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Studies of tilting mutation theory and derived equivalences

Research Project

Project/Area Number 15K17516
Research Category

Grant-in-Aid for Young Scientists (B)

Allocation TypeMulti-year Fund
Research Field Algebra
Research InstitutionTokyo Gakugei University

Principal Investigator

AIHARA Takuma  東京学芸大学, 教育学部, 講師 (40714150)

Project Period (FY) 2015-04-01 – 2019-03-31
Project Status Completed (Fiscal Year 2018)
Budget Amount *help
¥4,030,000 (Direct Cost: ¥3,100,000、Indirect Cost: ¥930,000)
Fiscal Year 2018: ¥780,000 (Direct Cost: ¥600,000、Indirect Cost: ¥180,000)
Fiscal Year 2017: ¥780,000 (Direct Cost: ¥600,000、Indirect Cost: ¥180,000)
Fiscal Year 2016: ¥1,170,000 (Direct Cost: ¥900,000、Indirect Cost: ¥270,000)
Fiscal Year 2015: ¥1,300,000 (Direct Cost: ¥1,000,000、Indirect Cost: ¥300,000)
Keywords傾理論 / 傾対象 / 傾変異 / 傾連結 / 傾離散 / ボンガルツ完備化 / 導来圏 / 導来同値 / τ傾加群 / τ傾有限 / 三角圏の次元 / 対称多元環 / ブラウアーグラフ多元環 / 前射影多元環
Outline of Final Research Achievements

The structure of the derived category over a finite dimensional algebra was investigated from the viewpoint of tilting mutation theory and derived equivalences. In particular, the tilting-discreteness of a given algebra was studied, and a powerful method of showing the tilting-discreteness was given. This leads to the following results: (1) any preprojective algebra of Dynkin type is tilting-discrete. (2) a Brauer graph algebra is tilting-discrete if and only if the Brauer graph has at most one odd-cycle and none of even-cycle.
Moreover, it was proved that the Bongartz completion for presilting objects is possible in a silting-discrete triangulated category.

Academic Significance and Societal Importance of the Research Achievements

傾変異理論は最近になって導入された理論であり、近年非常に注目を浴びている。また、傾離散性を満たす多元環もほとんど知られていなかった。本研究によって、傾離散性を満たすための必要十分条件を与えたことはとても意義がある。それは特に、数学的帰納法による手法を与えており、様々な状況下で利用できるため有用である。それを用いて、実際に傾離散性を満たす多元環のクラスを発見したことは価値があるといえる。今後も、この手法を用いることで傾離散性を満たす多元環の発見が期待される。
また、もともとは古典的な問題であったボンガルツ完備化問題に対しても、一般の傾理論に拡張してある十分条件を与えられたことは意義深い。

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

    (11 results)

All 2019 2018 2017 2016

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

  • [Journal Article] Remarks on dimensions of triangulated categories2019

    • Author(s)
      Aihara Takuma, Takahashi Ryo
    • Journal Title

      Journal of Algebra

      Volume: 521 Pages: 235-246

    • DOI

      10.1016/j.jalgebra.2018.12.001

    • Related Report
      2018 Annual Research Report
    • Peer Reviewed
  • [Journal Article] Algebras sharing the same τ-tilting poset with tree quiver algebras2018

    • Author(s)
      Takuma Aihara、Ryoichi Kase
    • Journal Title

      The Quarterly Journal Of Mathematics

      Volume: 印刷中

    • Related Report
      2017 Research-status Report
    • Peer Reviewed
  • [Journal Article] Classifying tilting complexes over preprojective algebras of Dynkin type2017

    • Author(s)
      Aihara Takuma、Mizuno Yuya
    • Journal Title

      Algebra & Number Theory

      Volume: 11 Issue: 6 Pages: 1287-1315

    • DOI

      10.2140/ant.2017.11.1287

    • Related Report
      2017 Research-status Report
    • Peer Reviewed
  • [Journal Article] Classification of two-term tilting complexes over Brauer graph algebras2017

    • Author(s)
      Adachi Takahide、Aihara Takuma、Chan Aaron
    • Journal Title

      Mathematische Zeitschrift

      Volume: 印刷中 Issue: 1-2 Pages: 1-36

    • DOI

      10.1007/s00209-017-2006-9

    • Related Report
      2017 Research-status Report
    • Peer Reviewed / Open Access / Int'l Joint Research
  • [Presentation] Silting-connected triangulated categories2018

    • Author(s)
      相原琢磨
    • Organizer
      第8回(非)可換代数とトポロジー
    • Related Report
      2017 Research-status Report
    • Invited
  • [Presentation] On the existence of silting objects2017

    • Author(s)
      相原琢磨
    • Organizer
      第62回代数学シンポジウム
    • Related Report
      2017 Research-status Report
    • Invited
  • [Presentation] Singularity categories and silting objects2017

    • Author(s)
      相原琢磨
    • Organizer
      第50回環論および表現論シンポジウム
    • Related Report
      2017 Research-status Report
  • [Presentation] Silting mutation theory: foundation and application I, II2017

    • Author(s)
      相原琢磨
    • Organizer
      第4回ワークショップ「非可換Gorenstein代数とその周辺」
    • Related Report
      2017 Research-status Report
    • Invited
  • [Presentation] Tilting mutation and flips of Brauer graph algebras2016

    • Author(s)
      Takuma Aihara
    • Organizer
      Workshop on Brauer graph algebras
    • Place of Presentation
      Stuttgart University (Germany)
    • Year and Date
      2016-03-22
    • Related Report
      2015 Research-status Report
    • Invited
  • [Presentation] Algebras sharing the same poset of support τ-tilting modules with tree quiver algebras2016

    • Author(s)
      加瀬遼一, 相原琢磨
    • Organizer
      第49回環論および表現論シンポジウム
    • Place of Presentation
      大阪府立大学(大阪府、堺市)
    • Related Report
      2016 Research-status Report
  • [Funded Workshop] Workshop on Brauer graph algebras2016

    • Place of Presentation
      Stuttgart University (Germany)
    • Year and Date
      2016-03-21
    • Related Report
      2015 Research-status Report

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

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