Syzygy problems by method of stable categories

• Project/Area Number: 18540044
• 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 Osaka Prefecture University, 理学系研究科, 准教授 (00347478)
• Co-Investigator(Kenkyū-buntansha): IRIE Kouyemon 大阪府立大学, 理学系研究科, 教授 (40151691) ; YOSHITOMI Kentaro 大阪府立大学, 総合教育研究機構, 講師 (10305609)
• Project Period (FY): 2006 – 2009
• Project Status: Completed (Fiscal Year 2009)
• Budget Amount: ¥3,630,000 (Direct Cost: ¥3,000,000、Indirect Cost: ¥630,000); Fiscal Year 2009: ¥910,000 (Direct Cost: ¥700,000、Indirect Cost: ¥210,000); Fiscal Year 2008: ¥910,000 (Direct Cost: ¥700,000、Indirect Cost: ¥210,000); Fiscal Year 2007: ¥910,000 (Direct Cost: ¥700,000、Indirect Cost: ¥210,000); Fiscal Year 2006: ¥900,000 (Direct Cost: ¥900,000)
• Keywords: 環論

Research Abstract

We mainly studied homotopy categories of chain complexes of projective modules because our standpoint is to use a triangle structure of homotopy categories to investigate stable categories. The results of this project are divided into the following three groups. The first one is on the symmetry caused by torsion pairs. We found a structure of torsion pairs with strong symmetry which leads us to a new triangle equivalence. The second one is on representability by monomorphisms (rbm) which measures the obstruction for the category to be triangulated.
The third one is on syzygies of modules with positive codimension. We obtained that every module is equivalent to some module with positive codimension up to first syzygy if and only if the ring is an integral domain.

Research Products

• [Journal Article] Morphisms represented by monomorphisms (2008)
  • Author(s): Kiriko Kato
  • Journal Title: Journal of Pure and Applied Algebra vol.208no.1 Pages: 261-283
  • Peer Reviewed
• [Journal Article] Syzygies of modules with positive codimension (2008)
  • Author(s): Kiriko Kato
  • Journal Title: Journal of Algebra vol.318no.1 Pages: 25-36
  • Peer Reviewed
• [Journal Article] Syzyzies of Modules with positive codimension (2007)
  • Author(s): Kiriko Kato
  • Journal Title: Journal of Algebra Vol.318, no.1 Pages: 25-36
  • Peer Reviewed
• [Journal Article] Morphisms represented by monomorphisms (2007)
  • Author(s): Kiriko Kato
  • Journal Title: Journal of Pure and Applred Algebra Vol.208, no.1 Pages: 261-283
  • Peer Reviewed
• [Journal Article] Syzygies of modules with positive codimension (2007)
  • Author(s): Kiriko KATO
  • Journal Title: Journal of Algebra 318 Pages: 25-36
  • Peer Reviewed
• [Journal Article] Morphisms representee by monomorphisms (2007)
  • Author(s): Kiriko KATO
  • Journal Title: Journal of Pure and Applied Algebra 208 Pages: 261-283
• [Journal Article] Syzygies of modules with positive codimension (2007)
  • Author(s): Kiriko KATO
  • Journal Title: Journal of Algebra (印刷中)
• [Journal Article] Stable suspension order of universal phantom maps and some stably indecomposable loop spaces, (2007)
  • Author(s): Kouemon IRIE
  • Journal Title: Journal of Mathematical Society of Japan 59 Pages: 97-112
• [Presentation] Recollement of Homotopy Category and Cohen-Macaulay Modules (2008)
  • Author(s): 加藤 希理子
  • Organizer: 研究集会「有限群と代数の表現論およびその周辺」
  • Place of Presentation: 京都大学数理解析研究所
  • Year and Date: 2008-11-21
• [Presentation] Recollement of homotopy categories and Cohen-Macaulay modules, Commutative Algebra (2008)
  • Author(s): Kiriko Kato
  • Organizer: Connections with Algebraic Topology and Representation Theory
  • Place of Presentation: 米国Nebraska大学
  • Year and Date: 2008-05-19
• [Presentation] Recollement of homotopy categories and Cohen-Macaulay modules (2008)
  • Author(s): 加藤希理子
  • Organizer: Commutative Algebra : Connections with Algebraic Topology and Representation Theory
  • Place of Presentation: 米国 Nebraska大学Lincoln校
  • Year and Date: 2008-05-18
• [Presentation] Recollement of Homotopy Category and Cohen-Macaulay Modules (2008)
  • Author(s): 加藤 希理子
  • Organizer: Conference“Commutative Algebra : Connections with algebraic topology and representation theory"
  • Place of Presentation: 米国Nebraska Lincoln大学
  • Year and Date: 2008-05-18
• [Presentation] Recollement of homotopy categories and Cohen-Macaulay modules (2008)
  • Author(s): 加藤希理子
  • Organizer: Workshop on Representation Theory of Algebra
  • Place of Presentation: 静岡大学
• [Presentation] Recollement of Homotopy Category and Cohen-Macaulay Modules (2008)
  • Author(s): 加藤 希理子
  • Organizer: Workshop on representation theory of algebras
  • Place of Presentation: 静岡大学
• [Presentation] Quotient category of homotopy category (2007)
  • Author(s): Kiriko Kato
  • Organizer: 第29回可換環論シンポジウム
  • Place of Presentation: 愛知県名古屋市
  • Year and Date: 2007-11-21
• [Presentation] Quotient Categories of Homotopy Categories (2007)
  • Author(s): 加藤 希理子
  • Organizer: 第29回可換環論シンポジウム
  • Place of Presentation: 愛知県名古屋市
  • Year and Date: 2007-11-21
• [Presentation] Triangulated categories over rings (2007)
  • Author(s): 加藤希理子
  • Organizer: 空間の代数的・幾何的モデルとその周辺
  • Place of Presentation: 信州大学
  • Year and Date: 2007-09-06
• [Presentation] Morphisms represented by monomorphisms (2006)
  • Author(s): Kiriko Kato
  • Organizer: Workshop on triangulated categories
  • Place of Presentation: 英国Leeds大学
  • Year and Date: 2006-08-15
• [Remarks]
  • URL: http://www.mi.s.osakafu-u.ac.jp/~kiriko/seminar/09JulRIMS/09JulRIMS.html