The Stacks project

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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