The Stacks project

Lemma 100.6.3. 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 93.18.7 and Lemma 93.18.10. 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 93.11.7). The modules $a_*a^*\mathcal{F}$ and $b_*b^*\mathcal{F}$ are locally quasi-coherent by Lemma 100.6.2. (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 93.11.8.

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 $\mathcal{F}|_{U_{\acute{e}tale}}$ 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 98.26.14 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 $(\mathit{Sch}/U_ i)_{fppf}$. 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 $(\mathit{Sch}/U)_{fppf}$ 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 $\mathcal{F}|_{U_{\acute{e}tale}}$ is quasi-coherent. $\square$


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