Lemma 33.25.3. Let $X \to \mathop{\mathrm{Spec}}(k)$ be a smooth morphism where $k$ is a field. Then $X$ is a regular scheme.

** Smooth over a field implies regular **

**Proof.**
(See also Lemma 33.12.6.) By Algebra, Lemma 10.139.3 every local ring $\mathcal{O}_{X, x}$ is regular. And because $X$ is locally of finite type over $k$ it is locally Noetherian. Hence $X$ is regular by Properties, Lemma 28.9.2.
$\square$

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Comment #1049 by Lenny Taelman on