# Maximal Cohen Macaulay modules over regular factor rings.

Hi,

my question is simple. Let (R,m) be a commutative regular local noetherian ring. Is it true that for every prime p \in Spec(R), the factor ring R/p has maximal cohen-macaulay R/p-module?

Best Regards, David

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David, why did you accept your own answer? – S. Carnahan Nov 19 '10 at 6:22
Sorry for it. I'm quite a newbie here. – David Nov 21 '10 at 22:14

## 3 Answers

If I remember correctly, certainly in the geometric case (that is $R$ is essentially of finite type over a field), Hochster proved the existence of Big Cohen-Macaulay modules. But the existence of finitely generated CM modules is open even in dimension three in the geometric case.

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I think this is certainly true if $p$ is generated by a regular sequence. In this case, $R/p$ is regular, local and hence Cohen Macaulay. Hence, $R/p$ is a maximal Cohen Macaulay module over $R/Ann(R/p)$. But $Ann(R/p)=p$ since $p$ is prime. So, $R/p$ is a maximal Cohen Macaulay $R/p$-module.

Essentially, all you need is for $R/p$ to be Cohen Macaulay. But probably weaker conditions might suffice (Edit: Also, $R/p$ is Cohen Macaulay when $R$ is regular, local, iff $p$ has height $1$)

The question is similar to small Cohen-Macaulay module conjecture where we ask the same question over a complete local ring (which I believe is open)

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Thank you for your answer. You are right. But what if p is not generated by a regular sequence? – David Nov 18 '10 at 16:40
I'm dealing with the situation when R/p is not Cohen-Macaulay ring. – David Nov 18 '10 at 16:44
I am not sure yet, but I will think about it. . – Timothy Wagner Nov 18 '10 at 16:45
I agree that this is quite close to small Cohen-Macaulay module conjecture. But I'm dealing with "special" rings of the shape R/p, where R is regular and p is prime. So I hope that this conjecture is solved for these rings. – David Nov 18 '10 at 17:02
I see that I left out part of what I meant to say above: since for the small MCMs conjecture we assume the ring is complete, it is a quotient of a regular ring by Cohen Structure. So your question is exactly the small MCMs conjecture for domains. – Graham Leuschke Nov 18 '10 at 19:00

thank you all again. now i understand that i have no chance to find these smcm modules if R has dim geq 5. let me ask you weaker question: for every p in Spec R (R regular local) is there an R-module Mp with projdimMp = ht p and p in Ass(Mp)? this should be weaker than the previous question. so any chance of this existence?

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$p$ contains a regular sequence $x$ of length equal to its height, since R is CM. Set $M=R/(x)$. – Graham Leuschke Nov 18 '10 at 20:09
Thank you Graham. Any references? – David Nov 18 '10 at 20:32
OK I think I have it. Thank you again. – David Nov 18 '10 at 21:27
This belongs as part of the question, or a comment on one of the other answers. – S. Carnahan Nov 19 '10 at 6:22
@David, if you go ahead and put dollar-sign symbols "\$" around your LaTeX keywords such as '\ge' (for *geq*=greater than or equal to) and remember to the backslash in front of those keywords, it will greatly improve the readability of your postings on this forum. Also, you can uncheck your own answer (this one) and put the content of what you've typed in this answer as an edited version of the question at the top of this page. Any comments about how you came to work on this question and your lines of reasoning would also be informative, and may get you more help. – sleepless in beantown Nov 23 '10 at 4:15