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Study of Invariant on Derived Categories over Algebras

Research Project

Project/Area Number 12640013
Research Category

Grant-in-Aid for Scientific Research (C)

Allocation TypeSingle-year Grants
Section一般
Research Field Algebra
Research InstitutionTokyo Gakugei University

Principal Investigator

MIYACHI Jun-ichi  Tokyo Gakugei University, Department of Mathematics, Associate Professor, 教育学部, 助教授 (50209920)

Co-Investigator(Kenkyū-buntansha) TOKUHIRO Yoshimi (KITAMURA Yoshimi)  Tokyo Gakugei University, Department of Mathematics, Professor, 教育学部, 教授 (00014811)
KURANO Kazuhiko  Tokyo Metropolitan University, Department of Mathematics, Associate Professor, 理学部, 助教授 (90205188)
Project Period (FY) 2000 – 2001
Project Status Completed (Fiscal Year 2001)
Budget Amount *help
¥1,100,000 (Direct Cost: ¥1,100,000)
Fiscal Year 2001: ¥1,100,000 (Direct Cost: ¥1,100,000)
KeywordsDerived category / Chain complex / Picard groups / Derived Picard groups / Hereditary algebras / Grothendieck groups / Chow groups / Frobenius algebras / 導来Picarrd群 / t-structure / torsion theory / compact object / intersection multiplicity / Dutta multiplicity / Roberts ring
Research Abstract

First, we study Picard groups and derived Picard groups of finite dimensional hereditary algebras. We obtain general results on the structure of Picard groups and derived Picard groups, as well as explicit calculations for the Dynkin and affine quivers, and for some wild quivers with multiple arrows. In addition we prove that when A is hereditary, the derived Picard group of A coincides with the full group of Minear triangle auto-equivalences of the derived category of A. Hence we can calculate the group of non-commutative projective spaces introduced by Kontsevich-Roseriberg.Second, we show that a compact object C in a triangulated category, which satisfies suitable conditions, induces a t-structure. Third, in an abelian category we show that a complex P of small projective objects of term length two, which satisfies suitable conditions, induces a torsion theory. In the case of module categories, using a torsion theory, we give equivalent conditions for P' to be a tilting complex. Finally, in the case of artin algebras, we give a one to one correspondence between tilting complexes of term length two and torsion theories with certain conditions.
Moreover, We have the following related results :
1) First, we define test modules to calculate Dutta multiplicities and study the relation. Second, we characterize Roberts rings by some Galois extensions. Third, it is shown that a Chow group A.(A) of A is determined by cycles and a rational equivalence with respect to certain G-graded ideals of a Noetherian ring that graded by a finitely generated Abelian group G.Finally,we prove that the induced map G_0(A) → G_0(A) by completion is injective if A is an excellent Noetherian local ring that satisfies one of the three conditions (K. Kurano).
2) It is shown that for a quasi-Frobenius extension A of a right non-singular ring B if A is a right self-injective ring, then so is B (Y. Kitamura).

Report

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

    (21 results)

All Other

All Publications (21 results)

  • [Publications] Jun-ichi Miyachi: "Derived Picard Groups of Finite Dimensional Hereditary Algebras"Compositio Math.. 129. 341-368 (2001)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2001 Final Research Report Summary
  • [Publications] Jun-ichi Miyachi: "On t-structures and Torsion Theories Induced by Compact Objects"J. Pure and Appl. Algebra. 167. 15-35 (2002)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2001 Final Research Report Summary
  • [Publications] Yoshimi Kitamura: "A note on quasi-Frobenius extensions"Arch. Math.. 77. 1-4 (2001)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2001 Final Research Report Summary
  • [Publications] Kazuhiko Kurano: "Test modules to calculate Dutta multiplicities"to appear in J. Algebra. 236. 216-235 (2001)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2001 Final Research Report Summary
  • [Publications] Kazuhiko Kurano: "On Roberts rings"to appear in Comm. Algebra.

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2001 Final Research Report Summary
  • [Publications] Kazuhiko Kurano: "On maps of Grothendieck groups induced by completion"J. Algebra. 236. 216-235 (2001)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2001 Final Research Report Summary
  • [Publications] Kazuhiko Kurano: "On Chow groups of G-graded rings"to appear in J. Math. Soc. Japan.

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2001 Final Research Report Summary
  • [Publications] Jun-ichi Miyachi: "Derived Picard groups of finite dimensional hereditary algebras"Composiitio Math.,. 129. 341-368 (2001)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2001 Final Research Report Summary
  • [Publications] Jun-ichi Miyachi: "On t-structures and torsion theories induced by compact objects"J.Pure and Appl. Algebra. 167. 15-35 (2002)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2001 Final Research Report Summary
  • [Publications] Yoshimi Kitamura: "A note on quasi-Frobenius extensions"Arch. Math. 77. 1-4 (2001)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2001 Final Research Report Summary
  • [Publications] Kazuhiko Kurano: "Test modules to calculate Dutta multiplicities"J. Algebra. 236. 216-235 (2001)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2001 Final Research Report Summary
  • [Publications] Kazuhiko Kurano: "On Roberts rings"J.Math.Soc.Japan. (to appear).

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2001 Final Research Report Summary
  • [Publications] Kazuhiko Kurano: "On maps of Grothendieck groups induced by completion"J.Algebra. (to appear).

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2001 Final Research Report Summary
  • [Publications] Kazuhiko Kurano: "On Chow groups of G-graded rings"Comm.Algebra. (to appear).

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2001 Final Research Report Summary
  • [Publications] Jun-ichi Miyachi: "Derived Picard Groups of Finite Dimensional Hereditary Algebras"Compositio Math.. 129. 341-368 (2001)

    • Related Report
      2001 Annual Research Report
  • [Publications] Kazuhiro Kurano: "On maps of Grothendieck groups induced by completion"J.Algebra. (to appear).

    • Related Report
      2001 Annual Research Report
  • [Publications] Kazuhiro Kurano: "On Chow groups of G-graded rings"Comm.Algebra. (to appear).

    • Related Report
      2001 Annual Research Report
  • [Publications] Yoshimi Kitamura: "A note on quasi-Frobenius extensions"Arch.Math. 77. 1-4 (2001)

    • Related Report
      2001 Annual Research Report
  • [Publications] Jun-ichi Miyachi: "On t-strtuctures and Torsion Theories Induced by Compact Objects"to appear in J.Pure and Appl.Algebra.

    • Related Report
      2000 Annual Research Report
  • [Publications] Kazuhiko Kurano: "Test modules to calculate Dutta Multiplicities"J.Algebra. 236. 216-235 (2001)

    • Related Report
      2000 Annual Research Report
  • [Publications] Kazuhiko Kurano: "On Roberts rings"to appear in J.Math.Soc.Japan.

    • Related Report
      2000 Annual Research Report

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Published: 2001-04-01   Modified: 2021-12-10  

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