Lemma 21.20.6 (0D6I)—The Stacks project

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Part 1: Preliminaries
Chapter 21: Cohomology on Sites
Section 21.20: Some properties of K-injective complexes
Lemma 21.20.6 (cite)

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Lemma 21.20.6. Let $f : (\mathcal{C}, \mathcal{O}_\mathcal {C}) \to (\mathcal{D}, \mathcal{O}_\mathcal {D})$ be a morphism of ringed sites corresponding to the continuous functor $u : \mathcal{D} \to \mathcal{C}$ . Let $K$ be in $D(\mathcal{O}_\mathcal {C})$ . Then $H^ i(Rf_*K)$ is the sheaf associated to the presheaf

$V \mapsto H^ i(u(V), K) = H^ i(V, Rf_*K)$

Proof. The equality $H^ i(u(V), K) = H^ i(V, Rf_*K)$ follows upon taking cohomology from the second statement in Lemma 21.20.5. Then the statement on sheafification follows from Lemma 21.20.3. $\square$

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