Study of ABness of commutative noetherian rings

• Project/Area Number: 23840043
• Research Category: Grant-in-Aid for Research Activity Start-up
• Allocation Type: Single-year Grants
• Research Field: Algebra
• Research Institution: Tokuyama College of Technology
• Principal Investigator: ARAYA Tokuji 徳山工業高等専門学校, 一般科目, 准教授 (70613222)
• Project Period (FY): 2011 – 2012
• Project Status: Completed (Fiscal Year 2012)
• Budget Amount: ¥3,250,000 (Direct Cost: ¥2,500,000、Indirect Cost: ¥750,000); Fiscal Year 2012: ¥1,560,000 (Direct Cost: ¥1,200,000、Indirect Cost: ¥360,000); Fiscal Year 2011: ¥1,690,000 (Direct Cost: ¥1,300,000、Indirect Cost: ¥390,000)
• Keywords: 可換環論 / ホモロジー / Gorenstein環 / AB環 / ホモロジカル次元 / Cohen-Macaulay環 / 極大Cohen-Macaulay加群 / 次数付環

Research Abstract

One of the aim of this study is to define a homological dimension which corresponds to the ABness of commutative noetherian rings, and to research its properties.
We define an "AB-dimension" which is closely related to the ABness of rings. We show that the "Auslander-Reiten conjecture" is true if the AB-dimension is finite.

Research Products

• [Journal Article] Exceptional Sequences over Path Algebras of Type An and Non-Crossing Spanning Trees. (2013)
  • Author(s): Tokuji Araya
  • Journal Title: Algebr. Represent. Theory Volume: 16, no. 1 Pages: 239-250
  • Peer Reviewed
• [Journal Article] Exceptional Sequences over Path Algebras of Type An and Non-crossing Spanning Trees (2013)
  • Author(s): Tokuji Araya
  • Journal Title: Algebr. Represent. Theory Volume: 16 Issue: 1 Pages: 239-250
  • DOI: 10.1007/s10468-011-9304-4
  • Peer Reviewed
• [Journal Article] Dimensions of triangularted categories with respect to subcateogries (2013)
  • Author(s): Tokuji Araya
  • Journal Title: 第34 回可換環論シンポジウム報告集 Pages: 146-149
• [Journal Article] Dimensions of triangularted categories with respect to subcateogries (2012)
  • Author(s): Tokuji Araya
  • Journal Title: 第34回可換環論シンポジウム報告集 Pages: 146-149
• [Journal Article] Dimensions of triangularted categories with respect to subcateogries (2012)
  • Author(s): Tokuji Araya
  • Journal Title: Proceedings of the 45th Symposium on Ring Theory and Representation Theory Pages: 12-15
• [Journal Article] (Sn)条件によるC-syzygyのある特徴づけ (2012)
  • Author(s): Tokuji Araya;Kei-ichiro lima
  • Journal Title: 第24回可換環論セミナー 報告集 Pages: 122-125
• [Journal Article] Auslander-Bridger型近似定理について (2012)
  • Author(s): Tokuji Araya; Kei-ichiro Iima
  • Journal Title: 第24回可換環論セミナー 報告集 Pages: 135-137
• [Journal Article] Auslander-Reiten conjecture for modules of finite AB-dimension (2012)
  • Author(s): Tokuji Araya
  • Journal Title: Proceedings of the 7th Japan-Vietnam joint seminar on commutative algebra Pages: 202-204
• [Journal Article] On the left perpendicular category of the modules of finite projective dimension (2012)
  • Author(s): Tokuji Araya; Kei-ichiro Iima; Ryo Takahashi
  • Journal Title: Comm. Algebra Volume: 40, no. 8 Pages: 2693-2699
  • Peer Reviewed
• [Journal Article] On the structure of Cohen-Macaulay modules over hypersurfaces (2012)
  • Author(s): Tokuji Araya; Kei-ichiro Iima; Ryo Takahashi
  • Journal Title: J. Algebra Volume: 361 Pages: 213-224
  • Peer Reviewed
• [Journal Article] (Sn) 条件によるC-syzygy のある特徴づけ (2012)
  • Author(s): 荒谷 督司，飯間 圭一郎
  • Journal Title: 第24 回可換環論セミナー報告集 Pages: 122-125
• [Journal Article] Auslander-Bridger 型近似定理について (2012)
  • Author(s): 荒谷 督司，飯間 圭一郎
  • Journal Title: 第24 回可換環論セミナー報告 Pages: 135-137
• [Journal Article] Auslander-Reiten conjecture for modules of finite AB-dimension (2012)
  • Author(s): Tokuji Araya
  • Journal Title: Proceedings of the 7th Japan-Vietnam joint seminar on commutative Pages: 202-204
• [Journal Article] On the left perpendicular category of the modules of finite projective dimension (2012)
  • Author(s): T. Araya; K.-i. Iima; R. Takahashi
  • Journal Title: Comm. Algebra Volume: 40.no. 8 Issue: 8 Pages: 2693-2699
  • DOI: 10.1080/00927872.2011.584926
  • Peer Reviewed
• [Journal Article] On the left perpendicular category of the modules of finite projective dimension (2011)
  • Author(s): Tokuji Araya, 他2名
  • Journal Title: Comm.Algebra Volume: (掲載決定)
  • Peer Reviewed
• [Journal Article] Exceptional sequences over path algebras of type An and non-crossing spanning trees (2011)
  • Author(s): Tokuji Araya
  • Journal Title: Algebr.Represent.Theory Volume: (掲載決定)
  • Peer Reviewed
• [Journal Article] On the structure of Cohen-Macaulay modules over hypersurfaces (2011)
  • Author(s): Tokuji Araya, 他2名
  • Journal Title: J.Algebra Volume: (掲載決定)
  • Peer Reviewed
• [Presentation] Auslander-Reiten予想について (2013)
  • Author(s): 荒谷 督司
  • Organizer: 第12回大和郡山セミナー
  • Place of Presentation: 奈良工業高等専門学校
  • Year and Date: 2013-02-03
• [Presentation] Graded Cohen-Macaulaymodules over Graded Cohen-Macaulay (2013)
  • Author(s): 荒谷 督司
  • Organizer: 第3回徳山環論セミナー
  • Place of Presentation: 徳山工業高等専門学校
  • Year and Date: 2013-01-13
• [Presentation] Dimensions oftriangularted categories with respectto subcateogries (2012)
  • Author(s): Tokuji Araya
  • Organizer: 第34回可換環論シンポジウム
  • Place of Presentation: 生産性国際交流センター
  • Year and Date: 2012-11-25
• [Presentation] Dimensions of triangulatedcategories with respect tosubcategories (2012)
  • Author(s): 荒谷 督司
  • Organizer: 2012年度日本数学会秋期総合分科会
  • Place of Presentation: 九州大学
  • Year and Date: 2012-09-20
• [Presentation] Dimensions of triangulated categories with respect tosubcategories (2012)
  • Author(s): Tokuji Aram
  • Organizer: 第45回環論および表現論シンポジウム
  • Place of Presentation: 信州大学
  • Year and Date: 2012-09-08
• [Presentation] Dimensions of triangulated categories with respect to subcategories (2012)
  • Author(s): Tokuji Araya
  • Organizer: XV International Conference on Representation of Algebras
  • Place of Presentation: Bielefeld University
  • Year and Date: 2012-08-14
• [Presentation] Dimensions of triangulatedcategories with respect tosubcategories (2012)
  • Author(s): 荒谷 督司
  • Organizer: Studio Phones Seminarat Kobe university
  • Place of Presentation: 神戸大学
  • Year and Date: 2012-07-16
• [Presentation] Torsionfreeness of modules with respect to a semidualizing module (2012)
  • Author(s): 飯間圭一郎
  • Organizer: 2012年度日本数学会年会
  • Place of Presentation: 東京理科大学
  • Year and Date: 2012-03-26
• [Presentation] On the approximations of Auslander-Bridger type (2012)
  • Author(s): 飯間圭一郎
  • Organizer: 2012年度日本数学会年会
  • Place of Presentation: 東京理科大学
  • Year and Date: 2012-03-26
• [Presentation] Dimensions of triangulatedcategories with respect tosubcategories 2 (2012)
  • Author(s): 荒谷 督司
  • Organizer: 2013年度日本数学会年会
  • Place of Presentation: 京都大学
  • Year and Date: 2012-03-21
• [Presentation] AB dimension (2012)
  • Author(s): Tokuji Araya
  • Organizer: Algebra seminar
  • Place of Presentation: University of Missouri(アメリカ)
  • Year and Date: 2012-03-15
• [Presentation] AB dimension (2012)
  • Author(s): Tokuji Araya
  • Organizer: Algebra seminar
  • Place of Presentation: University of Missouri(招待講演)
  • Year and Date: 2012-03-15
• [Presentation] Functor categories and derived dimensions (2012)
  • Author(s): 相原琢磨
  • Organizer: 第17回代数学若手研究会
  • Place of Presentation: 静岡大学
  • Year and Date: 2012-03-04
• [Presentation] 加群のtorsionfreenessとAuslander-Bridger型近似定理について (2012)
  • Author(s): 飯間圭一郎
  • Organizer: 第17回代数学若手研究会
  • Place of Presentation: 静岡大学
  • Year and Date: 2012-03-03
• [Presentation] Thick subcategories over graded simple singularities (2012)
  • Author(s): Tokuji Araya
  • Organizer: Commutative Algebra Seminar
  • Place of Presentation: University of Nebraska-Lincoln(アメリカ)
  • Year and Date: 2012-02-13
• [Presentation] Thick subcategories over graded simple singularities (2012)
  • Author(s): Tokuji Araya
  • Organizer: Commutative Algebra Seminar
  • Place of Presentation: University of Nebraska-Lincoln(招待講演)
  • Year and Date: 2012-02-13
• [Presentation] Auslander-Bridger型近似定理について (2012)
  • Author(s): 荒谷 督司
  • Organizer: 第24回可換環論セミナー
  • Place of Presentation: 霧島市市民サービスセンター
  • Year and Date: 2012-02-02
• [Presentation] (S_n)条件によるC-シジジーのある特徴づけ (2012)
  • Author(s): 飯間圭一郎
  • Organizer: 第24回可換環論セミナー
  • Place of Presentation: 霧島市市民サービスセンター
  • Year and Date: 2012-02-01
• [Presentation] Auslander-Reiten Conjecture for modules of finite Abdimension (2011)
  • Author(s): Tokuji Araya
  • Organizer: The 7th Japan-Vietnam joint seminar on commutative algebra
  • Place of Presentation: Qui Nhon University(ベトナム)
  • Year and Date: 2011-12-15
• [Presentation] Auslander-Reiten Conjecture for modules of finite Abdimension (2011)
  • Author(s): Tokuji Araya
  • Organizer: The 7th Japan-Vietnam joint seminar on commutative algebra
  • Place of Presentation: Qui Nhon University
  • Year and Date: 2011-12-15
• [Presentation] On the Auslander-Bridger type approximation of modules (2011)
  • Author(s): 飯間圭一郎
  • Organizer: 第33回可換環論シンポジウム
  • Place of Presentation: 浜名湖カリアック
  • Year and Date: 2011-11-09
• [Presentation] Dimensions of triangulated categories with respect to subcategories 2
  • Author(s): 荒谷 督司
  • Organizer: 2013年度日本数学会年会
  • Place of Presentation: 京都大学
• [Presentation] Auslander-Reiten 予想について
  • Author(s): 荒谷 督司
  • Organizer: 第12回大和郡山セミナー
  • Place of Presentation: 奈良工業高等専門学校
• [Presentation] Graded Cohen-Macaulay modules over Graded Cohen-Macaulay
  • Author(s): 荒谷 督司
  • Organizer: 第3回徳山環論セミナー
  • Place of Presentation: 徳山工業高等専門学校
• [Presentation] Dimensions of triangularted categories with respect to subcateogries
  • Author(s): Tokuji Araya
  • Organizer: 第34回可換環論シンポジウム
  • Place of Presentation: 生産性国際交流センター
• [Presentation] Dimensions of triangulated categories with respect to subcategories
  • Author(s): 荒谷 督司
  • Organizer: 2012年度日本数学会秋期総合分科会
  • Place of Presentation: 九州大学
• [Presentation] Dimensions of triangulated categories with respect to subcategories
  • Author(s): Tokuji Araya
  • Organizer: 第45回環論および表現論シンポジウム
  • Place of Presentation: 信州大学
• [Presentation] Dimensions of triangulated categories with respect to subcategories
  • Author(s): Tokuji Araya
  • Organizer: XV International Conference on Representation of Algebras
  • Place of Presentation: Bielefeld University (ドイツ)
• [Presentation] Dimensions of triangulated categories with respect to subcategories
  • Author(s): 荒谷 督司
  • Organizer: Studio Phones Seminar at Kobe university
  • Place of Presentation: 神戸大学
  • Invited