The Stacks project

Lemma 96.14.1. Let $S$ be a scheme. Let $\mathcal{X} \to (\mathit{Sch}/S)_{fppf}$ be a category fibred in groupoids which is representable by an algebraic space $F$. If $\mathcal{F}$ is in $\textit{LQCoh}(\mathcal{O}_\mathcal {X})$ then the restriction $\mathcal{F}|_{F_{\acute{e}tale}}$ (96.10.2.1) is quasi-coherent.

Proof. Let $U$ be a scheme étale over $F$. Then $\mathcal{F}|_{U_{\acute{e}tale}} = (\mathcal{F}|_{F_{\acute{e}tale}})|_{U_{\acute{e}tale}}$. This is clear but see also Remark 96.10.2. Thus the assertion follows from the definitions. $\square$

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