The Stacks project

Lemma 32.27.2. Let $k$ be a field. Let $f : Y \to X$ be a quasi-compact morphism of locally algebraic schemes over $k$. Let $X'$ be the normalization of $X$ in $Y$. If $Y$ is reduced, then $X' \to X$ is finite.

Proof. Since $Y$ is quasi-separated (by Properties, Lemma 27.5.4 and Morphisms, Lemma 28.14.6) the morphism $f$ is quasi-separated (Schemes, Lemma 25.21.13). Hence Morphisms, Definition 28.51.3 applies. The result follows from Morphisms, Lemma 28.51.14. This uses that locally algebraic schemes are locally Noetherian (hence have locally finitely many irreducible components) and that locally algebraic schemes are Nagata (Morphisms, Lemma 28.17.2). Some small details omitted. $\square$

Comments (0)

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Your email address will not be published.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.

In order to prevent bots from posting comments, we would like you to prove that you are human. You can do this by filling in the name of the current tag in the following input field. As a reminder, this is tag 0BXS. Beware of the difference between the letter 'O' and the digit '0'.