Lemma 96.25.2 (0H0F)—The Stacks project

Lemma 96.25.2. Let $p : \mathcal{X} \to (\mathit{Sch}/S)_{fppf}$ be a category fibred in groupoids. Let $\mathcal{F}$ be an $\mathcal{O}$-module on $\mathcal{X}_{affine}$. The following are equivalent

for every morphism $x \to x'$ of $\mathcal{X}_{affine}$ such that $p(x) \to p(x')$ is an étale morphism (of affine schemes), the map $\mathcal{F}(x') \otimes _{\mathcal{O}(x')} \mathcal{O}(x) \to \mathcal{F}(x)$ is an isomorphism,
$\mathcal{F}$ is a sheaf for the étale topology on $\mathcal{X}_{affine}$ and for every object $x$ of $\mathcal{X}_{affine}$ the restriction $x^*\mathcal{F}|_{U_{affine, {\acute{e}tale}}}$ is quasi-coherent where $U = p(x)$,
$\mathcal{F}$ corresponds to a locally quasi-coherent module on $\mathcal{X}$ via the equivalence (96.24.3.1) for the étale topology.

Proof. To make sense out of condition (2), recall that $U_{affine, {\acute{e}tale}}$ is the full subcategory of $U_{\acute{e}tale}$ consisting of affine objects, see Topologies, Definition 34.4.8. Similarly to the discussion in Section 96.9 the object $x$ of $\mathcal{X}_{affine}$ induces an equivalence $\mathcal{X}_{affine, {\acute{e}tale}}/x \to (\textit{Aff}/U)_{\acute{e}tale}$ of sites. Then $x^*\mathcal{F}$ is the sheaf of modules on $(\textit{Aff}/U)_{\acute{e}tale}$ corresponding to the restriction $\mathcal{F}|_{\mathcal{X}_{affine, {\acute{e}tale}}/x}$. Finally, using the continuous and cocontinuous inclusion functor $U_{affine, {\acute{e}tale}} \to (\textit{Aff}/U)_{\acute{e}tale}$ we can further restrict and obtain $x^*\mathcal{F}|_{U_{affine, {\acute{e}tale}}}$.

The equivalence of (1) and (2) follows from the remarks above and Descent, Lemma 35.11.2 applied to the restriction of $\mathcal{F}$ to $U_{affine, {\acute{e}tale}}$ for every object $x$ of $\mathcal{X}$ lying over an affine scheme $U$. The equivalence of (2) and (3) is immediate from the definitions and the fact that quasi-coherent modules on $U_{affine, {\acute{e}tale}}$ and $U_{\acute{e}tale}$ correspond (again by Descent, Lemma 35.11.2 for example). $\square$

The Stacks project

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