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The Stacks project

Lemma 10.72.6. Let R be a local Noetherian ring. Let 0 \to N' \to N \to N'' \to 0 be a short exact sequence of nonzero 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 10.72.5, 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 10.71.6. \square

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