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.
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. $\square$
Your email address will not be published.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).
All contributions are licensed under the GNU Free Documentation License.
Comments (3)
Comment #1038 by Jakob Scholbach on
Comment #2701 by Ariyan Javanpeykar on
Comment #2843 by Johan on
There are also: