Lemma 70.16.1. Let $k$ be a field. Let $X$ be an algebraic space smooth over $k$. Then $X$ is a regular algebraic space.

**Proof.**
Choose a scheme $U$ and a surjective étale morphism $U \to X$. The morphism $U \to \mathop{\mathrm{Spec}}(k)$ is smooth as a composition of an étale (hence smooth) morphism and a smooth morphism (see Morphisms of Spaces, Lemmas 65.39.6 and 65.37.2). Hence $U$ is regular by Varieties, Lemma 33.25.3. By Properties of Spaces, Definition 64.7.2 this means that $X$ is regular.
$\square$

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