Normality is local in the smooth topology.

Lemma 35.18.2. The property $\mathcal{P}(S) =$“$S$ is normal” 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. $\square$

Comment #914 by Matthieu Romagny on

Suggested slogan: Normality is local in the smooth topology

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