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 10.71.4. Let $R$ be a Noetherian ring, $I \subset R$ an ideal, and $M$ a finite nonzero $R$-module such that $IM \not= M$. Then $\text{depth}_ I(M) < \infty $.

Proof. Since $M/IM$ is nonzero we can choose $\mathfrak p \in \text{Supp}(M/IM)$ by Lemma 10.39.2. Then $(M/IM)_\mathfrak p \not= 0$ which implies $I \subset \mathfrak p$ and moreover implies $M_\mathfrak p \not= IM_\mathfrak p$ as localization is exact. Let $f_1, \ldots , f_ r \in I$ be an $M$-regular sequence. Then $M_\mathfrak p/(f_1, \ldots , f_ r)M_\mathfrak p$ is nonzero as $(f_1, \ldots , f_ r) \subset I$. As localization is flat we see that the images of $f_1, \ldots , f_ r$ form a $M_\mathfrak p$-regular sequence in $I_\mathfrak p$. Since this works for every $M$-regular sequence in $I$ we conclude that $\text{depth}_ I(M) \leq \text{depth}_{I_\mathfrak p}(M_\mathfrak p)$. The latter is $\leq \text{depth}(M_\mathfrak p)$ which is $< \infty $ by Lemma 10.71.3. $\square$


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