Proposition 103.8.1 (0771)—The Stacks project

Proposition 103.8.1. Summary of results on locally quasi-coherent modules having the flat base change property.

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 $\textit{Mod}(\mathcal{O}_\mathcal {X})$ .
The category $\textit{LQCoh}^{fbc}(\mathcal{O}_\mathcal {X})$ is a weak Serre subcategory of both $\textit{Mod}(\mathcal{O}_\mathcal {X})$ and $\textit{Mod}(\mathcal{X}_{\acute{e}tale}, \mathcal{O}_\mathcal {X})$ .
Pullback $f^*$ along any morphism of algebraic stacks $f : \mathcal{X} \to \mathcal{Y}$ induces a functor $f^* : \textit{LQCoh}^{fbc}(\mathcal{O}_\mathcal {Y}) \to \textit{LQCoh}^{fbc}(\mathcal{O}_\mathcal {X})$ .
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 $\textit{LQCoh}^{fbc}(\mathcal{O}_\mathcal {Y})$ .
The category $\textit{LQCoh}^{fbc}(\mathcal{O}_\mathcal {X})$ has colimits and they agree with colimits in $\textit{Mod}(\mathcal{X}_{\acute{e}tale}, \mathcal{O}_\mathcal {X})$ as well as in $\textit{Mod}(\mathcal{O}_\mathcal {X})$ .
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 $\textit{LQCoh}^{fbc}(\mathcal{O}_\mathcal {X})$ .
Given $\mathcal{F}$ of finite presentation and $\mathcal{G}$ in $\textit{LQCoh}^{fbc}(\mathcal{O}_\mathcal {X})$ then $\mathop{\mathcal{H}\! \mathit{om}}\nolimits _{\mathcal{O}_\mathcal {X}}(\mathcal{F}, \mathcal{G})$ is in $\textit{LQCoh}^{fbc}(\mathcal{O}_\mathcal {X})$ .

Proof. Part (1) is Sheaves on Stacks, Lemma 96.23.1.

Part (2) for the embedding $\textit{LQCoh}^{fbc}(\mathcal{O}_\mathcal {X}) \subset \textit{Mod}(\mathcal{X}_{\acute{e}tale}, \mathcal{O}_\mathcal {X})$ we have seen in the proof of Lemma 103.7.3. 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 $\mathop{\mathrm{Ker}}(\varphi )$ is the same whether computed in the étale or the fppf topology, we see that $\mathop{\mathrm{Ker}}(\varphi )$ 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 $H^1_{fppf}(x, \mathcal{F}_1) = H^1_{\acute{e}tale}(x, \mathcal{F}_1)$ by Sheaves on Stacks, Lemma 96.23.2 and because of locality of cohomology, see Cohomology on Sites, Lemma 21.7.3. This proves (2).

Part (3) follows from Lemma 103.7.2 and Sheaves on Stacks, Lemma 96.12.3.

Part (4)(b) for $R^ if_*\mathcal{F}$ computed in the étale cohomology follows from Lemma 103.7.3. Whereupon part (4)(a) follows from Sheaves on Stacks, Lemma 96.23.2 combined with (1) above.

Part (5) for the étale topology follows from Sheaves on Stacks, Lemma 96.12.4 and Lemma 103.7.2. 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 103.7.2 and Sheaves on Stacks, Lemma 96.12.4. $\square$

Comments (2)

Comment #3800 by Pieter Belmans on November 21, 2018 at 12:18

In the proof of (what is now) part (4)(b), you want to write "in the étale topology".

Comment #3918 by Johan on January 22, 2019 at 19:47

@#3800: Do you mean that in (4)(b) we are computing the pushforward in the \'etale topology? But by part (4)(a) this is the same as for the fppf topology which is the "default" topology, so I think no confusion can arise from the statement of the lemma as given. OK?

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