Deepening representation theory of orders by tilting theory

• Project/Area Number: 23K22384
• Project/Area Number (Other): 22H01113 (2022-2023)
• Research Category: Grant-in-Aid for Scientific Research (B)
• Allocation Type: Multi-year Fund (2024); Single-year Grants (2022-2023)
• Section: 一般
• Review Section: Basic Section 11010:Algebra-related
• Research Institution: The 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)
• Project Period (FY): 2024-04-01 – 2027-03-31
• Project Status: Granted (Fiscal Year 2024)
• Budget Amount: ¥17,290,000 (Direct Cost: ¥13,300,000、Indirect Cost: ¥3,990,000); Fiscal Year 2026: ¥3,380,000 (Direct Cost: ¥2,600,000、Indirect Cost: ¥780,000); Fiscal Year 2025: ¥3,380,000 (Direct Cost: ¥2,600,000、Indirect Cost: ¥780,000); Fiscal Year 2024: ¥3,380,000 (Direct Cost: ¥2,600,000、Indirect Cost: ¥780,000); Fiscal Year 2023: ¥3,380,000 (Direct Cost: ¥2,600,000、Indirect Cost: ¥780,000); Fiscal Year 2022: ¥3,770,000 (Direct Cost: ¥2,900,000、Indirect Cost: ¥870,000)
• Keywords: Cohen-Macaulay代数 / 収縮前射影代数 / 双対化加群 / 傾加群 / 箙 / ねじれ類 / 団代数 / ２項準傾複体 / 導来圏 / 安定性条件 / 団圏 / 特異圏 / Cohen-Macaulay加群 / Grothendieck群 / ねじれ対 / M-TF同値 / g扇 / 標準分解 / mixed団傾加群 / g-fan / silting object / cluster tilting object / AS Gorenstein代数 / 非可換超曲面 / dg圏 / Auslander対応 / 準傾対象 / 扇

Outline of Research at the Start

環とは加法、減法、乗法の与えられた集合であり、整数、有理数、実数、複素数などの数体系はもちろん、多項式や行列をはじめ様々な例があり、現代数学を支える重要な基礎概念の一つである。整環は、最も基本的な環のクラスの一つである。体上の有限次元代数およびCohen-Macaulay環と呼ばれる基本的な２つのクラスを共通に一般化したものであり、箙（クイバー）の道代数や有限群の群環をはじめ多くの重要な例がある。本研究計画では傾理論に基づいたアプローチにより、整環の表現論を深化させることを目的とする。

Outline of Annual Research Achievements

正則環、Gorenstein環、Cohen-Macaulay環は環論における重要な階層であり、その非可換類似は環の表現論の研究において重要である。Cohen-Macaulay環の非可換類似としては、整環(order)と呼ばれるクラスが良く調べられており、Auslander以降、Cohen-Macaulay表現論の基本的対象として定着している。一方これとは別に、90年代にAuslanderとReitenによって導入された、Cohen-Macaulay代数と呼ばれる有限次元代数のクラスがある。これは双対化加群と呼ばれる特別な傾加群を持つ代数なのだが、非自明な例は知られていなかった。Aaron Chan氏, Rene Marczinzik氏との共同研究(arXiv:2409.05603)で、Dynkin型の収縮前射影代数がCohen-Macaulay代数であることを示した。

可換ネター環Rと箙(quiver)Qに対し、道代数RQが定まる。木村雄太氏との共同研究(arXiv:2504.02371)で、Qがサイクルを持たない場合に、RQの２項準傾複体の加法同値類のなす半順序集合が、Qの団代数(cluster algebra)の団のなす半順序集合Clus(Q)と同型であることを示した。さらにQがDynkin型の場合に、有限生成RQ-加群のねじれ類のなす半順序集合が、Spec RからClus(Q)への順序を保つ写像全体のなす半順序集合と同型であることを示した。これはRが体を含む場合の以前の結果(Adv. Math. 2024)を最適化する。

上の結果を応用して、以前の木村氏との共同研究(arXiv:2303.02928)を改良した。この論文では、拡大Dynkin型の箙Qと特定の条件を満たす可換ネター環Rに対し、有限生成RQ-加群の圏のねじれ類のなす半順序集合を決定したのだが、そこで課していたRが体を含むという仮定を落とすことに成功した。
他の研究は省略する。

Current Status of Research Progress

2: Research has progressed on the whole more than it was originally planned.

Reason

新しいCohen-Macaulay代数のクラスを発見したのは、大きな進展である。箙の道代数のねじれ類に関しても、重要な結果を示すことができた。

Strategy for Future Research Activity

引き続き諸問題に対して、じっくりと取り組む予定である。

Research Products

• [Int'l Joint Research] University of Bonn/Universitaet zu Koeln(ドイツ)
• [Int'l Joint Research] Linkoeping University(スウェーデン)
• [Int'l Joint Research] NTNU(ノルウェー)
• [Int'l Joint Research] North Carolina State University(米国)
• [Int'l Joint Research] University of Bonn/Universitaet zu Koeln(ドイツ)
• [Int'l Joint Research] Universite de Picardie Jules Verne(フランス)
• [Int'l Joint Research] Uppsala University/Linkoeping University(スウェーデン)
• [Int'l Joint Research] NTNU(ノルウェー)
• [Int'l Joint Research]
• [Int'l Joint Research] North Carolina State University(米国)
• [Int'l Joint Research] University of Bonn/Universitaet zu Koeln(ドイツ)
• [Int'l Joint Research] Universite de Picardie Jules Verne(フランス)
• [Int'l Joint Research] Uppsala University/Linkoeping University(スウェーデン)
• [Int'l Joint Research] NTNU(ノルウェー)
• [Int'l Joint Research]
• [Journal Article] Periodic trivial extension algebras and fractionally Calabi-Yau algebras (2025)
  • Author(s): CHAN Aaron、DARPOE Erik、IYAMA Osamu、MARCZINZIK Rene
  • Journal Title: Annales Scientifiques de l'Ecole Normale Superieure Volume: 58 Pages: 463-510
  • DOI: 10.24033/asens.2610
  • Peer Reviewed / Int'l Joint Research
• [Journal Article] On $\tau$-tilting theory (Frontiers of Science Award) (2025)
  • Author(s): Takahide Adachi, Osamu Iyama, Idun Reiten
  • Journal Title: The Proceedings of International Congress of Basic Science 2024 Volume: -
  • Int'l Joint Research
• [Journal Article] Triangulations of Prisms and Preprojective Algebras of Type A (2024)
  • Author(s): Iyama Osamu、Williams Nicholas J
  • Journal Title: International Mathematics Research Notices Volume: 2024 Issue: 13 Pages: 10236-10254
  • DOI: 10.1093/imrn/rnae059
  • Peer Reviewed / Open Access / Int'l Joint Research
• [Journal Article] Semistable torsion classes and canonical decompositions in Grothendieck groups (2024)
  • Author(s): Asai Sota、Iyama Osamu
  • Journal Title: Proceedings of the London Mathematical Society Volume: 129 Issue: 5 Pages: 1-58
  • DOI: 10.1112/plms.12639
  • Peer Reviewed / Open Access
• [Journal Article] Auslander-Reiten theory in extriangulated categories (2024)
  • Author(s): Osamu Iyama, Hiroyuki Nakaoka, Yann Palu
  • Journal Title: Transactions of the American Mathematical Society, Series B Volume: 11 Issue: 8 Pages: 248-305
  • DOI: 10.1090/btran/159
  • Peer Reviewed / Open Access / Int'l Joint Research
• [Journal Article] Classifying subcategories of modules over Noetherian algebras (2024)
  • Author(s): Iyama Osamu、Kimura Yuta
  • Journal Title: Advances in Mathematics Volume: 446 Pages: 109631-109631
  • DOI: 10.1016/j.aim.2024.109631
  • Peer Reviewed / Int'l Joint Research
• [Journal Article] Periodic trivial extension algebras and fractionally Calabi-Yau algebras (2024)
  • Author(s): Aaron Chan, Erik Darpoe, Osamu Iyama, Rene Marczinzik
  • Journal Title: Ann. Sci. Ec. Norm. Super. (4) Volume: -
  • Peer Reviewed / Int'l Joint Research
• [Journal Article] Triangulations of prisms and preprojective algebras of type A (2024)
  • Author(s): Osamu Iyama, Nathan Williams
  • Journal Title: Int. Math. Res. Not. IMRN Volume: -
  • Peer Reviewed / Int'l Joint Research
• [Journal Article] Lattice theory of torsion classes: Beyond Tau-tilting theory (2023)
  • Author(s): Demonet Laurent、Iyama Osamu、Reading Nathan、Reiten Idun、Thomas Hugh
  • Journal Title: Transactions of the American Mathematical Society, Series B Volume: 10 Issue: 18 Pages: 542-612
  • DOI: 10.1090/btran/100
  • Peer Reviewed / Open Access / Int'l Joint Research
• [Journal Article] Representation Theory of Geigle-Lenzing Complete Intersections (2023)
  • Author(s): Herschend Martin、Iyama Osamu、Minamoto Hiroyuki、Oppermann Steffen
  • Journal Title: Memoirs of the American Mathematical Society Volume: 285 Issue: 1412 Pages: 1-141
  • DOI: 10.1090/memo/1412
  • Peer Reviewed / Int'l Joint Research
• [Journal Article] Representation Theory of Quivers and Finite-Dimensional Algebras (2023)
  • Author(s): Amiot Claire、Crawley-Boevey William、Iyama Osamu、Schroeer Jan
  • Journal Title: Oberwolfach Reports Volume: 20 Issue: 1 Pages: 397-486
  • DOI: 10.4171/owr/2023/7
  • Open Access / Int'l Joint Research
• [Journal Article] Representation theory of quivers and finite-dimensional algebras (2023)
  • Author(s): Claire Amiot, William Crawley-Boevey, Osamu Iyama, Jan Schroer
  • Journal Title: Oberwolfach Rep. Volume: 20
  • Open Access / Int'l Joint Research
• [Journal Article] Positive Fuss Catalan Numbers and Simple-Minded Systems in Negative Calabi-Yau Categories (2022)
  • Author(s): Iyama Osamu、Jin Haibo
  • Journal Title: International Mathematics Research Notices Volume: to appear Issue: 8 Pages: 6624-6647
  • DOI: 10.1093/imrn/rnab369
  • Peer Reviewed / Int'l Joint Research
• [Presentation] Cohen-Macaulay representations of Artin-Schelter Gorenstein algebras of dimension one (2025)
  • Author(s): Osamu Iyama
  • Organizer: Categorical and analytic invariants in algebraic, symplectic and complex geometry, IPMU
  • Int'l Joint Research / Invited
• [Presentation] Tilting theory revisited (Frontiers of Science Award) (2024)
  • Author(s): Osamu Iyama
  • Organizer: The 2024 International Congress of Basic Science, BIMSA, China
  • Int'l Joint Research / Invited
• [Presentation] Coxeter arrangements, Preprojective algebras and cDV singularities (2024)
  • Author(s): Osamu Iyama
  • Organizer: Bundles and lattices, 大阪大学
  • Int'l Joint Research / Invited
• [Presentation] Auslander-Reiten’s Cohen-Macaulay algebras and contracted preprojective algebras (2024)
  • Author(s): Osamu Iyama
  • Organizer: 第56回環論および表現論シンポジウム, 東京学芸大学
• [Presentation] Silting theory and Grothendieck groups（４回講演） (2024)
  • Author(s): Osamu Iyama
  • Organizer: Autumn School & Conference: New developments in representation theory of algebras: Algebraic, combinatorial and geometric aspects, OIST
  • Int'l Joint Research / Invited
• [Presentation] Auslander-Reiten’s Cohen-Macaulay algebras and contracted preprojective algebras (2024)
  • Author(s): Osamu Iyama
  • Organizer: Autumn School & Conference: New developments in representation theory of algebras: Algebraic, combinatorial and geometric aspects, OIST
  • Int'l Joint Research / Invited
• [Presentation] Cohen-Macaulay representations of Artin-Schelter Gorenstein algebras of dimension one (2024)
  • Author(s): Osamu Iyama
  • Organizer: 9th Isfahan Seminar on Representations of Algebras, online
  • Int'l Joint Research / Invited
• [Presentation] Periodic trivial extension algebras and fractionally Calabi-Yau algebras (2024)
  • Author(s): Osamu Iyama
  • Organizer: 9th Isfahan Seminar on Representations of Algebras, online
  • Int'l Joint Research / Invited
• [Presentation] Semistable torsion classes and canonical decompositions in Grothendieck groups (2024)
  • Author(s): Osamu Iyama
  • Organizer: Cluster Algebras and Its Applications, Mathematisches Forschungsinstitut Oberwolfach, Germany
  • Int'l Joint Research / Invited
• [Presentation] Tilting theory via g-fans in real Grothendieck groups（概説講演２回） (2023)
  • Author(s): Osamu Iyama
  • Organizer: Algebraic Lie Theory and Representation Theory 2023, 東京工業大学
  • Invited
• [Presentation] Singularity categories and cluster categories (2023)
  • Author(s): Osamu Iyama
  • Organizer: Current trends in categorically approach to algebraic and symplectic geometry 2, IPMU
  • Int'l Joint Research / Invited
• [Presentation] Tilting theory via g-fans in real Grothendieck groups (2023)
  • Author(s): Osamu Iyama
  • Organizer: MSJ-KMS Joint Meeting 2023, Sendai International Center
  • Int'l Joint Research / Invited
• [Presentation] 漸化式とグラフと団代数 (2023)
  • Author(s): Osamu Iyama
  • Organizer: 高校生と大学生のための金曜特別講座, online
  • Invited
• [Presentation] Auslander-Reiten理論特論（８回講演） (2023)
  • Author(s): Osamu Iyama
  • Organizer: online
• [Presentation] Tilting theory via g-fans in real Grothendieck groups (2023)
  • Author(s): Osamu Iyama
  • Organizer: The 44th Japan Symposium on Commutative Algebra, LecTore Hayama
  • Int'l Joint Research
• [Presentation] Representation theory of non-commutative Gorenstein rings in dimension one (2023)
  • Author(s): Osamu Iyama
  • Organizer: Representation Theory and Non-Commutative Geometry / ARTIG 3, Paderborn University, Germany
  • Int'l Joint Research / Invited
• [Presentation] Semistable torsion classes and canonical decompositions in Grothendieck groups（２回講演） (2023)
  • Author(s): Osamu Iyama
  • Organizer: Seminar on tau-tilting-finiteness and beyond, 名古屋大学
• [Book] McKay correspondence, mutation and related topics. Proceedings of the international conference held online July 17--August 14, 2020 (2023)
  • Author(s): Yukari Ito, Akira Ishii, Osamu Iyama
  • Total Pages: 533
  • Publisher: Mathematical Society of Japan, Tokyo
  • ISBN: 9784864970983
• [Remarks] Osamu Iyama
  • URL: https://www.ms.u-tokyo.ac.jp/~iyama/index.html
• [Funded Workshop] Perspectives in Tilting Theory and Related Topics (2025)
• [Funded Workshop] Spring Workshop on the Representation Theory of Algebras and related areas (2024)
• [Funded Workshop] International Conference on Representations of Algebras (ICRA 21, 2024) (2024)
• [Funded Workshop] Representation Theory of Quivers and Finite-Dimensional Algebras, Mathematisches Forschungsinstitut Oberwolfach, Germany (2024)
• [Funded Workshop] Silting in Representation Theory, Singularities, and Noncommutative Geometry, Casa Matematica Oaxaca (CMO), Mexico (2023)
• [Funded Workshop] McKay correspondence, Tilting theory and related topics, IPMU (2023)
• [Funded Workshop] Tokyo-Nagoya Algebra Seminar (online) (2023)
• [Funded Workshop] Silting in Representation Theory, Singularities, and Noncommutative Geometry, Casa Matematica Oaxaca (CMO), Mexico (2023)
• [Funded Workshop] McKay correspondence, Tilting theory and related topics, IPMU (2023)
• [Funded Workshop] Representation Theory of Quivers and Finite-Dimensional Algebras, Mathematisches Forschungsinstitut Oberwolfach, Germany (2023)
• [Funded Workshop] Tokyo-Nagoya Algebra Seminar (online) (2022)