
Lemma 95.12.1. 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 $\mathcal{X}_{\acute{e}tale}$. 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 93.32.2). The pullback of $\mathcal{F}$ to $(\mathit{Sch}/U_ j)_{\acute{e}tale}$ is still quasi-coherent, see Modules on Sites, Lemma 18.23.4. 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 88.18.10 there exists an étale covering $\{ x_ i \to x\} _{i \in I}$ such that each $x_ i$ lifts to an object $u_ i$ of $(\mathit{Sch}/U)_{\acute{e}tale}$. 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 $(\mathit{Sch}/V)_{\acute{e}tale}$ is quasi-coherent, for example by Modules on Sites, Lemma 18.23.3. By Sheaves on Stacks, Lemma 88.11.4 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 88.11.3 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 93.24.2 and 93.26.2). The pullback of $\mathcal{F}$ to $(\mathit{Sch}/U_ j)_{\acute{e}tale}$ is still locally quasi-coherent, see Sheaves on Stacks, Lemma 88.11.2. 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 88.18.10 there exists an fppf covering $\{ x_ i \to x\} _{i \in I}$ such that each $x_ i$ lifts to an object $u_ i$ of $(\mathit{Sch}/U)_{\acute{e}tale}$. 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 $(\mathit{Sch}/V)_{\acute{e}tale}$ is quasi-coherent, for example by Modules on Sites, Lemma 18.23.3. By Sheaves on Stacks, Lemma 88.11.3 we see that $\mathcal{F}$ is quasi-coherent. $\square$

Comment #3190 by anonymous on

Concerning part (1) of the proof: Lemmas 87.11.2 and 87.11.3 speak about fppf-sheaves and not about \'e{}tale sheaves.

Concerning both parts of the proof: when referring to Lemma 87.11.2 "is still locally quasi-coherent" should be "is still quasi-coherent".

Comment #3194 by on

Please see comment #3195. I will fix the other thing soonish.

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