The Stacks project

Lemma 95.25.1. 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

  1. 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,

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

  3. $\mathcal{F}$ 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,

  4. same as in (3) for the étale topology,

  5. same as in (3) for the smooth topology,

  6. same as in (3) for the syntomic topology,

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

  8. $\mathcal{F}$ corresponds to a quasi-coherent module on $\mathcal{X}$ via the equivalence (95.24.3.1).

Proof. To make sense out of part (2), recall that $\mathcal{X}_{affine}$ is a site gotten by endowing the category $\mathcal{X}_{affine}$ with the chaotic topology (Definition 95.24.1) and hence a sheaf of $\mathcal{O}$-modules $\mathcal{F}$ is the same thing as a presheaf of $\mathcal{O}$-modules. Conditions (1) and (2) are equivalent by Modules on Sites, Lemma 18.24.2. Observe that for $\tau \in \{ Zariski, {\acute{e}tale}, smooth, syntomic, fppf\} $ the presheaf $\mathcal{F}$ is a $\tau $-sheaf if and only if for all $x \in \mathop{\mathrm{Ob}}\nolimits (\mathcal{X}_{affine})$ the restriction to $\mathcal{X}_{affine}/x$ is a $\tau $-sheaf. Set $U = p(x)$. Similarly to the discussion in Section 95.9 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. In this way we see that the equivalence of (1) with (3) – (7) follows from Descent, Lemma 35.11.1 applied to each of these sites. The equivalence of (8) and (7) is immediate from the fact that “being quasi-coherent” is an intrinsic property of sheaves of modules, see Modules on Sites, Section 18.18 $\square$


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