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 $\text{depth}_ I(M)$.
Proof. Part (2) is a formal consequence of part (1). Let $x \in I$ be as in (1). By the short exact sequence $0 \to M \to M \to M/xM \to 0$ and Lemma 47.11.2 we see that $\text{depth}_ I(M/xM) \geq \text{depth}_ I(M) - 1$. On the other hand, if $x_1, \ldots , x_ r \in I$ is a regular sequence for $M/xM$, then $x, x_1, \ldots , x_ r$ is a regular sequence for $M$. Hence (1) holds. $\square$
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