The Stacks project

Serre's criterion for affineness.

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

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

  1. $X$ is quasi-compact,

  2. $X$ is quasi-separated, and

  3. $H^1(X, \mathcal{I}) = 0$ 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

Suggested slogan: Recognition theorem for affine schemes.

Comment #2701 by Ariyan Javanpeykar on

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

There are also:

  • 8 comment(s) on Section 30.3: Vanishing of cohomology

Post a comment

Your email address will not be published. Required fields are marked.

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).

Unfortunately JavaScript is disabled in your browser, so the comment preview function will not work.

All contributions are licensed under the GNU Free Documentation License.

In order to prevent bots from posting comments, we would like you to prove that you are human. You can do this by filling in the name of the current tag in the following input field. As a reminder, this is tag 01XG. Beware of the difference between the letter 'O' and the digit '0'.