Lemma 104.3.3 (07B3)—The Stacks project

Lemma 104.3.3. Let $\mathcal{X}$ be an algebraic stack. Notation as in Cohomology of Stacks, Lemma 103.14.2.

Let $\mathcal{H}$ be a quasi-coherent $\mathcal{O}_{\mathcal{X}_{lisse,{\acute{e}tale}}}$ -module on the lisse-étale site of $\mathcal{X}$ . For all $p \in \mathbf{Z}$ the sheaf $H^ p(Lg_!\mathcal{H})$ is a locally quasi-coherent module with the flat base change property on $\mathcal{X}$ .
Let $\mathcal{H}$ be a quasi-coherent $\mathcal{O}_{\mathcal{X}_{flat,fppf}}$ -module on the flat-fppf site of $\mathcal{X}$ . For all $p \in \mathbf{Z}$ the sheaf $H^ p(Lg_!\mathcal{H})$ is a locally quasi-coherent module with the flat base change property on $\mathcal{X}$ .

Proof. Pick a scheme $U$ and a surjective smooth morphism $x : U \to \mathcal{X}$ . By Modules on Sites, Definition 18.23.1 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 18.17.1). 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 96.9.2 and the discussion following.) After refining the covering we may assume each $U_ i$ is an affine scheme. Since each $f_ i$ is smooth (resp. flat) by Lemma 104.3.2 we see that $f_ i^{-1}Lg_!\mathcal{H} = Lg_{i, !}(f'_ i)^{-1}\mathcal{H}$ . Using Cohomology of Stacks, Lemma 103.8.2 we reduce the statement of the lemma to the case where $\mathcal{H}$ has a global presentation and where $\mathcal{X} = (\mathit{Sch}/X)_{fppf}$ for some affine scheme $X = \mathop{\mathrm{Spec}}(A)$ .

Say our presentation looks like

$\bigoplus \nolimits _{j \in J} \mathcal{O} \longrightarrow \bigoplus \nolimits _{i \in I} \mathcal{O} \longrightarrow \mathcal{H} \longrightarrow 0$

where $\mathcal{O} = \mathcal{O}_{\mathcal{X}_{lisse,{\acute{e}tale}}}$ (resp. $\mathcal{O} = \mathcal{O}_{\mathcal{X}_{flat,fppf}}$ ). Note that the site $\mathcal{X}_{lisse,{\acute{e}tale}}$ (resp. $\mathcal{X}_{flat,fppf}$ ) has a final object, namely $X/X$ which is quasi-compact (see Cohomology on Sites, Section 21.16). Hence we have

$\Gamma (\bigoplus \nolimits _{i \in I} \mathcal{O}) = \bigoplus \nolimits _{i \in I} A$

by Sites, Lemma 7.17.7. Hence the map in the presentation corresponds to a similar presentation

$\bigoplus \nolimits _{j \in J} A \longrightarrow \bigoplus \nolimits _{i \in I} A \longrightarrow M \longrightarrow 0$

of an $A$ -module $M$ . Moreover, $\mathcal{H}$ is equal to the restriction to the lisse-étale (resp. flat-fppf) site of the quasi-coherent sheaf $M^ a$ associated to $M$ . Choose a resolution

$\ldots \to F_2 \to F_1 \to F_0 \to M \to 0$

by free $A$ -modules. The complex

$\ldots \mathcal{O} \otimes _ A F_2 \to \mathcal{O} \otimes _ A F_1 \to \mathcal{O} \otimes _ A F_0 \to \mathcal{H} \to 0$

is a resolution of $\mathcal{H}$ by free $\mathcal{O}$ -modules because for each object $U/X$ of $\mathcal{X}_{lisse,{\acute{e}tale}}$ (resp. $\mathcal{X}_{flat,fppf}$ ) the structure morphism $U \to X$ is flat. Hence by construction the value of $Lg_!\mathcal{H}$ is

$\ldots \to \mathcal{O}_\mathcal {X} \otimes _ A F_2 \to \mathcal{O}_\mathcal {X} \otimes _ A F_1 \to \mathcal{O}_\mathcal {X} \otimes _ A F_0 \to 0 \to \ldots$

Since this is a complex of quasi-coherent modules on $\mathcal{X}_{\acute{e}tale}$ (resp. $\mathcal{X}_{fppf}$ ) it follows from Cohomology of Stacks, Proposition 103.8.1 that $H^ p(Lg_!\mathcal{H})$ is quasi-coherent. $\square$

The Stacks project

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