The Stacks project

Lemma 20.34.4. Let $(X, \mathcal{O}_ X)$ be a ringed space. Let $i : Z \to X$ be the inclusion of a closed subset. Then $R\Gamma (Z, - ) \circ R\mathcal{H}_ Z = R\Gamma _ Z(X, - )$ as functors $D(\mathcal{O}_ X) \to D(\mathcal{O}_ X(X))$.

Proof. Follows from the construction of right derived functors using K-injective resolutions, Lemma 20.34.3, and the fact that $\Gamma _ Z(X, -) = \Gamma (Z, -) \circ \mathcal{H}_ Z$. $\square$

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