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Commutative Ring Theory via Resolution of Singularities

Research Project

Project/Area Number 20K03522
Research Category

Grant-in-Aid for Scientific Research (C)

Allocation TypeMulti-year Fund
Section一般
Review Section Basic Section 11010:Algebra-related
Research InstitutionMeiji University

Principal Investigator

Watanabe Kei-ichi  明治大学, 研究・知財戦略機構(生田), 研究推進員(客員研究員) (10087083)

Project Period (FY) 2020-04-01 – 2023-03-31
Project Status Completed (Fiscal Year 2022)
Budget Amount *help
¥2,470,000 (Direct Cost: ¥1,900,000、Indirect Cost: ¥570,000)
Fiscal Year 2022: ¥780,000 (Direct Cost: ¥600,000、Indirect Cost: ¥180,000)
Fiscal Year 2021: ¥780,000 (Direct Cost: ¥600,000、Indirect Cost: ¥180,000)
Fiscal Year 2020: ¥910,000 (Direct Cost: ¥700,000、Indirect Cost: ¥210,000)
Keywords正規特異点 / 特異点解消 / 整閉イデアル / 還元数 / Gorenstein 環 / Hilbert-Kunz 重複度 / 楕円型特異点 / 数値半群 / elliptic singularity / reduction numbers / Hilbert-Kubz 重複度 / F-signature / Inverse polynomial / integrally closed ideal / normal Hilbert function / normal Rees algebra / elliptic ideal / numerical semigroup / inverse polynomial / normal reduction number / Girenstein 環 / 半群環 / Commutative Ring / Ideal Theory / 特異点の解消
Outline of Research at the Start

本研究で,特異点は代数多様体に現れるものを考える.特異点は,その点での関数のなす局所環で記述される.特異点の「悪さ」はその局所環に含まれるイデアルの性質で記述される事が多い.イデアルは特異点解消の空間で,例外曲線の「サイクル」で表現される.本研究の第1の目的は,特に2次元の正規特異点において,与えられた特異点に対してどのようなイデアルが存在するか.それらの性質がどのくらい特異点の性質に依存するかを記述する方法を確立する事である.
また第2の目的は正標数での特有の性質を用いて「良い」特異点を記述し,また渡辺が提唱した,F-閾値を用いてイデアルの重複度の上限を与える予想を証明する事である.

Outline of Final Research Achievements

I studied with Tomohiro Okuma (Yamagat Univ.) and Ken-ichi Yoshida (Nihon Univ.) integrally closed ideals in a 2-dimensional normal singularities. After our previous work on pg-ideals, we proposed a new class of ideals called "elliptic ideals" and showed nice properties of such ideals. This notion came up during the joint work with Okuma, Yoshida and M.-E. Rossi (Genova Univ.). This notion is useful to characterize "elliptic singularities". We also found a new example where the 2 notions of "normal reduction numbers" are different.

In theory of positive characteristics, I found a new lower bound of Hilbert-Kunz multiplicities (joint work with Yoshida, I. Smirnov et. al) and

Academic Significance and Societal Importance of the Research Achievements

可換環論は代数幾何学と深く結びついている,可換環論の最も重要な対象であるイデアルに対して,その環論的性質を幾何学的に解析する事は今まで行われて来なかった.本研究は,2次元の正規特異点に対して幾何学的な情報を用いて,環論的性質を導くもので,大変独自性が高い.実際,今まで知られていなかった,正規還元数を持つイデアルを幾何学的情報によって発見でき,またその代数的な表現を与えた事は大変大きな成果であった.
また,正標数の可換環論の手法を用いて,幾何学的な性質を与える事は将来幾何学的な情報をコンピューターで計算を可能にするために役に立つ可能性を持っている.

Report

(4 results)
  • 2022 Annual Research Report   Final Research Report ( PDF )
  • 2021 Research-status Report
  • 2020 Research-status Report
  • Research Products

    (12 results)

All 2022 2021 2020 Other

All Int'l Joint Research (1 results) Journal Article (6 results) (of which Int'l Joint Research: 6 results,  Peer Reviewed: 6 results,  Open Access: 2 results) Presentation (5 results) (of which Int'l Joint Research: 2 results,  Invited: 2 results)

  • [Int'l Joint Research] Dep. of Math. University of Genova(イタリア)

    • Related Report
      2020 Research-status Report
  • [Journal Article] Normal Hilbert coefficients and elliptic ideals in normal two-dimensional singularities2022

    • Author(s)
      Okuma, Tomohiro and Rossi, Maria Evelina and Watanabe, Kei-ichi and Yoshida, Ken-ichi
    • Journal Title

      Nagoya Mathematical Journal

      Volume: - Pages: 1-22

    • DOI

      10.1017/nmj.2022.5

    • Related Report
      2022 Annual Research Report
    • Peer Reviewed / Open Access / Int'l Joint Research
  • [Journal Article] Hilbert-Kunz density function for graded domains2022

    • Author(s)
      Trivedi Vijaylaxmi、Watanabe Kei-Ichi
    • Journal Title

      Journal of Pure and Applied Algebra

      Volume: 226 Issue: 2 Pages: 106835-106835

    • DOI

      10.1016/j.jpaa.2021.106835

    • Related Report
      2022 Annual Research Report 2021 Research-status Report
    • Peer Reviewed / Int'l Joint Research
  • [Journal Article] Lower bounds on Hilbert-Kunz multiplicities and maximal F-signatures2022

    • Author(s)
      JACK JEFFRIES, YUSUKE NAKAJIMA, ILYA SMIRNOV, KEI-ICHI WATANABE , KEN-ICHI YOSHIDA
    • Journal Title

      Mathematical Proceedings of the Cambridge Philosophical Society

      Volume: 174 Issue: 2 Pages: 247-271

    • DOI

      10.1017/s0305004122000238

    • Related Report
      2022 Annual Research Report
    • Peer Reviewed / Open Access / Int'l Joint Research
  • [Journal Article] The strong Rees property of powers of the maximal ideal and Takahashi-Dao's question,2021

    • Author(s)
      T. Puthenpurakal, K. Watanabe. K. YOshida
    • Journal Title

      Journal of Algebra

      Volume: 571 Pages: 297-315

    • DOI

      10.1016/j.jalgebra.2018.07.028

    • Related Report
      2021 Research-status Report 2020 Research-status Report
    • Peer Reviewed / Int'l Joint Research
  • [Journal Article] Hilbert-Kunz density functions and F-thresholds,2021

    • Author(s)
      V. Trivedi and K. Watanabe
    • Journal Title

      Journal of Algebra

      Volume: 567 Pages: 533-563

    • DOI

      10.1016/j.jalgebra.2020.09.025

    • Related Report
      2020 Research-status Report
    • Peer Reviewed / Int'l Joint Research
  • [Journal Article] Homogeneous prime elements in normal two-dimensional graded rings2020

    • Author(s)
      Anurag Singh; Ryo Takahashi; Kei-ichi Watanabe
    • Journal Title

      Journal of Algebra

      Volume: 0 Pages: 0-0

    • DOI

      10.1016/j.jalgebra.2018.07.012

    • Related Report
      2021 Research-status Report 2020 Research-status Report
    • Peer Reviewed / Int'l Joint Research
  • [Presentation] Inverse Polynomials of numerical semigroup rings2022

    • Author(s)
      Kei-ichi Watanabe
    • Organizer
      International Meeting on Numerical Semigroups, Roma, 2022,
    • Related Report
      2022 Annual Research Report
  • [Presentation] Elliptic ideals in 2 dimensional normal local rings2022

    • Author(s)
      Kei-ichi Watanabe
    • Organizer
      Commutative Algebra Seminar, University of Utah.
    • Related Report
      2022 Annual Research Report
  • [Presentation] Inverse polynomials of symmetric numerical semigroups2021

    • Author(s)
      Keiichi Watanabe
    • Organizer
      Virtual Commutative Algebra Seminar, IIT Bombay,
    • Related Report
      2021 Research-status Report
  • [Presentation] Normal Hilbert coefficients and elliptic ideals in normal 2-dimensional local domains2021

    • Author(s)
      Kei-ichi Watanabe
    • Organizer
      Workshop, "Fellowshipof the Ring", Mathematical Sciences Research Institute
    • Related Report
      2020 Research-status Report
    • Int'l Joint Research / Invited
  • [Presentation] NORMAL HILBERT COEFFICIENTS AND ELLIPTIC IDEALS IN NORMAL 2-DIMENSIONAL LOCAL DOMAINS2021

    • Author(s)
      Kei-ichi Watanabe
    • Organizer
      OIST Workshop: Quantum Math, Singularities and Applications
    • Related Report
      2020 Research-status Report
    • Int'l Joint Research / Invited

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Published: 2020-04-28   Modified: 2024-01-30  

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