Lemma 103.16.2 (07B0)—The Stacks project

Lemma 103.16.2. Let $\mathcal{X}$ be an algebraic stack. Notation as in 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 $\mathcal{X}$. Then $g_!\mathcal{H}$ is a quasi-coherent module on $\mathcal{X}$.
Let $\mathcal{H}$ be a quasi-coherent $\mathcal{O}_{\mathcal{X}_{flat,fppf}}$-module on the flat-fppf site of $\mathcal{X}$. 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 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.) Since each $f_ i$ is smooth (resp. flat) by Lemma 103.15.1 we see that $f_ i^{-1}g_!\mathcal{H} = g_{i, !}(f'_ i)^{-1}\mathcal{H}$. Using Lemma 103.16.1 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 $\mathcal{O} = \mathcal{O}_{\mathcal{X}_{lisse,{\acute{e}tale}}}$ (resp. $\mathcal{O} = \mathcal{O}_{\mathcal{X}_{flat,fppf}}$). Since $g_!$ commutes with arbitrary colimits (as a left adjoint functor, see Lemma 103.14.4 and Categories, Lemma 4.24.5) 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 103.14.5 shows that $g_!\mathcal{O} = \mathcal{O}_\mathcal {X}$. In case (2) we are done. In case (1) we apply Sheaves on Stacks, Lemma 96.11.4 to conclude. $\square$

Comments (4)

Comment #3188 by anonymous on February 10, 2018 at 22:09

The final statement "and we win" is a bit short. The last displayed sequence lives a priori in $\textit{Mod}(\mathcal{X}_\etale, \mathcal{O}_\mathcal{X})$ . To deduce that $g_!\mathcal{H}$ is a quasi-coherent module in $\textit{Mod}(\mathcal{O}_\mathcal{X})$ we need that this sequence already lives in $\textit{Mod}(\mathcal{O}_\mathcal{X})$ and is exact there. This follows from the results about colimits in tags 0771 and 06WN. It would be good to mention this explicitly.

Comment #3192 by Johan on February 11, 2018 at 15:19

In case (1) the sequence is exact as a sequence in modules with the etale topology. Hence the cokernel is a quasi-coherent on the big \'etale topos as the cokernel of a map of free modules. AHA, I see the problem: I should state in Section 96.11 explicitly that if you have a sheaf of $\mathcal{O}_\mathcal{X}$ -modules $\mathcal{F}$ on $\mathcal{X}$ in the Zariski, respectively \'etale topology and if $\mathcal{F}$ is quasi-coherent in the sense of Definition 18.23.1 on the (ringed) Zariski respectively \'etale topos, then actually it is quasi-coherent in the sense of Definition 96.11.1. This follows as in my previous comment from the fact that we know this to be true for schemes. OK?

Busy this week but will add this later. Thanks for the comment.

Comment #3199 by anonymous on February 14, 2018 at 08:52

This sounds like a very good idea. Many thanks!

Comment #3298 by Johan on May 17, 2018 at 15:16

See this commit.

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2 comment(s) on Section 103.16: Quasi-coherent modules and the lisse-étale and flat-fppf sites

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