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.
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. $\square$
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Your email address will not be published.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)