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.

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

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

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

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