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Study of graded rings associated to ideals and modules

Research Project

Project/Area Number 16540045
Research Category

Grant-in-Aid for Scientific Research (C)

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

Principal Investigator

GOTO Shiro  Meiji University, School of Science and Technology, Professor (50060091)

Co-Investigator(Kenkyū-buntansha) KURANO Kazuhiko  Meiji University, School of Science and Technology, Professor (90205188)
Project Period (FY) 2004 – 2006
Project Status Completed (Fiscal Year 2006)
Budget Amount *help
¥3,700,000 (Direct Cost: ¥3,700,000)
Fiscal Year 2006: ¥1,100,000 (Direct Cost: ¥1,100,000)
Fiscal Year 2005: ¥1,300,000 (Direct Cost: ¥1,300,000)
Fiscal Year 2004: ¥1,300,000 (Direct Cost: ¥1,300,000)
KeywordsCommutative algebra / Singularities / Blow-up algebra / Rees algebra / Buchsbaum ring / Cohen-Macaulay ring / Multiplicity / Gorenstein環 / 随伴次数環 / 巴系イデアルのsocle / 局所コホモロジー加群 / Cohen-Macaulay型 / Buchsbaum ring / Cohen-Macaulay ring / generalized Grassmannian algebra / Rees algebra of ideals / Rees algebra of modules
Research Abstract

This research aims at the analysis of several kinds of so-called blow-up algebras associated to ideals and modules, in order to clarify the structure of given ideals and modules as well as that of the base(local or graded) rings. Certain practical criteria for graded rings, say the Rees algebra R(I), the associated graded ring G(I), and the fiber cone F(I), to be Gorenstein, Cohen-Macaulay, and Buchsbaum rings are discovered. Especially, the leading ideal of a complete intersection of height two in a regular local ring is closely studied, which are researches jointly with W. Heinzer and Mee-Kyoung Kim. The head investigator is quite satisfied by these joint works. On the other hand, a joint research with R. Takahashi and N. Matsuoka about quasi-socle ideals in a Gorenstein local ring was started and the theme is now a one of the most attractive studies of the theory of reduction numbers of ideals and the ring-theoretic properties of associated graded rings. The results are already written into two papers(see below) and were submitted to some journal for publication.
[1] S. Goto, N. Matsuoka, and R. Takahashi, Quasi-socle ideals in Gorenstein local rings, Preprint 2006
[2] S. Goto, F. Hayasaka, and R. Takahashi, On vanishing of certain Ext modules, Preprint 2006

Report

(4 results)
  • 2006 Annual Research Report   Final Research Report Summary
  • 2005 Annual Research Report
  • 2004 Annual Research Report
  • Research Products

    (35 results)

All 2006 2005 2004 Other

All Journal Article (35 results)

  • [Journal Article] The leading ideal of a Complete intersection of height two2006

    • Author(s)
      S.Goto, W.Heinzer, and Mee-Kyoung Kim
    • Journal Title

      J.Algebra 298

      Pages: 238-247

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2006 Final Research Report Summary
  • [Journal Article] The singular Riemann-Roch theorem and Hilbert-Kunz functions2006

    • Author(s)
      K.Kurano
    • Journal Title

      J.Algebra 304

      Pages: 487-499

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2006 Final Research Report Summary
  • [Journal Article] The leading iedeal of a complete intersection of height two2006

    • Author(s)
      S. Goto, W. Heinzer, Mee-Kyoung Kim
    • Journal Title

      J. Algebra 298

      Pages: 238-247

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2006 Final Research Report Summary
  • [Journal Article] The leading ideal of a complete intersection of height two2006

    • Author(s)
      S.Goto, W.Heinzer, Mee-Kyoung Kim
    • Journal Title

      J.Alg. 298

      Pages: 238-247

    • Related Report
      2006 Annual Research Report
  • [Journal Article] The singular Riemann-Roch theorem and Hilbert-Kunz functions2006

    • Author(s)
      K.Kurano
    • Journal Title

      J.Alg. 304

      Pages: 487-499

    • Related Report
      2006 Annual Research Report
  • [Journal Article] Rees algebras of the Second syzygy module of the residue field of a regular local ring2005

    • Author(s)
      S.Goto, F.Hayasaka, K.Kurano, and Y.Nakamura
    • Journal Title

      Commutative Algebra and Algebraic Geometry, Contemp. Math., Amer.Math.Soc., Providence, RI, 2005 390

      Pages: 97-108

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2006 Final Research Report Summary
  • [Journal Article] When does the equality $I^2=QI$ hold true in Buchsbaum rings2005

    • Author(s)
      S.Goto and H.Sakurai
    • Journal Title

      the Special Volume on 'Commutative Algebra with a focus on gemetric and homological aspects', Proceedings of Sevilla, June 18-21, 2003 and Lisbon, June 23-27, 2003, Lecture Notes in Pure and Applied Mathematics 244

      Pages: 115-139

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2006 Final Research Report Summary
  • [Journal Article] Gorenstein graded rings associated to ideals2005

    • Author(s)
      S, Goto and S.-i.Iai
    • Journal Title

      J.Algebra 294

      Pages: 373-407

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2006 Final Research Report Summary
  • [Journal Article] Gorenstein graded rings associated to ideals2005

    • Author(s)
      S. Goto, S.-i. Iai
    • Journal Title

      J. Algebra 294

      Pages: 373-407

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2006 Final Research Report Summary
  • [Journal Article] When does the equality $I^2=QI$ hold true in Buchsbaum rings?2005

    • Author(s)
      S. Goto, H. Sakurai
    • Journal Title

      The Special Volume on Commutative Algebra with a focus on geometric and homological aspects', Proceedings of Sevilla, June 18-21, 2003 and Lisbon, June 23-27, 2003, Lecture Notes in Pure and Applied Mathematics Series' 244

      Pages: 115-139

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2006 Final Research Report Summary
  • [Journal Article] Rees algebras of the second syzygy module of the residue field of a regular local ring2005

    • Author(s)
      S. Goto, F. Hayasaka, K. Kurano, Y. Nakamura
    • Journal Title

      Commutative Algebra and Algebraic Geometry, Contemp. Math. Amer. Math. Soc., Providence, RI 390

      Pages: 97-108

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2006 Final Research Report Summary
  • [Journal Article] The singular Riemann-Roch theorem and Hilbert-Kunz functions2005

    • Author(s)
      K. Kurano
    • Journal Title

      J. Algebra 294

      Pages: 373-407

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2006 Final Research Report Summary
  • [Journal Article] When does the equality $I^2 = QI$ hold true in Buchsbaum nings ?2005

    • Author(s)
      後藤四郎, 櫻井秀人
    • Journal Title

      Marcel Dekker's Lecture Notes in Pure and Applied Mathematics Series 244

      Pages: 115-139

    • Related Report
      2005 Annual Research Report
  • [Journal Article] Gorenstein graded rings associated to ideals2005

    • Author(s)
      後藤四郎, 居相真一郎
    • Journal Title

      J.Algebra 294

      Pages: 373-407

    • Related Report
      2005 Annual Research Report
  • [Journal Article] Rees algebras of the second syzygy module of the residue field of a regular local ring2005

    • Author(s)
      後藤四郎, 早坂太, 蔵野和彦, 中村幸男
    • Journal Title

      Contemp.Math. 390

      Pages: 97-108

    • Related Report
      2005 Annual Research Report
  • [Journal Article] The reduction exponent of socle ideals associated to parameter ideals in a Buchsbaum local ring of multiplicity two2004

    • Author(s)
      S.Goto and H.Sakurai
    • Journal Title

      J.Math.Soc.Japan 56

      Pages: 1157-1168

    • NAID

      10013753523

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2006 Final Research Report Summary
  • [Journal Article] The reduction exponent of socle ideals associated to parameter ideals in a Buchsbaum local ring of multiplicity two2004

    • Author(s)
      S. Goto, H. Sakurai
    • Journal Title

      J. Math. Soc. Japan 56

      Pages: 1157-1168

    • NAID

      10013753523

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2006 Final Research Report Summary
  • [Journal Article] The reduction exponent of socle ideals associated to parameter ideals in a Buchsbaum local ring of multiplicity two2004

    • Author(s)
      S.Goto, H.Sakurai
    • Journal Title

      J.Math.Soc.Japan 56

      Pages: 1157-1168

    • NAID

      10013753523

    • Related Report
      2004 Annual Research Report
  • [Journal Article] On the parametric decomposition of powers of parameter ideals in a Noetherian local ring2004

    • Author(s)
      S.Goto, Y.Shimoda
    • Journal Title

      Tokyo J.Math. 27-1

      Pages: 125-135

    • Related Report
      2004 Annual Research Report
  • [Journal Article] Stanley-Reisner ideals whose powers have finite length cohomologies

    • Author(s)
      S, Goto and Y.Takayama
    • Journal Title

      Proc.Amer.Math.Soc. (to appear)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2006 Final Research Report Summary
  • [Journal Article] Buchsbaum homogeneous algebras with minimal multiplicity

    • Author(s)
      S.Goto and K.Yoshida
    • Journal Title

      J.Pure and Appl.Alg. (to appear)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2006 Final Research Report Summary
  • [Journal Article] The leading ideal of a complete intersection of height two,Part II

    • Author(s)
      S.Goto, W.Heinzer, and M.-K.Kim
    • Journal Title

      J.Algebra (to appear)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2006 Final Research Report Summary
  • [Journal Article] The leading ideal of a complete intersection of height two in a 2-dimensional regular local ring

    • Author(s)
      S.Goto, W.Heinzer, and M.-K.Kim
    • Journal Title

      Communications in Algebra (to appear)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2006 Final Research Report Summary
  • [Journal Article] Stanley-Reisner ideals whose powers have finite length cohomolgies

    • Author(s)
      S. Goto, Y. Takayama
    • Journal Title

      Proc. Amer. Math. Soc. (to appear)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2006 Final Research Report Summary
  • [Journal Article] Buchsbaum homogeneous algebras with minimal multiplicity

    • Author(s)
      S. Goro, K. Yoshida
    • Journal Title

      J. Pure and Appl. Alg. (to appear)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2006 Final Research Report Summary
  • [Journal Article] The leading ideal of a complete intersection of height two, Part II

    • Author(s)
      S. Goro, W. Heinzer, M.-K. Kim
    • Journal Title

      Communications in Algebra (to appear)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2006 Final Research Report Summary
  • [Journal Article] The leading ideal of a complete intersection of height two in a 2-dimensional regular local ring

    • Author(s)
      S. Goro, W. Heinzer, M. K.-Kim
    • Journal Title

      J. Algebra (to appear)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2006 Final Research Report Summary
  • [Journal Article] Stanley-Reisner ideals whose powers have finite length cohomologies

    • Author(s)
      S.Goto, Y.Tkakayama
    • Journal Title

      Proc.Amer.Math.Soc (to appear)

    • Related Report
      2006 Annual Research Report
  • [Journal Article] Buchsbaum homogeneous algebras with minimal multiplicity

    • Author(s)
      S.Goto, K.Yoshida
    • Journal Title

      J.Pure and Appl.Alg. (to appera)

    • Related Report
      2006 Annual Research Report
  • [Journal Article] The leading ideal of a complete intersection of height two,Part II

    • Author(s)
      S.Goto, W.Heinzer, M.K.Kim
    • Journal Title

      J.Alg. (to appear)

    • Related Report
      2006 Annual Research Report
  • [Journal Article] The leading ideal of a complete intersection of height two

    • Author(s)
      S.Goto, W.Heinzer, Mee-Kyoung Kim
    • Journal Title

      J.Algebra (to appear)

    • Related Report
      2005 Annual Research Report
  • [Journal Article] The singular Riemann-Roch theorem and Hilbert-Kunz functions

    • Author(s)
      蔵野和彦
    • Journal Title

      J.Algebra (to appear)

    • Related Report
      2005 Annual Research Report
  • [Journal Article] Gorenstein graded rings associated to ideals

    • Author(s)
      S.Goto, S.lai
    • Journal Title

      J.Algebra (to appear)

    • Related Report
      2004 Annual Research Report
  • [Journal Article] When does the equality $I^2 = QI$ hold true in Buchsbaum rings?

    • Author(s)
      S.Goto, H.Sakurai
    • Journal Title

      Marcel Dekker's Lecture Notes in Pure and Applied Mathematics Series (to appear)

    • Related Report
      2004 Annual Research Report
  • [Journal Article] Rees algebras of the second syzygy module of the residue field of a regular local ring

    • Author(s)
      S.Goto, F.Hayasaka, K.Kurano, Y.Nakamura
    • Journal Title

      Contem.Math. (to appear)

    • Related Report
      2004 Annual Research Report

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Published: 2004-04-01   Modified: 2016-04-21  

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