Lemma 21.10.2 (03AW)—The Stacks project

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Table of contents
Part 1: Preliminaries
Chapter 21: Cohomology on Sites
Section 21.10: Čech cohomology and cohomology
Lemma 21.10.2 (cite)

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Lemma 21.10.2. Let $\mathcal{C}$ be a site. Let $\mathcal{U} = \{ U_ i \to U\} _{i \in I}$ be a covering of $\mathcal{C}$ . Let $\mathcal{I}$ be an injective abelian sheaf, i.e., an injective object of $\textit{Ab}(\mathcal{C})$ . Then

$\check{H}^ p(\mathcal{U}, \mathcal{I}) = \left\{ \begin{matrix} \mathcal{I}(U) & \text{if} & p = 0 \\ 0 & \text{if} & p > 0 \end{matrix} \right.$

Proof. By Lemma 21.10.1 we see that $\mathcal{I}$ is an injective object in $\textit{PAb}(\mathcal{C})$ . Hence we can apply Lemma 21.9.6 (or its proof) to see the vanishing of higher Čech cohomology group. For the zeroth see Lemma 21.8.2. $\square$

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