On the Lefschetz properties of complete intersections in commutative algebra

• Project/Area Number: 23540052
• Research Category: Grant-in-Aid for Scientific Research (C)
• Allocation Type: Multi-year Fund
• Section: 一般
• Research Field: Algebra
• Research Institution: Niigata University (2013-2014); Ehime University (2011-2012)
• Principal Investigator: HARIMA Tadahito 新潟大学, 人文社会・教育科学系, 教授 (30258313)
• Co-Investigator(Kenkyū-buntansha): WATANABE Junzo 東海大学, 理学部, 特任教授 (40022727) ; ISOGAWA Satoru 熊本高等専門学校, 共通教育科, 教授 (80223056)
• Co-Investigator(Renkei-kenkyūsha): WACHI Akihito 北海道教育大学, 教育学部, 准教授 (30337018)
• Project Period (FY): 2011-04-28 – 2015-03-31
• Project Status: Completed (Fiscal Year 2014)
• Budget Amount: ¥4,940,000 (Direct Cost: ¥3,800,000、Indirect Cost: ¥1,140,000); Fiscal Year 2013: ¥1,560,000 (Direct Cost: ¥1,200,000、Indirect Cost: ¥360,000); Fiscal Year 2012: ¥1,560,000 (Direct Cost: ¥1,200,000、Indirect Cost: ¥360,000); Fiscal Year 2011: ¥1,820,000 (Direct Cost: ¥1,400,000、Indirect Cost: ¥420,000)
• Keywords: 完全交叉 / レフシェッツ性 / m-fullイデアル / completely m-fullイデアル / componentwise linearイデアル / Betti数 / Rees元 / アルティン環 / 可換環 / m-full イデアル / completely m-full イデアル / componentwise linearity / Rees性 / 可換環論 / 完全交叉環 / ゴレンスタイン環 / componentwise linear

Outline of Final Research Achievements

We studied the Lefschetz properties of complete intersections, and got the following results. 1. We gave a necessary and sufficient condition for a graded Artinian algebra defined by an m-full ideal to have the weak Lefschetz property in terms of graded Betti numbers. This is a generalization of a theorem of Wiebe for componentwise linear ideals. 2. We published a monograph on the Lefschetz properties of Artinian algebras. In this book we present recent results that were obtained and developed by authors over the last 10 years. 3. We showed that the class of completely m-full ideals coincides with the class of componentwise linear ideals in a polynomial ring over an infinite field.

Research Products

• [Journal Article] Rees property and its related properties of ranked partially ordered sets (2015)
  • Author(s): Satoru Isogawa
  • Journal Title: 熊本高等専門学校研究紀要 Volume: 6 Pages: 66-73
  • NAID: 130007992375
  • Acknowledgement Compliant
• [Journal Article] Rees property and its related properties of modules (2015)
  • Author(s): Satoru Isogawa
  • Journal Title: 熊本高等専門学校研究紀要 Volume: 6 Pages: 74-81
  • NAID: 130007992372
  • Acknowledgement Compliant
• [Journal Article] Completely m-full ideals and componentwise linear ideals (2015)
  • Author(s): Tadahito Harima and Junzo Watanabe
  • Journal Title: Mathematical Proceedings of the Cambridge Philosophical Society Volume: 158 Pages: 239-248
  • Peer Reviewed / Acknowledgement Compliant
• [Journal Article] A note on graded Betti numbers of completely m-full ideals (2014)
  • Author(s): Tadahito Harima, Satoru Isogawa and Junzo Watanabe
  • Journal Title: 新潟大学教育学部研究紀要 Volume: 7 Pages: 1-6
  • Acknowledgement Compliant
• [Journal Article] The weak Lefschetz property for m-full ideals and componentwise linear ideals (2013)
  • Author(s): Tadahito Harima and Junzo Watanabe
  • Journal Title: Illinois Journal of Mathematics Volume: -
  • Peer Reviewed
• [Journal Article] On the number of generators of the quotient of a complete intersecgion by a general linear form (2012)
  • Author(s): Junzo Watanabe
  • Journal Title: Proceedings of the School of Science, Tokai University Volume: 47 Pages: 1-10
• [Presentation] The quadratic complete intersections associated with the action of the symmetric group (2015)
  • Author(s): 渡辺純三
  • Organizer: Workshop on Lefschetz properties
  • Place of Presentation: Universitat Goettingen（ドイツ）
  • Year and Date: 2015-03-09
• [Presentation] Rees peoperty and its related properties of ranked partially ordered sets (2015)
  • Author(s): 五十川 読
  • Organizer: 第132回日本数学会九州支部例会
  • Place of Presentation: 福岡大学
  • Year and Date: 2015-01-22
• [Presentation] Completely m-full ideals and componentwise linear ideals (2014)
  • Author(s): 張間忠人、渡辺純三
  • Organizer: 日本数学会
  • Place of Presentation: 広島大学
  • Year and Date: 2014-09-26
• [Presentation] 強いレフシェッツ性条件 (2013)
  • Author(s): 渡邊純三
  • Organizer: 組合せ論と可換代数サマーセミナー
  • Place of Presentation: 下関市生涯学習プラザ
• [Presentation] A new class of Gorenstein algebras with the strong Lefschetz property (2013)
  • Author(s): 渡邊純三
  • Organizer: International conference on Commutative Algebra and its Interaction to Algebraic Geometry and Combinatorics
  • Place of Presentation: Hanoi, Vietnam
• [Presentation] Introduction to the paper of Gordan-Noether (Homogeneous polynomials whose Hessian determinat identically vanishes) (2012)
  • Author(s): Junzo Watanabe
  • Organizer: Aspects of SLP and WLP
  • Place of Presentation: Hawaii Tokai International College
• [Presentation] The weak Lefschetz property for m-full ideals and componentwise linear ideals (2012)
  • Author(s): 張間忠人, 渡辺純三
  • Organizer: 日本数学会
  • Place of Presentation: 東京理科大学
• [Book] The Lefschetz Properties (2013)
  • Author(s): T. Harima, T. Maeno, H. Morita, Y. Numata, A. Wachi, J. Watanabe
  • Total Pages: 250
  • Publisher: Springer (Lecture Notes in Mathematics, Vol. 2080)
• [Remarks] 新潟大学教育学部教員・研究室紹介
  • URL: http://www.ed.niigata-u.ac.jp/modules/faculty/index.php?content_id=12
• [Remarks]
  • URL: http://kenqweb.office.ehime-u.ac.jp/Profiles/0003/0002304/profile.html