Lemma 28.5.3 (01OX)—The Stacks project

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Part 2: Schemes
Chapter 28: Properties of Schemes
Section 28.5: Noetherian schemes
Lemma 28.5.3 (cite)

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Lemma 28.5.3. Any immersion $Z \to X$ with $X$ locally Noetherian is quasi-compact.

Proof. A closed immersion is clearly quasi-compact. A composition of quasi-compact morphisms is quasi-compact, see Topology, Lemma 5.12.2. Hence it suffices to show that an open immersion into a locally Noetherian scheme is quasi-compact. Using Schemes, Lemma 26.19.2 we reduce to the case where $X$ is affine. Any open subset of the spectrum of a Noetherian ring is quasi-compact (for example combine Algebra, Lemma 10.31.5 and Topology, Lemmas 5.9.2 and 5.12.13). $\square$

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