Lemma 47.11.3 (0BUW)—The Stacks project

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Table of contents
Part 3: Topics in Scheme Theory
Chapter 47: Dualizing Complexes
Section 47.11: Depth
Lemma 47.11.3 (cite)

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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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