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Representation Theory of Algebras

代数表示论

基本信息

批准号：
10640037
负责人：
SUMIOKA Takeshi
金额：
$ 2.05万
依托单位：
Osaka City University
依托单位国家：
日本
项目类别：
Grant-in-Aid for Scientific Research (C)
财政年份：
1998
资助国家：
日本
起止时间：
1998 至 1999
项目状态：
已结题

项目摘要

In ring and representation theory, Morita duality is applied in various field and is a very important research task. In 1969, as a detail version of Morita duality, Fuller gave characterizations of indecomposable indicative ideals over right artinian rings with a relation of two projective ideals, and in1992, Baba and Oshiro extended these results to semiprimary rings. In our researches, we extended some results by Fuller and Baba-Oshiro related to projective ideals to a theory for modules by using a notion "pairs of modules" which was introduced by Morita and Tachikawa. Applying these results, we gave a condition for modules in pairs with annihilator condition to have finite Goldie dimension and gave a characterization for finitely cogenerated injective modules. These results not only extend projective ideals to modules but also clarify essence of properties, and more developments are expected.On the other hand, the Auslander-Reiten theory is one of important tools in studying the representation theory of Artin algebras. In order to apply this Auslander-Reiten theory for the representation theory of finite groups, we have considered Auslander-Reiten quivers of finite groups. In 1995, Erdmann proved that if the block of a finite group over a field is of wild representation type, then any connected component of the stable Auslander-Reiten quiver of this block has tree class AィイD2∝ィエD2. In this project we have showed that if the group ring of a finite p-group over a complete discrete valuation ring is of wild representation type, then the tree class of the connected component of the stable Auslander-Reiten quiver of this group ring containing the trivial lattice is AィイD2∝ィエD2. Also we obtained some relation between almost split sequences in the case of modular representation and those in the case of integral representation.

在环和表示论中，森田对偶性被应用于各个领域，是一项非常重要的研究任务。 1969年，作为森田对偶性的详细版本，富勒给出了具有两个射影理想关系的右阿天环上不可分解指示理想的表征，1992年，Baba和Oshiro将这些结果扩展到半主环。在我们的研究中，我们使用 Morita 和 Tachikawa 引入的“模块对”概念，将 Fuller 和 Baba-Oshiro 的一些与射影理想相关的结果扩展到模块理论。应用这些结果，我们给出了具有有限 Goldie 维数的成对模的条件，并给出了有限共生单射模的表征。这些结果不仅将射影理想扩展到模，而且阐明了性质的本质，值得期待更多的发展。另一方面，Auslander-Reiten理论是研究Artin代数表示论的重要工具之一。为了将 Auslander-Reiten 理论应用于有限群的表示论，我们考虑了有限群的 Auslander-Reiten 颤动。 1995年，Erdmann证明，如果域上的有限群的块是野表示类型，则该块的稳定Auslander-Reiten颤动的任何连通分量都具有树类AィイD2∝ィD2。在本项目中，我们证明，如果完全离散估价环上的有限 p 群的群环是狂野表示类型，则包含平凡格的该群环的稳定 Auslander-Reiten 颤动的连通分量的树类为 AiiD2∝ィD2。我们还获得了模表示情况下的几乎分裂序列和积分表示情况下的几乎分裂序列之间的一些关系。