Lemma 35.15.1. The property $\mathcal{P}(S) =$“$S$ is reduced” is local in the smooth topology.

Proof. We will use Lemma 35.12.2. First we note that “being reduced” is local in the Zariski topology. This is clear from the definition, see Schemes, Definition 26.12.1. Next, we show that if $S' \to S$ is a smooth morphism of affines and $S$ is reduced, then $S'$ is reduced. This is Algebra, Lemma 10.163.7. Finally, we show that if $S' \to S$ is a surjective smooth morphism of affines and $S'$ is reduced, then $S$ is reduced. This is Algebra, Lemma 10.164.2. Thus (1), (2) and (3) of Lemma 35.12.2 hold and we win. $\square$

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).