The Stacks project

Lemma 33.38.2. Let $X$ be an affine scheme all of whose local rings are Noetherian of dimension $\leq 1$. Then any quasi-compact open $U \subset X$ is affine.

Proof. Denote $j : U \to X$ the inclusion morphism. Let $\mathcal{F}$ be a quasi-coherent $\mathcal{O}_ U$-module. By Lemma 33.38.1 the higher direct images $R^ pj_*\mathcal{F}$ are zero. The $\mathcal{O}_ X$-module $j_*\mathcal{F}$ is quasi-coherent (Schemes, Lemma 26.24.1). Hence it has vanishing higher cohomology groups by Cohomology of Schemes, Lemma 30.2.2. By the Leray spectral sequence Cohomology, Lemma 20.13.6 we have $H^ p(U, \mathcal{F}) = 0$ for all $p > 0$. Thus $U$ is affine, for example by Cohomology of Schemes, Lemma 30.3.1. $\square$

Comments (0)

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Your email address will not be published.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 09N9. Beware of the difference between the letter 'O' and the digit '0'.