The Stacks project

Proof. We will check (1), (2) and (3) of Lemma 35.15.2. As a smooth morphism is flat of finite presentation (Morphisms, Lemmas 29.34.9 and 29.34.8) we have already checked this for “being locally Noetherian” in the proof of Lemma 35.16.1. We will use this without further mention in the proof. First we note that $\mathcal{P}$ is local in the Zariski topology. This is clear from the definition, see Properties, Definition 28.12.1. Next, we show that if $S' \to S$ is a smooth morphism of affines and $S$ has $\mathcal{P}$, then $S'$ has $\mathcal{P}$. This is Algebra, Lemmas 10.163.5 (use Morphisms, Lemma 29.34.2, Algebra, Lemmas 10.137.4 and 10.140.3). Finally, we show that if $S' \to S$ is a surjective smooth morphism of affines and $S'$ has $\mathcal{P}$, then $S$ has $\mathcal{P}$. This is Algebra, Lemma 10.164.6. Thus (1), (2) and (3) of Lemma 35.15.2 hold and we win. $\square$