The Stacks project

Lemma 74.16.8. 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 74.16.2 and Morphisms of Spaces, Lemmas 65.28.4 and 65.39.2. $\square$

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