Lemma 73.11.10. The property $\mathcal{P}(f) =$“$f$ is locally of finite presentation” is fpqc local on the base.

**Proof.**
We will use Lemma 73.10.4 to prove this. Assumptions (1) and (2) of that lemma follow from Morphisms of Spaces, Lemma 66.28.4. Let $Z' \to Z$ be a surjective flat morphism of affine schemes over $S$. Let $f : X \to Z$ be a morphism of algebraic spaces, and assume that the base change $f' : Z' \times _ Z X \to Z'$ is locally of finite presentation. We have to show that $f$ is locally of finite presentation. Let $U$ be a scheme and let $U \to X$ be surjective and étale. By Morphisms of Spaces, Lemma 66.28.4 again, it is enough to show that $U \to Z$ is locally of finite presentation. Since $f'$ is locally of finite presentation, and since $Z' \times _ Z U$ is a scheme étale over $Z' \times _ Z X$ we conclude (by the same lemma again) that $Z' \times _ Z U \to Z'$ is locally of finite presentation. As $\{ Z' \to Z\} $ is an fpqc covering we conclude that $U \to Z$ is locally of finite presentation by Descent, Lemma 35.23.11 as desired.
$\square$

## 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Your email address will not be published.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)