Lemma 35.11.2 (0GZX)—The Stacks project

Lemma 35.11.2. Let $S$ be a scheme. Let $\tau \in \{ Zariski, {\acute{e}tale}\}$ . Let $\mathcal{F}$ be a presheaf of $\mathcal{O}_{affine}$ -modules on $S_{affine, \tau }$ . The following are equivalent

for every morphism $U \to U'$ of $S_{affine, \tau }$ the map $\mathcal{F}(U') \otimes _{\mathcal{O}(U')} \mathcal{O}(U) \to \mathcal{F}(U)$ is an isomorphism,
$\mathcal{F}$ is a sheaf on $S_{affine, \tau }$ and a quasi-coherent module on the ringed site $(S_{affine, \tau }, \mathcal{O}_{affine})$ in the sense of Modules on Sites, Definition 18.23.1,
$\mathcal{F}$ corresponds to a quasi-coherent module on $S_\tau$ via the equivalence (35.11.1.1),
$\mathcal{F}$ comes from a unique quasi-coherent $\mathcal{O}_ S$ -module $\mathcal{G}$ by the procedure described in Section 35.8.

Proof. Let us prove this in the case of the étale topology.

Assume (1) holds. To show that $\mathcal{F}$ is a sheaf, let $\mathcal{U} = \{ U_ i \to U\} _{i = 1, \ldots , n}$ be a covering of $S_{affine, {\acute{e}tale}}$ . The sheaf condition for $\mathcal{F}$ and $\mathcal{U}$ , by our assumption on $\mathcal{F}$ . reduces to showing that

$0 \to \mathcal{F}(U) \to \prod \mathcal{F}(U) \otimes _{\mathcal{O}(U)} \mathcal{O}(U_ i) \to \prod \mathcal{F}(U) \otimes _{\mathcal{O}(U)} \mathcal{O}(U_ i \times _ U U_ j)$

is exact. This is true because $\mathcal{O}(U) \to \prod \mathcal{O}(U_ i)$ is faithfully flat (by Lemma 35.9.1 and the fact that coverings in $S_{affine, {\acute{e}tale}}$ are standard étale coverings) and we may apply Lemma 35.3.6. Next, we show that $\mathcal{F}$ is quasi-coherent on $S_{affine, {\acute{e}tale}}$ . Namely, for $U$ in $S_{affine, {\acute{e}tale}}$ , set $R = \mathcal{O}(U)$ and choose a presentation

$\bigoplus \nolimits _{k \in K} R \longrightarrow \bigoplus \nolimits _{l \in L} R \longrightarrow \mathcal{F}(U) \longrightarrow 0$

by free $R$ -modules. By property (1) and the right exactness of tensor product we see that for every morphism $U' \to U$ in $S_{affine, {\acute{e}tale}}$ we obtain a presentation

$\bigoplus \nolimits _{k \in K} \mathcal{O}(U') \longrightarrow \bigoplus \nolimits _{l \in L} \mathcal{O}(U') \longrightarrow \mathcal{F}(U') \longrightarrow 0$

In other words, we see that the restriction of $\mathcal{F}$ to the localized category $S_{affine, etale}/U$ has a presentation

$\bigoplus \nolimits _{k \in K} \mathcal{O}_{affine}|_{S_{affine, {\acute{e}tale}}/U} \longrightarrow \bigoplus \nolimits _{l \in L} \mathcal{O}_{affine}|_{S_{affine, {\acute{e}tale}}/U} \longrightarrow \mathcal{F}|_{S_{affine, {\acute{e}tale}}/U} \longrightarrow 0$

as required to show that $\mathcal{F}$ is quasi-coherent. With apologies for the horrible notation, this finishes the proof that (1) implies (2).

Since the notion of a quasi-coherent module is intrinsic (Modules on Sites, Lemma 18.23.2) we see that the equivalence (35.11.1.1) induces an equivalence between categories of quasi-coherent modules. Thus we have the equivalence of (2) and (3).

The equivalence of (3) and (4) follows from Proposition 35.8.9.

Let us assume (4) and prove (1). Namely, let $\mathcal{G}$ be as in (4). Let $h : U \to U' \to S$ be a morphism of $S_{affine, {\acute{e}tale}}$ . Denote $f : U \to S$ and $f' : U' \to S$ the structure morphisms, so that $f = f' \circ h$ . We have $\mathcal{F}(U') = \Gamma (U', (f')^*\mathcal{G})$ and $\mathcal{F}(U) = \Gamma (U, f^*\mathcal{G}) = \Gamma (U, h^*(f')^*\mathcal{G})$ . Hence (1) holds by Schemes, Lemma 26.7.3.

We omit the proof in the case of the Zariski topology. $\square$

The Stacks project

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