I will warmly appreciate any body who could give me, even, a hint or a reference on the following problem. I'm really confused with that!
Let R be a commutative noetherian ring and let $\Lambda$ be an R-algebra which is finitely generated as $R$-module. If $\Lambda$ is a Gorenstein ring in the sense that it has finite self-injective dimension, say at most $n$, both on the left and right, I can prove that ${\rm Hom}_R(\Lambda, R)$ has finite projective dimension both as left and right $\Lambda$-module.
Having accepted this fact, now my question is: is it possible to construct a projective resolution of ${\rm Hom}_R(\Lambda, R)$ of the form $$ 0\rightarrow P_n\rightarrow\cdots\rightarrow P_1\rightarrow P_0\rightarrow {\rm Hom}_R(\Lambda, R)\rightarrow 0$$ with the property that all the modules are projective $\Lambda-\Lambda$-bimodules and all the maps are homomorphisms of bimodules?
I'm asking this question because ${\rm Hom}_R(\Lambda, R)$ is a $\Lambda-\Lambda$-bimodule, which is finitely generated and has finite projective dimension on both sides.