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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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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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  • $\begingroup$ Thank you for your answer. You are right. But what if p is not generated by a regular sequence? $\endgroup$
    – David
    Commented Nov 18, 2010 at 16:40
  • $\begingroup$ I'm dealing with the situation when R/p is not Cohen-Macaulay ring. $\endgroup$
    – David
    Commented Nov 18, 2010 at 16:44
  • $\begingroup$ I am not sure yet, but I will think about it. . $\endgroup$
    – Timothy Wagner
    Commented Nov 18, 2010 at 16:45
  • $\begingroup$ 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. $\endgroup$
    – David
    Commented Nov 18, 2010 at 17:02
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    $\begingroup$ 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. $\endgroup$
    – Graham Leuschke
    Commented Nov 18, 2010 at 19:00

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