Lemma 28.10.5 (0AAX)—The Stacks project

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Table of contents
Part 2: Schemes
Chapter 28: Properties of Schemes
Section 28.10: Dimension
Lemma 28.10.5 (cite)

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Lemma 28.10.5. A locally Noetherian scheme of dimension $0$ is a disjoint union of spectra of Artinian local rings.

Proof. A Noetherian ring of dimension $0$ is a finite product of Artinian local rings, see Algebra, Proposition 10.60.7. Hence an affine open of a locally Noetherian scheme $X$ of dimension $0$ has discrete underlying topological space. This implies that the topology on $X$ is discrete. The lemma follows easily from these remarks. $\square$

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