Lemma 30.19.2 (02O6)—The Stacks project

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Table of contents
Part 2: Schemes
Chapter 30: Cohomology of Schemes
Section 30.19: Higher direct images of coherent sheaves
Lemma 30.19.2 (cite)

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Lemma 30.19.2. Let $S = \mathop{\mathrm{Spec}}(A)$ with $A$ a Noetherian ring. Let $f : X \to S$ be a proper morphism. Let $\mathcal{F}$ be a coherent $\mathcal{O}_ X$-module. Then $H^ i(X, \mathcal{F})$ is a finite $A$-module for all $i \geq 0$.

Proof. This is just the affine case of Proposition 30.19.1. Namely, by Lemmas 30.4.5 and 30.4.6 we know that $R^ if_*\mathcal{F}$ is the quasi-coherent sheaf associated to the $A$-module $H^ i(X, \mathcal{F})$ and by Lemma 30.9.1 this is a coherent sheaf if and only if $H^ i(X, \mathcal{F})$ is an $A$-module of finite type. $\square$

Comments (2)

Comment #8142 by Rachel Webb on February 27, 2023 at 13:07

Small typo: I think we want "Then $H^i(X, \mathcal{F})$ is a finite $A$ -module . . . ''

Comment #8234 by Stacks Project on March 03, 2023 at 15:08

Thanks and fixed here.

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