The Stacks project

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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