Hilbert function of local rings

• Project/Area Number: 15K04820
• Research Category: Grant-in-Aid for Scientific Research (C)
• Allocation Type: Multi-year Fund
• Section: 一般
• Research Field: Algebra
• Research Institution: Yamaguchi University
• Principal Investigator: Ozeki Kazuho 山口大学, 大学院創成科学研究科, 准教授 (70445849)
• Project Period (FY): 2015-04-01 – 2018-03-31
• Project Status: Completed (Fiscal Year 2017)
• Budget Amount: ¥3,510,000 (Direct Cost: ¥2,700,000、Indirect Cost: ¥810,000); 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: ¥1,430,000 (Direct Cost: ¥1,100,000、Indirect Cost: ¥330,000)
• Keywords: ヒルベルト函数 / ヒルベルト係数 / Rees代数 / 随伴次数環 / Sally加群 / コーエン・マコ―レイ局所環 / 随伴次数間 / Cohen-Macaulay環 / 可換環論 / 巴系イデアル

Outline of Final Research Achievements

The purpose of this research is to explore the behavior of the Hilbert functions　of ideals in Cohen-Macaulay local rings.
I explored the relationship between the first Hilbert coefficient and the structure of the associated graded ring of integrally closed ideals in a Cohen-Macaulay local ring.
I also gave a characterization for the almost minimal value of the first Hilbert coefficient in the case of the normal filtration in an analytically unramified Cohen-Macaulay local ring. I strongly used some structure theorems of Sally modules in these results.

Research Products

• [Int'l Joint Research] ジェノバ大学(イタリア)
• [Int'l Joint Research] ジェノバ大学数学科(イタリア)
• [Journal Article] The structure of the Sally module of integrally closed ideals (2017)
  • Author(s): Kazuho Ozeki and Maria Evelina Rossi
  • Journal Title: Nagoya Mathematical Journal Volume: 227 Pages: 49-76
  • DOI: 10.1017/nmj.2016.47
  • Peer Reviewed / Int'l Joint Research
• [Journal Article] The structure of the Sally modules of integrally closed ideals (2017)
  • Author(s): Kazuho Ozeki
  • Journal Title: Proceedings of the 49th Symposium on Ring Theory and Representation Theory Volume: - Pages: 135-142
• [Journal Article] The first Hilbert coefficient and Buchsbaumness of associated graded rings (2017)
  • Author(s): Kazuho Ozeki
  • Journal Title: 第38回可換環論シンポジウム報告集 Volume: - Pages: 152-161
• [Journal Article] Sectional genera of parameter ideals (2016)
  • Author(s): Shiro Goto and Kazuho Ozeki
  • Journal Title: Journal of Algebra Volume: 446 Pages: 58-76
  • DOI: 10.1016/j.jalgebra.2015.09.005
  • Peer Reviewed
• [Journal Article] The second Hilbert coefficients and the homological torsions of parameters (2015)
  • Author(s): Shiro Goto, Kazuho Ozeki
  • Journal Title: Journal of Pure and Applied Algebra Volume: 219 Issue: 5 Pages: 1685-1703
  • DOI: 10.1016/j.jpaa.2014.07.006
  • Peer Reviewed
• [Presentation] A study of the second normal Hilbert coefficient (2018)
  • Author(s): Kazuho Ozeki
  • Organizer: Mini-Workshop on Commutative Algebra, 北海道教育大学札幌駅前サテライト（北海道札幌市）
  • Invited
• [Presentation] The structure of the Sally modules of integrally closed ideals (2017)
  • Author(s): Kazuho Ozeki
  • Organizer: Algebra & Geometry Seminar
  • Place of Presentation: Genova University (Genova, Italy)
  • Year and Date: 2017-03-22
  • Invited
• [Presentation] The first Hilbert coefficient and Buchsbaumness of associated graded rings (2017)
  • Author(s): Kazuho Ozeki
  • Organizer: International Workshop on Commutative Algebra
  • Place of Presentation: Thai Nguyen University (Thai Nguyen, Vietnam)
  • Year and Date: 2017-01-05
  • Int'l Joint Research / Invited
• [Presentation] Almost minimal normal Hilbert coefficients (2017)
  • Author(s): Kazuho Ozeki
  • Organizer: International Conference on Commutative Algebra and its interaction to Combinatorics, Discrete Geometry and Singularity Theory, Hanoi and Ha Long (Vietnam)
  • Int'l Joint Research / Invited
• [Presentation] Almost minimal normal Hilbert coefficients (2017)
  • Author(s): Kazuho Ozeki
  • Organizer: International School and Workshop on Commutative Algebra, Ton Duc Thang University, Ho Chi Minh City (Vietnam)
  • Int'l Joint Research / Invited
• [Presentation] The first Hilbert coefficient and Buchsbaumness of associated graded rings (2016)
  • Author(s): Kazuho Ozeki
  • Organizer: 第38回可換環論シンポジウム
  • Place of Presentation: IPC生産性国際交流センター（神奈川県葉山市）
  • Year and Date: 2016-11-21
• [Presentation] The first two Hilbert coefficients and Buchsbaumness of associated graded rings (2016)
  • Author(s): Kazuho Ozeki
  • Organizer: Japan-Vietnam Workshop on Commutative Algebra, -by and for young mathematicians- Local rings Combinatorics, and Representation Theory
  • Place of Presentation: 明治大学生田キャンパス（神奈川県川崎市）
  • Year and Date: 2016-09-19
  • Int'l Joint Research / Invited
• [Presentation] The structure of the Sally modules of integrally closed ideals (2016)
  • Author(s): Kazuho Ozeki
  • Organizer: The 49th Symposium on Ring Theory and Representation Theory
  • Place of Presentation: 大阪府立大学中百舌鳥キャンパス（大阪府堺市）
  • Year and Date: 2016-08-31
• [Presentation] 整閉イデアルのSally加群の構造について (2016)
  • Author(s): 大関一秀
  • Organizer: 2016年度日本数学会年会
  • Place of Presentation: 筑波大学（茨城県つくば市）
  • Year and Date: 2016-03-16
• [Presentation] The structure of the Sally modules and the first Hilbert coefficient of ideals (2016)
  • Author(s): Kazuho Ozeki
  • Organizer: Workshop on Commutative Algebra- Part II, Local Algebra and Representation Theory-
  • Place of Presentation: VIASM's villa (Tuan Chau, Vietnam)
  • Year and Date: 2016-01-06
  • Int'l Joint Research / Invited
• [Presentation] The structure of the Sally module of integrally closed ideals (2015)
  • Author(s): 大関一秀
  • Organizer: Ring Theory and Representation Theory Seminar
  • Place of Presentation: 名古屋大学多元数理科学研究科（愛知県名古屋市）
  • Year and Date: 2015-12-14
• [Presentation] The structure of the filtration of Sally modules (2015)
  • Author(s): Kazuho Ozeki and Maria Evelina Rossi
  • Organizer: 第37回可換環論シンポジウム
  • Place of Presentation: 倉敷シーサイドホテル（岡山県倉敷市）
  • Year and Date: 2015-11-20