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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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