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 10.71.6. Let $R$ be a local Noetherian ring. Let $0 \to N' \to N \to N'' \to 0$ be a short exact sequence of finite $R$-modules.

  1. $\text{depth}(N) \geq \min \{ \text{depth}(N'), \text{depth}(N'')\} $

  2. $\text{depth}(N'') \geq \min \{ \text{depth}(N), \text{depth}(N') - 1\} $

  3. $\text{depth}(N') \geq \min \{ \text{depth}(N), \text{depth}(N'') + 1\} $

Proof. Use the characterization of depth using the Ext groups $\mathop{\mathrm{Ext}}\nolimits ^ i(\kappa , N)$, see Lemma 10.71.5, and use the long exact cohomology sequence

\[ \begin{matrix} 0 \to \mathop{\mathrm{Hom}}\nolimits _ R(\kappa , N') \to \mathop{\mathrm{Hom}}\nolimits _ R(\kappa , N) \to \mathop{\mathrm{Hom}}\nolimits _ R(\kappa , N'') \\ \phantom{0\ } \to \mathop{\mathrm{Ext}}\nolimits ^1_ R(\kappa , N') \to \mathop{\mathrm{Ext}}\nolimits ^1_ R(\kappa , N) \to \mathop{\mathrm{Ext}}\nolimits ^1_ R(\kappa , N'') \to \ldots \end{matrix} \]

from Lemma 10.70.6. $\square$


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