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Study on modules over commutative rings by categorical methods

Research Project

Project/Area Number 26287008
Research Category

Grant-in-Aid for Scientific Research (B)

Allocation TypePartial Multi-year Fund
Section一般
Research Field Algebra
Research InstitutionOkayama University

Principal Investigator

Yoshino Yuji  岡山大学, 自然科学研究科, 教授 (00135302)

Project Period (FY) 2014-04-01 – 2019-03-31
Project Status Completed (Fiscal Year 2018)
Budget Amount *help
¥9,880,000 (Direct Cost: ¥7,600,000、Indirect Cost: ¥2,280,000)
Fiscal Year 2018: ¥1,950,000 (Direct Cost: ¥1,500,000、Indirect Cost: ¥450,000)
Fiscal Year 2017: ¥1,950,000 (Direct Cost: ¥1,500,000、Indirect Cost: ¥450,000)
Fiscal Year 2016: ¥1,950,000 (Direct Cost: ¥1,500,000、Indirect Cost: ¥450,000)
Fiscal Year 2015: ¥1,950,000 (Direct Cost: ¥1,500,000、Indirect Cost: ¥450,000)
Fiscal Year 2014: ¥2,080,000 (Direct Cost: ¥1,600,000、Indirect Cost: ¥480,000)
Keywords導来圏 / 三角圏 / Cohen-Macaulay 加群 / 次数付き微分加群 / 非有界複体 / 安定圏 / シジジー / DG代数とDG加群 / 代数学 / 可換環論 / 鎖複体
Outline of Final Research Achievements

New mathematics called homological algebra has been established in the middle of the 20th century and in the process of this establishment, many problems and conjectures have arisen, which remain unsolved yet. Now in this new century, these problems are regarded as in the framework of the categories. The most interesting objective is to study finitely generated modules over commutative rings that are given by finitely many elements with finitely many relations. However, in this study, we consider the problems once in larger categories called unbounded derived categories that naturally involve infinitely generated modules. Eventually through these methods, we succeeded in deriving several results for finitely generated modules.

Academic Significance and Societal Importance of the Research Achievements

数学の基礎研究であるので特に社会的意義を主張するには及ばない。しかしながら学術的には、半世紀以上未解決であった代数学の問題に対して、それを解決するための糸口が掴めたという点では大きな学術的意義があると考えている。
とくに今回の研究の全般を通して行った導来圏やホモトピー圏などの研究は、問題の解決という意図とは別に、それ自体が数学的に美しい体系を提供しているように思っている。

Report

(6 results)
  • 2018 Annual Research Report   Final Research Report ( PDF )
  • 2017 Annual Research Report
  • 2016 Annual Research Report
  • 2015 Annual Research Report
  • 2014 Annual Research Report
  • Research Products

    (19 results)

All 2019 2018 2017 2016 2015 2014 Other

All Int'l Joint Research (3 results) Journal Article (9 results) (of which Int'l Joint Research: 2 results,  Peer Reviewed: 9 results,  Open Access: 8 results) Presentation (6 results) (of which Int'l Joint Research: 3 results,  Invited: 4 results) Funded Workshop (1 results)

  • [Int'l Joint Research] American Institute of Mathematics/Georgia Southern University(米国)

    • Related Report
      2018 Annual Research Report
  • [Int'l Joint Research] Georgia Southern University(米国)

    • Related Report
      2017 Annual Research Report
  • [Int'l Joint Research] American Institute of Mathematics(米国)

    • Related Report
      2016 Annual Research Report
  • [Journal Article] Degenerations over (A∞)-singularities and construction of degenerations over commutative rings2019

    • Author(s)
      Hiramatsu Naoya、Takahashi Ryo、Yoshino Yuji
    • Journal Title

      Journal of Algebra

      Volume: 525 Pages: 374-389

    • DOI

      10.1016/j.jalgebra.2018.12.031

    • Related Report
      2018 Annual Research Report
    • Peer Reviewed / Open Access
  • [Journal Article] Primary decompositions in Abelian R-categories2018

    • Author(s)
      Kenichi Sato and Yuji Yoshino
    • Journal Title

      Mathematical Journal of Okayama University

      Volume: 60 Pages: 91-108

    • NAID

      120006469110

    • Related Report
      2018 Annual Research Report
    • Peer Reviewed / Open Access
  • [Journal Article] Noncommutative resolutions using syzygies2018

    • Author(s)
      Dao Hailong、Iyama Osamu、Iyengar Srikanth B.、Takahashi Ryo、Wemyss Michael、Yoshino Yuji
    • Journal Title

      Bulletin of the London Mathematical Society

      Volume: 51 Issue: 1 Pages: 43-48

    • DOI

      10.1112/blms.12210

    • Related Report
      2018 Annual Research Report
    • Peer Reviewed / Open Access / Int'l Joint Research
  • [Journal Article] A local duality principle in derived categories of commutative Noetherian rings2018

    • Author(s)
      Tsutom Nakamura and Yuji Yoshino
    • Journal Title

      Journal of Pure and Applied Algebra

      Volume: 222 Issue: 9 Pages: 2580-2595

    • DOI

      10.1016/j.jpaa.2017.10.008

    • Related Report
      2018 Annual Research Report 2017 Annual Research Report
    • Peer Reviewed / Open Access
  • [Journal Article] Weak liftings of DG modules2018

    • Author(s)
      Saeed Nasseh and Yuji Yoshino
    • Journal Title

      Journal of Algebra

      Volume: 502 Pages: 233-248

    • DOI

      10.1016/j.jalgebra.2018.01.024

    • Related Report
      2018 Annual Research Report 2017 Annual Research Report
    • Peer Reviewed / Open Access / Int'l Joint Research
  • [Journal Article] Localization functors and cosupport in derived categories of commutative Noetherian rings2018

    • Author(s)
      Tsutom Nakamura and Yuji Yoshino
    • Journal Title

      Pacific Journal of Math.

      Volume: 296 Issue: 2 Pages: 405-435

    • DOI

      10.2140/pjm.2018.296.405

    • Related Report
      2018 Annual Research Report
    • Peer Reviewed / Open Access
  • [Journal Article] Primary decompositions in Abelian R-categories2018

    • Author(s)
      Kenichi Sato, Yuji Yoshino
    • Journal Title

      Mathematical Journal of Okayama University

      Volume: 60

    • NAID

      120006469110

    • Related Report
      2017 Annual Research Report
    • Peer Reviewed / Open Access
  • [Journal Article] An Auslander-Reiten principle in derived categories2017

    • Author(s)
      Ono Maiko、Yoshino Yuji
    • Journal Title

      Journal of Pure and Applied Algebra

      Volume: 221 Issue: 6 Pages: 1268-1278

    • DOI

      10.1016/j.jpaa.2016.09.009

    • Related Report
      2017 Annual Research Report
    • Peer Reviewed / Open Access
  • [Journal Article] A remark on vanishing of chain complexes2015

    • Author(s)
      Yuji Yoshino
    • Journal Title

      Acta Mathematica Vietnamica

      Volume: 40 Issue: 1 Pages: 173-177

    • DOI

      10.1007/s40306-015-0121-4

    • Related Report
      2014 Annual Research Report
    • Peer Reviewed
  • [Presentation] The homotopy category of unbounded complexes of projective modules2018

    • Author(s)
      Yuji Yoshino
    • Organizer
      Workshop “Stable Cohomology: Foundations and Applications” at Snowbird, Utah
    • Related Report
      2018 Annual Research Report
    • Int'l Joint Research / Invited
  • [Presentation] Sable complex theory2017

    • Author(s)
      Yuji Yoshino
    • Organizer
      京都大学数理解析研究所共同研究第39回可換環論シンポジウム
    • Related Report
      2017 Annual Research Report
    • Int'l Joint Research / Invited
  • [Presentation] AR principle と local duality の一般化について2016

    • Author(s)
      中村力、吉野雄二
    • Organizer
      第28回可換環論セミナー
    • Place of Presentation
      岡山理科大学
    • Year and Date
      2016-01-24
    • Related Report
      2015 Annual Research Report
  • [Presentation] The principle of Auslander-Reiten duality2015

    • Author(s)
      Maiko Ono, Yuji Yoshino
    • 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
    • Year and Date
      2015-11-18
    • Related Report
      2015 Annual Research Report
  • [Presentation] A little more about Auslander-Reiten duality and conjecture for MCM2015

    • Author(s)
      Yuji Yoshino
    • Organizer
      Conference on derived categories of finite dimensional algebras
    • Place of Presentation
      静岡大学理学部
    • Year and Date
      2015-09-11
    • Related Report
      2015 Annual Research Report
    • Int'l Joint Research / Invited
  • [Presentation] *torsion free and *reflexive complexes2014

    • Author(s)
      Yuji Yoshino
    • Organizer
      Commutative Algebra and Singularity Theory
    • Place of Presentation
      立山国際ホテル(富山県)
    • Year and Date
      2014-07-28 – 2014-08-01
    • Related Report
      2014 Annual Research Report
    • Invited
  • [Funded Workshop] 第37回可換環論シンポジウム2015

    • Place of Presentation
      倉敷シーサイドホテル
    • Year and Date
      2015-11-18
    • Related Report
      2015 Annual Research Report

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Published: 2014-04-04   Modified: 2022-02-21  

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