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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