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 10.65.9. Let $R$ be a ring. Let $M$ be an $R$-module. Let $\mathfrak p$ be a prime ideal of $R$ which is finitely generated. Then

\[ \mathfrak p \in \text{Ass}(M) \Leftrightarrow \mathfrak p \in \text{WeakAss}(M). \]

In particular, if $R$ is Noetherian, then $\text{Ass}(M) = \text{WeakAss}(M)$.

Proof. Write $\mathfrak p = (g_1, \ldots , g_ n)$ for some $g_ i \in R$. It is enough the prove the implication “$\Leftarrow $” as the other implication holds in general, see Lemma 10.65.6. Assume $\mathfrak p \in \text{WeakAss}(M)$. By Lemma 10.65.2 there exists an element $m \in M_{\mathfrak p}$ such that $I = \{ x \in R_{\mathfrak p} \mid xm = 0\} $ has radical $\mathfrak pR_{\mathfrak p}$. Hence for each $i$ there exists a smallest $e_ i > 0$ such that $g_ i^{e_ i}m = 0$ in $M_{\mathfrak p}$. If $e_ i > 1$ for some $i$, then we can replace $m$ by $g_ i^{e_ i - 1} m \not= 0$ and decrease $\sum e_ i$. Hence we may assume that the annihilator of $m \in M_{\mathfrak p}$ is $(g_1, \ldots , g_ n)R_{\mathfrak p} = \mathfrak p R_{\mathfrak p}$. By Lemma 10.62.15 we see that $\mathfrak p \in \text{Ass}(M)$. $\square$


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