On the Lefschetz property of complete intersections

• Project/Area Number: 15K04812
• Research Category: Grant-in-Aid for Scientific Research (C)
• Allocation Type: Multi-year Fund
• Section: 一般
• Research Field: Algebra
• Research Institution: Niigata University
• Principal Investigator: Harima Tadahito 新潟大学, 人文社会科学系, 教授 (30258313)
• Co-Investigator(Kenkyū-buntansha): 和地 輝仁 北海道教育大学, 教育学部, 准教授 (30337018) ; 五十川 読 熊本高等専門学校, 共通教育科(八代キャンパス), 教授 (80223056)
• Research Collaborator: Watanabe Junzo
• Project Period (FY): 2015-04-01 – 2019-03-31
• Project Status: Completed (Fiscal Year 2018)
• Budget Amount: ¥4,420,000 (Direct Cost: ¥3,400,000、Indirect Cost: ¥1,020,000); Fiscal Year 2018: ¥1,040,000 (Direct Cost: ¥800,000、Indirect Cost: ¥240,000); Fiscal Year 2017: ¥1,040,000 (Direct Cost: ¥800,000、Indirect Cost: ¥240,000); Fiscal Year 2016: ¥1,040,000 (Direct Cost: ¥800,000、Indirect Cost: ¥240,000); Fiscal Year 2015: ¥1,300,000 (Direct Cost: ¥1,000,000、Indirect Cost: ¥300,000)
• Keywords: 可換環 / 完全交叉環 / アルティン環 / ゴレンスタイン環 / レフシェッツ性 / 対称式 / 終結式 / 可換環論 / 完全交叉 / Macaulay dual generator / m-fullness / componentwise m-fullness / Macaulayの双対元 / completely m-fullness / componentwise linearity / EGH予想 / Sperner性 / matching性 / filterwise m-fullness

Outline of Final Research Achievements

We studied the Lefschetz property of complete intersections. Main results of this research are the followings: 1. Any quadratic complete intersection with certain action of the symmetric group has the strong Lefschetz property. 2. Suppose that the EGH Conjecture is true for a complete intersection A. Then A has the Sperner property. 3. All complete intersections defined by products of general linear forms have the strong Lefschetz property. 4. We gave a characterization of the Macaulay dual generators for quadratic complete intersections. 5. We gave another proof of some known results on power sum symmetric polynomials in three variables.

Academic Significance and Societal Importance of the Research Achievements

完全交叉のレフシェッツ性に関する研究は、コンピュータサイエンスとも関連のある多項式環論の基礎研究の一つである。また、レフシェッツ性は、線形写像の最強のジョルダン分解を求める問題とも関連しており、今後、線形写像のレフシェッツ性は、代数学の基本的な事項として位置付けられるのではないだろうか。

Research Products

• [Journal Article] The resultants of quadratic binomial complete intersections (2019)
  • Author(s): Tadahito Harima, Akihito Wachi, Junzo Watanabe
  • Journal Title: Journal of Commutative Algebra Volume: -
  • Peer Reviewed
• [Journal Article] The strong Lefschetz property for complete intersections defined by products of linear forms (2019)
  • Author(s): Tadahito Harima, Akihito Wachi, Junzo Watanabe
  • Journal Title: Archiv der Mathematik Volume: -
  • Peer Reviewed
• [Journal Article] Lefschetz properties for complete intersection ideals generated by products of linear forms (2018)
  • Author(s): Martina Juhnke-Kubitzke, Rosa Miro-Roig, Satoshi Murai, Akihito Wachi
  • Journal Title: Proceedings of the American Mathematical Society Volume: 146 Pages: 3249-3256
  • Peer Reviewed / Int'l Joint Research
• [Journal Article] Regular sequences of power sums in the polynomial ring in three variables (2018)
  • Author(s): Satoru Isogawa, Tadahito Harima
  • Journal Title: Research Reports of National Institute of Technology (Kumamoto College) Volume: 10 Pages: 60-66
  • NAID: 130007974307
• [Journal Article] EGH Conjecture and the Sperner Property of complete intersections (2017)
  • Author(s): Tadahito Harima, Akihito Wachi, Junzo Watanabe
  • Journal Title: Proceedings of the American Mathematical Society Volume: 145 Pages: 1497-1503
  • Peer Reviewed
• [Journal Article] Componentwise m-full modules (2017)
  • Author(s): Satoru Isogawa
  • Journal Title: Research Reports of NIT, Kumamoto College Volume: 9 Pages: 91-99
  • NAID: 130007978240
• [Journal Article] Strongly m-full modules (2017)
  • Author(s): Satoru Isogawa
  • Journal Title: Research Reports of NIT, Kumamoto College Volume: 9 Pages: 100-108
  • NAID: 130007978230
• [Journal Article] Weyr structure of matrices and relevance to commutative finite-dimensional algebras (2017)
  • Author(s): Kevin O'Meara, Junzo Watanabe
  • Journal Title: Linear algebras and its applications Volume: 532 Pages: 364-386
  • Peer Reviewed
• [Journal Article] The quadratic complete intersections associated with the action of the symmetric group (2015)
  • Author(s): Tadahito Harima, Akihito Wachi, Junzo Watanabe
  • Journal Title: Illinois Journal of Mathematics Volume: 59 Pages: 99-113
  • Peer Reviewed / Acknowledgement Compliant
• [Presentation] 一次式の積で定義される完全交叉の強いレフシェッツ性について (2018)
  • Author(s): 張間 忠人、和地 輝仁、渡辺 純三
  • Organizer: 日本数学会２０１８年度秋期総合分科会
• [Presentation] Regular sequences of power sums (2018)
  • Author(s): 五十川 読、張間 忠人
  • Organizer: 第１３９回日本数学会九州支部例会
• [Presentation] A characterization of the Macaulay dual generators for quadratic complete intersections (2017)
  • Author(s): Akihito Wachi
  • Organizer: Workshop on Lefschetz Properties in Algebra, Geometry and Combinatorics, Mittag-Leffler Insitute (Sweeden)
  • Int'l Joint Research
• [Presentation] The Weyr form seems a better tool for Artinitan algebras than the Jordan form (2017)
  • Author(s): Junzo Watanabe
  • Organizer: Lefschetz Property in Algebra, Geometry and Combinatorics, Mittag-Leffler Insitute (Sweeden)
  • Int'l Joint Research
• [Presentation] The resultants of quadratic binomials (2017)
  • Author(s): 渡辺純三
  • Organizer: 第39回可換環論シンポジウム（京都数理解析研究所）
• [Presentation] Componentwise m-full modules over a standard graded algebra (2017)
  • Author(s): 五十川 読
  • Organizer: 第137回日本数学会九州支部例会（熊本大学）
• [Presentation] 次数付き０次元ゴレンスタイン環のレフシェッツ性について (2017)
  • Author(s): 渡辺純三
  • Organizer: 日本数学会春期総合分科会（東京大学）、企画特別講演
  • Invited
• [Presentation] The EGH conjecture and the Sperner property of complete intersections (2016)
  • Author(s): 張間忠人、和地輝仁、渡辺純三
  • Organizer: 日本数学会秋季総合分科会
  • Place of Presentation: 関西大学（大阪府吹田市）
  • Year and Date: 2016-09-16
• [Presentation] 対称群が作用する0 次元完全交叉環の強いレフシェッツ性 (2016)
  • Author(s): 和地輝仁、張間忠人、渡辺純三
  • Organizer: 日本数学会秋季総合分科会
  • Place of Presentation: 関西大学（大阪府吹田市）
  • Year and Date: 2016-09-16
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• [Remarks] 新潟大学 研究者総覧
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• [Remarks] 新潟大学研究者総覧ホームページ
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