Lemma 37.18.3. A smooth morphism is normal.
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 normal by Varieties, Lemma 33.25.4. Thus $f$ is normal by Lemma 37.18.2. $\square$
Post a comment
Your email address will not be published. Required fields are marked.
In your comment you can use Markdown and LaTeX style mathematics (enclose it like
$\pi$). A preview option is available if you wish to see how it works out (just click on the eye in the toolbar).
All contributions are licensed under the GNU Free Documentation License.