Lemma 35.30.1. The property $\mathcal{P}(f)=$“$f$ is smooth” is smooth local on the source.

## 35.30 Properties of morphisms local in the smooth topology on the source

Here are some properties of morphisms that are smooth local on the source. Note also the (in some respects stronger) result on descending smoothness via flat morphisms, Lemma 35.14.5.

**Proof.**
Combine Lemma 35.26.4 with Morphisms, Lemma 29.34.2 (local for Zariski on source and target), Morphisms, Lemma 29.34.4 (pre-composing), and Lemma 35.14.4 (part (4)).
$\square$

## 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Your email address will not be published.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).

Unfortunately JavaScript is disabled in your browser, so the comment preview function will not work.

All contributions are licensed under the GNU Free Documentation License.

## Comments (2)

Comment #6204 by Matthieu Romagny on

Comment #6345 by Johan on