The Stacks project

Lemma 75.16.8. Let $S$ be a scheme. Let $f : X \to Y$ be a morphism of algebraic spaces over $S$. The following are equivalent:

  1. The morphism $f$ is étale, and

  2. the morphism $f$ is locally of finite presentation and formally étale.

Proof. Follows from the case of schemes, see More on Morphisms, Lemma 37.8.9 and étale localization, see Lemma 75.16.2 and Morphisms of Spaces, Lemmas 66.28.4 and 66.39.2. $\square$

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