The Stacks project

Lemma 67.39.5. An étale morphism of algebraic spaces is locally quasi-finite.

Proof. Let $X \to Y$ be an étale morphism of algebraic spaces, see Properties of Spaces, Definition 66.16.2. By Properties of Spaces, Lemma 66.16.3 we see this means there exists a diagram as in Lemma 67.22.1 with $h$ étale and surjective vertical arrow $a$. By Morphisms, Lemma 29.36.6 $h$ is locally quasi-finite. Hence $X \to Y$ is locally quasi-finite by definition. $\square$

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