Lemma 29.53.15. Let $f : X \to S$ be a morphism. Assume that

$S$ is a Nagata scheme,

$f$ is of finite type,

$X$ is reduced.

Then the normalization $\nu : S' \to S$ of $S$ in $X$ is finite.

Lemma 29.53.15. Let $f : X \to S$ be a morphism. Assume that

$S$ is a Nagata scheme,

$f$ is of finite type,

$X$ is reduced.

Then the normalization $\nu : S' \to S$ of $S$ in $X$ is finite.

**Proof.**
This is a special case of Lemma 29.53.14. Namely, (2) holds as the finite type morphism $f$ is quasi-compact by definition and quasi-separated by Lemma 29.15.7. Condition (3) holds because $X$ is locally Noetherian by Lemma 29.15.6. Finally, condition (4) holds because a finite type morphism induces finitely generated residue field extensions.
$\square$

Your email address will not be published.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)