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 10.118.11. Let $R$ be a domain with fraction field $K$. Let $M$ be an $R$-submodule of $K^{\oplus r}$. Assume $R$ is Noetherian of dimension $1$. For any nonzero $x \in R$ we have $\text{length}_ R(M/xM) < \infty $.

Proof. Since $R$ has dimension $1$ we see that $x$ is contained in finitely many primes $\mathfrak m_ i$, $i = 1, \ldots , n$, each maximal. Since $R$ is Noetherian we see that $R/xR$ is Artinian, see Proposition 10.59.6. Hence $R/xR$ is a quotient of $\prod R/\mathfrak m_ i^{e_ i}$ for certain $e_ i$ because that $\mathfrak m_1^{e_1} \ldots \mathfrak m_ n^{e_ n} \subset (x)$ for suitably large $e_ i$ as $R/xR$ is Artinian (see Section 10.52). Hence $M/xM$ similarly decomposes as a product $\prod (M/xM)_{\mathfrak m_ i} = \prod M/(\mathfrak m_ i^{e_ i}, x)M$ of its localizations at the $\mathfrak m_ i$. By Lemma 10.118.9 applied to $M_{\mathfrak m_ i}$ over $R_{\mathfrak m_ i}$ we see each $M_{\mathfrak m_ i}/xM_{\mathfrak m_ i} = (M/xM)_{\mathfrak m_ i}$ has finite length over $R_{\mathfrak m_ i}$. It easily follows that $M/xM$ has finite length over $R$. $\square$

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