Lemma 96.11.3 (06WI)—The Stacks project

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Table of contents
Part 7: Algebraic Stacks
Chapter 96: Sheaves on Algebraic Stacks
Section 96.11: Quasi-coherent modules
Lemma 96.11.3 (cite)

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Lemma 96.11.3. Let $p : \mathcal{X} \to (\mathit{Sch}/S)_{fppf}$ be a category fibred in groupoids. Let $\mathcal{F}$ be a sheaf of $\mathcal{O}_\mathcal {X}$ -modules. Then $\mathcal{F}$ is quasi-coherent if and only if $x^*\mathcal{F}$ is a quasi-coherent sheaf on $(\mathit{Sch}/U)_{fppf}$ for every object $x$ of $\mathcal{X}$ with $U = p(x)$ .

Proof. By Lemma 96.11.2 the condition is necessary. Conversely, since $x^*\mathcal{F}$ is just the restriction to $\mathcal{X}_{fppf}/x$ we see that it is sufficient directly from the definition of a quasi-coherent sheaf (and the fact that the notion of being quasi-coherent is an intrinsic property of sheaves of modules, see Modules on Sites, Section 18.18). $\square$

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