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Representation Theory of Algebras

Research Project

Project/Area Number 10640037
Research Category

Grant-in-Aid for Scientific Research (C)

Allocation TypeSingle-year Grants
Section一般
Research Field Algebra
Research InstitutionOsaka City University

Principal Investigator

SUMIOKA Takeshi (1999)  Osaka City University, Faculty of Science, Associate Professor, 理学部, 助教授 (90047366)

浅芝 秀人 (1998)  大阪市立大学, 理学部, 助教授 (70175165)

Co-Investigator(Kenkyū-buntansha) KAWATA Shigeto  Osaka City University, Faculty of Science, Associate Professor, 理学部, 助教授 (50195103)
ASASHIBA Hideto  Osaka City University, Faculty of Science, Associate Professor, 理学部, 助教授 (70175165)
TSUSHIMA Yukio  Osaka City University, Faculty of Science, Professor, 理学部, 教授 (80047240)
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)
住岡 武  大阪市立大学, 理学部, 助教授 (90047366)
Project Period (FY) 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)
KeywordsMorita duality / annihilator condition / injective module / Goldie dimension / representation theory / finite group / Auslander-Reiten theory / almost split sequence / 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 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.

Report

(3 results)
  • 1999 Annual Research Report   Final Research Report Summary
  • 1998 Annual Research Report
  • Research Products

    (18 results)

All Other

All Publications (18 results)

  • [Publications] 森本真理,住岡 武: "On dual pairs and simple-injective modules"Journal of Algebra. (印刷中).

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      1999 Final Research Report Summary
  • [Publications] 森本真理,住岡 武: "Semicolocal pairs and finitely cogenerated injective modules"Osaka Journal of Mathematics. (印刷中).

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      1999 Final Research Report Summary
  • [Publications] 住岡 武,渡嘉敷尚: "Some remarks on (M,N)-injective modules"Procedings of the 32th Symposium on Ring Theory and Representation Theary. (印刷中).

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      1999 Final Research Report Summary
  • [Publications] 井上孝之,河田成人: "On Auslander-Reiten conponennts and trivial modules for integral group rings of P-groups"Journal of Algebra. 203. 374-384 (1998)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      1999 Final Research Report Summary
  • [Publications] 河田 成人: "On standard Auslander-Reiten sequences for fonite groups"Archiv der Mathematik. (印刷中).

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      1999 Final Research Report Summary
  • [Publications] 河田 成人: "On Auslander-Reiten Sequences for irreducible lattices over integral group rings"Proceedings of the 32^<th> Symposium on Ring Theory and Representation theory. (印刷中).

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      1999 Final Research Report Summary
  • [Publications] M.Morimoto and T.Sumioka: "On dual pairs and simple-injective modules"J.Algebra. to appear.

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      1999 Final Research Report Summary
  • [Publications] M.Morimoto and T.Sumioka: "Semicolocal pairs and finitely cogenerated injective modules"Osaka J.Math.. to appear.

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      1999 Final Research Report Summary
  • [Publications] T.Sumioka and T.Tokashiki: "Some remarks on (M,N)-injective modules"Proceedings of the 32th Symposium on Ring Theory and Representation Theory. to appear.

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      1999 Final Research Report Summary
  • [Publications] T.Inoue and S.Kawata: "On Auslander-Reiten components and trivial modules for integral group rings of p-groups"Arch.Math.. to appear.

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      1999 Final Research Report Summary
  • [Publications] S.Kawata: "On standard Auslander-Reiten sequences for finite groups"Arch.Math.. to appear.

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      1999 Final Research Report Summary
  • [Publications] S.Kawata: "On Auslander-Reiten sequences for irreducible lattices over integral group rings"Proceedings of the 32th Symposium on Ring Theory and Representation Theory. to appear.

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      1999 Final Research Report Summary
  • [Publications] 森本真理,住岡武: "On dual pairs and simple - injective modules"Journal of Algebra. (発表予定).

    • Related Report
      1999 Annual Research Report
  • [Publications] 森本真理,住岡武: "Semicolocal pairs and finitely cogenerated injective modules"Osaka Journal of Mathematics. (発表予定).

    • Related Report
      1999 Annual Research Report
  • [Publications] 住岡武,渡嘉敷尚: "Some remarks on (M,N)-injective modules"Proceedings of the 32th Symposium on Ring Theory and Representation Theory. (発表予定).

    • Related Report
      1999 Annual Research Report
  • [Publications] 井上孝之,河田成人: "On Auslander-Reiten Components and trivial modules for integral group rings of p-groups"Journal of Algebra. 203. 374-384 (1998)

    • Related Report
      1999 Annual Research Report
  • [Publications] 河田成人: "On standard Auslander-Reiten sequences for finite groups"Archiv der Mathematik. (発表予定).

    • Related Report
      1999 Annual Research Report
  • [Publications] 河田成人: "On Auslander-Reiten sequences for irreducible lattices over integral group rings"Proceedings of the 32th Symposium on Ring Theory and Representation Theory. (発表予定).

    • Related Report
      1999 Annual Research Report

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Published: 1998-04-01   Modified: 2016-04-21  

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