global dimension, then we have $$R^i\Gamma_\mfa(M)\cong \operatorname{Tor}_{p-i}^A(M,D)$$ for $M\in\operatorname{Mod}A$ and $i\ge0$, where $p=\Pty(R,G)$ and $D=R^p\Gamma_\mfa(A)$.

We will list some examples in Sections \[sec-quot\], \[sec6\] and \[sec7\] which satisfy Setup \[set\]. In case the algebra $R$ is commutative, then most conditions in Setup \[set\] are automatically satisfied.

Theorem \[thm0.1\] is obtained via an equivalence of quotient categories associated to the ideals $\mfa$ and $\mfr$. Let $\operatorname{Mod}A$ be the category of right $A$-modules, and let $\operatorname{Tor}_\mfa A$ be the full subcategory of $\operatorname{Mod}A$ consisting of $\mfa$-torsion modules. Since $R$ is noetherian and $G$ is finite, $A$ is noetherian as well. Thus $\operatorname{Tor}_\mfa A$ is a Serre subcategory of $\operatorname{Mod}A$. Hence we obtain an abelian category $\QMod_\mfa A:=\frac{\operatorname{Mod}A}{\operatorname{Tor}_\mfa A}$. Similarly, let $\mfb=\mfr\otimes\kk G\subseteq B=R*G$, we have $\QMod_\mfb B:=\frac{\operatorname{Mod}B}{\operatorname{Tor}_\mfb B}$. The following equivalence of quotient categories holds (for the terminology, see Section \[sec-quot\]).

\[Theorem 6.4\] Let $R$ and $G$ be as above. Assume that $\mfa$ and $\mfr$ are cofinal, and $\mfa$ has the right AR-property.

-   The functor $\operatorname{Hom}_A(R,-):\operatorname{Mod}A\longrightarrow \operatorname{Mod}B$ induces a functor $$\operatorname{Hom}_\mathcal{A}(\mathcal{R},-):\QMod_\mfa A\longrightarrow \QMod_\mfb B.$$

-   The functors $-\otimes_\mathcal{B}\mathcal{R}$ and $\operatorname{Hom}_\mathcal{A}(\mathcal{R},-)$ are quasi-inverse to each other.

If $R$ is a commutative algebra, then the cofinal condition in the above theorem is indeed 