Lemma 59.51.6 (03Q8)—The Stacks project

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Table of contents
Part 3: Topics in Scheme Theory
Chapter 59: Étale Cohomology
Section 59.51: Colimits
Lemma 59.51.6 (cite)

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Lemma 59.51.6. Let $f: X\to Y$ be a morphism of schemes and $\mathcal{F}\in \textit{Ab}(X_{\acute{e}tale})$ . Then $R^ pf_*\mathcal{F}$ is the sheaf associated to the presheaf

$(V \to Y) \longmapsto H_{\acute{e}tale}^ p(X \times _ Y V, \mathcal{F}|_{X \times _ Y V}).$

More generally, for $K \in D(X_{\acute{e}tale})$ we have that $R^ pf_*K$ is the sheaf associated to the presheaf

$(V \to Y) \longmapsto H_{\acute{e}tale}^ p(X \times _ Y V, K|_{X \times _ Y V}).$

Proof. This lemma is valid for topological spaces, and the proof in this case is the same. See Cohomology on Sites, Lemma 21.7.4 for the case of a sheaf and see Cohomology on Sites, Lemma 21.20.3 for the case of a complex of abelian sheaves. $\square$

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