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The Stacks project

Lemma 69.5.3. 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 66.6.3), 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 67.8.10 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 66.6.3). 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 66.29.7. \square


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