Lemma 30.3.2 (01XG)—The Stacks project

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Table of contents
Part 2: Schemes
Chapter 30: Cohomology of Schemes
Section 30.3: Vanishing of cohomology
Lemma 30.3.2 (cite)

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Serre's criterion for affineness.

[Serre-criterion], [II, Theorem 5.2.1, EGA]

Lemma 30.3.2. Let $X$ be a scheme. Assume that

$X$ is quasi-compact,
$X$ is quasi-separated, and
$H^1(X, \mathcal{I}) = 0$ for every quasi-coherent sheaf of ideals $\mathcal{I}$ of finite type.

Then $X$ is affine.

Proof. By Properties, Lemma 28.22.3 every quasi-coherent sheaf of ideals is a directed colimit of quasi-coherent sheaves of ideals of finite type. By Cohomology, Lemma 20.19.1 taking cohomology on $X$ commutes with directed colimits. Hence we see that $H^1(X, \mathcal{I}) = 0$ for every quasi-coherent sheaf of ideals on $X$ . In other words we see that Lemma 30.3.1 applies. $\square$

Comments (3)

Comment #1038 by Jakob Scholbach on September 16, 2014 at 13:03

Suggested slogan: Recognition theorem for affine schemes.

Comment #2701 by Ariyan Javanpeykar on August 01, 2017 at 23:25

A reference for Serre's criterion: EGA II, Chapter 5.2

Comment #2843 by Johan on October 04, 2017 at 13:50

Thanks, fixed here.

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