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 10.118.10. Let $R \to S$ be a homomorphism of domains inducing an injection of fraction fields $K \subset L$. If $R$ is Noetherian local of dimension $1$ and $[L : K] < \infty $ then

  1. each prime ideal $\mathfrak n_ i$ of $S$ lying over the maximal ideal $\mathfrak m$ of $R$ is maximal,

  2. there are finitely many of these, and

  3. $[\kappa (\mathfrak n_ i) : \kappa (\mathfrak m)] < \infty $ for each $i$.

Proof. Pick $x \in \mathfrak m$ nonzero. Apply Lemma 10.118.9 to the submodule $S \subset L \cong K^{\oplus n}$ where $n = [L : K]$. Thus the ring $S/xS$ has finite length over $R$. It follows that $S/\mathfrak m S$ has finite length over $\kappa (\mathfrak m)$. In other words, $\dim _{\kappa (\mathfrak m)} S/\mathfrak m S$ is finite (Lemma 10.51.6). Thus $S/\mathfrak mS$ is Artinian (Lemma 10.52.2). The structural results on Artinian rings implies parts (1) and (2), see for example Lemma 10.52.6. Part (3) is implied by the finiteness established above. $\square$


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