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Transactions of the American Mathematical Society
Transactions of the American Mathematical Society
ISSN 1088-6850(online) ISSN 0002-9947(print)

 

Grothendieck groups of quotient singularities


Author: Eduardo do Nascimento Marcos
Journal: Trans. Amer. Math. Soc. 332 (1992), 93-119
MSC: Primary 19A31; Secondary 13A50, 14B05
MathSciNet review: 1033235
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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 $ \vert G\vert$ 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$.


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DOI: http://dx.doi.org/10.1090/S0002-9947-1992-1033235-8
PII: S 0002-9947(1992)1033235-8
Article copyright: © Copyright 1992 American Mathematical Society