The Stacks project

Lemma 33.25.4. Let $X \to \mathop{\mathrm{Spec}}(k)$ be a smooth morphism where $k$ is a field. Then $X$ is geometrically regular, geometrically normal, and geometrically reduced over $k$.

Proof. (See also Lemma 33.12.6.) Let $k'$ be a finite purely inseparable extension of $k$. It suffices to prove that $X_{k'}$ is regular, normal, reduced, see Lemmas 33.12.3, 33.10.3, and 33.6.5. By Morphisms, Lemma 29.32.5 the morphism $X_{k'} \to \mathop{\mathrm{Spec}}(k')$ is smooth too. Hence it suffices to show that a scheme $X$ smooth over a field is regular, normal, and reduced. We see that $X$ is regular by Lemma 33.25.3. Hence Properties, Lemma 28.9.4 guarantees that $X$ is normal. $\square$


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