## 29.18 Nagata schemes, reprise

See Properties, Section 28.13 for the definitions and basic properties of Nagata and universally Japanese schemes.

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. $\square$

Lemma 29.18.2. The following types of schemes are Nagata.

1. Any scheme locally of finite type over a field.

2. Any scheme locally of finite type over a Noetherian complete local ring.

3. Any scheme locally of finite type over $\mathbf{Z}$.

4. Any scheme locally of finite type over a Dedekind ring of characteristic zero.

5. And so on.

Proof. By Lemma 29.18.1 we only need to show that the rings mentioned above are Nagata rings. For this see Algebra, Proposition 10.162.16. $\square$

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