The Stacks project

Lemma 68.5.3. Let $S$ be a scheme. Let $X$ be a quasi-compact and quasi-separated algebraic space over $S$. Let $I$ be a directed set and let $(\mathcal{F}_ i, \varphi _{ii'})$ be a system over $I$ of $\mathcal{O}_ X$-modules. Let $\mathcal{G}$ be an $\mathcal{O}_ X$-module of finite presentation. Then we have

\[ \mathop{\mathrm{colim}}\nolimits _ i \mathop{\mathrm{Hom}}\nolimits _ X(\mathcal{G}, \mathcal{F}_ i) = \mathop{\mathrm{Hom}}\nolimits _ X(\mathcal{G}, \mathop{\mathrm{colim}}\nolimits _ i \mathcal{F}_ i). \]

In particular, $\mathop{\mathrm{Hom}}\nolimits _ X(\mathcal{G}, -)$ commutes with filtered colimits in $\mathit{QCoh}(\mathcal{O}_ X)$.

Proof. The displayed equality is a special case of Modules on Sites, Lemma 18.27.12. In order to apply it, we need to check the hypotheses of Sites, Lemma 7.17.8 part (4) for the site $X_{\acute{e}tale}$. In order to do this, we will check hypotheses (2)(a), (2)(b), (2)(c) of Sites, Remark 7.17.9. Namely, let $\mathcal{B} \subset \mathop{\mathrm{Ob}}\nolimits (X_{\acute{e}tale})$ be the set of affine objects. Then

  1. Since $X$ is quasi-compact, there exists a $U \in \mathcal{B}$ such that $U \to X$ is surjective (Properties of Spaces, Lemma 65.6.3), hence $h_ U^\# \to *$ is surjective.

  2. For $U \in \mathcal{B}$ every étale covering $\{ U_ i \to U\} _{i \in I}$ of $U$ can be refined by a finite étale covering $\{ U_ j \to U\} _{j = 1, \ldots , m}$ with $U_ j \in \mathcal{B}$ (Topologies, Lemma 34.4.4).

  3. For $U, U' \in \mathop{\mathrm{Ob}}\nolimits (X_{\acute{e}tale})$ we have $h_ U^\# \times h_{U'}^\# = h_{U \times _ X U'}^\# $. If $U, U' \in \mathcal{B}$, then $U \times _ X U'$ is quasi-compact because $X$ is quasi-separated, see Morphisms of Spaces, Lemma 66.8.10 for example. Hence we can find a surjective étale morphism $U'' \to U \times _ X U'$ with $U'' \in \mathcal{B}$ (Properties of Spaces, Lemma 65.6.3). In other words, we have morphisms $U'' \to U$ and $U'' \to U'$ such that the map $h_{U''}^\# \to h_ U^\# \times h_{u'}^\# $ is surjective.

For the final statement, observe that the inclusion functor $\mathit{QCoh}(\mathcal{O}_ X) \to \textit{Mod}(\mathcal{O}_ X)$ commutes with colimits and that finitely presented modules are quasi-coherent. See Properties of Spaces, Lemma 65.29.7. $\square$

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