The Stacks project

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

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

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

Proof. Part (2) is a formal consequence of part (1). Let $x \in R$ be as in (1). By the short exact sequence $0 \to M \to M \to M/xM \to 0$ and Lemma 10.71.6 we see that the depth drops by at most 1. On the other hand, if $x_1, \ldots , x_ r \in \mathfrak m$ is a regular sequence for $M/xM$, then $x, x_1, \ldots , x_ r$ is a regular sequence for $M$. Hence we see that the depth drops by at least 1. $\square$


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