The Stacks project

\begin{equation*} \DeclareMathOperator\Coim{Coim} \DeclareMathOperator\Coker{Coker} \DeclareMathOperator\Ext{Ext} \DeclareMathOperator\Hom{Hom} \DeclareMathOperator\Im{Im} \DeclareMathOperator\Ker{Ker} \DeclareMathOperator\Mor{Mor} \DeclareMathOperator\Ob{Ob} \DeclareMathOperator\Sh{Sh} \DeclareMathOperator\SheafExt{\mathcal{E}\mathit{xt}} \DeclareMathOperator\SheafHom{\mathcal{H}\mathit{om}} \DeclareMathOperator\Spec{Spec} \newcommand\colim{\mathop{\mathrm{colim}}\nolimits} \newcommand\lim{\mathop{\mathrm{lim}}\nolimits} \newcommand\Qcoh{\mathit{Qcoh}} \newcommand\Sch{\mathit{Sch}} \newcommand\QCohstack{\mathcal{QC}\!\mathit{oh}} \newcommand\Cohstack{\mathcal{C}\!\mathit{oh}} \newcommand\Spacesstack{\mathcal{S}\!\mathit{paces}} \newcommand\Quotfunctor{\mathrm{Quot}} \newcommand\Hilbfunctor{\mathrm{Hilb}} \newcommand\Curvesstack{\mathcal{C}\!\mathit{urves}} \newcommand\Polarizedstack{\mathcal{P}\!\mathit{olarized}} \newcommand\Complexesstack{\mathcal{C}\!\mathit{omplexes}} \newcommand\Pic{\mathop{\mathrm{Pic}}\nolimits} \newcommand\Picardstack{\mathcal{P}\!\mathit{ic}} \newcommand\Picardfunctor{\mathrm{Pic}} \newcommand\Deformationcategory{\mathcal{D}\!\mathit{ef}} \end{equation*}

Lemma 10.123.1. Let $A \subset B$ be an extension of domains. Assume

  1. $A$ is a local Noetherian ring of dimension $1$,

  2. $A \to B$ is of finite type, and

  3. 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 10.120.2. We have equality if and only if $A \to B$ is finite.

Proof. The ring $B$ is semi-local by Lemma 10.112.2. Let $B'$ be the integral closure of $A$ in $B$. By Lemma 10.122.14 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 10.112.2. Let $\mathfrak p_ j$, $j = 1, \ldots , m$ be the other maximal ideals of $C$ (the “missing points”). By Lemma 10.120.8 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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