Lemma 66.24.3 (04ZG)—The Stacks project

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Part 4: Algebraic Spaces
Chapter 66: Properties of Algebraic Spaces
Section 66.24: Noetherian spaces
Lemma 66.24.3 (cite)

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Lemma 66.24.3. Let $S$ be a scheme. Let $X$ be an algebraic space over $S$. If $X$ is Noetherian, then $|X|$ is a sober Noetherian topological space.

Proof. A quasi-separated algebraic space has an underlying sober topological space, see Lemma 66.15.1. It is Noetherian by Lemma 66.24.2. $\square$

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