The Stacks project

Lemma 21.20.2. Let $(\mathcal{C}, \mathcal{O})$ be a ringed site. Let $U \in \mathop{\mathrm{Ob}}\nolimits (\mathcal{C})$. For $K$ in $D(\mathcal{O})$ we have $H^ p(U, K) = H^ p(\mathcal{C}/U, K|_{\mathcal{C}/U})$.

Proof. Let $\mathcal{I}^\bullet $ be a K-injective complex of $\mathcal{O}$-modules representing $K$. Then

\[ H^ q(U, K) = H^ q(\Gamma (U, \mathcal{I}^\bullet )) = H^ q(\Gamma (\mathcal{C}/U, \mathcal{I}^\bullet |_{\mathcal{C}/U})) \]

by construction of cohomology. By Lemma 21.20.1 the complex $\mathcal{I}^\bullet |_{\mathcal{C}/U}$ is a K-injective complex representing $K|_{\mathcal{C}/U}$ and the lemma follows. $\square$

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