The Stacks project

65.26 Nagata spaces

See Properties of Spaces, Section 64.7 for the definition of a Nagata algebraic space.

Lemma 65.26.1. Let $S$ be a scheme. Let $f : X \to Y$ be a morphism of algebraic spaces over $S$. If $Y$ is Nagata and $f$ locally of finite type then $X$ is Nagata.

Proof. Let $V$ be a scheme and let $V \to Y$ be a surjective ├ętale morphism. Let $U$ be a scheme and let $U \to X \times _ Y V$ be a surjective ├ętale morphism. If $Y$ is Nagata, then $V$ is a Nagata scheme. If $X \to Y$ is locally of finite type, then $U \to V$ is locally of finite type. Hence $V$ is a Nagata scheme by Morphisms, Lemma 29.18.1. Then $X$ is Nagata by definition. $\square$

Lemma 65.26.2. 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 65.26.1. Thus the others hold as well, see Morphisms, Lemma 29.18.2. $\square$


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