Lemma 67.26.1. 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.
67.26 Nagata spaces
See Properties of Spaces, Section 66.7 for the definition of a Nagata algebraic space.
Proof. Let $V$ be a scheme and let $V \to Y$ be a surjective étale morphism. Let $U$ be a scheme and let $U \to X \times _ Y V$ be a surjective étale 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 67.26.2. The following types of algebraic spaces are Nagata.
Any algebraic space locally of finite type over a Nagata scheme.
Any algebraic space locally of finite type over a field.
Any algebraic space locally of finite type over a Noetherian complete local ring.
Any algebraic space locally of finite type over $\mathbf{Z}$.
Any algebraic space locally of finite type over a Dedekind ring of characteristic zero.
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. $\square$
Post a comment
Your email address will not be published. Required fields are marked.
In your comment you can use Markdown and LaTeX style mathematics (enclose it like $\pi$). A preview option is available if you wish to see how it works out (just click on the eye in the toolbar).
All contributions are licensed under the GNU Free Documentation License.
Comments (0)