Lemma 65.28.9. Let $S$ be a scheme. Let $f : X \to Y$ and $Y \to Z$ be morphisms of algebraic spaces over $S$. If $X$ is locally of finite presentation over $Z$, and $Y$ is locally of finite type over $Z$, then $f$ is locally of finite presentation.

**Proof.**
Choose a scheme $W$ and a surjective étale morphism $W \to Z$. Then choose a scheme $V$ and a surjective étale morphism $V \to W \times _ Z Y$. Finally choose a scheme $U$ and a surjective étale morphism $U \to V \times _ Y X$. By definition $U$ is locally of finite presentation over $W$ and $V$ is locally of finite type over $W$. By Morphisms, Lemma 29.21.11 the morphism $U \to V$ is locally of finite presentation. Hence $f$ is locally of finite presentation.
$\square$

## 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Your email address will not be published.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)