The Stacks project

Lemma 10.72.2. Let $R$ be a ring, $I \subset R$ an ideal, and $M$ a finite $R$-module. Then $\text{depth}_ I(M)$ is equal to the supremum of the lengths of sequences $f_1, \ldots , f_ r \in I$ such that $f_ i$ is a nonzerodivisor on $M/(f_1, \ldots , f_{i - 1})M$.

Proof. Suppose that $IM = M$. Then Lemma 10.20.1 shows there exists an $f \in I$ such that $f : M \to M$ is $\text{id}_ M$. Hence $f, 0, 0, 0, \ldots $ is an infinite sequence of successive nonzerodivisors and we see agreement holds in this case. If $IM \not= M$, then we see that a sequence as in the lemma is an $M$-regular sequence and we conclude that agreement holds as well. $\square$

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