The Stacks project

Lemma 67.26.2. The following types of algebraic spaces are Nagata.

  1. Any algebraic space locally of finite type over a Nagata scheme.

  2. Any algebraic space locally of finite type over a field.

  3. Any algebraic space locally of finite type over a Noetherian complete local ring.

  4. Any algebraic space locally of finite type over $\mathbf{Z}$.

  5. Any algebraic space locally of finite type over a Dedekind ring of characteristic zero.

  6. And so on.

Proof. The first property holds by Lemma 67.26.1. Thus the others hold as well, see Morphisms, Lemma 29.18.2. $\square$

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