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Study of toric varieties in ring theory

Research Project

Project/Area Number 16540037
Research Category

Grant-in-Aid for Scientific Research (C)

Allocation TypeSingle-year Grants
Section一般
Research Field Algebra
Research InstitutionNippon Institute of Technology

Principal Investigator

ETO Kazufumi  Nippon Institute of Technology, Faculty of Engineering, Associate Professor (30271357)

Co-Investigator(Kenkyū-buntansha) KAWASAKI Ken-ichiroh  Nara University of Education, Faculty of Education, Associate Professor (60288040)
Project Period (FY) 2004 – 2007
Project Status Completed (Fiscal Year 2007)
Budget Amount *help
¥3,540,000 (Direct Cost: ¥3,300,000、Indirect Cost: ¥240,000)
Fiscal Year 2007: ¥1,040,000 (Direct Cost: ¥800,000、Indirect Cost: ¥240,000)
Fiscal Year 2006: ¥800,000 (Direct Cost: ¥800,000)
Fiscal Year 2005: ¥800,000 (Direct Cost: ¥800,000)
Fiscal Year 2004: ¥900,000 (Direct Cost: ¥900,000)
Keywordstonic ideal / lattice ideal / complete intersection / almost complete intersection / binomial / primary decomposition / set-theoretic complete intersection / polynomial ring
Research Abstract

We consider a problem whether every affine monomial curve is a set-theoretic complete intersection, which is derived from a famous problem, asked by Kronecker, whether every affine algebraic curve is a set-theoretic complete intersection. We successfully give an equivalent condition in which the defining ideal of a monomial curve in affine N-space is generated by N-2 binomials and one polynomial up to radical, where N>2. The reason why we consider this condition is that most of earlier examples of set-theoretic complete intersection monomial curves satisfy this. Applying this condition, the monomial curve defined by 17, 19, 25 and 27 does not satisfy this condition. Consequently, its defining ideal is not generated by two binomials and a polynomial up to radical. On the other hand, we can prove that it is a set-theoretic complete intersection. Indeed, it is generated by one binomial and two polynomials up to radical. Further, we can extend this result as follows: If a monomial curve associated with a balanced semigroup, then it is a set-theoretic complete intersection. A balanced semigroup is an additive semigroup defined by four natural numbers, say a, b, c and d, with a+d=b+c Thus the previous example is associated with a balanced semigroup. In addition, we may prove the same result for monomial curves associated with extended balanced semigroups.

Report

(5 results)
  • 2007 Annual Research Report   Final Research Report Summary
  • 2006 Annual Research Report
  • 2005 Annual Research Report
  • 2004 Annual Research Report
  • Research Products

    (47 results)

All 2008 2007 2006 2005 2004 2002

All Journal Article (26 results) (of which Peer Reviewed: 7 results) Presentation (21 results)

  • [Journal Article] The monomial curves associated with balanced semigroups are set-theoretic complete intersections2008

    • Author(s)
      Kazufumi Eto
    • Journal Title

      J.Algebra 319

      Pages: 1355-1367

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2007 Annual Research Report 2007 Final Research Report Summary
    • Peer Reviewed
  • [Journal Article] The monomial curves associated with balanced semigroups are set-theoretic complete intersections2008

    • Author(s)
      Kazufumi, Eto
    • Journal Title

      J. Algebra 319

      Pages: 1355-1367

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2007 Final Research Report Summary
  • [Journal Article] A Counterexample of the Theorem of Erdogdu2007

    • Author(s)
      衛藤 和文
    • Journal Title

      日本工業大学研究報告 37

      Pages: 577-577

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2007 Final Research Report Summary
  • [Journal Article] 集合論的完全交叉について(I)2007

    • Author(s)
      衛藤 和文
    • Journal Title

      日本工業大学研究報告 37

      Pages: 529-530

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2007 Annual Research Report 2007 Final Research Report Summary
  • [Journal Article] A counterexample of the theorem of Erdogdu2007

    • Author(s)
      Kazufumi, Eto
    • Journal Title

      Report of researches Nippon Inst. of Tech 37

      Pages: 577-577

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2007 Final Research Report Summary
  • [Journal Article] On set-theoretic complete intersections2007

    • Author(s)
      Kazufumi, Eto
    • Journal Title

      Report of researches Nippon Inst. of Tech 37

      Pages: 529-530

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2007 Final Research Report Summary
  • [Journal Article] On finiteness properties of local cohomology modules over Cohen-Macaulay local rings2007

    • Author(s)
      Ken-ichiroh, Kawasaki
    • Journal Title

      RIMS Kokyuroku 37

      Pages: 529-530

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2007 Final Research Report Summary
  • [Journal Article] A Counterexample of the Theorem of Erdogdu2007

    • Author(s)
      Kazufumi Eto
    • Journal Title

      日本工業大学研究報告 37

      Pages: 577-577

    • Related Report
      2007 Annual Research Report
  • [Journal Article] Monomial curves associated with balanced semigroups2006

    • Author(s)
      衛藤 和文
    • Journal Title

      数理解析研講究録 1478

      Pages: 5-8

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2007 Final Research Report Summary
    • Peer Reviewed
  • [Journal Article] Set-theoretic complete intersection lattice ideals in monoid rings2006

    • Author(s)
      Kazufumi Eto
    • Journal Title

      J.Algebra 299

      Pages: 689-706

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2007 Final Research Report Summary
    • Peer Reviewed
  • [Journal Article] An example of set-theoretic complete intersection lattice ideal2006

    • Author(s)
      Kazufumi Eto
    • Journal Title

      Tokyo J.Math 29

      Pages: 319-324

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2007 Final Research Report Summary
    • Peer Reviewed
  • [Journal Article] コーエン・マコーレー局所環上の局所コホモロジー加群の有限性に関する性質について2006

    • Author(s)
      川崎 謙一郎
    • Journal Title

      数理解析研講究録 1478

      Pages: 1-4

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2007 Final Research Report Summary
    • Peer Reviewed
  • [Journal Article] On the finiteness of Bass numbers of generalized local cohomology modules2006

    • Author(s)
      Satoshi Kawakami, Ken-ichiroh Kawasaki
    • Journal Title

      Toyama Math.J. 29

      Pages: 59-64

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2007 Annual Research Report 2007 Final Research Report Summary
    • Peer Reviewed
  • [Journal Article] Monomial curves associated with balanced semigroups2006

    • Author(s)
      Kazufumi, Eto
    • Journal Title

      RIMS Kokyuroku 1478

      Pages: 5-8

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2007 Final Research Report Summary
  • [Journal Article] Set-theoretic complete intersection lattice ideals in monoid rings2006

    • Author(s)
      Kazufumi, Eto
    • Journal Title

      J. Algebra 299

      Pages: 689-706

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2007 Final Research Report Summary
  • [Journal Article] An example of set-theoretic complete intersection lattice ideal2006

    • Author(s)
      Kazufumi, Eto
    • Journal Title

      Tokyo J. Math 29

      Pages: 319-324

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2007 Final Research Report Summary
  • [Journal Article] On the finiteness of Bass numbers of generalized local cohomology modules2006

    • Author(s)
      Satoshi, Kawakami, Ken-ichiroh, Kawasaki
    • Journal Title

      Math. J. Toyama Univ 29

      Pages: 59-64

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2007 Final Research Report Summary
  • [Journal Article] Monomial Curves Associated with Balanced Semigroups2006

    • Author(s)
      衛藤和文
    • Journal Title

      数理解析研講究録 1478

      Pages: 5-8

    • Related Report
      2006 Annual Research Report
  • [Journal Article] Set-theroretic Complete Intersection Lattice Ideals in Monoid Rings2006

    • Author(s)
      Kazufumi Eto
    • Journal Title

      J.Algebra 299・2

      Pages: 689-706

    • Related Report
      2006 Annual Research Report
  • [Journal Article] An Example of Set-theoretic Complete Intersection Lattice Ideal2006

    • Author(s)
      Kazufumi Eto
    • Journal Title

      J.Algebra 29・2

      Pages: 319-324

    • Related Report
      2006 Annual Research Report
  • [Journal Article] Almost complete intersection lattice ideals2005

    • Author(s)
      衛藤 和文
    • Journal Title

      日本工業大学研究報告 35

      Pages: 237-248

    • NAID

      40006987071

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2007 Final Research Report Summary
  • [Journal Article] Almost complete intersection lattice ideals2005

    • Author(s)
      Kazufumi, Eto
    • Journal Title

      Report of researches Nippon Inst. of Tech 35

      Pages: 237-248

    • NAID

      40006987071

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2007 Final Research Report Summary
  • [Journal Article] Almost complete intersection lattice ideals2005

    • Author(s)
      Kazufumi Eto
    • Journal Title

      日本工業大学研究報告 35・2

      Pages: 3-14

    • NAID

      40006987071

    • Related Report
      2005 Annual Research Report
  • [Journal Article] When is a binomial ideal equal to a lattice ideal up to radical? II2004

    • Author(s)
      Kazufumi Eto
    • Journal Title

      Math.J.Toyama Univ 27

      Pages: 111-123

    • NAID

      110004679104

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2007 Final Research Report Summary
    • Peer Reviewed
  • [Journal Article] When is a binomial ideal equal to a lattice ideal up to radical? II2004

    • Author(s)
      Kazufumi, Eto
    • Journal Title

      Math., J. Toyama Univ 27

      Pages: 111-123

    • NAID

      110004679104

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2007 Final Research Report Summary
  • [Journal Article] When is a binomial ideal equal to a lattice ideal? II2004

    • Author(s)
      Eto, K.
    • Journal Title

      Math.J.Toyama.Univ. 27

      Pages: 111-123

    • Related Report
      2004 Annual Research Report
  • [Presentation] あるmonomial curveが集合論的完全交叉であることの証明2008

    • Author(s)
      衛藤 和文
    • Organizer
      第20回可換環論セミナー
    • Place of Presentation
      勝浦簡易保険保養センター
    • Year and Date
      2008-01-28
    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2007 Annual Research Report 2007 Final Research Report Summary
  • [Presentation] A proof of a set-theoretic complete intersection monomial curve2008

    • Author(s)
      Kazufumi, Eto
    • Organizer
      20-th Seminar on Commutative Algebra
    • Place of Presentation
      Katsuura
    • Year and Date
      2008-01-28
    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2007 Final Research Report Summary
  • [Presentation] Several results on finiteness properties of local cohomology modules over Cohn-Macaulay local rings2007

    • Author(s)
      川崎 謙一郎
    • Organizer
      第29回可換環論シンポジウム
    • Place of Presentation
      愛知厚生年金会館
    • Year and Date
      2007-11-19
    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2007 Final Research Report Summary
  • [Presentation] Several results on finiteness properties of local cohomology modules over Cohen-Macaulay local rings2007

    • Author(s)
      川崎 謙一郎
    • Organizer
      第29回可換環論シンポジウム
    • Place of Presentation
      愛知厚生年金会館
    • Year and Date
      2007-11-19
    • Related Report
      2007 Annual Research Report
  • [Presentation] Several results on finiteness properties of local cohomology modules over Cohen-Macaulay local rings2007

    • Author(s)
      Ken-ichiroh, Kawasaki
    • Organizer
      29-th Symposium on Commutative Algebra
    • Place of Presentation
      Nagoya
    • Year and Date
      2007-11-17
    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2007 Final Research Report Summary
  • [Presentation] Set-theoretic complete intersectionについて2007

    • Author(s)
      衛藤 和文
    • Organizer
      第19回可換環論セミナー
    • Place of Presentation
      九州大学
    • Year and Date
      2007-02-01
    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2007 Final Research Report Summary
  • [Presentation] On set-theoretic complete intersections2007

    • Author(s)
      Kazufumi, Eto
    • Organizer
      19-th Seminar on Commutative Algebra
    • Place of Presentation
      Kyushu University
    • Year and Date
      2007-02-01
    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2007 Final Research Report Summary
  • [Presentation] あるmonomial curveが集合論的完全交叉である証明に関しての注意2006

    • Author(s)
      衛藤 和文
    • Organizer
      第18回可換環論セミナー
    • Place of Presentation
      かでる2・7
    • Year and Date
      2006-02-02
    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2007 Final Research Report Summary
  • [Presentation] On finiteness properties of local cohomology modules over Cohen-Macaulay local rings2006

    • Author(s)
      川崎 謙一郎
    • Organizer
      第18回可換環論セミナー
    • Place of Presentation
      かでる2・7
    • Year and Date
      2006-02-02
    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2007 Final Research Report Summary
  • [Presentation] Note on a proof of a set-theoretic complete intersection monomial curve2006

    • Author(s)
      Kazufumi, Eto
    • Organizer
      18-th Seminar on Commutative Algebra
    • Place of Presentation
      Sapporo
    • Year and Date
      2006-02-02
    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2007 Final Research Report Summary
  • [Presentation] Set-theoretic complete intersection monomial curves2005

    • Author(s)
      衛藤 和文
    • Organizer
      第27回可換環論シンポジウム
    • Place of Presentation
      インテック大山研修センター
    • Year and Date
      2005-11-15
    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2007 Final Research Report Summary
  • [Presentation] Set-theoretic complete intersection monomial curves2005

    • Author(s)
      Kazufumi, Eto
    • Organizer
      27-th Symposium on Commutative Algebra
    • Place of Presentation
      Kyoto
    • Year and Date
      2005-11-15
    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2007 Final Research Report Summary
  • [Presentation] Monomial curves associated with balanced semigroups2005

    • Author(s)
      衛藤 和文
    • Organizer
      RIMS共同研究「可換Banach環と関連分野研究集会」
    • Place of Presentation
      京都大学数理研究所
    • Year and Date
      2005-10-31
    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2007 Final Research Report Summary
  • [Presentation] コーエン・マコーレー局所環上の局所コホモロジー加群の有限性に関する性質について2005

    • Author(s)
      川崎 謙一郎
    • Organizer
      RIMS共同研究「可換Banach環と関連分野研究集会」
    • Place of Presentation
      京都大学数理研究所
    • Year and Date
      2005-10-31
    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2007 Final Research Report Summary
  • [Presentation] Monomial curves associated with balanced semigroups2005

    • Author(s)
      Kazufumi, Eto
    • Organizer
      Communication in commutative Banach algebra and several field of mathematics
    • Place of Presentation
      RIMS, Kyoto Univ
    • Year and Date
      2005-10-31
    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2007 Final Research Report Summary
  • [Presentation] On finiteness properties of local cohomology modules over Cohen-Macaulay local rings2005

    • Author(s)
      Ken-ichiroh, Kawasaki
    • Organizer
      Communication in commutative Banach, algebra and several field of mathematics
    • Place of Presentation
      RIMS, Kyoto Univ
    • Year and Date
      2005-10-31
    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2007 Final Research Report Summary
  • [Presentation] Set-theoretic complete intersection lattice ideals on lattice ideals2005

    • Author(s)
      衛藤 和文
    • Organizer
      日本数学会 2005年度秋季総合分科会
    • Place of Presentation
      岡山大学
    • Year and Date
      2005-09-21
    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2007 Final Research Report Summary
  • [Presentation] Set-theoretic complete intersection lattice ideals on lattice ideals2005

    • Author(s)
      Kazufumi, Eto
    • Organizer
      MSJ meetings, September
    • Place of Presentation
      Okayama University
    • Year and Date
      2005-09-21
    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2007 Final Research Report Summary
  • [Presentation] lattice idealの集合論的完全交叉に関する逆問題2005

    • Author(s)
      衛藤 和文
    • Organizer
      第17回可換環論セミナー
    • Place of Presentation
      キャンパスプラザ京都
    • Year and Date
      2005-01-28
    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2007 Final Research Report Summary
  • [Presentation] An inverse problem concerning with set-theoretic complete intersection lattice ideals2005

    • Author(s)
      Kazufumi, Eto
    • Organizer
      17-th Seminar on Commutative Algebra
    • Place of Presentation
      Kyoto
    • Year and Date
      2005-01-28
    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2007 Final Research Report Summary
  • [Presentation] On finiteness properties of local cohomology modules over Cohen-Macaulay local rings2002

    • Author(s)
      Ken-ichiroh, Kawasaki
    • Organizer
      18-th Seminar on Commutative Algebra
    • Place of Presentation
      Sapporo
    • Year and Date
      2002-02-02
    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2007 Final Research Report Summary

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

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