Lemma 21.20.3 (0BKV)—The Stacks project

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Table of contents
Part 1: Preliminaries
Chapter 21: Cohomology on Sites
Section 21.20: Some properties of K-injective complexes
Lemma 21.20.3 (cite)

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Lemma 21.20.3. Let $(\mathcal{C}, \mathcal{O})$ be a ringed site. Let $K$ be an object of $D(\mathcal{O})$ . The sheafification of

$U \mapsto H^ q(U, K) = H^ q(\mathcal{C}/U, K|_{\mathcal{C}/U})$

is the $q$ th cohomology sheaf $H^ q(K)$ of $K$ .

Proof. The equality $H^ q(U, K) = H^ q(\mathcal{C}/U, K|_{\mathcal{C}/U})$ holds by Lemma 21.20.2. Choose a K-injective complex $\mathcal{I}^\bullet$ representing $K$ . Then

$H^ q(U, K) = \frac{\mathop{\mathrm{Ker}}(\mathcal{I}^ q(U) \to \mathcal{I}^{q + 1}(U))}{\mathop{\mathrm{Im}}(\mathcal{I}^{q - 1}(U) \to \mathcal{I}^ q(U))}.$

by our construction of cohomology. Since $H^ q(K) = \mathop{\mathrm{Ker}}(\mathcal{I}^ q \to \mathcal{I}^{q + 1})/ \mathop{\mathrm{Im}}(\mathcal{I}^{q - 1} \to \mathcal{I}^ q)$ the result is clear. $\square$

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