The Stacks project

73.5 Quasi-coherent modules and affines

Let $S$ be a scheme. Let $X$ be an algebraic space over $S$. Recall that $X_{affine, {\acute{e}tale}}$ is the full subcategory of $X_{\acute{e}tale}$ whose objects are affine turned into a site by declaring the coverings to be the standard ├ętale coverings. See Properties of Spaces, Definition 65.18.5. By Properties of Spaces, Lemma 65.18.6 we have an equivalence of topoi $g : \mathop{\mathit{Sh}}\nolimits (X_{affine, {\acute{e}tale}}) \to \mathop{\mathit{Sh}}\nolimits (X_{\acute{e}tale})$ whose pullback functor is given by restriction. Recall that $\mathcal{O}_ X$ denotes the structure sheaf on $X_{\acute{e}tale}$. Then we obtain an equivalence

73.5.0.1
\begin{equation} \label{spaces-descent-equation-alternative-small-ringed} (\mathop{\mathit{Sh}}\nolimits (X_{affine, {\acute{e}tale}}), \mathcal{O}_ X|_{X_{affine, {\acute{e}tale}}}) \longrightarrow (\mathop{\mathit{Sh}}\nolimits (X_{\acute{e}tale}), \mathcal{O}_ X) \end{equation}

of ringed topoi. We will often write $\mathcal{O}_ X$ in stead of $\mathcal{O}_ X|_{X_{affine, {\acute{e}tale}}}$. Having said this we can compare quasi-coherent modules as well.

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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