The Stacks project

Lemma 59.22.3 (Locality of cohomology). Let $\mathcal{C}$ be a site, $\mathcal{F}$ an abelian sheaf on $\mathcal{C}$, $U$ an object of $\mathcal{C}$, $p > 0$ an integer and $\xi \in H^ p(U, \mathcal{F})$. Then there exists a covering $\mathcal{U} = \{ U_ i \to U\} _{i \in I}$ of $U$ in $\mathcal{C}$ such that $\xi |_{U_ i} = 0$ for all $i \in I$.

Proof. Choose an injective resolution $\mathcal{F} \to \mathcal{I}^\bullet $. Then $\xi $ is represented by a cocycle $\tilde{\xi } \in \mathcal{I}^ p(U)$ with $d^ p(\tilde{\xi }) = 0$. By assumption, the sequence $\mathcal{I}^{p - 1} \to \mathcal{I}^ p \to \mathcal{I}^{p + 1}$ in exact in $\textit{Ab}(\mathcal{C})$, which means that there exists a covering $\mathcal{U} = \{ U_ i \to U\} _{i \in I}$ such that $\tilde{\xi }|_{U_ i} = d^{p - 1}(\xi _ i)$ for some $\xi _ i \in \mathcal{I}^{p-1}(U_ i)$. Since the cohomology class $\xi |_{U_ i}$ is represented by the cocycle $\tilde{\xi }|_{U_ i}$ which is a coboundary, it vanishes. For more details see Cohomology on Sites, Lemma 21.7.3. $\square$

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