Project/Area Number  10640037 
Research Category 
GrantinAid for Scientific Research (C).

Section  一般 
Research Field 
Algebra

Research Institution  Osaka City University 
Principal Investigator 
SUMIOKA Takeshi Osaka City University, Faculty of Science, Associate Professor, 理学部, 助教授 (90047366)

CoInvestigator(Kenkyūbuntansha) 
ASASHIBA Hideto Osaka City University, Faculty of Science, Associate Professor, 理学部, 助教授 (70175165)
OKUYAMA Tetsuro Hokkaido University of Education, Faculty of Education, Professor, 旭川校・教育学部, 教授 (60128733)
KADO Jiro Osaka City University, Faculty of Science, Lecturer, 理学部, 講師 (10117939)
TSUSHIMA Yukio Osaka City University, Faculty of Science, Professor, 理学部, 教授 (80047240)
KAWATA Shigeto Osaka City University, Faculty of Science, Associate Professor, 理学部, 助教授 (50195103)

Project Fiscal Year 
1998 – 1999

Project Status 
Completed(Fiscal Year 1999)

Budget Amount *help 
¥3,200,000 (Direct Cost : ¥3,200,000)
Fiscal Year 1999 : ¥1,500,000 (Direct Cost : ¥1,500,000)
Fiscal Year 1998 : ¥1,700,000 (Direct Cost : ¥1,700,000)

Keywords  Morita duality / annihilator condition / injective module / Goldie dimension / representation theory / finite group / AuslanderReiten theory / almost split sequence / 森田同値性 / 零化条件 / 入射加群 / ゴルディー次元 / 表現論 / 有限群 / アウスランダー・レイテン理論 / 概分裂列 / 導来同値 / ブルーエ予想 / 多元環 / 繰り返し圏 / 自己入射的 / 中山自己同型 / 双列 / derived 
Research Abstract 
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 BabaOshiro 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 AuslanderReiten theory is one of important tools in studying the representation theory of Artin algebras. In order to apply this AuslanderReiten theory for the representation theory of finite groups, we have considered AuslanderReiten 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 AuslanderReiten 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 pgroup over a complete discrete valuation ring is of wild representation type, then the tree class of the connected component of the stable AuslanderReiten 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.
