The Stacks project

Lemma 29.18.1. Let $f : X \to S$ be a morphism. If $S$ is Nagata and $f$ locally of finite type then $X$ is Nagata. If $S$ is universally Japanese and $f$ locally of finite type then $X$ is universally Japanese.

Proof. For “universally Japanese” this follows from Algebra, Lemma 10.162.4. For “Nagata” this follows from Algebra, Proposition 10.162.15. $\square$

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