Lemma 69.19.2. Notation and assumptions as in Lemma 69.19.1. Let $g : X \to W$ correspond to $h : Y \to V$ via the equivalence. Then $g$ is quasi-compact, quasi-separated, separated, locally of finite presentation, of finite presentation, locally of finite type, of finite type, proper, integral, finite, and add more here if and only if $h$ is so.

**Proof.**
If $g$ is quasi-compact, quasi-separated, separated, locally of finite presentation, of finite presentation, locally of finite type, of finite type, proper, finite, so is $h$ as a base change of $g$ by Morphisms of Spaces, Lemmas 66.8.4, 66.4.4, 66.28.3, 66.23.3, 66.40.3, 66.45.5. Conversely, let $P$ be a property of morphisms of algebraic spaces which is étale local on the base and which holds for the identity morphism of any algebraic space. Since $\{ W^0 \to W, V \to W\} $ is an étale covering, to prove that $g$ has $P$ it suffices to show that $h$ has $P$. Thus we conclude using Morphisms of Spaces, Lemmas 66.8.8, 66.4.12, 66.28.4, 66.23.4, 66.40.2, 66.45.3.
$\square$

## 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Your email address will not be published.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).

Unfortunately JavaScript is disabled in your browser, so the comment preview function will not work.

All contributions are licensed under the GNU Free Documentation License.

## Comments (0)