Construction of Auslander-Reiten quivers for a class of group rings
Construction of Auslander-Reiten quivers for a class of group rings
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用于一类群环的 Auslander-Reiten 箭袋的构造
DOI：
10.1007/bf01162006
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发表时间：
1983
影响因子：
0.8
通讯作者：
Ernst Dieterich
中科院分区：
文献类型：
--
作者：
Ernst Dieterich
考虑由完全离散赋值环R和有限p-群G给出的群环A= RG。本文的第一个目的是对所有有限表示型群环A构造Auslander-Reiten群环（A）（定义见[16]）。当G是奇素数阶p的循环格且p的分歧度为2时，以及当G是3阶循环格且3的分歧度为3时，这将特别地产生一种分类所有不可分解A-格的新方法，从而为已有的众多证明增添另一贡献（例如固有的[1，7-9，11，-13，15]），这些群环是有限表示型。“格上的订单和整数群环是众所周知的复杂对象”（[43]）。本文的第二个目的是指出构造d（LW）=~ 4（A）的方法，这种方法尽可能避免了在A-格范畴内的计算，而利用了相关范畴中更容易获得的信息。(In这一点，我也看了[173]。在这个方向上的第一种方法是基于这样的观察，即在适当的假设下，在E4]中引入的表示等价o：~ i-，~ i，导致箭图d~（Sfy）和d（~ U）之间的同构。这里，d~（5~ i）表示~（5~ i）的全子空间，它的点是& aI中不可分解A-格的同构类。因此，我们的想法是在第一步中描述d（~），并在第二步中完成d~（Sfl）到d（~）。这种策略在以下每种情况下都能很好地工作：（i）（ii）（iii）（iv）（在这里，
Let us consider group rings A= RG which are given by a complete discrete valuation ring R and a finite p-group G. This papers first objective is to construct the Auslander-Reiten quiver d (A)(see [16] for definitions) for all group rings A of finite representation type. In case G is cyclic of odd prime order p and the ramification degree of p is 2, as well as in case G is cyclic of order 3 and the ramification degree of 3 is 3, this will in particular yield a new way of classifying all indecomposable A-lattices, thus adding another contribution to the numerous already existing proofs (eg inherent in [1, 7-9, 11,-13, 15]) that these group rings are of finite representation type." Lattices over orders and integer group rings are notoriously complicated objects"([43). This remark generally also applies to the calculation of Auslander-Reiten sequences in the category of A-lattices 5~= a &~ Therefore this papers second objective is to indicate ways of constructing d (LW)=~ 4 (A) which avoid calculations inside aS~ as far as possible, yet use informations from related categories which are easier accessible.(In this respect, see also [173). The first approach in this direction is based on the observation that-under suitable hypotheses-the representation equivalence o~:~ i-,~,, introduced in E4], induces an isomorphism between the quivers d~(Sfy) and d (~ U). Here, by d~(5~ i) we mean the full subquiver of~(~) which has as points the isomorphism-classes of indecomposable A-lattices in &a I. Therefore the idea is to describe d (~) in the first step, and to complete d~(Sfl) to d (~) in the second step. This strategy works nicely in each of the following cases:(i)(ii)(iii)(iv)(Here, uation