The Stacks project

Lemma 94.11.4. Let $p : \mathcal{X} \to (\mathit{Sch}/S)_{fppf}$ be a category fibred in groupoids. Let $\mathcal{F}$ be a presheaf of modules on $\mathcal{X}$. The following are equivalent

  1. $\mathcal{F}$ is an object of $\textit{Mod}(\mathcal{X}_{Zar}, \mathcal{O}_\mathcal {X})$ and $\mathcal{F}$ is a quasi-coherent module on $(\mathcal{X}_{Zar}, \mathcal{O}_\mathcal {X})$ in the sense of Modules on Sites, Definition 18.23.1,

  2. $\mathcal{F}$ is an object of $\textit{Mod}(\mathcal{X}_{\acute{e}tale}, \mathcal{O}_\mathcal {X})$ and $\mathcal{F}$ is a quasi-coherent module on $(\mathcal{X}_{\acute{e}tale}, \mathcal{O}_\mathcal {X})$ in the sense of Modules on Sites, Definition 18.23.1, and

  3. $\mathcal{F}$ is a quasi-coherent module on $\mathcal{X}$ in the sense of Definition 94.11.1.

Proof. Assume either (1), (2), or (3) holds. Let $x$ be an object of $\mathcal{X}$ lying over the scheme $U$. Recall that $x^*\mathcal{F} = x^{-1}\mathcal{F}$ is just the restriction to $\mathcal{X}/x = (\mathit{Sch}/U)_\tau $ where $\tau = fppf$, $\tau = {\acute{e}tale}$, or $\tau = Zar$, see Section 94.9. By the definition of quasi-coherent modules on a ringed site this restriction is quasi-coherent provided $\mathcal{F}$ is. By Descent, Proposition 35.8.11 we see that $x^*\mathcal{F}$ is the sheaf associated to a quasi-coherent $\mathcal{O}_ U$-module and is therefore a quasi-coherent module in the fppf, ├ętale, and Zariski topology; here we also use Descent, Lemma 35.8.1 and Definition 35.8.2. Since this holds for every object $x$ of $\mathcal{X}$, we see that $\mathcal{F}$ is a sheaf in any of the three topologies. Moreover, we find that $\mathcal{F}$ is quasi-coherent in any of the three topologies directly from the definition of being quasi-coherent and the fact that $x$ is an arbitrary object of $\mathcal{X}$. $\square$


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