Lemma 20.12.4 (09SZ)—The Stacks project

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Table of contents
Part 1: Preliminaries
Chapter 20: Cohomology of Sheaves
Section 20.12: Flasque sheaves
Lemma 20.12.4 (cite)

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Lemma 20.12.4. Let $(X, \mathcal{O}_ X)$ be a ringed space. Let $\mathcal{F}$ be a sheaf of $\mathcal{O}_ X$ -modules. Let $\mathcal{U} : U = \bigcup U_ i$ be an open covering. If $\mathcal{F}$ is flasque, then $\check{H}^ p(\mathcal{U}, \mathcal{F}) = 0$ for $p > 0$ .

Proof. The presheaves $\underline{H}^ q(\mathcal{F})$ used in the statement of Lemma 20.11.5 are zero by Lemma 20.12.3. Hence $\check{H}^ p(U, \mathcal{F}) = H^ p(U, \mathcal{F}) = 0$ by Lemma 20.12.3 again. $\square$

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