Lemma 35.18.2. The property $\mathcal{P}(S) =$“$S$ is normal” is local in the smooth topology.

** Normality is local in the smooth topology. **

**Proof.**
We will use Lemma 35.15.2. First we show “being normal” is local in the Zariski topology. This is clear from the definition, see Properties, Definition 28.7.1. Next, we show that if $S' \to S$ is a smooth morphism of affines and $S$ is normal, then $S'$ is normal. This is Algebra, Lemma 10.163.9. Finally, we show that if $S' \to S$ is a surjective smooth morphism of affines and $S'$ is normal, then $S$ is normal. This is Algebra, Lemma 10.164.3. Thus (1), (2) and (3) of Lemma 35.15.2 hold and we win.
$\square$

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## Comments (1)

Comment #914 by Matthieu Romagny on