Lemma 96.25.1. 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 $\mathcal{X}_{affine}$ the map $\mathcal{F}(x') \otimes _{\mathcal{O}(x')} \mathcal{O}(x) \to \mathcal{F}(x)$ is an isomorphism,

$\mathcal{F}$ is a quasi-coherent module on $(\mathcal{X}_{affine}, \mathcal{O})$ in the sense of Modules on Sites, Definition 18.23.1,

$\mathcal{F}$ is a sheaf for the Zariski topology on $\mathcal{X}_{affine}$ and a quasi-coherent module on $(\mathcal{X}_{affine, Zar}, \mathcal{O})$ in the sense of Modules on Sites, Definition 18.23.1,

same as in (3) for the étale topology,

same as in (3) for the smooth topology,

same as in (3) for the syntomic topology,

same as in (3) for the fppf topology, and

$\mathcal{F}$ corresponds to a quasi-coherent module on $\mathcal{X}$ via the equivalence (96.24.3.1).

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