96.25 Quasi-coherent modules and affines
Let $p : \mathcal{X} \to (\mathit{Sch}/S)_{fppf}$ be a category fibred in groupoids. In Section 96.24 we have associated to this a ringed site $(\mathcal{X}_{affine}, \mathcal{O})$.
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).
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 96.24.1) 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 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. 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
$\square$
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).
$\square$
Comments (0)