Classification of the Auslander-Reiten quivers of local Gorenstein orders and a characterization of the simple curve singularities
Classification of the Auslander-Reiten quivers of local Gorenstein orders and a characterization of the simple curve singularities
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局部 Gorenstein 阶的 Auslander-Reiten 颤动的分类和简单曲线奇点的表征
DOI：
10.1016/0022-4049(86)90115-5
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发表时间：
1986
影响因子：
0.8
通讯作者：
A. Wiedemann
中科院分区：
文献类型：
--
作者：
A. Wiedemann
在本文中，我们给出了有限格型的完全Dedekind域R（即a是自身上的一个内射不可分解左格，并且具有同构的有限多个不可分解左格）上所有局部Gorenstein阶a的有限Auslander-Reiten振动的完整列表[7,19]。对于这个列表中的每个平移颤振F，我们明确地指出了一个以F作为其Auslander-Reiten颤振的Gorenstein阶a。此外，在每种情况下，我们都描述了不可分解的a格。特别地，该列表包含平面简单曲线奇点的Auslander-Reiten颤振，其完全局部环可以看作是复数[9]上单变量幂级数环上的Gorenstein阶。我们简单地回顾一下这些奇点的描述，这些奇点与上述平移颤振[1,6,21,22]有关：考虑线性作用于幂级数环CRU， VB的有限非平凡子群SL2 (C)的不变量环。它有三个生成器X， Y， Z，满足一个关系f (X， Y)+ z2 = 0，该关系在原点附近定义了一个曲面，其原点为孤立奇点。以这种方式出现的奇点作为有限群的商奇点，通常被称为有理双点或克莱因奇点。众所周知，这些奇异点的解析图是Dynkin图/ n，[Dn, n: 6,: 7,: 8[6]。则与平面Z= 0的交点为格鲁埃尔- kn6rrer[14]表征的简化平面曲线奇点[1]：
In this paper we give a complete list of all finite Auslander-Reiten quivers of local Gorenstein orders A over a complete Dedekind domain R of finite lattice type (ie A is an injective indecomposable left lattice over itself and has-up to isomorphismonly finitely many indecomposable left lattices)[7, 19]. For each translation quiver F in this list, we indicate explicitly a Gorenstein order A with F as its Auslander-Reiten quiver. Moreover, in each case we describe the indecomposable A-lattices.In particular, this list contains the Auslander-Reiten quivers of the plane simple curve singularities whose complete local rings can be viewed as Gorenstein orders over the power series ring in one variable over the complex numbers [9]. Briefly we recall a description of these singularities which turns out to be of interest in connection with the above translation quivers [1, 6, 21, 22]: Consider the ring of invariants of a finite nontrivial subgroup of SL2 (C) acting linearly on the power series ring CRU, VB. It has three generators X, Y, Z satisfying one relation f (X, Y)+ Z 2= 0 which defines in the neighbourhood of the origin a surface with the origin as an isolated singularity. The singularities occuring in this way as quotient singularity of a finite group are usually known as rational double points or Kleinian singularities. It is well known that the resolution graph of these singularities are the Dynkin diagrams/A n,[Dn, N: 6,: 7,: 8 [6]. Then the intersection with the plane Z= 0 is a reduced simple plane curve singularity [1] characterized by Greuel-Kn6rrer [14]: