Lemma 27.5.4. A locally Noetherian scheme is quasi-separated.

**Proof.**
By Schemes, Lemma 25.21.6 we have to show that the intersection $U \cap V$ of two affine opens of $X$ is quasi-compact. This follows from Lemma 27.5.3 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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