Computation of ideal qutients by transformation of acyclic complexes and its application

• Project/Area Number: 17K05192
• Research Category: Grant-in-Aid for Scientific Research (C)
• Allocation Type: Multi-year Fund
• Section: 一般
• Research Field: Algebra
• Research Institution: Chiba University
• Principal Investigator: Nishida Koji 千葉大学, 大学院理学研究院, 教授 (60228187)
• Project Period (FY): 2017-04-01 – 2024-03-31
• Project Status: Completed (Fiscal Year 2023)
• Budget Amount: ¥4,420,000 (Direct Cost: ¥3,400,000、Indirect Cost: ¥1,020,000); Fiscal Year 2021: ¥780,000 (Direct Cost: ¥600,000、Indirect Cost: ¥180,000); Fiscal Year 2020: ¥780,000 (Direct Cost: ¥600,000、Indirect Cost: ¥180,000); Fiscal Year 2019: ¥780,000 (Direct Cost: ¥600,000、Indirect Cost: ¥180,000); Fiscal Year 2018: ¥780,000 (Direct Cost: ¥600,000、Indirect Cost: ¥180,000); Fiscal Year 2017: ¥1,300,000 (Direct Cost: ¥1,000,000、Indirect Cost: ¥300,000)
• Keywords: 可換環 / シンボリック冪 / イデアル商 / 非輪状複体 / ヒルベルト係数 / シンボリックリース環 / Huneke の判定法

Outline of Final Research Achievements

The purpose of this research was to establish efficient methods for computing symbolic powers of ideals and apply them for studying symbolic Rees algebras. We first revised a technique for getting free resolutions of quotients of ideals transforming acyclic complexes which give free resolutions of ideals. Next, in the case where the rings are graded, studying the Hilbert coefficients of the quotient rings by homogeneous ideals which may coincide with the required symbolic powers, we found a method for checking whether the expected coincidences hold or not.

Academic Significance and Societal Importance of the Research Achievements

シンボリックリース環のネータ性はHilbertの第14問題と密接に関連しており、特に、体上の多項式環のイデアルに付随するシンボリックリース環で、非ネータなものを構成することが重要である。その為には、具体的に与えられたイデアルのシンボリック冪を計算する必要があるのだが、その様な計算を実行する実用的な手順はあまり知られていなかった。本研究では、多様なイデアルに対して適用可能な手法を提示することができ、この成果を未解決問題の解明にも活かせるのではないかと期待している。

Research Products

• [Journal Article] Finitely generated symbolic Rees rings of ideals defining certain finite sets of points in P^2 (2021)
  • Author(s): Kai Keisuke、Nishida Koji
  • Journal Title: Journal of Algebra Volume: 587 Pages: 20-35
  • DOI: 10.1016/j.jalgebra.2021.07.024
  • Peer Reviewed
• [Journal Article] Infinitely generated symbolic Rees rings of space monomial curves having negative curves (2019)
  • Author(s): Kazuhiko Kurano and Koji Nishida
  • Journal Title: Michigan Mathematical Journal Volume: 68 Pages: 409-445
  • Peer Reviewed
• [Journal Article] Infinitely generated sybolic Rees rings of space monomial curves having negative curves (2018)
  • Author(s): Kazuhiko Kurano and Koji Nishida
  • Journal Title: Michigan Mathematical Journal Volume: 印刷中
• [Presentation] On the Hilbert coefficients of graded modules over graded rings (2022)
  • Author(s): 西田康二
  • Organizer: 第43回 可換環論シンポジウム
  • Int'l Joint Research / Invited
• [Presentation] 次数付環上の次数付加群のヒルベルト係数について (2022)
  • Author(s): 西田康二
  • Organizer: 可換環論の新しい融合セミナー
  • Invited
• [Presentation] Noetherian symbolic Rees rings of finite sets of points in P^2 (2019)
  • Author(s): 西田康二
  • Organizer: 第41回可換環論シンポジウム
• [Presentation] Finitely generated symbolic Rees rings defined by certain finite sets of points in P^2 (2019)
  • Author(s): 西田康二
  • Organizer: 東京可換環論セミナー
• [Presentation] On the symbolic Rees rings for Fermat ideals (2019)
  • Author(s): Koji Nishida
  • Organizer: 1147th AMS Meeting, Special Session on Commutative Algebra
  • Int'l Joint Research
• [Presentation] シンボリックリース環のネータ性について (2018)
  • Author(s): 西田康二
  • Organizer: 第63回代数学シンポジウム
  • Invited
• [Presentation] On the symbolic Rees rings for Fermat ideals (2017)
  • Author(s): 西田 康二
  • Organizer: 第39回可換環論シンポジウム