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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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