Lemma 20.7.4 (01E5)—The Stacks project

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Table of contents
Part 1: Preliminaries
Chapter 20: Cohomology of Sheaves
Section 20.7: Locality of cohomology
Lemma 20.7.4 (cite)

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Lemma 20.7.4. Let $f : X \to Y$ be a morphism of ringed spaces. Let $\mathcal{F}$ be an $\mathcal{O}_ X$ -module. Let $V \subset Y$ be an open subspace. Denote $g : f^{-1}(V) \to V$ the restriction of $f$ . Then we have

$R^ pg_*(\mathcal{F}|_{f^{-1}(V)}) = (R^ pf_*\mathcal{F})|_ V$

There is a similar statement for the derived image $Rf_*\mathcal{F}^\bullet$ where $\mathcal{F}^\bullet$ is a bounded below complex of $\mathcal{O}_ X$ -modules.

Proof. First proof. Apply Lemmas 20.7.3 and 20.7.1 to see the displayed equality. Second proof. Choose an injective resolution $\mathcal{F} \to \mathcal{I}^\bullet$ and use that $\mathcal{F}|_{f^{-1}(V)} \to \mathcal{I}^\bullet |_{f^{-1}(V)}$ is an injective resolution also. $\square$

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