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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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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