Lemma 35.27.1. The property $\mathcal{P}(f)=$“$f$ is smooth” is smooth local on the source.

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

Here are some properties of morphisms that are smooth local on the source. ote also the (in some respects stronger) result on descending smoothness via flat morphisms, Lemma 35.11.5.

**Proof.**
Combine Lemma 35.23.4 with Morphisms, Lemma 29.34.2 (local for Zariski on source and target), Morphisms, Lemma 29.34.4 (pre-composing), and Lemma 35.11.4 (part (4)).
$\square$

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## Comments (1)

Comment #6204 by Matthieu Romagny on