Lemma 10.69.6 (061S)—The Stacks project

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Table of contents
Part 1: Preliminaries
Chapter 10: Commutative Algebra
Section 10.69: Quasi-regular sequences
Lemma 10.69.6 (cite)

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Lemma 10.69.6. Let $(R, \mathfrak m)$ be a local Noetherian ring. Let $M$ be a nonzero finite $R$-module. Let $f_1, \ldots , f_ c \in \mathfrak m$ be an $M$-quasi-regular sequence. Then $f_1, \ldots , f_ c$ is an $M$-regular sequence.

Proof. Set $J = (f_1, \ldots , f_ c)$. Let us show that $f_1$ is a nonzerodivisor on $M$. Suppose $x \in M$ is not zero. By Krull's intersection theorem there exists an integer $r$ such that $x \in J^ rM$ but $x \not\in J^{r + 1}M$, see Lemma 10.51.4. Then $f_1 x \in J^{r + 1}M$ is an element whose class in $J^{r + 1}M/J^{r + 2}M$ is nonzero by the assumed structure of $\bigoplus J^ nM/J^{n + 1}M$. Whence $f_1x \not= 0$.

Now we can finish the proof by induction on $c$ using Lemma 10.69.5. $\square$

Comments (2)

Comment #2656 by Ko Aoki on July 23, 2017 at 05:00

I think "By Krull's intersection theorem" is more straightforward than "By the Artin-Rees lemma". Actually the lemma cited in the sentence is Krull's intersection theorem.

Comment #2672 by Johan on July 28, 2017 at 17:45

OK, yes, I agree. Fixed here.

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