The Stacks project

Lemma 35.15.5. The property $\mathcal{P}(S) =$“$S$ is Nagata” is local in the smooth topology.

Proof. We will check (1), (2) and (3) of Lemma 35.12.2. First we note that being Nagata is local in the Zariski topology. This is Properties, Lemma 28.13.6. Next, we show that if $S' \to S$ is a smooth morphism of affines and $S$ is Nagata, then $S'$ is Nagata. This is Morphisms, Lemma 29.18.1. Finally, we show that if $S' \to S$ is a surjective smooth morphism of affines and $S'$ is Nagata, then $S$ is Nagata. This is Algebra, Lemma 10.164.7. Thus (1), (2) and (3) of Lemma 35.12.2 hold and we win. $\square$


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