Lemma 27.10.5. 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 10.59.6. 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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