Lemma 65.39.5. An étale morphism of algebraic spaces is locally quasi-finite.

**Proof.**
Let $X \to Y$ be an étale morphism of algebraic spaces, see Properties of Spaces, Definition 64.16.2. By Properties of Spaces, Lemma 64.16.3 we see this means there exists a diagram as in Lemma 65.22.1 with $h$ étale and surjective vertical arrow $a$. By Morphisms, Lemma 29.35.6 $h$ is locally quasi-finite. Hence $X \to Y$ is locally quasi-finite 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Your email address will not be published.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).

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)