Torsion structures in triangulated categories

• Project/Area Number: 22540053
• Research Category: Grant-in-Aid for Scientific Research (C)
• Allocation Type: Single-year Grants
• Section: 一般
• Research Field: Algebra
• Research Institution: Osaka Prefecture University
• Principal Investigator: KATO Kiriko 大阪府立大学, 理学(系)研究科(研究院), 准教授 (00347478)
• Project Period (FY): 2010-04-01 – 2014-03-31
• Project Status: Completed (Fiscal Year 2013)
• Budget Amount: ¥4,290,000 (Direct Cost: ¥3,300,000、Indirect Cost: ¥990,000); Fiscal Year 2013: ¥1,040,000 (Direct Cost: ¥800,000、Indirect Cost: ¥240,000); Fiscal Year 2012: ¥1,040,000 (Direct Cost: ¥800,000、Indirect Cost: ¥240,000); Fiscal Year 2011: ¥1,040,000 (Direct Cost: ¥800,000、Indirect Cost: ¥240,000); Fiscal Year 2010: ¥1,170,000 (Direct Cost: ¥900,000、Indirect Cost: ¥270,000)
• Keywords: 環論 / 圏論 / ホモロジー代数 / 三角圏 / 導来圏 / t-structure / recollement / Cohen-Macaulay加群 / 特異圏 / 捩れ対

Research Abstract

We focused on torsion pairs in triangulated categories. Torsion pairs are basic framework in representation theory. They enable us to decompose a given triangulated category into subcategories. We aimed at studying mechanism of existing "good" torsion pairs and applying them. We have obtained the following results: (1) discovery of a "triangle of recollements" which induces high symmetry and a new triangle equivalence as an application. (2) introducing homotopy and derived categories of "n-complexes" and triangle equivalence of these categories to homotopy and derived categories of normal complexes over a suitable extension ring. (3) one to one correspondence of weak torsion pairs to torsion pairs in a quotient category.

Research Products

• [Journal Article] Recollement of homotopy categories and Cohen-Macaulay modules (2011)
  • Author(s): Osamu Iyama , Kiriko Kato and Jun-ichi Miyachi
  • Journal Title: Journal of K-theory Volume: Vol.8, no.3 Pages: 507-542
  • Peer Reviewed
• [Journal Article] Symmetric Auslander and Bass Categories (2011)
  • Author(s): Peter J rgensen and Kiriko Kato
  • Journal Title: Mathematical Proceedings of the Cambridge Philosophical Society Volume: Vol.50, no.2
  • Peer Reviewed
• [Journal Article] Recollement of homotopy categories and Cohen-Macaulay modules (2011)
  • Author(s): O.Iyama, K.Kato, J.Miyachi
  • Journal Title: J.K-Theory Volume: 8 Issue: 3 Pages: 507-542
  • DOI: 10.1017/is011003007jkt143
  • Peer Reviewed
• [Journal Article] Symmetric Auslander and Bass Categories (2011)
  • Author(s): Peter Jorgensen, Kiriko Kato
  • Journal Title: Mathematical Proceedings of the Cambridge Philosophical Society Volume: 50-2 Pages: 227-240
  • Peer Reviewed
• [Presentation] Triangulated categories of extensions and triangles of recollements (2013)
  • Author(s): Kiriko Kato
  • Organizer: American Mathematical Society, The 2013 Fall Southeastern Sectional Meeting
  • Place of Presentation: University of Louisville, KY, USA
  • Year and Date: 2013-10-05
• [Presentation] Triangulated categories of extensions and triangles of recollements (2013)
  • Author(s): Kiriko Kato
  • Organizer: American Mathematical Society, Fall Southeastern Sectional Meeting
  • Place of Presentation: 米国ケンタッキー州ルイビル大学
  • Invited
• [Presentation] Triangulated categories of extensions and triangles of recollements (2013)
  • Author(s): 加藤 希理子
  • Organizer: 第４６回環論および表現論シンポジウム
  • Place of Presentation: 東京理科大学神楽坂キャンパス
• [Presentation] 三角圏とホモロジー代数 (2012)
  • Author(s): 加藤希理子
  • Organizer: 研究集会:(非)可換代数とトポロジー
  • Place of Presentation: 信州大学
• [Presentation] 三角圏とホモロジー代数 (2012)
  • Author(s): 加藤希理子
  • Organizer: 研究集会:(非)可換代数とトポロジー
  • Place of Presentation: 信州大学理学部(招待講演)
• [Presentation] Recollements of homotopy category and Cohen-Macaulay modules (2010)
  • Author(s): Kiriko Kato
  • Organizer: International Conference on Representations of Algebras XIV
  • Year and Date: 2010-08-15
• [Presentation] Recollements of homotopy categories and Cohen-Macaulay modules (2010)
  • Author(s): Kiriko Kato
  • Organizer: International Conference on representations of algebras XIV
  • Place of Presentation: 国立オリンピック記念青少年総合センター
  • Year and Date: 2010-08-15
• [Remarks]
  • URL: http://www.mi.s.osakafu-u.ac.jp/~kiriko/research/research.html
• [Remarks] 三角圏とホモロジー代数
  • URL: http://www.mi.s.osakafu-u.ac.jp/~kiriko/research/ss2012.pdf
• [Remarks]
  • URL: http://www.mi.s.osakafu-u.ac.jp/~kiriko/
• [Remarks]
  • URL: http://www.mi.s.osakafu-u.ac.jp/~kiriko/