Lemma 37.20.3. A smooth morphism is regular.

**Proof.**
Let $f : X \to Y$ be a smooth morphism. As $f$ is locally of finite presentation, see Morphisms, Lemma 29.34.8 the fibres $X_ y$ are locally of finite type over a field, hence locally Noetherian. Moreover, $f$ is flat, see Morphisms, Lemma 29.34.9. Finally, the fibres $X_ y$ are smooth over a field (by Morphisms, Lemma 29.34.5) and hence geometrically regular by Varieties, Lemma 33.25.4. Thus $f$ is regular by Lemma 37.20.2.
$\square$

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