The Stacks project

Lemma 72.16.1. 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 67.39.6 and 67.37.2). Hence $U$ is regular by Varieties, Lemma 33.25.3. By Properties of Spaces, Definition 66.7.2 this means that $X$ is regular. $\square$

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