The Stacks project

Lemma 29.53.14. Let $f : X \to S$ be a morphism. Assume that

  1. $S$ is a Nagata scheme,

  2. $f$ is quasi-compact and quasi-separated,

  3. quasi-compact opens of $X$ have finitely many irreducible components,

  4. if $x \in X$ is a generic point of an irreducible component, then the field extension $\kappa (f(x)) \subset \kappa (x)$ is finitely generated, and

  5. $X$ is reduced.

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

Proof. There is an immediate reduction to the case $S = \mathop{\mathrm{Spec}}(R)$ where $R$ is a Nagata ring by assumption (1). We have to show that the integral closure $A$ of $R$ in $\Gamma (X, \mathcal{O}_ X)$ is finite over $R$. Since $f$ is quasi-compact by assumption (2) we can write $X = \bigcup _{i = 1, \ldots , n} U_ i$ with each $U_ i$ affine. Say $U_ i = \mathop{\mathrm{Spec}}(B_ i)$. Each $B_ i$ is reduced by assumption (5) and has finitely many minimal primes $\mathfrak q_{i1}, \ldots , \mathfrak q_{im_ i}$ by assumption (3) and Algebra, Lemma 10.26.1. We have

\[ \Gamma (X, \mathcal{O}_ X) \subset B_1 \times \ldots \times B_ n \subset \prod \nolimits _{i = 1, \ldots , n} \prod \nolimits _{j = 1, \ldots , m_ i} (B_ i)_{\mathfrak q_{ij}} \]

the second inclusion by Algebra, Lemma 10.25.2. We have $\kappa (\mathfrak q_{ij}) = (B_ i)_{\mathfrak q_{ij}}$ by Algebra, Lemma 10.25.1. Hence the integral closure $A$ of $R$ in $\Gamma (X, \mathcal{O}_ X)$ is contained in the product of the integral closures $A_{ij}$ of $R$ in $\kappa (\mathfrak q_{ij})$. Since $R$ is Noetherian it suffices to show that $A_{ij}$ is a finite $R$-module for each $i, j$. Let $\mathfrak p_{ij} \subset R$ be the image of $\mathfrak q_{ij}$. As $\kappa (\mathfrak p_{ij}) \subset \kappa (\mathfrak q_{ij})$ is a finitely generated field extension by assumption (4), we see that $R \to \kappa (\mathfrak q_{ij})$ is essentially of finite type. Thus $R \to A_{ij}$ is finite by Algebra, Lemma 10.162.2. $\square$


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Your email address will not be published.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 0AVK. Beware of the difference between the letter 'O' and the digit '0'.