Lemma 66.39.12. 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 65.16.3 we see this means there exists a diagram as in Lemma 66.22.1 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. $\square$

In your comment you can use Markdown and LaTeX style mathematics (enclose it like $\pi$). A preview option is available if you wish to see how it works out (just click on the eye in the toolbar).