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Homological structure and representation of Noetherian algebras

Research Project

Project/Area Number 11640020
Research Category

Grant-in-Aid for Scientific Research (C)

Allocation TypeSingle-year Grants
Section一般
Research Field Algebra
Research InstitutionShinshu University

Principal Investigator

NISHIDA Kenji  Shinshu University, Science, Professor, 理学部, 教授 (70125392)

Co-Investigator(Kenkyū-buntansha) NINOMIYA Yasushi  Shinshu University, Science, Professor, 理学部, 教授 (40092887)
MUKAI Juno  Shinshu University, Science, Professor, 理学部, 教授 (50029675)
IWANAGA Yasuo  Shinshu University, Education, Professor, 教育学部, 教授 (80015825)
FUJITA H.  University of Tsukuba, Mathematical Science, Instructor, 数学系, 講師 (60143161)
HANAKI Akihide  Shinshu University, Science, Assistant Professor, 理学部, 助教授 (50262647)
山崎 愛一  京都大学, 総合人間学部, 助教授 (10283590)
Project Period (FY) 1999 – 2001
Project Status Completed (Fiscal Year 2001)
Budget Amount *help
¥3,100,000 (Direct Cost: ¥3,100,000)
Fiscal Year 2001: ¥1,000,000 (Direct Cost: ¥1,000,000)
Fiscal Year 2000: ¥1,000,000 (Direct Cost: ¥1,000,000)
Fiscal Year 1999: ¥1,100,000 (Direct Cost: ¥1,100,000)
KeywordsNoetherian algebra / Gorenstein Algebra / Gorenstein dimension / dualizing module / order / アルティン多元環 / ネータ環 / バス予想 / 有限入射次元 / 完全加群
Research Abstract

We consider a theory of non-commutative Noetherian algebras as commutative ring theory. Mainly, we consider an algebra which is finitely generated as a module over a commutative Noetherian ring. We define a Gorenstein algebra by the Cousin complex and give a characterization of a Gorenstein algebra by the Bass number. Indeed, the fact concerning Bass numbers holds completely similar to that of commutative rings. We extend the results due to R. Y. Sharp to our algebras, namely, under some assumption, we give a category equivalence between the category of finite modules of finite injective dimension and that of finite modules of finite protective dimension. We give a criterion that a module over an order or artin algebra is a dualizing modulen and apply it to a Cohen-Macaulay isolated singularities. We study a global dimension of a tiled oreder. Especially, we give a theory to deal with a tiled order having a large global dimension.

Report

(4 results)
  • 2001 Annual Research Report   Final Research Report Summary
  • 2000 Annual Research Report
  • 1999 Annual Research Report
  • Research Products

    (27 results)

All Other

All Publications (27 results)

  • [Publications] S.Goto, K.Nishida: "Finite modules of finite injective dimension over a Noetherian algebra"J.London Math.Soc.. (2)63. 319-335 (2001)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2001 Final Research Report Summary
  • [Publications] S.Goto, K.Nishida: "Towards a theory of Bass numbers with application to Gorenstein algebras"Colloquium Mathematicum. 91・2. 191-253 (2002)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2001 Final Research Report Summary
  • [Publications] K.Nishida: "Dualizing modules for orders and ortin algebras"Algebras and Representation Theory. (to appear).

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2001 Final Research Report Summary
  • [Publications] Y.Ninomiya: "Nilpotency indices of the radical of tinitep-solvable group algbras, I"J.Anstral.Math.Soc.. 71. 117-133 (2001)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2001 Final Research Report Summary
  • [Publications] A.Hanaki: "Semi simplicity of adjacencyalgevrasogassociation schemes"J. of Alavbra. 225. 124-129 (2000)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2001 Final Research Report Summary
  • [Publications] J.Mukai: "Self-homotopy of a suspended real 4-profective space"Contemporary Mathematics. 274. 241-255 (2001)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2001 Final Research Report Summary
  • [Publications] Goto,S. Nishida,K.: "Finite modules of finite injective dimension over a Noetherian algebra"J. London Math. Soc.. ( 2 ) 63. 319-335 (2001)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2001 Final Research Report Summary
  • [Publications] Goto,S., Nishida,K.: "Towards a theory of Bass numbers with application to Gorenstein algebras"Colloquium Mathematicum. 91 no. 2. 191-253 (2002)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2001 Final Research Report Summary
  • [Publications] Nishida,K.: "Dualizing modules for orders and artin algebras"Algebras and Representation Theory. to appear.

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2001 Final Research Report Summary
  • [Publications] Ninomiya,Y.: "Nilpotency indices of the radical of finite p-solvable group algebras, I"J. Anstral, Math. Soc. 71. 117-133 (2001)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2001 Final Research Report Summary
  • [Publications] Hanaki,A.: "Semisimplicity of adjacency algebras of association shemes"J. Algebra. 225. 124-129 (2000)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2001 Final Research Report Summary
  • [Publications] Mukai,J.: "Self-homotopy of a suspended real 4-projective space"Comtemporary Mathematics. 274. 241-255 (2001)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2001 Final Research Report Summary
  • [Publications] Kenji Nishida: "Dnalizing modules for orders and Artin alzebras"Alzebras and Representation Theory, to appear. (2002)

    • Related Report
      2001 Annual Research Report
  • [Publications] Kenji Nishida, Shiro Goto: "Towards or theory of Buss numbers with applications to Gorenstein alzebras"Colloquium Mathematioum. 91・2. 191-253 (2002)

    • Related Report
      2001 Annual Research Report
  • [Publications] Yasushi Ninomiya: "Nilpoteney indices of the radical of fimtip-soloable group alzebras IV"J. Fan. Sci. Shinshu Univ.. 36・1. 9-28 (2001)

    • Related Report
      2001 Annual Research Report
  • [Publications] Yasushi Ninomiya: "Nilpoteney indices of the radical of fimtip-soloable group alzebras III"J. Fan. Sci. Shinshu Univ.. 36・1. 1-8 (2001)

    • Related Report
      2001 Annual Research Report
  • [Publications] Juno Mukai: "Self-homotopy of a sreppendedreal 4-projective space"Contemporary Mathematics. 274. 241-255 (2001)

    • Related Report
      2001 Annual Research Report
  • [Publications] Akihide Hanaki: "Locality of a modular adjucenay alzelbras of am association schone of prime jaower order"Archiv der Mathematik. (to appear).

    • Related Report
      2001 Annual Research Report
  • [Publications] Kenji Nishida,Shiro Goto: "Finite modules of finite injective dimension over a Noetherian algebras"J.London Math, Soc.. (to appear).

    • Related Report
      2000 Annual Research Report
  • [Publications] Yasuo Iwanaga : "Note on Waka matsu's generalized tilting modules"Journal of the Faculty of Education, Shinshu Univ.. 99. 155-166 (2000)

    • Related Report
      2000 Annual Research Report
  • [Publications] Yasusi Ninomiya : "Nilpotency indices of the radicals of finite β-solvabe group algebras, I"J.Austral.Math, Soc.,SerA, . (to appear).

    • Related Report
      2000 Annual Research Report
  • [Publications] Akihide Hanaki,Izumi Miyamoto: "Classification of primitive association schemes with small vertices"Kyashu J.Math. 54・7. 81-86 (2000)

    • Related Report
      2000 Annual Research Report
  • [Publications] Akihide Hanaki: "Semisimplicity of adjacency algebras of association shemes"J.Algebra. 225. 124-129 (2000)

    • Related Report
      2000 Annual Research Report
  • [Publications] Kenji Nishida: "Dualizing modules for orders and artin algebras "Algebras and Representation Theory. (to appear).

    • Related Report
      2000 Annual Research Report
  • [Publications] Kenji Nishida, Shiro Goto: "Minimal injective resolutins of Cohen-Macaulay isolated singularities"Archiv der Mathematik. 73. 249-255 (1999)

    • Related Report
      1999 Annual Research Report
  • [Publications] Kenji Nishida, Shiro Goto: "Catenarity in module-finite algebras"Proc. Amer. Math. Soc.. 127. 3495-3502 (1999)

    • Related Report
      1999 Annual Research Report
  • [Publications] Akihide Hanaki: "Semisimplicity of adjacency algebras of association Sthemes"Journal of Algebra. 225. 124-129 (2000)

    • Related Report
      1999 Annual Research Report

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Published: 1999-04-01   Modified: 2016-04-21  

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