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 10.118.1. Let $R$ be a local Noetherian domain with fraction field $K$. Assume $R$ is not a field. Then there exist $R \subset R' \subset K$ with

  1. $R'$ local Noetherian of dimension $1$,

  2. $R \to R'$ a local ring map, i.e., $R'$ dominates $R$, and

  3. $R \to R'$ essentially of finite type.

Proof. Choose any valuation ring $A \subset K$ dominating $R$ (which exist by Lemma 10.49.2). Denote $v$ the corresponding valuation. Let $x_1, \ldots , x_ r$ be a minimal set of generators of the maximal ideal $\mathfrak m$ of $R$. We may and do assume that $v(x_ r) = \min \{ v(x_1), \ldots , v(x_ r)\} $. Consider the ring

\[ S = R[x_1/x_ r, x_2/x_ r, \ldots , x_{r - 1}/x_ r] \subset K. \]

Note that $\mathfrak mS = x_ rS$ is a principal ideal. Note that $S \subset A$ and that $v(x_ r) > 0$, hence we see that $x_ rS \not= S$. Choose a minimal prime $\mathfrak q$ over $x_ rS$. Then $\text{height}(\mathfrak q) = 1$ by Lemma 10.59.10 and $\mathfrak q$ lies over $\mathfrak m$. Hence we see that $R' = S_{\mathfrak q}$ is a solution. $\square$


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