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Study on the Lefschetz properties and generic initial ideals of complete intersections

Research Project

Project/Area Number 20540035
Research Category

Grant-in-Aid for Scientific Research (C)

Allocation TypeSingle-year Grants
Section一般
Research Field Algebra
Research InstitutionHokkaido University of Education

Principal Investigator

HARIMA Tadahito  Hokkaido University of Education, 教育学部, 准教授 (30258313)

Co-Investigator(Kenkyū-buntansha) WATANABE Junzo  東海大学, 理学部, 教授 (40022727)
WACHI Akihito  北海道教育大学釧路校, 教育学部, 准教授 (30337018)
ISOGAWA Satoru  熊本高等専門学校, 共通教育科, 教授 (80223056)
Project Period (FY) 2008 – 2010
Project Status Completed (Fiscal Year 2010)
Budget Amount *help
¥4,030,000 (Direct Cost: ¥3,100,000、Indirect Cost: ¥930,000)
Fiscal Year 2010: ¥1,300,000 (Direct Cost: ¥1,000,000、Indirect Cost: ¥300,000)
Fiscal Year 2009: ¥1,300,000 (Direct Cost: ¥1,000,000、Indirect Cost: ¥300,000)
Fiscal Year 2008: ¥1,430,000 (Direct Cost: ¥1,100,000、Indirect Cost: ¥330,000)
Keywords完全交叉 / レフシェッツ性 / ジェネリックイニシャルイデアル / 単項式イデアルゴレンスタイン環 / ジョルダン標準形 / 可換環 / ゴレンスタイン環 / リース元 / 単項式イデアル / Rees元 / 強いレフシェッツ性 / ヘッセ行列 / rev-lexイデアル
Research Abstract

The Lefschetz properties are basic and important properties for Artinian graded algebras. In this research, we studied the Lefschetz properties and generic initial ideals of complete intersections. As a result, we obtained the following. 1. We gave a sharp upper bound on the Betti numbers of Artinian algebras with the k-Lefsctez property and a fixed Hilbert function. 2. We found generic initial ideals of some complete intersections in four variables. 3. The tensor product of strongest Jordan decompositions is again strongest. This is a generalization of the fact which states that the strong Lefschetz property is preserved by tensor product. 4. We gave some examples of Gorenstein algebras not having the weak Lefschetz property.

Report

(4 results)
  • 2010 Annual Research Report   Final Research Report ( PDF )
  • 2009 Annual Research Report
  • 2008 Annual Research Report
  • Research Products

    (20 results)

All 2010 2009 2008 Other

All Journal Article (12 results) (of which Peer Reviewed: 12 results) Presentation (3 results) Remarks (5 results)

  • [Journal Article] Generic initial ideals of some monomial complete intersections in four variables2010

    • Author(s)
      Tadahito Harima, Sho Sakaki, Akihito Wachi
    • Journal Title

      Archiv der Mathematik 69巻

      Pages: 129-137

    • Related Report
      2010 Final Research Report
    • Peer Reviewed
  • [Journal Article] Generic initial ideals of some monomial complete intersections in four variables2010

    • Author(s)
      Tadahito Harima, Sho Sakaki, Akihito Wachi
    • Journal Title

      Archiv der Mathematik 94

      Pages: 129-137

    • Related Report
      2009 Annual Research Report
    • Peer Reviewed
  • [Journal Article] Generic initial ideals, graded Betti numbers, and k -Lefschetz properties2009

    • Author(s)
      Tadahito Harima, Akihito Wachi
    • Journal Title

      Communications in Algebra 37巻

      Pages: 4012-4025

    • Related Report
      2010 Final Research Report
    • Peer Reviewed
  • [Journal Article] Lefschetz elements of Artinian Gorenstein algebras and Hessians of homogeneous polynomials2009

    • Author(s)
      Toshiaki Maeno, Junzo Watanabe
    • Journal Title

      Illinois Journal of Mathematics 53巻

      Pages: 591-603

    • Related Report
      2010 Final Research Report
    • Peer Reviewed
  • [Journal Article] Regular sequences of symmetric polynomials2009

    • Author(s)
      Aldo Conca, Christian Krattenthaler, Junzo Watanabe
    • Journal Title

      Rendiconti del Seminario Matematico della Universita di Padova 121巻

      Pages: 179-199

    • Related Report
      2010 Final Research Report
    • Peer Reviewed
  • [Journal Article] Zero-dimensional Gorenstein algebras with the action of the symmetric group2009

    • Author(s)
      Hideaki Morita, Akihito Wachi, Junzo Watanabe
    • Journal Title

      Rendiconti del Seminario Matematico della Universita di Padova 121巻

      Pages: 45-71

    • Related Report
      2010 Final Research Report
    • Peer Reviewed
  • [Journal Article] The Dilworth lattice of Artinian rings2009

    • Author(s)
      Hidemi Ikeda, Junzo Watanabe
    • Journal Title

      Journal of Commutative Algebra 1巻

      Pages: 315-326

    • Related Report
      2010 Final Research Report
    • Peer Reviewed
  • [Journal Article] A note on the Capelli identities for symmetric pairs of Hermitian type2009

    • Author(s)
      Kyo Nishiyama, Akihito Wachi
    • Journal Title

      Infinite dimensional harmonic analysis IV

      Pages: 223-254

    • Related Report
      2010 Final Research Report
    • Peer Reviewed
  • [Journal Article] Generic Initial Ideals, Graded Betti Numbers, and k-Lefschetz properties2009

    • Author(s)
      Tadahito Harima, Akihito Wachi
    • Journal Title

      Communications in Algebra 37

      Pages: 4012-4025

    • Related Report
      2009 Annual Research Report
    • Peer Reviewed
  • [Journal Article] The determinants of certain matirces arising from the Boolean lattice2008

    • Author(s)
      Masao Hara, Junzo Watanabe
    • Journal Title

      Discrete Mathematics 308巻

      Pages: 5815-5822

    • Related Report
      2010 Final Research Report
    • Peer Reviewed
  • [Journal Article] Intersection of harmonics and Capelli identities for symmetric pairs2008

    • Author(s)
      Soo Teck Lee, Kyo Nishiyama, Akihito Wachi
    • Journal Title

      Journal of the Mathematical Society of Japan 60巻

      Pages: 955-982

    • NAID

      10024905179

    • Related Report
      2010 Final Research Report
    • Peer Reviewed
  • [Journal Article] Strong Lefschetz elements of the coinvariant rings of finite Coxeter groups

    • Author(s)
      Toshiaki Maeno, Yasuhide Numata, Akihito Wachi
    • Journal Title

      Algebras and Representation Theory (Springer)

    • NAID

      130008004821

    • Related Report
      2010 Final Research Report
    • Peer Reviewed
  • [Presentation] Generic initial ideals of some complete intersections in four variables2010

    • Author(s)
      張間忠人、和地輝仁
    • Organizer
      日本数学会
    • Place of Presentation
      名古屋大学
    • Year and Date
      2010-09-23
    • Related Report
      2010 Final Research Report
  • [Presentation] Generic initial ideals of some complete intersections in four variables2010

    • Author(s)
      張間忠人・和地輝仁
    • Organizer
      日本数学会 秋季総合分科会
    • Place of Presentation
      名古屋大学
    • Year and Date
      2010-09-23
    • Related Report
      2010 Annual Research Report
  • [Presentation] 完全交叉のレフシェッツ性問題について2008

    • Author(s)
      張間忠人
    • Organizer
      第53回代数学シンポジウム
    • Place of Presentation
      いわて県民情報交流センター
    • Year and Date
      2008-08-05
    • Related Report
      2010 Final Research Report 2008 Annual Research Report
  • [Remarks] ホームベジ等

    • URL

      http://kenqweb.office.ehime-u.ac.jp/Profiles/0003/0002304/thesesl.html

    • Related Report
      2010 Final Research Report
  • [Remarks]

    • URL

      http://kenqweb.office.ehime-u.ac.jp/Profiles/0003/0002304/gakkail.html

    • Related Report
      2010 Final Research Report
  • [Remarks]

    • URL

      http://kenqweb.office.ehime-u.ac.jp/Profiles/0003/0002304/thesesl.html

    • Related Report
      2010 Annual Research Report
  • [Remarks]

    • URL

      http://kenqweb.office.ehime-u.ac.jp/Profiles/0003/0002304/gakkail.html

    • Related Report
      2010 Annual Research Report
  • [Remarks]

    • URL

      http://kengweb.office.ehime-u.ac.jp/Profiles/0003/0002304/profile.html

    • Related Report
      2009 Annual Research Report

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Published: 2008-04-01   Modified: 2016-04-21  

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