The Stacks project

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. $\square$

Comments (2)

Comment #881 by on

The words "is an isomorphism" are missing in the statement of the Lemma.

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