Rational singularities and almost split sequences
Rational singularities and almost split sequences
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DOI：
10.1090/s0002-9947-1986-0816307-7
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发表时间：
1986-02
影响因子：
1.3
通讯作者：
M. Auslander
中科院分区：
文献类型：
--
作者：
M. Auslander
本文的主要目的是将几乎可裂序列与奇异性理论联系起来，证明了由GL（2，C）的有限子群G的有限维表示所构造的McKay代数同构于G的商奇异性的自反模的AR代数，其中C是复数。在过去的十年中，几乎可裂序列在有限维代数和经典序的表示论中扮演着越来越重要的角色（参见（5和3，8）这些上下文中的基本存在定理）。虽然我们知道它们存在于高维情形中已有一段时间了（3），但我们并不清楚它们与奇点理论的关系。本文的主要目的是将几乎可裂序列与奇异性理论联系起来，证明了由GL（2，C）的有限子群G的有限维复表示所构造的McKay代数同构于G的商奇异性R上的自反模的AR代数，其中C是复数。与有限维代数的情形一样，自反R-模的AR-模也是用自反R-模的几乎可裂序列来定义的。在GcSL（2，C）的情况下，McKay观察到McKay代数的基本图，去掉平凡模后，同构于相关奇点的去奇异化图。Knorrer、Gonzalez-Sprinberg-Verdier（6）和Artin-Verdier（1）对这一现象给出了各种解释，它们沿着这条路，在（1）中最明确地建立了不可分解的自反R-模与去奇异化图的节点之间的自然一一对应。但是为什么几乎分裂序列描述去奇异化图的边缘仍然有待解释。已作出努力，使本文件尽可能自成一体。特别是，几乎分裂序列的先验知识是不需要的。在描述本文的六个部分的内容之前，我们确定一些符号。本文中G是一个有限群，k是G的特征不整除阶的代数闭域，V是G的二维k-表示。设S = k（（X，Y）），即形式幂级数的k-代数，二维表示V给出G在S上的线性作用作为一个k-代数自同构群.我们用S（G）表示由这个作用给出的斜群环。
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-dimen- sional 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 fimte-dimensional complex represen- tations 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 R-modules is defined in terms of the almost split sequences of reflexive R-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 desingulariza tion graph of the associated singularity. Various explanations of this phenomenon have been given by Knorrer, Gonzalez-Sprinberg-Verdier (6) and Artin-Verdier (1), which along the way have established, most explicitly in (1), a natural one-to-one correspondence between the indecomposable reflexive R-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 V a two-dimensional k-representation of G. Setting S = k((X, Y)), the k-algebra of formal power series, the two-dimen- sional representation V gives a linear action of G on S as a group of k-algebra automorphisms. We denote by S(G) the skew group ring given by this action.