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The Stacks project

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 37.21.2. \square

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