The Stacks project

69.5 Colimits and cohomology

The following lemma in particular applies to diagrams of quasi-coherent sheaves.

Lemma 69.5.1. Let $S$ be a scheme. Let $X$ be an algebraic space over $S$. If $X$ is quasi-compact and quasi-separated, then

\[ \mathop{\mathrm{colim}}\nolimits _ i H^ p(X, \mathcal{F}_ i) \longrightarrow H^ p(X, \mathop{\mathrm{colim}}\nolimits _ i \mathcal{F}_ i) \]

is an isomorphism for every filtered diagram of abelian sheaves on $X_{\acute{e}tale}$.

Proof. This follows from Cohomology on Sites, Lemma 21.16.1. Namely, let $\mathcal{B} \subset \mathop{\mathrm{Ob}}\nolimits (X_{spaces, {\acute{e}tale}})$ be the set of quasi-compact and quasi-separated spaces étale over $X$. Note that if $U \in \mathcal{B}$ then, because $U$ is quasi-compact, the collection of finite coverings $\{ U_ i \to U\} $ with $U_ i \in \mathcal{B}$ is cofinal in the set of coverings of $U$ in $X_{spaces, {\acute{e}tale}}$. By Morphisms of Spaces, Lemma 67.8.10 the set $\mathcal{B}$ satisfies all the assumptions of Cohomology on Sites, Lemma 21.16.1. Since $X \in \mathcal{B}$ we win. $\square$


Lemma 69.5.2. Let $S$ be a scheme. Let $f : X \to Y$ be a quasi-compact and quasi-separated morphism of algebraic spaces over $S$. Let $\mathcal{F} = \mathop{\mathrm{colim}}\nolimits \mathcal{F}_ i$ be a filtered colimit of abelian sheaves on $X_{\acute{e}tale}$. Then for any $p \geq 0$ we have

\[ R^ pf_*\mathcal{F} = \mathop{\mathrm{colim}}\nolimits R^ pf_*\mathcal{F}_ i. \]

Proof. Recall that $R^ pf_*\mathcal{F}$ is the sheaf on $Y_{spaces, {\acute{e}tale}}$ associated to $V \mapsto H^ p(V \times _ Y X, \mathcal{F})$, see Cohomology on Sites, Lemma 21.7.4 and Properties of Spaces, Lemma 66.18.8. Recall that the colimit is the sheaf associated to the presheaf colimit. Hence we can apply Lemma 69.5.1 to $H^ p(V \times _ Y X, -)$ where $V$ is affine to conclude (because when $V$ is affine, then $V \times _ Y X$ is quasi-compact and quasi-separated). Strictly speaking this also uses Properties of Spaces, Lemma 66.18.6 to see that there exist enough affine objects. $\square$

The following lemma tells us that finitely presented modules behave as expected in quasi-compact and quasi-separated algebraic spaces.

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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