group action on a noetherian algebra
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Introduction
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Let $R$ be a noetherian algebra with finite GK-dimension, and let $G$ be a finite group acting on $R$. In general, it is not easy to study the structure and the representations of a group action. However, one possible way to understand the group action is through the McKay correspondence, which establishes certain correspondence among the representations of the group $G$, of the skew group algebra $R*G$, and of the invariant subalgebra $R^G$ (see [@B] for a good survey of the classical McKay correspondence, or [@Mo; @CKWZ1; @CKWZ2] for noncommutative McKay correspondence). An important step for establishing noncommutative McKay correspondence is the Auslander Theorem in the noncommutative case (cf. [@BHZ; @BHZ2; @GKMW]). To prove certain noncommutative version of the Auslander theorem, Bao-Zhang and the first named author introduced an invariant $\Pty(R,G)$, called the pertinency of the group action. In some sense, the pertinency $\Pty(R,G)$ determines whether the Auslander theorem holds or not (cf. [@BHZ]).

The invariant subalgebra $R^G$ is not regular in general. For instance, if $R$ is Artin-Schelter regular and the homological determinant of $G$ is trivial, then $R^G$ is Artin-Schelter Gorenstein [@JZ]. So, studying the representations of the noncommutative 