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 $A$.
$\square$
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