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Let (R,m) be a CM ring of dimension d, and M a finite R- module. Is pd M always finite?

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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.

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