Rational Singularities and Almost Split Sequences

The main aim of this paper is to relate almost split sequences to singularity theory by showing that the McKay quiver built from the finite-dimensional representations of a finite subgroup G of GL(2, C), where C is the complex numbers, is isomorphic to the AR quiver of the reflexive modules of the quotient singularity associated with G.Over the past decade, almost split sequences have been playing an increasingly important role in the representation theories of finite-dimensional algebras and classical orders (see 5 and 3, 8 for basic existence theorems in these contexts).While they have been known for some time to exist in higher-dimensional situations 3, it has not been at all clear how they related to singularity theory, if at all.The main aim of this paper is to relate almost split sequences to singularity theory by showing that the McKay quiver built from the finite-dimensional complex representations of a finite subgroup G of GL(2, C), where C is the complex numbers, is isomorphic to the AR quiver of the reflexive modules over the quotient singularity R associated with G.As in the case of finite-dimensional algebras, the AR quiver of reflexive -modules is defined in terms of the almost split sequences of reflexive T?-modules.In the case G c SL(2, C), McKay observed that the underlying graph of the McKay quiver, with the trivial module removed, is isomorphic to the desingularization graph of the associated singularity.Various explanations of this phenomenon have been given by Knrrer, Gonzalez-Sprinberg-Verdier6 and Artin-Verdier 1, which along the way have established, most explicitly in 1, a natural one-to-one correspondence between the indecomposable reflexive 7\-modules and the nodes of the desingularization graph.But why the almost split sequences describe the edge of the desingularization graph still remains to be explained.An effort has been made to make this paper as self-contained as possible.In particular, no prior knowledge of almost split sequences is required.Before describing the contents of the six sections of this paper we fix some notation.Throughout this paper G is a finite group, k an algebraically closed field of characteristic not dividing the order of G and F a two-dimensional Ac-representation of G. Setting S = k[X, Y], the Ac-algebra of formal power series, the two-dimensional representation V gives a linear action of G on S as a group of Ac-algebra automorphisms.We denote by SG the skew group ring given by this action.