Let $R$ be a local Cohen-Macaulay Noetherian ring. A maximal Cohen-Macaulay module or mCM-module is an $R$-module $M$ of finite type such that $\text{dim }M = \text{depth }M =d$ A module $M$ is of finite projective dimension if it has a finite resolution by projectives.
When is a MCM module M of finite projective dimension?
I suppose that the answer is not "always", so a counter-example, or an indirect proof of existence of such a counter-example, would be welcome. A nice criterion for an mCM module to be of finite pd would also be very useful. In particular, for what kind of CM local rings is it true that all mCM modules have finite projective dimension? Regular rings, and equidimensional ring of dim 1 (see below) have this property.
What I (think I) know:
(i) If R is not CM, the question still makes sense, but the answer is never. Indeed, a local ring with a CM module of finite projective dimension is itself CM.
(ii) Back to the situation of the question, where $R$ is CM, I think that if $R$ has dimension 1, and say has equal characteristic, then the answer is "always": all mCM modules have finite projective dimension. For we can assume R complete by Noether's normalization, construing a subring $K[[X]]$ in $R$ such that $R$ is finite over $K[[X]]$. In this case, a finite-type module $M$ on $R$ is MCM on $R$ if and only if it is MCM on $K[[X]]$, that is if and only if it is free on $k[[X]]$. Now for any mCM module $N$ over $R$, $M \otimes_R N$ is torsion-free over $k[[X]]$, hence free over $k[[X]]$ (since this ring is a PID), hence an MCM $R$-module. But Cor 3.5 of [Ioshida] says that this lies that M is perfect, meaning that it has finite projective dimension in addition of being CM in the terminology of the cited article.
[Ioshida] Ken-Ichi Ioshida, Tensor products of perfect modules and maximal surjective, Journal of Pure and Applied Algebra 123 (1998) 313-326
