The Stacks project

Lemma 67.49.2. Let $S$ be a scheme. Let $X$ be a locally Noetherian algebraic space over $S$. Then $X$ satisfies the equivalent conditions of Lemma 67.49.1.

Proof. If $U \to X$ is étale and $U$ is a scheme, then $U$ is a locally Noetherian scheme, see Properties of Spaces, Section 66.7. A locally Noetherian scheme has a locally finite set of irreducible components (Divisors, Lemma 31.26.1). Thus we conclude that $X$ passes condition (2) of the lemma. $\square$

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