Lemma 10.124.1 (02MM)—The Stacks project

Lemma 10.124.1. Let $A \subset B$ be an extension of domains. Assume

$A$ is a local Noetherian ring of dimension $1$ ,
$A \to B$ is of finite type, and
the induced extension $L/K$ of fraction fields is finite.

Then $B$ is semi-local. Let $x \in \mathfrak m_ A$ , $x \not= 0$ . Let $\mathfrak m_ i$ , $i = 1, \ldots , n$ be the maximal ideals of $B$ . Then

$[L : K]\text{ord}_ A(x) \geq \sum \nolimits _ i [\kappa (\mathfrak m_ i) : \kappa (\mathfrak m_ A)] \text{ord}_{B_{\mathfrak m_ i}}(x)$

where $\text{ord}$ is defined as in Definition 10.121.2. We have equality if and only if $A \to B$ is finite.

Proof. The ring $B$ is semi-local by Lemma 10.113.2. Let $B'$ be the integral closure of $A$ in $B$ . By Lemma 10.123.14 we can find a finite $A$ -subalgebra $C \subset B'$ such that on setting $\mathfrak n_ i = C \cap \mathfrak m_ i$ we have $C_{\mathfrak n_ i} \cong B_{\mathfrak m_ i}$ and the primes $\mathfrak n_1, \ldots , \mathfrak n_ n$ are pairwise distinct. The ring $C$ is semi-local by Lemma 10.113.2. Let $\mathfrak p_ j$ , $j = 1, \ldots , m$ be the other maximal ideals of $C$ (the “missing points”). By Lemma 10.121.8 we have

$\text{ord}_ A(x^{[L : K]}) = \sum \nolimits _ i [\kappa (\mathfrak n_ i) : \kappa (\mathfrak m_ A)] \text{ord}_{C_{\mathfrak n_ i}}(x) + \sum \nolimits _ j [\kappa (\mathfrak p_ j) : \kappa (\mathfrak m_ A)] \text{ord}_{C_{\mathfrak p_ j}}(x)$

hence the inequality follows. In case of equality we conclude that $m = 0$ (no “missing points”). Hence $C \subset B$ is an inclusion of semi-local rings inducing a bijection on maximal ideals and an isomorphism on all localizations at maximal ideals. So if $b \in B$ , then $I = \{ x \in C \mid xb \in C\}$ is an ideal of $C$ which is not contained in any of the maximal ideals of $C$ , and hence $I = C$ , hence $b \in C$ . Thus $B = C$ and $B$ is finite over $A$ . $\square$

The Stacks project

Comments (0)

Post a comment