find pertinent sequences which lead to determine elements in the radical.

The pertinency of a group action $\Pty(R,G)$ introduced in [@BHZ] is equal to $\operatorname{GKdim}(R)-\operatorname{GKdim}(R/\mfr(R,G))$ (cf. Section \[sec3\]). The condition $\Pty(R,G)=1$ is related to Shephard-Todd-Chevalley’s Theorem (see Propositions \[prop2-3\] and \[prop2-4\]). As it was shown in [@BHZ] that the condition $\Pty(R,G)\ge2$ is equivalent to the Auslander Theorem when $R$ is GK-Cohen-Macaulay. We recover some known results in Section \[set4\] by using pertinent sequences .

Set $A:=R^G$, $\mfr=\mfr(R,G)$ and $\mfa=A\cap \mfr$. We mainly focus on the local cohomology of $A$-modules associated to the ideal $\mfa$. In Section \[sec5\], we established some isomorphisms of the local cohomology (cf. Theorem \[thm-locdual\]). As one of our main results, we have the following theorem (for the terminology, see Section \[sec7\]).

\[thm0.1\] Let $G$ be a finite group, and let $R$ be a noetherian left $G$-module algebra with finite GK-dimension. Assume that $R$ satisfies Setup \[set\]. If $R$ is $G$-Cohen-Macaulay, then $A$ is $\mfa$-Cohen-Macaulay of dimension $\Pty(R,G)$.

As a corollary, we have the following equivalence of local cohomology, where $R^i\Gamma_\mfa$ is the $i$th local cohomology associated to the ideal $\mfa$.

Let $R$ be a $G$-Cohen-Macaulay algebra. Assume that $R$ satisfies Setup \[set\]. If $R$ has finite 