On the Grothendieck group of a quotient singularity defined by a finite abelian group (Q1188274)

Let \(A=\mathbb{C} [[x_ 1,\dots,x_ n]]\) be the formal power series ring in \(n\) variables over the complex numbers, and \(G\) a finite abelian group acting linearly and faithfully on \(A\). The aim of this paper is to study the Grothendieck group \(K_ 0\pmod R\) of the category of finitely generated modules over the invariant ring \(R=A^ G\). \textit{M. Auslander} and \textit{I. Reiten} [J. Pure Appl. Algebra 39, 1-51 (1986; Zbl 0576.18008)] showed that \(K_ 0\pmod R\) is finitely generated by at most \(c(G)\) elements, where \(c(G)\) denotes the class number of \(G\). In particular, \(K_ 0\pmod R\) is a factor group of \(\mathbb{Z} [G^*]\) where \(G^*\) denotes the character group of \(G\). The authors prove that \(K_ 0\pmod R \simeq \mathbb{Z} [G^*]/K\), where \(K\) is defined as in the paper of \textit{J. Herzog} and \textit{H. Sanders} [Singularities, representation of algebras, and vector bundles, Proc. Symp., Lambrecht 1985, Lect. Notes Math. 1273, 134-149 (1987; Zbl 0652.14016)].