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.

## 29.18 Nagata schemes, reprise

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

**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.

Any scheme locally of finite type over a field.

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

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

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

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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