(In the following, a (not necessarily commutative) ring $R$ is *Gorenstein* if it has finite injective dimension as a module over itself on either side, and a finitely generated (right) $R$-module is *maximal Cohen-Macaulay* (MCM) if $\text{Ext}\_R^i(M,R)=0$ for all $i \geqslant1$.)


In the paper [Maximal Cohen-Macaulay Modules and Tate-Cohomology Over Gorenstein Rings](https://tspace.library.utoronto.ca/handle/1807/16682), Buchweitz demonstrates that for a noetherian, Gorenstein ring the stabilized derived category of $R$ is equivalent to the stable module category of $R$, restricted to MCM objects, written $\underline{\text{MCM}}(R)$.


It is clear that the stabilized derived category of a noetherian, Gorenstein ring $R$ is trivial if $\text{gldim } R < \infty$, but (without making use of the above equivalence) I can't see why $\underline{\text{MCM}}(R)$ should be trivial in this case. It seems that an equivalent problem would be to show that for MCM modules $M$ and $N$, we have $\text{Hom}\_R(M,N) = NM^\*$, where $M^\* = \text{Hom}\_R(M,R)$, but I haven't made much progress with this either.



Showing that the stable module category of a ring $R$ restricted to maximal Cohen-Macaulay objects is trivial if $\text{gldim } R < \infty$
If $M$ is a maximal Cohen-Macaulay module for a Noetherian Gorenstein ring $R$, then it follows easily by induction on the projective dimension of $N$ that $\operatorname{Ext}^i\_R(M,N)=0$ for all $i>0$ if $N$ is finitely generated of finite projective dimension.


So if $R$ also has finite global dimension, then $\operatorname{Ext}^i(M,N)=0$ for $i>0$ for all finitely generated modules $N$, and so $M$ is projective.


So when $R$ has finite global dimension, all $MCM$ modules are projective, so the stable module category of $MCM$ modules is trivial.


