since $\alpha\neq 2$. Therefore, $\mathfrak{R}(R,G)$ is finite dimensional. Hence $\Pty(R,G)=3$ as $\operatorname{GKdim}(R)=3$ (cf. [@KMP]).

The pertinency of a permutation group acting on $\kk_{-1}[x_1,\dots, x_n]$ or on a down-up algebra was recently computed by Gaddis, Kirkman, Moore and Won using different methods [@GKMW].

Local cohomology {#sec5}
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In this section, $A$ is always a noetherian algebra, and $\mfa$ is an ideal of $A$. Denote by $\operatorname{Mod}A$ the category of right $A$-modules, by $\operatorname{Mod}A^\circ$ the category of left $A$-modules, and by $\operatorname{Mod}A^e$ the category of $A$-$A$-bimodules. Let $\Gamma_\mfa=\underrightarrow{\lim}\operatorname{Hom}_A(A/\mfa^n,-):\operatorname{Mod}A\longrightarrow \operatorname{Mod}A$. For $M\in\operatorname{Mod}A$, if $\Gamma_\mfa(M)=M$, then we say that $M$ is an [*$\mfa$-torsion module*]{}. If $\Gamma_\mfa(M)=0$, then we say that $M$ is [*$\mfa$-torsion-free*]{}.

We say that $\mfa$ has the [*right Artin-Rees (AR) property*]{} if one of the following equivalent conditions holds [@MR 4.2.3]

-   For every right ideal $\mfb$ of $A$, $\mfb\cap \mfa^n\subseteq \mfb\mfa$ for some $n$.

-   For every finitely generated right $A$-module $M$, and every submodule $N\subseteq M$, $N\cap M\mfa^n\subseteq N\mfa$ for some $n$.

-   For every finitely generated right $A$-module $M$, and every submodule $N\subseteq M$, and for every integer $s\ge0$, there is an integer $n>0$ such that $N\cap M\mfa^n\subseteq N\mfa^s$.

\[lem-torinj\] Let $A$ be a noetherian algebra, and $\mfa$ an 