Lemma 38.11.4. Let $f : X \to S$ be a morphism of schemes which is flat and proper. Let $U \subset S$ be a dense open such that $X_ U \to U$ is finite. If also either $f$ is locally of finite presentation or $U \subset S$ is retrocompact, then $f$ is finite.

**Proof.**
By Lemma 38.11.3 the fibres of $f$ have dimension zero. Hence $f$ is quasi-finite (Morphisms, Lemma 29.29.5) whence has finite fibres (Morphisms, Lemma 29.20.10). Hence $f$ is finite by More on Morphisms, Lemma 37.44.1.
$\square$

## 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Your email address will not be published.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)