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Study on Cohen-Macaulay modules by degeneration theory and its applications to representation types of algebras

Research Project

Project/Area Number 15K17527
Research Category

Grant-in-Aid for Young Scientists (B)

Allocation TypeMulti-year Fund
Research Field Algebra
Research InstitutionKure National College of Technology

Principal Investigator

Hiramatsu Naoya  呉工業高等専門学校, 自然科学系分野, 准教授 (20612039)

Project Period (FY) 2015-04-01 – 2019-03-31
Project Status Completed (Fiscal Year 2018)
Budget Amount *help
¥2,860,000 (Direct Cost: ¥2,200,000、Indirect Cost: ¥660,000)
Fiscal Year 2017: ¥780,000 (Direct Cost: ¥600,000、Indirect Cost: ¥180,000)
Fiscal Year 2016: ¥910,000 (Direct Cost: ¥700,000、Indirect Cost: ¥210,000)
Fiscal Year 2015: ¥1,170,000 (Direct Cost: ¥900,000、Indirect Cost: ¥270,000)
Keywordsコーエン・マコーレー加群 / 退化 / 表現型 / グロタンディック群 / 行列表現 / 極大コーエン・マコーレー加群 / 有限表現型 / 加算CM表現型 / 三角圏 / 加算表現型 / 可算無限表現型 / Cohen-Macaulay加群 / 安定圏 / Gorenstein環
Outline of Final Research Achievements

We give the sufficient condition to make the stable hom relation a partial order. We give the description of stable degenerations of Cohen-Macaulay modules over several hypersurfaces by using the stable hom relation. We show that a Gorenstein ring is of finite representation type if the Auslander-Reiten sequences generate the relations for Grothendieck groups. We also give a necessary condition of degeneration via matrix representations, and completely determine degenerations of indecomposable Cohen-Macaulay modules over hypersurface of countable A representation type of dimension 1 and 2.

Academic Significance and Societal Importance of the Research Achievements

極大コーエン・マコーレー加群の退化に関する成果は, これまで計算が困難であった退化の関係の記述を安定射集合の次元や行列表現の考察に帰着することで容易にし, 多くの具体的な例を提供することができる。またグロタンディック群に関する結果はAuslanderの予想の部分的な解決を与えているばかりか, グロタンディック群を定義する関係の様子から有限表現型、また特異点(孤立特異点)の様子を記述することができたので, 表現型理論・特異点論の観点からも意義があると考えられる。

Report

(5 results)
  • 2018 Annual Research Report   Final Research Report ( PDF )
  • 2017 Research-status Report
  • 2016 Research-status Report
  • 2015 Research-status Report
  • Research Products

    (20 results)

All 2019 2018 2017 2016 2015 Other

All Journal Article (8 results) (of which Peer Reviewed: 3 results,  Open Access: 3 results,  Acknowledgement Compliant: 1 results) Presentation (11 results) (of which Int'l Joint Research: 3 results,  Invited: 2 results) Remarks (1 results)

  • [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] Degenerations of Cohen-Macaulay modules via matrix representations2018

    • Author(s)
      平松直哉
    • Journal Title

      Proceedings of the 50th Symposium on Ring Theory and Representation Theory

      Volume: 50 Pages: 54-59

    • Related Report
      2017 Research-status Report
  • [Journal Article] On the stable hom relation and stable degenerations of Cohen?Macaulay modules2017

    • Author(s)
      Hiramatsu Naoya
    • Journal Title

      Journal of Pure and Applied Algebra

      Volume: 222 Issue: 9 Pages: 2609-2625

    • DOI

      10.1016/j.jpaa.2017.10.010

    • Related Report
      2017 Research-status Report
    • Peer Reviewed
  • [Journal Article] Relations for Grothendieck groups of Gorenstein rings2017

    • Author(s)
      Naoya Hiramatsu
    • Journal Title

      Proceedings of the American Mathematical Society

      Volume: 145 Issue: 2 Pages: 559-562

    • DOI

      10.1090/proc/13255

    • Related Report
      2016 Research-status Report
    • Peer Reviewed / Acknowledgement Compliant
  • [Journal Article] On relations for Grothendieck groups of Gorenstein rings2017

    • Author(s)
      Naoya Hiramatsu
    • Journal Title

      Proceedings of The 38th symposium on commutative Algebra in Japan

      Volume: 38 Pages: 112-115

    • Related Report
      2016 Research-status Report
    • Open Access
  • [Journal Article] On the relations for Grothendieck groups of Cohen-Macaulay modules over Gorenstein rings2017

    • Author(s)
      Naoya Hiramatsu
    • Journal Title

      Proceedings of the 49-th Symposium on Ring Theory and Representation Theory

      Volume: 49 Pages: 28-30

    • Related Report
      2016 Research-status Report
    • Open Access
  • [Journal Article] Stable degenerations of Cohen-Macaulay modules over simple singularities of type (A_n)2016

    • Author(s)
      Naoya Hiramatsu
    • Journal Title

      Proceedings of the 48th Symposium on Ring Theory and Representation Theory

      Volume: 48 Pages: 33-39

    • Related Report
      2015 Research-status Report
  • [Journal Article] On stable degenerations of Cohen-Macaulay modules over simple singularities of type (A_n)2016

    • Author(s)
      Naoya Hiramatsu
    • Journal Title

      Proceedings of The 37th symposium on commutative Algebra in Japan

      Volume: 37 Pages: 137-143

    • Related Report
      2015 Research-status Report
  • [Presentation] Degenerations of Cohen- Macaulay modules via matrix representations2018

    • Author(s)
      Naoya Hiramatsu
    • Organizer
      The Workshop and 18th International Conference on Representations of Algebras
    • Related Report
      2018 Annual Research Report
    • Int'l Joint Research
  • [Presentation] 0n degenerations of Cohen-Macaulay modules via matrix representations2017

    • Author(s)
      平松直哉
    • Organizer
      第39回可換環論シンポジウム
    • Related Report
      2017 Research-status Report
  • [Presentation] Degenerations of Cohen-Macaulay modules via matrix representations2017

    • Author(s)
      平松直哉
    • Organizer
      第50回環論および表現論シンポジューム
    • Related Report
      2017 Research-status Report
  • [Presentation] Degenerations of Cohen-Macaulay modules2017

    • Author(s)
      Naoya Hiramatsu
    • Organizer
      KU Algebra Seminar
    • Place of Presentation
      カンザス大学(アメリカ)
    • Related Report
      2016 Research-status Report
    • Invited
  • [Presentation] On relations for Grothendieck groups of Gorenstein rings2016

    • Author(s)
      Naoya Hiramatsu
    • Organizer
      第38回可換環論シンポジウム(The 9th Japan-Vietnam Joint Seminar on Commutative Algebra と合同開催)
    • Place of Presentation
      IPC 生産性国際交流センター(神奈川県・葉山町)
    • Related Report
      2016 Research-status Report
    • Int'l Joint Research
  • [Presentation] On the relations for Grothendieck groups of Cohen-Macaulay modules over Gorenstein rings2016

    • Author(s)
      平松直哉
    • Organizer
      第49回環論および表現論シンポジューム
    • Place of Presentation
      大阪府立大学(大阪府・堺市)
    • Related Report
      2016 Research-status Report
  • [Presentation] Relations for Grothendieck groups of Gorenstein rings2016

    • Author(s)
      Naoya Hiramatsu
    • Organizer
      XVth International Conference on Representations of Algebras and Workshop
    • Place of Presentation
      シラキュース大学(アメリカ)
    • Related Report
      2016 Research-status Report
    • Int'l Joint Research
  • [Presentation] On the contravariantly determinedness of a stable category of Cohen-Macaulay modules2015

    • Author(s)
      平松直哉
    • Organizer
      Ring Theory and Representation Theory Seminar in 2015
    • Place of Presentation
      名古屋大学
    • Year and Date
      2015-11-27
    • Related Report
      2015 Research-status Report
    • Invited
  • [Presentation] Grothendeick群のリレーションに関するButler-Auslanderの定理の逆について2015

    • Author(s)
      平松直哉
    • Organizer
      岡山大学理学部代数セミナー
    • Place of Presentation
      岡山大学
    • Year and Date
      2015-11-26
    • Related Report
      2015 Research-status Report
  • [Presentation] On stable degenerations of Cohen-Macaulay modules over simple singularities of type (An)2015

    • Author(s)
      平松直哉
    • 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
      2015-11-22
    • Related Report
      2015 Research-status Report
  • [Presentation] Stable degenerations of Cohen-Macaulay modules over simple singularities of type (An)2015

    • Author(s)
      平松直哉
    • Organizer
      第48回環論および表現論シンポジューム
    • Place of Presentation
      名古屋大学
    • Year and Date
      2015-09-10
    • Related Report
      2015 Research-status Report
  • [Remarks] Naoya HIRAMATSU

    • URL

      https://www.kure-nct.ac.jp/department/g/original/hiramatsu/index.html

    • Related Report
      2018 Annual Research Report

URL: 

Published: 2015-04-16   Modified: 2020-03-30  

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