Lemma 28.5.4 (01OY)—The Stacks project

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Part 2: Schemes
Chapter 28: Properties of Schemes
Section 28.5: Noetherian schemes
Lemma 28.5.4 (cite)

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Lemma 28.5.4. A locally Noetherian scheme is quasi-separated.

Proof. By Schemes, Lemma 26.21.6 we have to show that the intersection $U \cap V$ of two affine opens of $X$ is quasi-compact. This follows from Lemma 28.5.3 above on considering the open immersion $U \cap V \to U$ for example. (But really it is just because any open of the spectrum of a Noetherian ring is quasi-compact.) $\square$

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