The Stacks project

Proof. We will use Lemma 35.15.2. 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 35.15.2 hold and we win. $\square$