The Stacks project

Lemma 73.5.1. Let $S$ be a scheme. Let $X$ be an algebraic space over $S$. Let $\mathcal{F}$ be a presheaf of $\mathcal{O}_ X$-modules on $X_{affine, {\acute{e}tale}}$. The following are equivalent

  1. for every morphism $U \to U'$ of $X_{affine, {\acute{e}tale}}$ the map $\mathcal{F}(U') \otimes _{\mathcal{O}_ X(U')} \mathcal{O}_ X(U) \to \mathcal{F}(U)$ is an isomorphism,

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

  3. $\mathcal{F}$ corresponds to a quasi-coherent module on $X$ via the equivalence (73.5.0.1),

Proof. 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 $X_{affine, {\acute{e}tale}}$. The sheaf condition for $\mathcal{F}$ and $\mathcal{U}$, by our assumption on $\mathcal{F}$, reduces to showing that

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

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

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

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

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

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 (73.5.0.1) induces an equivalence between categories of quasi-coherent modules. Thus we have the equivalence of (2) and (3).

Let us assume (3) and prove (1). Namely, let $\mathcal{G}$ be a quasi-coherent module on $X$ corresponding to $\mathcal{F}$. Let $h : U \to U' \to X$ be a morphism of $X_{affine, {\acute{e}tale}}$. Denote $f : U \to X$ and $f' : U' \to X$ 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. $\square$


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