Let (R,m) be a CM ring of dimension d, and M a finite R- module. Is pd M always finite?
Remember to vote up questions/answers you find interesting or helpful (requires 15 reputation points)
|
0
|
|
|
|
|
4
|
No. Put $k=R/m$. Then $k$ is of finite projective dimension if and only if $R$ is regular. This is the famous theorem of Auslander-Buchsbaum, Serre (see for instance Bruns-Herzog Cohen-Macaulay rings theorem 2.2.7 for a proof). So the residual field of a non-regular CM ring will give a counter-example. |
||
|
|
