(In the following, a 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, 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.

(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, 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.

(In the following, a ring $R$ is Gorenstein if it has finite injective dimension as a module over itself on either side, and a finitely generated $R$-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, 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.

(In the following, a 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, 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.