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2020 Fiscal Year Annual Research Report

整環の表現論の総合的研究

Research Project

Project/Area Number 16H03923
Research InstitutionThe University of Tokyo

Principal Investigator

伊山 修  東京大学, 大学院数理科学研究科, 教授 (70347532)

Co-Investigator(Kenkyū-buntansha) 高橋 亮  名古屋大学, 多元数理科学研究科, 准教授 (40447719)
毛利 出  静岡大学, 理学部, 教授 (50436903)
Project Period (FY) 2016-04-01 – 2021-03-31
Keywordsg扇 / Grothendieck群 / 導来圏 / 特異圏 / 団圏 / 準傾複体 / Auslander正則環 / 自明拡大代数
Outline of Annual Research Achievements

埴原と共同(arXiv:2209.14090)で、次数付きGorenstein環Rと有限次元代数Aに対し、Rの次数付き特異圏とAの導来圏の同値から、Rの特異圏とAの団圏の同値が得られることを示し、種々の先行研究を統一的に説明した。
木村と共同(2106.00469, 2303.02928)で(一般には非可換)ネター代数上の加群圏の分類理論を構築した。ねじれ自由類とSerre部分圏を分類し、いくつかの特別な場合にねじれ類を分類した。
Chan, Marczinzikと共同(2210.06180)で、dominantと呼ばれるAuslander正則環のクラスを導入し、mixed団傾加群による特徴付けを与え、さらにKoszul双対性で閉じていることを示した。Marczinzikと共同(Adv. Math.2022)で、有限束Pに対し、Pの隣接代数AがAuslander正則環であることとPが分配束であることの同値性を示した。Chan, Darpoe, Marczinzikと共同(2012.11927)で、有限次元代数Aの自明拡大代数が周期的であることと、Aの導来圏が分数的Calabi-Yauであることの同値性を示した。
青木, 東谷, 加瀬, 水野と共同(2203.15213)で、有限次元代数の実Grothendieck群内に定まるg扇の基礎理論を構築した。ランク2の完備なg扇は符号同一性で特徴付けられることを示した(2301.01498)。淺井と共同(2112.14908)で、有限次元代数Aの実Grothendieck群に対し、標準分解から定まる錐の内点は全てTF同値であることを示し、Aが遺伝的またはE-tameの場合には錐の内部が一つの同値類であることを示した。Williamsと共同(2208.12957)で、A型前射影代数の2項準傾複体とprismの三角形分割の一対一対応を与えた。

Research Progress Status

令和2年度が最終年度であるため、記入しない。

Strategy for Future Research Activity

令和2年度が最終年度であるため、記入しない。

  • Research Products

    (40 results)

All 2023 2022 2021 2020 Other

All Int'l Joint Research (6 results) Journal Article (10 results) (of which Int'l Joint Research: 10 results,  Peer Reviewed: 10 results,  Open Access: 2 results) Presentation (22 results) (of which Int'l Joint Research: 14 results,  Invited: 20 results) Remarks (1 results) Funded Workshop (1 results)

  • [Int'l Joint Research] University of Glasgow/University of East Anglia/Lancaster University(英国)

    • Country Name
      UNITED KINGDOM
    • Counterpart Institution
      University of Glasgow/University of East Anglia/Lancaster University
  • [Int'l Joint Research] Nanjing University/NUIST(中国)

    • Country Name
      CHINA
    • Counterpart Institution
      Nanjing University/NUIST
  • [Int'l Joint Research] North Carolina State University/University of Kansas(米国)

    • Country Name
      U.S.A.
    • Counterpart Institution
      North Carolina State University/University of Kansas
  • [Int'l Joint Research] University of Toronto Scarborough/Universite du Quebec a Montreal(カナダ)

    • Country Name
      CANADA
    • Counterpart Institution
      University of Toronto Scarborough/Universite du Quebec a Montreal
  • [Int'l Joint Research] NTNU/Linkoeping University(ノルウェー)

    • Country Name
      NORWAY
    • Counterpart Institution
      NTNU/Linkoeping University
  • [Int'l Joint Research]

    • # of Other Countries
      3
  • [Journal Article] Positive Fuss-Catalan numbers and Simple-minded systems in negative Calabi-Yau categories2023

    • Author(s)
      Osamu Iyama, Haibo Jin
    • Journal Title

      Int. Math. Res. Not.

      Volume: - Pages: -

    • Peer Reviewed / Int'l Joint Research
  • [Journal Article] Lattice theory of torsion classes: Beyond $\tau$-tilting theory2023

    • Author(s)
      Laurent Demonet, Osamu Iyama, Nathan Reading, Idun Reiten, Hugh Thomas
    • Journal Title

      Trans. Amer. Math. Soc. Ser. B

      Volume: - Pages: -

    • Peer Reviewed / Open Access / Int'l Joint Research
  • [Journal Article] Representation theory of Geigle-Lenzing complete intersections2023

    • Author(s)
      Martin Herschend, Osamu Iyama, Hiroyuki Minamoto, Steffen Oppermann
    • Journal Title

      Mem. Amer. Math. Soc.

      Volume: - Pages: -

    • Peer Reviewed / Int'l Joint Research
  • [Journal Article] Auslander-Reiten theory in extriangulated categories2023

    • Author(s)
      Osamu Iyama, Hiroyuki Nakaoka, Yann Palu
    • Journal Title

      Trans. Amer. Math. Soc.

      Volume: - Pages: -

    • Peer Reviewed / Int'l Joint Research
  • [Journal Article] Distributive lattices and Auslander regular algebras2022

    • Author(s)
      Osamu Iyama, Rene Marczinzik
    • Journal Title

      Adv. Math.

      Volume: 398 Pages: 108233

    • DOI

      10.1016/j.aim.2022.108233

    • Peer Reviewed / Int'l Joint Research
  • [Journal Article] Tilting theory for Gorenstein rings in dimension one2020

    • Author(s)
      Ragnar-Olaf Buchweitz, Osamu Iyama, Kota Yamaura
    • Journal Title

      Forum Math. Sigma

      Volume: 8 Pages: e36

    • DOI

      10.1017/fms.2020.28

    • Peer Reviewed / Open Access / Int'l Joint Research
  • [Journal Article] Classifying $\tau$-tilting modules over the Auslander algebra of $K[x]/(x^n)$2020

    • Author(s)
      Osamu Iyama, Xiaojin Zhang
    • Journal Title

      J. Math. Soc. Japan

      Volume: 72 Pages: 731--764

    • DOI

      10.2969/jmsj/75117511

    • Peer Reviewed / Int'l Joint Research
  • [Journal Article] Quotients of triangulated categories and Equivalences of Buchweitz, Orlov and Amiot-Guo-Keller2020

    • Author(s)
      Osamu Iyama, Dong Yang
    • Journal Title

      Amer. J. Math.

      Volume: 142 Pages: 1641--1659

    • DOI

      10.1353/ajm.2020.0041

    • Peer Reviewed / Int'l Joint Research
  • [Journal Article] Gorenstein modifications and Q-Gorenstein rings2020

    • Author(s)
      Hailong Dao, Osamu Iyama, Ryo Takahashi, Michael Wemyss
    • Journal Title

      J. Algebraic Geom.

      Volume: 29 Pages: 729--751

    • DOI

      10.1090/jag/760

    • Peer Reviewed / Int'l Joint Research
  • [Journal Article] Higher preprojective algebras, Koszul algebras and superpotentials2020

    • Author(s)
      Joseph Grant, Osamu Iyama
    • Journal Title

      Compos. Math.

      Volume: 156 Pages: 2588--2627

    • DOI

      10.1112/S0010437X20007538

    • Peer Reviewed / Int'l Joint Research
  • [Presentation] Complete g-fans of rank 22023

    • Author(s)
      Osamu Iyama
    • Organizer
      Flash Talks in Representation Theory, Norwegian University of Science and Technology
    • Int'l Joint Research / Invited
  • [Presentation] Fans in tilting theory: rank 2 case2023

    • Author(s)
      Osamu Iyama
    • Organizer
      Representation Theory of Quivers and Finite-Dimensional Algebras, Mathematisches Forschungsinstitut Oberwolfach
    • Int'l Joint Research
  • [Presentation] Singularity categories and cluster categories2022

    • Author(s)
      Osamu Iyama
    • Organizer
      Geometric and homological methods in representation theory, Lancaster University
    • Int'l Joint Research / Invited
  • [Presentation] Cohen-Macaulay representations, Singularity categories and Cluster categories2022

    • Author(s)
      Osamu Iyama
    • Organizer
      東京可換環論セミナー
    • Invited
  • [Presentation] 団代数と傾理論2022

    • Author(s)
      Osamu Iyama
    • Organizer
      Encounter with mathematics, 第75回Cluster Algebras, 中央大学
    • Invited
  • [Presentation] Singularity categories and Cluster categories2022

    • Author(s)
      Osamu Iyama
    • Organizer
      Interactions between Algebraic Geometry and Noncommutative Algebra, Mathematisches Forschungsinstitut Oberwolfach
    • Int'l Joint Research / Invited
  • [Presentation] Auslander-Gorenstein rings revisited(3回講演)2022

    • Author(s)
      Osamu Iyama
    • Organizer
      Online workshop on higher dimensional homological algebras, IPM-Isfahan and University of Isfahan
    • Int'l Joint Research / Invited
  • [Presentation] Singularity categories and cluster categories2022

    • Author(s)
      Osamu Iyama
    • Organizer
      Bridges between representation theory and algebraic geometry: Singularities, friezes and cluster categories, University of Leeds
    • Int'l Joint Research / Invited
  • [Presentation] 団代数と傾理論2022

    • Author(s)
      Osamu Iyama
    • Organizer
      東京理科大学談話会
    • Invited
  • [Presentation] Quiver representations and Cohen-Macaulay representations2022

    • Author(s)
      Osamu Iyama
    • Organizer
      Algebra days in Paris in honor of Bernhard Keller, Institut Henri Poincare
    • Int'l Joint Research / Invited
  • [Presentation] Realizing stable categories of Cohen-Macaulay modules as cluster categories2022

    • Author(s)
      Osamu Iyama
    • Organizer
      第43回可換環論シンポジウム, 大阪大学
  • [Presentation] 特異傾理論入門:Quiver表現とCohen-Macaulay表現2022

    • Author(s)
      Osamu Iyama
    • Organizer
      名古屋大学集中講義
    • Invited
  • [Presentation] 傾理論と前射影代数2021

    • Author(s)
      Osamu Iyama
    • Organizer
      京都大学数学教室・集中講義
    • Invited
  • [Presentation] 特異圏の傾理論2021

    • Author(s)
      Osamu Iyama
    • Organizer
      京都大学数学教室・談話会
    • Invited
  • [Presentation] (3回講演)Introduction to silting theory, Polytopes in silting theory, Silting theory of Noetherian algebras2021

    • Author(s)
      Osamu Iyama
    • Organizer
      Network on Silting Theory, Inaugural lecture series
    • Int'l Joint Research / Invited
  • [Presentation] Periodic trivial extension algebras and fractionally Calabi-Yau algebras2021

    • Author(s)
      Osamu Iyama
    • Organizer
      FD seminar
    • Int'l Joint Research / Invited
  • [Presentation] Tilting theory of contracted preprojective algebras and cluster tilting theory of cDV singularities2021

    • Author(s)
      Osamu Iyama
    • Organizer
      Braids, mutations and twists in algebra and geometry, Uppsala University
    • Int'l Joint Research / Invited
  • [Presentation] Wall-chamber structure in Grothendieck group and canonical decomposition2021

    • Author(s)
      Osamu Iyama
    • Organizer
      Cluster Algebras and Related Topics, Morningside Center of Mathematics
    • Int'l Joint Research / Invited
  • [Presentation] Tilting theory of contracted preprojective algebras and cDV singularities2020

    • Author(s)
      Osamu Iyama
    • Organizer
      Paris algebra seminar
    • Int'l Joint Research / Invited
  • [Presentation] Simple-minded systems in cluster categories and generalized cluster complexes2020

    • Author(s)
      Osamu Iyama
    • Organizer
      Cluster algebras 2020
    • Int'l Joint Research / Invited
  • [Presentation] Contracted preprojective algebras, Coxeter groups and cDV singularities2020

    • Author(s)
      Osamu Iyama
    • Organizer
      Birthday Colloquium for Bill Crawley-Boevey, ICMS
    • Int'l Joint Research / Invited
  • [Presentation] 傾理論とその仲間たち2020

    • Author(s)
      Osamu Iyama
    • Organizer
      東京大学大学院数理科学研究科・談話会
    • Invited
  • [Remarks] Osamu Iyama

    • URL

      https://www.ms.u-tokyo.ac.jp/~iyama/index.html

  • [Funded Workshop] Tokyo-Nagoya Algebra Seminar (online)2020

URL: 

Published: 2023-12-25  

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