Lemma 10.103.3. Let $R$ be a Noetherian local ring. Let $M$ be a Cohen-Macaulay module over $R$. Suppose $g \in \mathfrak m$ is such that $\dim (\text{Supp}(M) \cap V(g)) = \dim (\text{Supp}(M)) - 1$. Then (a) $g$ is a nonzerodivisor on $M$, and (b) $M/gM$ is Cohen-Macaulay of depth one less.

Proof. Choose a $M$-regular sequence $f_1, \ldots , f_ d$ with $d = \dim (\text{Supp}(M))$. If $g$ is good with respect to $(M, f_1, \ldots , f_ d)$ we win by Lemma 10.103.2. In particular the lemma holds if $d = 1$. (The case $d = 0$ does not occur.) Assume $d > 1$. Choose an element $h \in R$ such that (i) $h$ is good with respect to $(M, f_1, \ldots , f_ d)$, and (ii) $\dim (\text{Supp}(M) \cap V(h, g)) = d - 2$. To see $h$ exists, let $\{ \mathfrak q_ j\}$ be the (finite) set of minimal primes of the closed sets $\text{Supp}(M)$, $\text{Supp}(M)\cap V(f_1, \ldots , f_ i)$, $i = 1, \ldots , d - 1$, and $\text{Supp}(M) \cap V(g)$. None of these $\mathfrak q_ j$ is equal to $\mathfrak m$ and hence we may find $h \in \mathfrak m$, $h \not\in \mathfrak q_ j$ by Lemma 10.15.2. It is clear that $h$ satisfies (i) and (ii). From Lemma 10.103.2 we conclude that $M/hM$ is Cohen-Macaulay. By (ii) we see that the pair $(M/hM, g)$ satisfies the induction hypothesis. Hence $M/(h, g)M$ is Cohen-Macaulay and $g : M/hM \to M/hM$ is injective. By Lemma 10.68.4 we see that $g : M \to M$ and $h : M/gM \to M/gM$ are injective. Combined with the fact that $M/(g, h)M$ is Cohen-Macaulay this finishes the proof. $\square$

Comment #2957 by Dario Weißmann on

I think there is a shorter proof: By assumption $g$ is not contained in any of the minimal primes of the support of $M$. Thus not contained in any of the minimal associated primes, Lemma 10.62.6. By Lemma 10.102.7 there are no embedded associated primes. Applying Lemma 10.62.9 we see that $g$ is a nonzerodivisor on $M$. Lemma 10.102.5 finishes the proof.

Of course the results in this chapter would need to be rearranged, but Lemmas 10.102.5 and 10.102.7 only use results from earlier sections anyway.

Comment #2958 by Dario Weißmann on

I think this would also make Lemma 10.102.2 and its notation obsolete.

Comment #3084 by on

Dear Dario, yes this lemma is left over from the attempt I made to write about Cohen-Macaulay modules with very little general theory about depth and regular sequences. The steps are in 10.103.2, 10.103.3, and 10.103.4 using the notion of a good element. I do still think it is somewhat fun that this can be done, so I am going to leave it as is for now.

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