pd finite for finite module over local CM ring? - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-18T06:33:10Z http://mathoverflow.net/feeds/question/87407 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/87407/pd-finite-for-finite-module-over-local-cm-ring pd finite for finite module over local CM ring? Andrei 2012-02-03T08:35:35Z 2012-02-03T08:52:55Z <p>Let (R,m) be a CM ring of dimension d, and M a finite R- module. Is pd M always finite?</p> http://mathoverflow.net/questions/87407/pd-finite-for-finite-module-over-local-cm-ring/87411#87411 Answer by Olivier for pd finite for finite module over local CM ring? Olivier 2012-02-03T08:52:55Z 2012-02-03T08:52:55Z <p>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. </p>