits (as a left adjoint functor, see Lemma 0789 and Categories, Lemma 0038) we conclude that there exists an exact sequence \[\bigoplus\nolimits_{j \in J} g_!\mathcal{O} \longrightarrow \bigoplus\nolimits_{i \in I} g_!\mathcal{O} \longrightarrow g_!\mathcal{H} \longrightarrow 0\] Lemma 078A shows that \(g_!\mathcal{O} = \mathcal{O}_\mathcal{X}\). In case (2) we are done. In case (1) we apply Sheaves on Stacks, Lemma 0EM8 to conclude.

Lemma

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

1. With \(g\) as in Lemma 0788 for the lisse-étale site we have

   1. the functors \(g^{-1}\) and \(g_!\) define mutually inverse functors \[\xymatrix{ \QCoh(\mathcal{O}_\mathcal{X}) \ar@<1ex>[r]^-{g^{-1}} & \QCoh(\mathcal{X}_{lisse,\etale}, \mathcal{O}_{\mathcal{X}_{lisse,\etale}}) \ar@<1ex>[l]^-{g_!} }\]

   2. if \(\mathcal{F}\) is in \(\textit{LQCoh}^{fbc}(\mathcal{O}_\mathcal{X})\) then \(g^{-1}\mathcal{F}\) is in \(\QCoh(\mathcal{O}_{\mathcal{X}_{lisse,\etale}})\) and

   3. \(Q(\mathcal{F}) = g_!g^{-1}\mathcal{F}\) where \(Q\) is as in Lemma 0778.

2. With \(g\) as in Lemma 0788 for the flat-fppf site we have

   1. the functors \(g^{-1}\) and \(g_!\) define mutually inverse functors \[\xymatrix{ \QCoh(\mathcal{O}_\mathcal{X}) \ar@<1ex>[r]^-{g^{-1}} & \QCoh(\mathcal{X}_{flat,fppf}, \mathcal{O}_{\mathcal{X}_{flat,fppf}}) \ar@<1ex>[l]^-{g_!} }\]

   2. if \(\mathcal{F}\) is in \(\textit{LQCoh}^{fbc}(\mathcal{O}_\mathcal{X})\) then \(g^{-1}\mathcal{F}\) is in \(\QCoh(\mathcal{O}_{\mathcal{X}_{flat,fppf}})\) and

   3. \(Q(\mathcal{F}) = g_!g^{-1}\mathcal{F}\) where \(Q\) is as in Lemma 0778.

Proof

Pullback by any morphism of ringed topoi preserves categories of quasi-coherent modules, see Modules on Sites, Lemma 03DO. Hence \(g^{-1}\) preserves the categories of quasi-coherent modules; here we use that \(\QCoh(\mathcal{O}_\mathcal{X}) = \QCoh(\mathcal{X}_\etale, \mathcal{O}_\mathcal{X})\) by Sheaves on Stacks, Lemma 0EM8. The same is true for \(g_!\) by Lemma 07B0. We know that \(\mathcal{H} \to g^{-1}g_!\mathcal{H}\) is an isomorphism by Lemma 0788. Conversely, if \(\mathcal{F}\) is in \(\QCoh(\mathcal{O}_\mathcal{X})\) then the map \(g_!g^{-1}\mathcal{F} \to \mathcal{F}\) is a map of quasi-coherent modules on \(\mathcal{X}\) whose restriction to any scheme smooth over \(\mathcal{X}\) is an isomorphism. Then the discussion in Sheaves on Stacks, Sections 06WS and 06WU (comparing with quasi-coherent modules on presentations) shows it is an isomorphism. This proves (1)(a) and (2)(a).

Let \(\mathcal{F}\) be an object of \(\textit{LQCoh}^{fbc}(\mathcal{O}_\mathcal{X})\). By Lemma 0779 the kernel and cokernel of the map \(Q(\mathcal{F}) \to \mathcal{F}\) are parasitic. Hence by Lemma 07AR and since \(g^* = g^{-1}\) is exact, we conclude \(g^*Q(\mathcal{F}) \to g^*\mathcal{F}\) is an isomorphism. Thus \(g^*\mathcal{F}\) is quasi-coherent. This proves (1)(b) and (2)(b). Finally, (1)(c) and (2)(c) follow because \(g_!g^*Q(\mathcal{F}) \to Q(\mathcal{F})\) is an isomorphism by our arguments above.

Lemma

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

1. \(\QCoh(\mathcal{O}_{\mathcal{X}_{lisse,\etale}})\) is a weak Serre subcategory of \(\textit{Mod}(\mathcal{O}_{\mathcal{X}_{lisse,\etale}})\).

2. \(\QCoh(\mathcal{O}_{\mathcal{X}_{flat,fppf}})\) is a weak Serre subcategory of \(\textit{Mod}(\mathcal{O}_{\mathcal{X}_{flat,fppf}})\).

Proof

We will verify conditions (1), (2), (3), (4) of Homology, Lemma 0754.

Since \(0\) is a quasi-coherent module on any ringed site we see that (1) holds.

By definition \(\QCoh(\mathcal{O})\) is a strictly full subcategory \(\textit{Mod}(\mathcal{O})\), so (2) holds.

Let \(\varphi : \mathcal{G} \to \mathcal{F}\) be a morphism of quasi-coherent modules on \(\mathcal{X}_{lisse,\etale}\) or \(\mathcal{X}_{flat,fppf}\). We have \(g^*g_!\mathcal{F} = \mathcal{F}\) and similarly for \(\mathcal{G}\) and \(\varphi\), see Lemma 0789. By Lemma 07B0 we see that \(g_!\mathcal{F}\) and \(g_!\mathcal{G}\) are quasi-coherent \(\mathcal{O}_\mathcal{X}\)-modules. By Sheaves on Stacks, Lemma 06WV we have that \(\Coker(g_!\varphi)\) is a quasi-coherent module on \(\mathcal{X}\) (and the cokernel in the category of quasi-coherent modules on \(\mathcal{X}\)). Since \(g^*\) is exact (see Lemma 0788) \(g^*\Coker(g_!\varphi) = \Coker(g^*g_!\varphi) = \Coker(\varphi)\) is quasi-coherent too (see Lemma 07B1). By Proposition 0771 the kernel \(\Ker(g_!\varphi)\) is in \(\textit{LQCoh}^{fbc}(\mathcal{O}_\mathcal{X})\). Since \(g^*\) is exact, we have \(g^*\Ker(g_!\varphi) = \Ker(g^*g_!\varphi) = \Ker(\varphi)\). Since \(g^*\) maps objects of \(\textit{LQCoh}^{fbc}(\mathcal{O}_\mathcal{X})\) to quasi-coherent modules by Lemma 07B1 we conclude that \(\Ker(\varphi)\) is quasi-coherent as well. This proves (3).

Finally, suppose that \[0 \to \mathcal{F} \to \mathcal{E} \to \mathcal{G} \to 0\] is an extension of \(\mathcal{O}_{\mathcal{X}_{lisse,\etale}}\)-modules (resp. \(\mathcal{O}_{\mathcal{X}_{flat,fppf}}\)-modules) with \(\mathcal{F}\) and \(\mathcal{G}\) quasi-coherent. To prove (4) and finish the proof we have to show that \(\mathcal{E}\) is quasi-coherent on \(\mathcal{X}_{lisse,\etale}\) (resp. \(\mathcal{X}_{flat,fppf}\)). Let \(U\) be an object of \(\mathcal{X}_{lisse,\etale}\) (resp. \(\mathcal{X}_{flat,fppf}\); we think of \(U\) as a scheme smooth (resp. flat) over \(\mathcal{X}\). We have to show that the restriction of \(\mathcal{E}\) to \(U_{lisse,\etale}\) (resp. \(=U_{flat,fppf}\)) is quasi-coherent. Thus we may assume that \(\mathcal{X} = U\) is a scheme. Because \(\mathcal{G}\) is quasi-coherent on \(U_{lisse,\etale}\) (resp. \(U_{flat,fppf}\)), we may assume, after replacing \(U\) by the members of an étale (resp. fppf) covering, that \(\mathcal{G}\) has a presentation \[\bigoplus\nolimits_{j \in J} \mathcal{O} \longrightarrow \bigoplus\nolimits_{i \in I} \mathcal{O} \longrightarrow \mathcal{G} \longrightarrow 0\] on \(U_{lisse,\etale}\) (resp. \(U_{flat,fppf}\)) where \(\mathcal{O}\) is the structure sheaf on the site. We may also assume \(U\) is affine. Since \(\mathcal{F}\) is quasi-coherent, we have \[H^1(U_{lisse,\etale}, \mathcal{F}) = 0, \quad\text{resp.}\quad H^1(U_{flat,fppf}, \mathcal{F}) = 0\] Namely, \(\mathcal{F}\) is the pullback of a quasi-coherent module \(\mathcal{F}'\) on the big site of \(U\) (by Lemma 07B1), cohomology of \(\mathcal{F}\) and \(\mathcal{F}'\) agree (by Lemma 0GR0), and we know that the cohomology of \(\mathcal{F}'\) on the big site of the affine scheme \(U\) is zero (to get this in the current situation you have to combine Descent, Propositions 03DX and 03DW with Cohomology of Schemes, Lemma 01XB). Thus we can lift the map \(\bigoplus_{i \in I} \mathcal{O} \to \mathcal{G}\) to \(\mathcal{E}\). A diagram chase shows that we obtain an exact sequence \[\bigoplus\nolimits_{j \in J} \mathcal{O} \to \mathcal{F} \oplus \bigoplus\nolimits_{i \in I} \mathcal{O} \to \mathcal{E} \to 0\] By (3) proved above, we conclude that \(\mathcal{E}\) is quasi-coherent as desired.

Coherent sheaves on locally Noetherian stacks

This section is the analogue of Cohomology of Spaces, Section 07U9. We have defined the notion of a coherent module on any ringed topos in Modules on Sites, Section 03DK. However, for any algebraic stack \(\mathcal{X}\) the category of coherent \(\mathcal{O}_\mathcal{X}\)-modules is zero, essentially because the site \(\mathcal{X}\) contains too many non-Noetherian objects (even if \(\mathcal{X}\) is itself locally Noetherian). Instead, we will define coherent modules using the following lemma.

Lemma

Let \(\mathcal{X}\) be a locally Noetherian algebraic stack. Let \(\mathcal{F}\) be an \(\mathcal{O}_\mathcal{X}\)-module. The following are equivalent

1. \(\mathcal{F}\) is a quasi-coherent, finite type \(\mathcal{O}_\mathcal{X}\)-module,

2. \(\mathcal{F}\) is an \(\mathcal{O}_\mathcal{X}\)-module of finite presentation,

3. \(\mathcal{F}\) is quasi-coherent and for any morphism \(f : U \to \mathcal{X}\) where \(U\) is a locally Noetherian algebraic space, the pullback \(f^*\mathcal{F}|_{U_\etale}\) is coherent, and

4. \(\mathcal{F}\) is quasi-coherent and there exists an algebraic space \(U\) and a morphism \(f : U \to \mathcal{X}\) which is locally of finite type, flat, and surjective, such that the pullback \(f^*\mathcal{F}|_{U_\etale}\) is coherent.

Proof

Let \(f : U \to \mathcal{X}\) be as in (4). Then \(U\) is locally Noetherian (Morphisms of Stacks, Lemma 06R6) and we see that the statement of the lemma makes sense. Additionally, \(f\) is locally of finite presentation by Morphisms of Stacks, Lemma 0DQJ. Let \(x\) be an object of \(\mathcal{X}\) lying over the scheme \(V\). In order to prove (2) we have to show that, after replacing \(V\) by the members of an fppf covering of \(V\), the restriction \(x^*\mathcal{F}\) has a global finite presentation on \(\mathcal{X}/x \cong (\Sch/V)_{fppf}\). The projection \(W = U \times_\mathcal{X} V \to V\) is locally of finite presentation, flat, and surjective. Hence we may replace \(V\) by the members of an étale covering of \(W\) by schemes and assume we have a morphism \(h : V \to U\) with \(f \circ h = x\). Since \(\mathcal{F}\) is quasi-coherent, we see that the restriction \(x^*\mathcal{F}\) is the pullback of \(h_{small}^*(f^*\mathcal{F})|_{U_\etale}\) by \(\pi_V\), see Sheaves on Stacks, Lemma 0GQD. Since \(f^*\mathcal{F}|_{U_\etale}\) locally in the étale topology has a finite presentation by assumption, we conclude (4) \(\Rightarrow\) (2).

Part (2) implies (1) for any ringed topos (immediate from the definition). The properties “finite type” and “quasi-coherent” are preserved under pullback by any morphism of ringed topoi, see Modules on Sites, Lemma 03DO. Hence (1) implies (3), see Cohomology of Spaces, Lemma 07UB. Finally, (3) trivially implies (4).

Definition

Let \(\mathcal{X}\) be a locally Noetherian algebraic stack. An \(\mathcal{O}_\mathcal{X}\)-module \(\mathcal{F}\) is called coherent if \(\mathcal{F}\) satisfies one (and hence all) of the equivalent conditions of Lemma 0GR5. The category of coherent \(\mathcal{O}_\mathcal{X}\)-modules is denote \(\textit{Coh}(\mathcal{O}_\mathcal{X})\).

Lemma

Let \(\mathcal{X}\) be a locally Noetherian algebraic stack. The module \(\mathcal{O}_\mathcal{X}\) is coherent, any invertible \(\mathcal{O}_\mathcal{X}\)-module is coherent, and more generally any finite locally free \(\mathcal{O}_\mathcal{X}\)-module is coherent.

Proof

Follows from the definition and Cohomology of Spaces, Lemma 07UB.

Lemma

Let \(f : \mathcal{X} \to \mathcal{Y}\) be a morphism of locally Noetherian algebraic stacks. Then \(f^*\) sends coherent modules on \(\mathcal{Y}\) to coherent modules on \(\mathcal{X}\).

Proof

Immediate from the definition and the fact that pullback for any morphism of ringed topoi preserves finitely presented modules, see Modules on Sites, Lemma 03DO.

Lemma

Let \(\mathcal{X}\) be a locally Noetherian algebraic stack. The category of coherent \(\mathcal{O}_\mathcal{X}\)-modules is abelian. If \(\varphi : \mathcal{F} \to \mathcal{G}\) is a map of coherent \(\mathcal{O}_\mathcal{X}\)-modules, then

1. the cokernel \(\Coker(\varphi)\) computed in \(\textit{Mod}(\mathcal{O}_\mathcal{X})\) is a coherent \(\mathcal{O}_\mathcal{X}\)-module,

2. the image \(\Im(\varphi)\) computed in \(\textit{Mod}(\mathcal{O}_\mathcal{X})\) is a coherent \(\mathcal{O}_\mathcal{X}\)-module, and

3. the kernel \(\Ker(\varphi)\) computed in \(\textit{Mod}(\mathcal{O}_\mathcal{X})\) may not be coherent, but it is in \(\textit{LQCoh}^{fbc}(\mathcal{O}_\mathcal{X})\) and \(Q(\Ker(\varphi))\) is coherent and is the kernel of \(\varphi\) in \(\textit{Coh}(\mathcal{O}_\mathcal{X})\).

The inclusion functor \(\textit{Coh}(\mathcal{O}_\mathcal{X}) \to \QCoh(\mathcal{O}_\mathcal{X})\) is exact.

Proof

The rules given for taking kernels, images, and cokernels in \(\textit{Coh}(\mathcal{O}_\mathcal{X})\) agree with the prescription for quasi-coherent modules in Remark 0GQL. Hence the lemma will follow if we can show that the quasi-coherent modules \(\Coker(\varphi)\), \(\Im(\varphi)\), and \(Q(\Ker(\varphi))\) are coherent. By Lemma 0GR5 it suffices to prove this after restricting to \(U_\etale\) for some surjective smooth morphism \(f : U \to \mathcal{X}\). The functor \(\mathcal{F} \mapsto f^*\mathcal{F}|_{U_\etale}\) is exact. Hence \(f^*\Coker(\varphi)\) and \(f^*\Im(\varphi)\) are the cokernel and image of a map between coherent \(\mathcal{O}_U\)-modules hence coherent as desired. The functor \(\mathcal{F} \mapsto f^*\mathcal{F}|_{U_\etale}\) kills parasitic modules by Lemma 0774. Hence \(f^*Q(\Ker(\varphi))|_{U_\etale} = f^*\Ker(\varphi)|_{U_\etale}\) by part (2) of Lemma 0779. Thus we conclude that \(Q(\Ker(\varphi))\) is coherent in the same way.

Lemma

Let \(\mathcal{X}\) be a locally Noetherian algebraic stack. 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 \(\mathcal{F}_3\) coherent, then \(\mathcal{F}_2\) is coherent.

Proof

By Sheaves on Stacks, Lemma 06WV part (7) we see that \(\mathcal{F}_2\) is quasi-coherent. Then we can check that \(\mathcal{F}_2\) is coherent by restricting to \(U_\etale\) for some \(U \to \mathcal{X}\) surjective and smooth. This follows from Cohomology of Spaces, Lemma 07UC. Some details omitted.

Coherent modules form a Serre subcategory of the category of quasi-coherent \(\mathcal{O}_\mathcal{X}\)-modules. This does not hold for modules on a general ringed topos.

Lemma

Let \(\mathcal{X}\) be a locally Noetherian algebraic stack. Then \(\textit{Coh}(\mathcal{O}_\mathcal{X})\) is a Serre subcategory of \(\QCoh(\mathcal{O}_\mathcal{X})\). Let \(\varphi : \mathcal{F} \to \mathcal{G}\) be a map of quasi-coherent \(\mathcal{O}_\mathcal{X}\)-modules. We have

1. if \(\mathcal{F}\) is coherent and \(\varphi\) surjective, then \(\mathcal{G}\) is coherent,

2. if \(\mathcal{F}\) is coherent, then \(\Im(\varphi)\) is coherent, and

3. if \(\mathcal{G}\) coherent and \(\Ker(\varphi)\) parasitic, then \(\mathcal{F}\) is coherent.

Proof

Choose a scheme \(U\) and a surjective smooth morphism \(f : U \to \mathcal{X}\). Then the functor \(f^* : \QCoh(\mathcal{O}_\mathcal{X}) \to \QCoh(\mathcal{O}_U)\) is exact (Lemma 076X) and moreover by definition \(\textit{Coh}(\mathcal{O}_\mathcal{X})\) is the full subcategory of \(\QCoh(\mathcal{O}_\mathcal{X})\) consisting of objects \(\mathcal{F}\) such that \(f^*\mathcal{F}\) is in \(\textit{Coh}(\mathcal{O}_U)\). The statement that \(\textit{Coh}(\mathcal{O}_\mathcal{X})\) is a Serre subcategory of \(\QCoh(\mathcal{O}_\mathcal{X})\) follows immediately from this and the corresponding fact for \(U\), see Cohomology of Spaces, Lemmas 07UC and 07UD. We omit the proof of (1), (2), and (3). Hint: compare with the proof of Lemma 0GR9.

Let \(\mathcal{X}\) be a locally Noetherian algebraic stack. Let \(U\) be an algebraic space and let \(f : U \to \mathcal{X}\) be surjective, locally of finite presentation, and flat. Observe that \(U\) is locally Noetherian (Morphisms of Stacks, Lemma 06R6). Let \((U, R, s, t, c)\) be the groupoid in algebraic spaces and \(f_{can} : [U/R] \to \mathcal{X}\) the isomorphism constructed in Algebraic Stacks, Lemma 04T4 and Remark 04WY. As in Sheaves on Stacks, Section 06WU we obtain equivalences \[\QCoh(\mathcal{O}_\mathcal{X}) \cong \QCoh(\mathcal{O}_{[U/R]}) \cong \QCoh(U, R, s, t, c)\] where the second equivalence is Sheaves on Stacks, Proposition 06WT. Recall that in Groupoids in Spaces, Section 0GPN we have defined the full subcategory \[\textit{Coh}(U, R, s, t, c) \subset \QCoh(U, R, s, t, c)\] of coherent modules as those \((\mathcal{G}, \alpha)\) such that \(\mathcal{G}\) is a coherent \(\mathcal{O}_U\)-module.

Lemma

In the situation discussed above, the equivalence \(\QCoh(\mathcal{O}_\mathcal{X}) \cong \QCoh(U, R, s, t, c)\) sends coherent sheaves to coherent sheaves and vice versa, i.e., induces an equivalence \(\textit{Coh}(\mathcal{O}_\mathcal{X}) \cong \textit{Coh}(U, R, s, t, c)\).

Proof

This is immediate from the definition of coherent \(\mathcal{O}_\mathcal{X}\)-modules. For bookkeeping purposes: the material above uses Morphisms of Stacks, Lemma 06R6, Algebraic Stacks, Lemma 04T4 and Remark 04WY, Sheaves on Stacks, Section 06WU, Sheaves on Stacks, Proposition 06WT, and Groupoids in Spaces, Section 0GPN.

Lemma

Let \(\mathcal{X}\) be a locally Noetherian algebraic stack. Let \(\mathcal{F}\) and \(\mathcal{G}\) be coherent be \(\mathcal{O}_\mathcal{X}\)-modules. Then the internal hom \(hom(\mathcal{F}, \mathcal{G})\) constructed in Lemma 0GQN is a coherent \(\mathcal{O}_\mathcal{X}\)-module.

Proof

Let \(U \to \mathcal{X}\) be a smooth surjective morphism from a scheme. By item (0GQT) in Section 0GQS we see that the restriction of \(hom(\mathcal{F}, \mathcal{G})\) to \(U\) is the Hom sheaf of the restrictions. Hence this lemma follows from the case of algebraic spaces, see Cohomology of Spaces, Lemma 07UE.

Coherent sheaves on Noetherian stacks

This section is the analogue of Cohomology of Spaces, Section 07UI.

Lemma

Let \(\mathcal{X}\) be a Noetherian algebraic stack. Every quasi-coherent \(\mathcal{O}_\mathcal{X}\)-module is the filtered colimit of its coherent submodules.

Proof

Let \(\mathcal{F}\) be a quasi-coherent \(\mathcal{O}_\mathcal{X}\)-module. If \(\mathcal{G}, \mathcal{H} \subset \mathcal{F}\) are coherent \(\mathcal{O}_\mathcal{X}\)-submodules then the image of \(\mathcal{G} \oplus \mathcal{H} \to \mathcal{F}\) is another coherent \(\mathcal{O}_\mathcal{X}\)-submodule which contains both of them, see Lemma 0GRB. In this way we see that the system is directed. Hence it now suffices to show that \(\mathcal{F}\) can be written as a filtered colimit of coherent modules, as then we can take the images of these modules in \(\mathcal{F}\) to conclude there are enough of them.

Let \(U\) be an affine scheme and \(U \to \mathcal{X}\) a surjective smooth morphism (Properties of Stacks, Lemma 04YC). Set \(R = U \times_\mathcal{X} U\) so that \(\mathcal{X} = [U/R]\) as in Algebraic Stacks, Lemma 04T5. By Lemma 0GRC we have \(\QCoh(\mathcal{O}_X) = \QCoh(U, R, s, t, c)\) and \(\textit{Coh}(\mathcal{O}_X) = \textit{Coh}(U, R, s, t, c)\). In this way we reduce to the problem of proving the corresponding thing for \(\QCoh(U, R, s, t, c)\). This is Groupoids in Spaces, Lemma 0GPS; we check its assumptions in the next paragraph.

We urge the reader to skip the rest of the proof. The affine scheme \(U\) is Noetherian; this follows from our definition of \(\mathcal{X}\) being locally Noetherian, see Properties of Stacks, Definition 04YG and Remark 04YH. The projection morphisms \(s, t : R \to U\) are smooth (see reference given above) and quasi-separated and quasi-compact (Morphisms of Stacks, Lemma 075S). In particular, \(R\) is a quasi-compact and quasi-separated algebraic space smooth over \(U\) and hence Noetherian (Morphisms of Spaces, Lemma 04ZL).