Lemma 35.18.3. The property $\mathcal{P}(S) =$“$S$ is locally Noetherian and $(R_ k)$” is local in the smooth topology.

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. $\square$

Comment #1658 by Lucas Braune on

The link to Algebra, Lemma 10.158.5 (Tag 0352, on local descent of Serre's property S_k) should instead point to Algebra, Lemma 10.159.6 (Tag 0353, on local descent of Serre's property R_k). That being said, this chapter is a pretty nice read!

Comment #1659 by Lucas Braune on

Oops, I meant the link should point to Algebra, Lemma 10.158.6 (Tag 0353).

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