Lemma 20.32.6 (0D5X)—The Stacks project

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Table of contents
Part 1: Preliminaries
Chapter 20: Cohomology of Sheaves
Section 20.32: Some properties of K-injective complexes
Lemma 20.32.6 (cite)

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Lemma 20.32.6. Let $f : (X, \mathcal{O}_ X) \to (Y, \mathcal{O}_ Y)$ be a morphism of ringed spaces. Let $K$ be in $D(\mathcal{O}_ X)$ . Then $H^ i(Rf_*K)$ is the sheaf associated to the presheaf

$V \mapsto H^ i(f^{-1}(V), K) = H^ i(V, Rf_*K)$

Proof. The equality $H^ i(f^{-1}(V), K) = H^ i(V, Rf_*K)$ follows upon taking cohomology from the second statement in Lemma 20.32.5. Then the statement on sheafification follows from Lemma 20.32.3. $\square$

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