The Stacks project

Lemma 67.39.12. A locally finitely presented, flat, unramified morphism of algebraic spaces is étale.

Proof. Let $X \to Y$ be a locally finitely presented, flat, unramified morphism of algebraic spaces. By Properties of Spaces, Lemma 66.16.3 we see this means there exists a diagram as in Lemma 67.22.1 with $h$ locally finitely presented, flat, unramified and surjective vertical arrow $a$. By Morphisms, Lemma 29.36.16 $h$ is étale. Hence $X \to Y$ is étale by definition. $\square$

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