Lemma 10.71.7. Let $R$ be a local Noetherian ring and $M$ a nonzero finite $R$-module.

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

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

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