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 10.50.5. Let $R$ be a Noetherian ring. Let $I \subset R$ be an ideal. Let $M$ be a finite $R$-module. Let $N = \bigcap _ n I^ n M$.

  1. For every prime $\mathfrak p$, $I \subset \mathfrak p$ there exists a $f \in R$, $f \not\in \mathfrak p$ such that $N_ f = 0$.

  2. If $I$ is contained in the Jacobson radical of $R$, then $N = 0$.

Proof. Proof of (1). Let $x_1, \ldots , x_ n$ be generators for the module $N$, see Lemma 10.50.1. For every prime $\mathfrak p$, $I \subset \mathfrak p$ we see that the image of $N$ in the localization $M_{\mathfrak p}$ is zero, by Lemma 10.50.4. Hence we can find $g_ i \in R$, $g_ i \not\in \mathfrak p$ such that $x_ i$ maps to zero in $N_{g_ i}$. Thus $N_{g_1g_2\ldots g_ n} = 0$.

Part (2) follows from (1) and Lemma 10.22.1. $\square$


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