On Auslander-Reiten components and simple modules for finite group algebras
On Auslander-Reiten components and simple modules for finite group algebras
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关于有限群代数的 Auslander-Reiten 分量和简单模
DOI：
10.18910/7367
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发表时间：
1997
影响因子：
0.4
通讯作者：
Shigeto Kawata
中科院分区：
文献类型：
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作者：
Shigeto Kawata
设G是有限群，k是特征p > 0的域，B是群代数kG的块.设θ是B的稳定Auslander-Reiten图的连通分支（简称AR-分支）. Erdmann证明了如果B是kG的野生块，则θ的树类是AOQ [6]。本文研究了简单模在B的Auslander-Reiten图中的位置.设Λ是对称代数，M是单Λ-模.则终止于Ω~M的Auslander-Reiten序列A（Ω~M）的形式为0 -> ΩM -> HM®PM -> Ω~ M -» 0，其中Ω是Heller算子，PM是M的投射覆盖，HM是PM的心RadPM/SocPM（见[1，命题4.11]），这种类型的序列将被称为标准序列。因此，如果包含M的AR-分支θ的树类是A^，则M位于θ的末端当且仅当HM是不可分解的。在第一节中，我们考虑了一般对称代数，如果某个具有树类AQQ的AR-分支包含一个不位于其AR-分支末端的单模，会发生什么。在第二节中，我们给出了B中所有单模都位于AR-分支端点的条件。符号几乎是标准的。这里考虑的所有模都是k上有限维的。关于这里使用的一些基本事实和术语，我们参考[2]和[5]。
Let G be a finite group, k a field of characteristic p > 0 and B a block of the group algebra kG. Let θ be a connected component (AR-component for short) of the stable Auslander-Reiten quiver of B. Erdmann showed that if B is a wild block of kG, then the tree class of θ is AOQ [6]. In this note we investigate where simple modules lie in the Auslander-Reiten quiver of B. Let Λ be a symmetric algebra and M a simple Λ-module. Then the Auslander-Reiten sequence A(Ω~M) terminating in Ω~M is of the form 0 -> ΩM -> HM®PM -> Ω~ M -» 0, where Ω is the Heller operator, PM is the projective cover of M and HM is the heart RadPM/SocPM of PM (see [1, Proposition 4.11]), and sequences of this type will be called standard sequences. Therefore if the tree class of the AR-component θ containing M is A^, then M lies at the end of θ if and only if HM is indecomposable. In Section 1, we consider for general symmetric algebras what happens if some AR-component with tree class AQQ contains a simple module not lying at the end of its AR-component. In Section 2 we give certain conditions which imply that all simple modules in B lie at the ends of AR-components. The notation is almost standard. All the modules considered here are finite dimensional over k. Concerning some basic facts and terminologies used here, we refer to [2] and [5].