Lemma 65.24.4. Let $S$ be a scheme. Let $X$ be a Noetherian algebraic space over $S$. Let $\overline{x}$ be a geometric point of $X$. Then $\mathcal{O}_{X, \overline{x}}$ is a Noetherian local ring.

Proof. Choose an étale neighbourhood $(U, \overline{u})$ of $\overline{x}$ where $U$ is a scheme. Then $\mathcal{O}_{X, \overline{x}}$ is the strict henselization of the local ring of $U$ at $u$, see Lemma 65.22.1. By our definition of Noetherian spaces the scheme $U$ is locally Noetherian. Hence we conclude by More on Algebra, Lemma 15.45.3. $\square$

Comment #7745 by Mingchen on

Just to be more precise, I think you only know U is locally Noetherian, not Noetherian.

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).