Lemma 47.11.2. Let A be a Noetherian ring. Let 0 \to N' \to N \to N'' \to 0 be a short exact sequence of finite A-modules. Let I \subset A be an ideal.
\text{depth}_ I(N) \geq \min \{ \text{depth}_ I(N'), \text{depth}_ I(N'')\}
\text{depth}_ I(N'') \geq \min \{ \text{depth}_ I(N), \text{depth}_ I(N') - 1\}
\text{depth}_ I(N') \geq \min \{ \text{depth}_ I(N), \text{depth}_ I(N'') + 1\}
Proof. Assume IN \not= N, IN' \not= N', and IN'' \not= N''. Then we can use the characterization of depth using the Ext groups \mathop{\mathrm{Ext}}\nolimits ^ i(A/I, N), see Lemma 47.11.1, and use the long exact cohomology sequence
from Algebra, Lemma 10.71.6. This argument also works if IN = N because in this case \mathop{\mathrm{Ext}}\nolimits ^ i_ A(A/I, N) = 0 for all i. Similarly in case IN' \not= N' and/or IN'' \not= N''. \square
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