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2000 Fiscal Year Final Research Report Summary

Good filtrations and rings of invariants

Research Project

Project/Area Number 10640017
Research Category

Grant-in-Aid for Scientific Research (C)

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

Principal Investigator

HASHIMORO Mitsuyasu  Nagoya University, Graduate School of Mathematics, Associate Professor, 大学院・多元数理科学研究科, 助教授 (10208465)

Co-Investigator(Kenkyū-buntansha) YOSHIDA Ken-ichi  Nagoya University, Graduate School of Mathematics, Research Assistant, 大学院・多元数理科学研究科, 助手 (80240802)
OKADA Soichi  Nagoya University, Graduate School of Mathematics, Associate Professor, 大学院・多元数理科学研究科, 助教授 (20224016)
HAYASHI Takahiro  Nagoya University, Graduate School of Mathematics, Associate Professor, 大学院・多元数理科学研究科, 助教授 (60208618)
Project Period (FY) 1998 – 2000
Keywordsgood filtrations / reductive group / ring of invariants / duality theorem / equivariant module / Gorenstein property / F-rationality / strong F-regularity
Research Abstract

Through the research done in the last academic year, more or less we achieved the objective on the fundamental research on homological behavior of equivariant modules. In this academic year, we published a monograph in English including the technical part of that homological research. The monograph includes a theory which unifies the Cohen-Macaulay approximation theory over a Cohen-Macaulay ring with canonical modules by Auslander-Buchweitz and the theory of Δ-good approximations over quasi-hereditary algebras by Ringel.
Moreover, continued from the last academic year, we were trying to enrich the theorem which asserts that if a connected reductive group G acts on a polynomial algebra S linearly, and if S admits good filtrations as a G-module, then the ring of invariants S^G is strongly F-regular. Since the condition that a polynomial algebra admits good filtrations is always true in characteristic zero, and the condition is Zariski open, S^G is of strongly F-regular type in characteristic zero. As the strong F-regular type property is stronger than rationality of singularities, the theorem is not included in Boutot's well-known theorem. On the other hand, in order to study Gorenstein property of invariant subrings in positive characteristics, it is necessary to investigate the behavior of canonical modules. For this purpose, we proved the Grothendieck duality theorem with respect to equivariant proper morphisms, and constructed the equivariant version of twisted inverse pseudofunctors which are necessary to state the equivariant duality theorem. Moreover, utilizing them, we partly succeeded in modifying the results on Gorenstein property of invariant subrings in characteristic zero by Knop to positive characteristics. These results were announced at domestic and international meetings, and the abstracts were published. Moreover, we got some results on behavior of F-rationality with respect to flat morphisms, and it has been published.

  • Research Products

    (14 results)

All Other

All Publications (14 results)

  • [Publications] Mitsuyasu Hashimoto: "Good filtrations of symmetric algebras and strong F-regularity of invariant subrings"Math.Z.. 236. 605-623 (2001)

    • Description
      「研究成果報告書概要(和文)」より
  • [Publications] Takahiro Hayashi.: "Face algebras and unitarity of SU(N)_L-TQFT"Commun.Math.Phys.. 203. 211-247 (1999)

    • Description
      「研究成果報告書概要(和文)」より
  • [Publications] Takahiro Hayashi: "Coribbon Hopf (face) algebras generated by lattice models"J.Algebra. 233. 614-641 (2000)

    • Description
      「研究成果報告書概要(和文)」より
  • [Publications] Soichi Okada: "Applications of minor-summation formulas to rectangular-shaped representations of classical groups"J.Algebra. 205. 337-367 (1998)

    • Description
      「研究成果報告書概要(和文)」より
  • [Publications] Ken-ichi Yoshida: "Tensor products of perfect modules and maximal surjective Buchsbaum modules"J.Pure Appl.Algebra. 123. 313-326 (1998)

    • Description
      「研究成果報告書概要(和文)」より
  • [Publications] Ken-ichi Yoshida: "A generalization of linear Buchsbaum modules in terms of homological degree"Comm.Algebra. 26・3. 931-945 (1998)

    • Description
      「研究成果報告書概要(和文)」より
  • [Publications] Mitsuyasu Hashimoto: "Auslander-Buchweitz Approximations of Equivariant Modules"Cambridge University Press. 281 (2000)

    • Description
      「研究成果報告書概要(和文)」より
  • [Publications] Mitsuyasu Hashimoto: "Good filtrations of symmetric algebras and strong F-regularity of invariant subrings."Math.Z.. 236. 605-623 (2001)

    • Description
      「研究成果報告書概要(欧文)」より
  • [Publications] Takahiro Hayashi: "Face algebras and unitarity of SU (N)_L-TQFT.Commun."Math.Phys.. 203. 211-247 (1999)

    • Description
      「研究成果報告書概要(欧文)」より
  • [Publications] Takahiro Hayashi: "Coribbon Hopf (face) algebras generated by lattice models"J.Algebra. 233. 614-641 (2000)

    • Description
      「研究成果報告書概要(欧文)」より
  • [Publications] Soichi Okada: "Applications of minor-summation formulas to rectangular-shaped representations of classical groups"J.Algebra. 205. 337-367 (1998)

    • Description
      「研究成果報告書概要(欧文)」より
  • [Publications] Ken-ichi Yoshida: "Tensor products of perfect modules and maximal sur-jective Buchsbaum modules."J.Pure Appl.Algebra. 123. 313-326 (1998)

    • Description
      「研究成果報告書概要(欧文)」より
  • [Publications] Ken-ichi Yoshida: "A generalization of linear Buchsbaum modules in terms of homological degree."Comm.Algebra. 26-3. 931-945 (1998)

    • Description
      「研究成果報告書概要(欧文)」より
  • [Publications] Mitsuyasu Hashimoto: "Auslander-Buchweitz Approximations of Equiv ariant Modules. London Mathematical Society Lecture Note Series vol.282"Cambridge University Press, Cambridge. (2000)

    • Description
      「研究成果報告書概要(欧文)」より

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Published: 2002-03-26  

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