Lemma 47.11.3. Let $A$ be a Noetherian ring, let $I \subset A$ be an ideal, and let $M$ a finite $A$-module with $IM \not= M$.

If $x \in I$ is a nonzerodivisor on $M$, then $\text{depth}_ I(M/xM) = \text{depth}_ I(M) - 1$.

Any $M$-regular sequence $x_1, \ldots , x_ r$ in $I$ can be extended to an $M$-regular sequence in $I$ of length $\text{depth}_ I(M)$.

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