Grothendieck groups of quotient singularities

Abstract:

Given a quotient singularity $R = {S^G}$ where $S = {\mathbf {C}}[[{x_1}, \ldots ,{x_n}]]$ is the formal power series ring in $n$-variables over the complex numbers ${\mathbf {C}}$, there is an epimorphism of Grothendieck groups $\psi :{G_0}(S[G]) \to {G_0}(R)$, where $S[G]$ is the skew group ring and $\psi$ is induced by the fixed point functor. The Grothendieck group of $S[G]$ carries a natural structure of a ring, isomorphic to ${G_0}({\mathbf {C}}[G])$. We show how the structure of ${G_0}(R)$ is related to the structure of the ramification locus of $V$ over $V/G$, and the action of $G$ on it. The first connection is given by showing that $\operatorname {Ker}\;\psi$ is the ideal generated by $[{\mathbf {C}}]$ if and only if $G$ acts freely on $V$. That this is sufficient has been proved by Auslander and Reiten in [4]. To prove the necessity we show the following: Let $U$ be an integrally closed domain and $T$ the integral closure of $U$ in a finite Galois extension of the field of quotients of $U$ with Galois group $G$. Suppose that $|G|$ is invertible in $U$, the inclusion of $U$ in $T$ is unramified at height one prime ideals and $T$ is regular. Then ${G_0}(T[G]) \cong Z$ if and only if $U$ is regular. We analyze the situation $V = {V_1}{\coprod } _{\mathbf {C}[G]}{V_2}$ where $G$ acts freely on ${V_1},{V_1} \ne 0$. We prove that for a quotient singularity $R,{G_0}(R) \cong {G_0}(R[[t]])$. We also study the structure of ${G_0}(R)$ for some cases with $\dim R = 3$.