# 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

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.

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)

• 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