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.
72.16 Spaces smooth over fields
This section is the analogue of Varieties, Section 33.25.
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
Lemma 72.16.2. Let k be a field. Let X be an algebraic space smooth over \mathop{\mathrm{Spec}}(k). The set of x \in |X| which are image of morphisms \mathop{\mathrm{Spec}}(k') \to X with k' \supset k finite separable is dense in |X|.
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 we can apply Varieties, Lemma 33.25.6 to see that the closed points of U whose residue fields are finite separable over k are dense. This implies the lemma by our definition of the topology on |X|. \square
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