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Zero-dimensional Gorenstein algebra with the strong Lefschets properties

Research Project

Project/Area Number 24540050
Research Category

Grant-in-Aid for Scientific Research (C)

Allocation TypeMulti-year Fund
Section一般
Research Field Algebra
Research InstitutionTokai University

Principal Investigator

WATANABE Junzo  東海大学, 理学部, 教授 (40022727)

Project Period (FY) 2012-04-01 – 2015-03-31
Project Status Completed (Fiscal Year 2014)
Budget Amount *help
¥5,070,000 (Direct Cost: ¥3,900,000、Indirect Cost: ¥1,170,000)
Fiscal Year 2014: ¥1,300,000 (Direct Cost: ¥1,000,000、Indirect Cost: ¥300,000)
Fiscal Year 2013: ¥1,300,000 (Direct Cost: ¥1,000,000、Indirect Cost: ¥300,000)
Fiscal Year 2012: ¥2,470,000 (Direct Cost: ¥1,900,000、Indirect Cost: ¥570,000)
KeywordsLefschetz Properties / 0次元ゴレンスタイン環 / 完全交叉環 / Sperner 性 / ヒルベルト関数 / 次数付き環 / コホモロジー環 / 巾ゼロ元 / アルティン可換環 / 強いレフシェッツ条件 / 弱いレフシェッツ条件 / Sperner 理論 / 平坦拡大定理 / 部分環定理 / Tolgiatti Theorem / アルティン環 / 有限半順序集合 / 不変式部分環 / ゴレンスタイン環 / レフシェッツ条件 / シュペルナ性 / 高次ヘシアン / Hard Lefschetz Theorem
Outline of Final Research Achievements

In September, 2013, I published a book entitled "The Lefschetz Properties," with 5 Japanese coauthors. It took 10 years to write this book. In the course of writing the book, it turned out that the topic has many ties to other areas of algebra. In September 2012, I organized a workshop in Hawaii Tokai International College, where 13 researchers from Japan, USA, Sweden and Germany, gathered and we reported the latest results and exchanged ideas. In March, 2015, I organized a workshop in Goettingen, Germany, for the second time, under a cooperation with German and American and Spanish researchers, where more or less 25 researchers came together and we exchanged ideas and new results. It is scheduled to have such a workshop at Banff, Canada in 2016 for the third time.

Report

(4 results)
  • 2014 Annual Research Report   Final Research Report ( PDF )
  • 2013 Research-status Report
  • 2012 Research-status Report
  • Research Products

    (17 results)

All 2015 2014 2013 2012

All Journal Article (5 results) (of which Peer Reviewed: 5 results,  Acknowledgement Compliant: 1 results) Presentation (10 results) (of which Invited: 9 results) Book (2 results)

  • [Journal Article] Completely m-full ideals and componentwise linear ideals2015

    • Author(s)
      Tadahito Harima, Junzo Watanabe
    • Journal Title

      Mathematical Proc. Cambridge Philos. Soc.

      Volume: 158 Issue: 2 Pages: 239-248

    • DOI

      10.1017/s0305004114000632

    • Related Report
      2014 Annual Research Report
    • Peer Reviewed / Acknowledgement Compliant
  • [Journal Article] The set of solutions of a system of linear partial differential equations2015

    • Author(s)
      Toyohiro Akamatsu, Junzo Watanabe
    • Journal Title

      Proceedings of the School of Science of Tokai University

      Volume: 50

    • NAID

      120005677492

    • Related Report
      2014 Annual Research Report
    • Peer Reviewed
  • [Journal Article] On ideals with the Rees property2013

    • Author(s)
      Migliore, Juan;Miro-Roig, Rosa Maria;Murai, Satoshi; Nagel, Uwe; Watanabe, Junzo
    • Journal Title

      Arch. Math.

      Volume: 101 Pages: 445-454

    • Related Report
      2013 Research-status Report
    • Peer Reviewed
  • [Journal Article] On the Theory of Gordan-Noether on Homogeneous Forms with Zero Hessian2013

    • Author(s)
      Watanabe, Junzo
    • Journal Title

      Proc. Sch. Sci. Tokai Univ.

      Volume: 48 Pages: 1-21

    • NAID

      120005677484

    • Related Report
      2013 Research-status Report
    • Peer Reviewed
  • [Journal Article] The weak Lefschetz property for m-full ideals and componentwise linear ideals2013

    • Author(s)
      Tadahito HARIMA and Junzo WATANABE
    • Journal Title

      Illinois Journal of Mathematics

      Volume: 印刷中

    • Related Report
      2012 Research-status Report
    • Peer Reviewed
  • [Presentation] Quadratic complete intersections associated with the action of the symmetric group2015

    • Author(s)
      Junzo Watanabe
    • Organizer
      Workshop: Lefschetz Properties
    • Place of Presentation
      Universitaet Goettingen
    • Year and Date
      2015-03-07
    • Related Report
      2014 Annual Research Report
    • Invited
  • [Presentation] 解説:ゴルダン・ネーターの論文"Ueber die algebraischen Formen, deren Hesse’sche Determinante identisch verschwindet"2014

    • Author(s)
      渡辺純三
    • Organizer
      可換環論と表現論 吉野雄二先生還暦記念研究集会
    • Place of Presentation
      岡山大学理学部本館
    • Year and Date
      2014-10-11 – 2014-10-13
    • Related Report
      2014 Annual Research Report
    • Invited
  • [Presentation] Completely m-full ideals and componentwise linear ideals2014

    • Author(s)
      張間忠人、渡辺純三
    • Organizer
      日本数学会秋期総合分科会
    • Place of Presentation
      広島大学
    • Year and Date
      2014-09-25 – 2014-09-28
    • Related Report
      2014 Annual Research Report
  • [Presentation] 2次式で生成される完全交叉環のレフッシェッツについて2014

    • Author(s)
      渡辺純三
    • Organizer
      組合せ論と可換代数サマーセミナー
    • Place of Presentation
      広島大学
    • Year and Date
      2014-09-22 – 2014-09-24
    • Related Report
      2014 Annual Research Report
    • Invited
  • [Presentation] A conjecture: All complete intersections have the strong Lefschetz properties in characteristic zero.2014

    • Author(s)
      Junzo Watanabe
    • Organizer
      Algebra Seminar, Nebraska State University at Lincoln
    • Place of Presentation
      Nebraska State University at Lincoln
    • Year and Date
      2014-04-14 – 2014-04-18
    • Related Report
      2014 Annual Research Report
    • Invited
  • [Presentation] The Sperner property of poset and the strong Lefschetz property of Artinian Gorenstiein rings2014

    • Author(s)
      Junzo Watanabe
    • Organizer
      Colloquium in Department of Mathematics, Nebraska State University at Lincoln
    • Place of Presentation
      Nebraska State University at Lincloln
    • Year and Date
      2014-04-14 – 2014-04-18
    • Related Report
      2014 Annual Research Report
    • Invited
  • [Presentation] 概説:強いレフシェッツ条件2013

    • Author(s)
      渡辺 純三
    • Organizer
      組合せ論と可換代数サマーセミナー
    • Place of Presentation
      下関生涯学習プラザ
    • Related Report
      2013 Research-status Report
    • Invited
  • [Presentation] A new class of Gorenstein algebras with the strong Lefschetz property2013

    • Author(s)
      Watanabe, Junzo
    • Organizer
      Commutative Algebra and Its Interaction to Algebraic Geometry and Combinatorics
    • Place of Presentation
      Hanoi, Vietnam
    • Related Report
      2013 Research-status Report
    • Invited
  • [Presentation] 「予想:完全交叉は強いレフシェッツ性をもつ」はどこまで分かったか?2013

    • Author(s)
      渡辺 純三
    • Organizer
      第35回可換環論シンポジウム
    • Place of Presentation
      京都大学数理解析研究所
    • Related Report
      2013 Research-status Report
    • Invited
  • [Presentation] Introduction to the paper of Gordan-Noether--Homogeneous polynomials whose Hessian determinat identically vanishes2012

    • Author(s)
      Junzo WATANABE
    • Organizer
      Aspects of WLP and SLP
    • Place of Presentation
      Hawaii Tokai International College
    • Related Report
      2012 Research-status Report
    • Invited
  • [Book] The Lefschetz Properties2013

    • Author(s)
      Harima, Tadahito; Maeno, Toshiaki; Morita, Hideaki; Numata, Yasuhide; Wachi, Akihito; Watanabe, Junzo.
    • Total Pages
      250
    • Publisher
      Springer
    • Related Report
      2013 Research-status Report
  • [Book] Lefschetz Properties2013

    • Author(s)
      J. Watanabe, A. Wachi, T. Harima, T. Maeno, H. Morita, Y. Numata
    • Total Pages
      247
    • Publisher
      Springer
    • Related Report
      2012 Research-status Report

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Published: 2013-05-31   Modified: 2019-07-29  

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