Lemma 66.24.4 (08AH)—The Stacks project

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Table of contents
Part 4: Algebraic Spaces
Chapter 66: Properties of Algebraic Spaces
Section 66.24: Noetherian spaces
Lemma 66.24.4 (cite)

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Lemma 66.24.4. 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 66.22.1. By our definition of Noetherian spaces the scheme $U$ is locally Noetherian. Hence we conclude by More on Algebra, Lemma 15.45.3. $\square$

Comments (2)

Comment #7745 by Mingchen on September 10, 2022 at 06:47

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

Comment #7990 by Stacks Project on January 14, 2023 at 12:16

Thanks and fixed here.

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