The Stacks project

Lemma 74.5.1. 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 (,

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 ( 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. $\square$

Comments (0)

Post a comment

Your email address will not be published. Required fields are marked.

In your comment you can use Markdown and LaTeX style mathematics (enclose it like $\pi$). A preview option is available if you wish to see how it works out (just click on the eye in the toolbar).

Unfortunately JavaScript is disabled in your browser, so the comment preview function will not work.

All contributions are licensed under the GNU Free Documentation License.

In order to prevent bots from posting comments, we would like you to prove that you are human. You can do this by filling in the name of the current tag in the following input field. As a reminder, this is tag 0H04. Beware of the difference between the letter 'O' and the digit '0'.