# 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

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