Lemma 30.19.2 (02O6)—The Stacks project

Loading web-font TeX/Math/Italic

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)

numberstags

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.

There are also:

2 comment(s) on Section 30.19: Higher direct images of coherent sheaves

In your comment you can use Markdown and LaTeX style mathematics (enclose it like $\pi$ ). A preview option is available if you wish to see how it works out (just click on the eye in the toolbar).

Comments (2)

Post a comment