Lemma 29.15.7 (01T7)—The Stacks project

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Part 2: Schemes
Chapter 29: Morphisms of Schemes
Section 29.15: Morphisms of finite type
Lemma 29.15.7 (cite)

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Lemma 29.15.7. Let $f : X \to S$ be locally of finite type with $S$ locally Noetherian. Then $f$ is quasi-separated.

Proof. In fact, it is true that $X$ is quasi-separated, see Properties, Lemma 28.5.4 and Lemma 29.15.6 above. Then apply Schemes, Lemma 26.21.13 to conclude that $f$ is quasi-separated. $\square$

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