Studies of categories of finitely generated modules over Gorenstein local rings

• Project/Area Number: 19740008
• Research Category: Grant-in-Aid for Young Scientists (B)
• Allocation Type: Single-year Grants
• Research Field: Algebra
• Research Institution: Shinshu University
• Principal Investigator: TAKAHASHI Ryo Shinshu University, 理学部, 准教授 (40447719)
• Project Period (FY): 2007 – 2009
• Project Status: Completed (Fiscal Year 2009)
• Budget Amount: ¥3,830,000 (Direct Cost: ¥3,200,000、Indirect Cost: ¥630,000); Fiscal Year 2009: ¥1,300,000 (Direct Cost: ¥1,000,000、Indirect Cost: ¥300,000); Fiscal Year 2008: ¥1,430,000 (Direct Cost: ¥1,100,000、Indirect Cost: ¥330,000); Fiscal Year 2007: ¥1,100,000 (Direct Cost: ¥1,100,000)
• Keywords: Gorenstein環 / Cohen-Macaulay環 / 極大Cohen-Macaulay加群 / 全反射加群 / 分解部分圏 / ゴレンシュタイン環 / コーエンマコーレー環 / 極大コーエンマコーレー加群 / 三角圏 / thick部分圏 / 安定圏 / 反変有限分解部分圏

Research Abstract

(1) 全反射加群の圏の反変有限性が局所環のGorenstein性を特徴付けることを示した。これはAuslander-Buchweitzの定理の逆に相当する。これを用いて全反射加群の個数の有限性による単純特異点の特徴付けも得た。また,Gorenstein Hensel局所環上の反変有限分解部分圏を完全に分類した。
(2) 正則環の局所化のD加群構造に関するAlvarez-Montaner-Blickle-Lyubeznikの定理と局所コホモロジー加群の素因子に関するHuneke-Sharpの定理を有限F表現型の環に拡張した。

Research Products

• [Journal Article] On existence of embeddings into modules of finite homological dimensions (2010)
  • Author(s): Ryo Takahashi; Siamak Yassemi; Yuji Yoshino
  • Journal Title: Proceedings of the American Mathematical Society 138,no.7 Pages: 2265-2268
  • NAID: 120007114208
  • Peer Reviewed
• [Journal Article] Homological aspects of semidualizing modules (2010)
  • Author(s): Ryo Takahashi; Diana White
  • Journal Title: Mathematica Scandinavica 106,no.1 Pages: 5-22
  • Peer Reviewed
• [Journal Article] Homological aspects of semidualizing modules (2010)
  • Author(s): Ryo Takahashi ; Diana White
  • Journal Title: Mathematica Scandinavica 106 Pages: 5-22
  • Peer Reviewed
• [Journal Article] On existence of embeddings into modules of finite homological dimensions (2010)
  • Author(s): Ryo Takahashi ; Siamak Yassemi ; Yuji Yoshino
  • Journal Title: Proceedings of the American Mathematical Society 138 Pages: 2265-2268
  • NAID: 120007114208
  • Peer Reviewed
• [Journal Article] On localizing subcategories of derived categories (2009)
  • Author(s): Ryo Takahashi
  • Journal Title: Journal of Mathematics of Kyoto University 49,no.4 Pages: 771-783
  • Peer Reviewed
• [Journal Article] A generalization of a theorem of Foxby (2009)
  • Author(s): Tokuji Araya; Ryo Takahashi
  • Journal Title: Archiv der Mathematik (Basel) 93,no.2 Pages: 123-127
  • Peer Reviewed
• [Journal Article] Modules in resolving subcategories which are free on the punctured spectrum (2009)
  • Author(s): Ryo Takahashi
  • Journal Title: Pacific Journal of Mathematics 241,no.2 Pages: 347-367
  • Peer Reviewed
• [Journal Article] Local cohomology based on a nonclosed support defined by a pair of ideals (2009)
  • Author(s): Ryo Takahashi; Yuji Yoshino; Takeshi Yoshizawa
  • Journal Title: Journal of Pure and Applied Algebra 213,no.4 Pages: 582-600
  • Peer Reviewed
• [Journal Article] Local cohomology based on a nonclosed support defined by a pair of ideals (2009)
  • Author(s): Ryo Takahashi ; Yuji Yoshino ; Takeshi Yoshizawa
  • Journal Title: Journal of Pure and Applied Algebra 213 Pages: 582-600
  • Peer Reviewed
• [Journal Article] Modules in resolving subcategories which are free on the punctured spectrum (2009)
  • Author(s): Ryo Takahashi
  • Journal Title: Pacific Journal of Mathematics 241 Pages: 347-367
  • Peer Reviewed
• [Journal Article] A generalization of a theorem of Foxby (2009)
  • Author(s): Tokuji Araya ; Ryo Takahashi
  • Journal Title: Archiv der Mathematik (Basel) 93 Pages: 123-127
  • Peer Reviewed
• [Journal Article] On localizing subcategories of derived categories (2009)
  • Author(s): Ryo Takahashi
  • Journal Title: Journal of Mathematics of Kyoto University 49 Pages: 771-783
  • Peer Reviewed
• [Journal Article] Finite Gorenstein representation type implies simple singularty (2009)
  • Author(s): Lars Winther Christensen, Greg Piepmeyer, Janet Striuli and Ryo Takahashi
  • Journal Title: Advance in Mathematics (掲載決定済)
  • Peer Reviewed
• [Journal Article] Classifying subcategories of modules over a commutative noetherian ring (2008)
  • Author(s): Ryo Takahashi
  • Journal Title: Journal of the London Mathematical Society (2) 78,no.3 Pages: 767-782
  • Peer Reviewed
• [Journal Article] On G-regular local rings (2008)
  • Author(s): Ryo Takahashi
  • Journal Title: Communications in Algebra 36,no.12 Pages: 4472-4491
  • Peer Reviewed
• [Journal Article] On vanishing of certain Ext modules (2008)
  • Author(s): Shiro Goto; Futoshi Hayasaka; Ryo Takahashi
  • Journal Title: Journal of Mathematical Society of Japan 60,no.4 Pages: 1045-1064
  • NAID: 10024905254
  • Peer Reviewed
• [Journal Article] Finite Gorenstein representation type implies simple singularity (2008)
  • Author(s): Lars Winther Christensen; Greg Piepmeyer; Janet Striuli; Ryo Takahashi
  • Journal Title: Advances in Mathematics 218,no.4 Pages: 1012-1026
  • Peer Reviewed
• [Journal Article] D-modules over rings with finite F-representation type (2008)
  • Author(s): Shunsuke Takagi; Ryo Takahashi
  • Journal Title: Mathematical Research Letters 15,no.3 Pages: 563-581
  • Peer Reviewed
• [Journal Article] Direct summands of syzygy modules of the residue class field (2008)
  • Author(s): Ryo Takahashi
  • Journal Title: Nagoya Mathematical Journal 189 Pages: 1-25
  • Peer Reviewed
• [Journal Article] On quasi-socle ideals in Gorenstein local rings (2008)
  • Author(s): Shiro Goto; Ryo Takahashi; Naoyuki Matsuoka
  • Journal Title: Journal of Pure and Applied Algebra 212,no.5 Pages: 969-980
  • Peer Reviewed
• [Journal Article] On quasi-socle ideals in Gorenstein local rings (2008)
  • Author(s): Shiro Goto ; Naoyuki Matsuoka ; Rvo Takahashi
  • Journal Title: Journal of Pure and Applied Algebra 212 Pages: 969-980
  • Peer Reviewed
• [Journal Article] D-modules over rings with finite F-representation type (2008)
  • Author(s): Shunsuke Takagi ; Ryo Takahashi
  • Journal Title: Mathematical Research Letters 15 Pages: 563-581
  • Peer Reviewed
• [Journal Article] Finite Gorenstein representation type implies simple singularity (2008)
  • Author(s): Lars Winther Christensen ; Greg Piepmeyer ; Janet Striuli ; Ryo Takahashi
  • Journal Title: Advances in Mathematics 218 Pages: 1012-1026
  • Peer Reviewed
• [Journal Article] On vanishing of certain Ext modules (2008)
  • Author(s): Shiro Goto ; Futoshi Hayasaka ; Ryo Takahashi
  • Journal Title: Journal of Mathematical Society of Japan 60 Pages: 1045-1064
  • NAID: 10024905254
  • Peer Reviewed
• [Journal Article] On G-regular local rings (2008)
  • Author(s): Ryo Takahashi
  • Journal Title: Communications in Algebra 36 Pages: 4472-4491
  • Peer Reviewed
• [Journal Article] Classifying subcategories of modules over a commutative noetherian ring (2008)
  • Author(s): Ryo Takahashi
  • Journal Title: Journal of the London Mathematical Society (2) 78 Pages: 767-782
  • Peer Reviewed
• [Journal Article] Quasi-socle ideals in a Gorenstein local ring (2008)
  • Author(s): Shiro Goto, Naoyuki Matsuoka and Ryo Takahashi
  • Journal Title: Journal of Pure and Applied Algebra 212,no.5 Pages: 969-980
  • Peer Reviewed
• [Journal Article] On G-regular local rings (2008)
  • Author(s): Ryo Takahashi
  • Journal Title: Communications in Algebra (掲載決定済)
  • Peer Reviewed
• [Journal Article] D-modules over rings with finite F-representation type (2008)
  • Author(s): Shunsuke Takagi and Ryo Taka hashi
  • Journal Title: Mathematical Research Letters (掲載決定済)
  • Peer Reviewed
• [Journal Article] An uncountably infinite number of indecomposable totally reflexive modules (2007)
  • Author(s): Ryo Takahashi
  • Journal Title: Nagoya Mathematical Journal 187 Pages: 35-48
  • Peer Reviewed
• [Journal Article] A characterization of modules locally of finite injective dimension (2007)
  • Author(s): Ryo Takahashi
  • Journal Title: Proceedings of the American Mathematical Society 135,no.11 Pages: 3461-3464
  • Peer Reviewed
• [Journal Article] Totally reflexive modules constructed from smooth projective curves of genus g>=2 (2007)
  • Author(s): Ryo Takahashi; Kei-ichi Watanabe
  • Journal Title: Archiv der Mathematik (Basel) 89,no.1 Pages: 60-67
  • Peer Reviewed
• [Journal Article] On the number of indecomposable totally reflexive modules (2007)
  • Author(s): Ryo Takahashi
  • Journal Title: Bulletin of the London Mathematical Society 39,no.3 Pages: 487-492
  • Peer Reviewed
• [Journal Article] On the number of indecomposable totally reflexive modules (2007)
  • Author(s): Ryo Takahashi
  • Journal Title: The Bulletin of the London Mathematical Society 39,no.3 Pages: 487-492
  • Peer Reviewed
• [Journal Article] Totally reflexive modules constructed from smooth projective curves of genus g \geq 2 (2007)
  • Author(s): Ryo Takahashi and Kei-ichi Watanabe
  • Journal Title: Archiv der Mathematik(Basel) 89,no.1 Pages: 60-67
  • Peer Reviewed
• [Journal Article] An uncountably infinite number of indecomposable totally reflexive mod ules (2007)
  • Author(s): Ryo Takahashi
  • Journal Title: Nagoya Mathematical Journal 135,no.11 Pages: 35-48
  • Peer Reviewed
• [Journal Article] Contravariantly finite resolving subcategories over commutative rings
  • Author(s): Ryo Takahashi
  • Journal Title: American Journal of Mathematics (to appear)
  • Peer Reviewed
• [Journal Article] Contravariantly finite resolving subcategories over commutative rings
  • Author(s): Ryo Takahashi
  • Journal Title: American Journal of Mathematics (印刷中)
  • Peer Reviewed
• [Journal Article] Homological aspects of semidualizing modules
  • Author(s): Ryo Takahashi ; Diana White
  • Journal Title: Mathematica Scandinavica (印刷中)
  • Peer Reviewed
• [Journal Article] Modules in resolving subcategories which are free on the punctured spectrum
  • Author(s): Ryo Takahashi
  • Journal Title: Pacific Journal of Mathematics (印刷中)
  • Peer Reviewed
• [Journal Article] A generalization of a theorem of Foxby
  • Author(s): Tokuji Araya ; Ryo Takahashi
  • Journal Title: Archiv der Mathematik (Basel) (印刷中)
  • Peer Reviewed
• [Presentation] 自明でない拡大閉部分圏をもつ環について (2010)
  • Author(s): 高橋亮
  • Organizer: 第15回明治大学可換環論セミナー
  • Place of Presentation: 明治大学
  • Year and Date: 2010-02-27
• [Presentation] On locally free Cohen-Macaulay modules (2010)
  • Author(s): Ryo Takahashi
  • Organizer: The 5th Japan-Vietnam Joint Seminar on Commutative Algebra
  • Place of Presentation: Institute of Mathematics, Hanoi
  • Year and Date: 2010-01-06
• [Presentation] On locally free Cohen-Macaulay modules (2010)
  • Author(s): Ryo Takahashi
  • Organizer: The 5th Japan-Vietnam Joint Seminar on Commutative Algebra
  • Place of Presentation: Institute of Mathematics, Hanoi, Vietnam, January
• [Presentation] Thick subcategories of stable categories of Cohen-Macaulay modules (2009)
  • Author(s): Ryo Takahashi
  • Organizer: The inaugural Pacific Rim Mathematical Association (PRIMA) Congress
  • Place of Presentation: University of New South Wales
  • Year and Date: 2009-07-08
• [Presentation] 超曲面上の極大Cohen-Macaulay加群について (2009)
  • Author(s): 高橋亮
  • Organizer: 明治大学大学院特別講義
  • Place of Presentation: 明治大学
  • Year and Date: 2009-05-16
• [Presentation] 極大Cohen-Macaulay加群の安定圏のthick部分圏とその応用 (2009)
  • Author(s): 高橋亮
  • Organizer: 平成19年度文部科学省採択:組織的な大学院教育改革推進プログラム(GP)「社会に数理科学を発信する次世代型人材創発」2009年度MTS数理科学教育・理論数理コース:第三回明治大学可換環論セミナー
  • Place of Presentation: 明治大学
  • Year and Date: 2009-04-11
• [Presentation] コーエン・マコーレー加群の安定圏のthick部分圏についてII (2009)
  • Author(s): 高橋亮
  • Organizer: 岡山大学セミナー
  • Place of Presentation: 岡山大学
  • Year and Date: 2009-03-31
• [Presentation] 射影次元が有限な加群への埋め込み (2009)
  • Author(s): 高橋亮 ; Siamak Yassemi ; 吉野雄二
  • Organizer: 日本数学会代数学分科会
  • Place of Presentation: 東京大学
  • Year and Date: 2009-03-29
• [Presentation] Walks in resolving subcategories (2009)
  • Author(s): 高橋亮
  • Organizer: The 4th Japan-Vietnam Joint Seminar on Commutative Algebra by and for young mathematicians, Meiji Institute for Advanced Study of Mathematical Sciences
  • Place of Presentation: 明治大学
  • Year and Date: 2009-02-20
• [Presentation] コーエン・マコーレー加群の安定圏のthick部分圏について (2009)
  • Author(s): 高橋亮
  • Organizer: 岡山大学セミナー
  • Place of Presentation: 岡山大学
  • Year and Date: 2009-02-05
• [Presentation] Embeddings into modules of finite projective dimension (2009)
  • Author(s): 高橋亮
  • Organizer: 第21回可換環論セミナー
  • Place of Presentation: 釧路市生涯学習センター
  • Year and Date: 2009-01-28
• [Presentation] 非自明な拡大閉部分圏の存在性について (2009)
  • Author(s): 高橋亮
  • Organizer: 岡山大学代数セミナー
  • Place of Presentation: 岡山大学
• [Presentation] Cohen-Macaulay加群の安定圏のthick部分圏の分類について (2009)
  • Author(s): 高橋亮
  • 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
• [Presentation] 可換環論の発展-ホモロジカル予想を中心として- (2009)
  • Author(s): 高木俊輔;高橋亮
  • Organizer: 第54回代数学シンポジウム
  • Place of Presentation: 明治大学
• [Presentation] Thick subcategories of stable categories of Cohen-Macaulay modules, The inaugural Pacific Rim Mathematical Association (PRIMA) Congress (2009)
  • Author(s): Ryo Takahashi
  • Organizer: University of New South Wales
  • Place of Presentation: Sydney, Australia
• [Presentation] Thick subcategories of stable categories of Cohen-Macaulay modules (2009)
  • Author(s): Ryo Takahashi
  • Organizer: Summer Seminar on ring theory
  • Place of Presentation: Osaka Prefecture University
• [Presentation] 極大Cohen-Macaulay加群の安定圏のthick部分圏とその応用, 平成19年度文部科学省採択:組織的な大学院教育改革推進プログラム(gp)「社会に数理科学を発信する次世代型人材創発」2009年度mts数理科学教育・理論数理コース (2009)
  • Author(s): 高橋亮
  • Organizer: 第三回明治大学可換環論セミナー
  • Place of Presentation: 明治大学
• [Presentation] Walks in resolving subcategories (2009)
  • Author(s): Ryo Takahashi
  • Organizer: The 4th Japan-Vietnam Joint Seminar on Commutative Algebra by and for young mathematicians, Meiji Institute for Advanced Study of Mathematical Sciences
  • Place of Presentation: Japan
• [Presentation] Modules in resolving subcategories (2008)
  • Author(s): 高橋亮
  • Organizer: 第30回可換環論シンポジウム
  • Place of Presentation: 国民宿舎虹の松原ホテル
  • Year and Date: 2008-11-20
• [Presentation] Resolving subcategories of modules over commutative rings I (2008)
  • Author(s): 高橋亮
  • Organizer: 多元環の表現論ワークショップ
  • Place of Presentation: 静岡大学
  • Year and Date: 2008-09-03
• [Presentation] Resolving subcategories of modules over commutative rings II (2008)
  • Author(s): 高橋亮
  • Organizer: 多元環の表現論ワークショップ
  • Place of Presentation: 静岡大学
  • Year and Date: 2008-09-03
• [Presentation] On modules in resolving subcategories over commutative rings (2008)
  • Author(s): 高橋亮
  • Organizer: Mini-Symposium on ring theory in connection with representation theory and homological algebra
  • Place of Presentation: 大阪府立大学
  • Year and Date: 2008-07-17
• [Presentation] 分解部分圏の中の加群の動きについて (2008)
  • Author(s): 高橋亮
  • Organizer: 明治大学可換環論セミナー
  • Place of Presentation: 明治大学
  • Year and Date: 2008-06-14
• [Presentation] Approximating modules by resoloving subcategories (2008)
  • Author(s): Ryo Takahashi
  • Organizer: International Conference Towards Establishing a Cutting-Edge Platform for Commutative Ring The ory in the 21st Century
  • Place of Presentation: Port Opening Memorial Hall,Yokohama,Japan
  • Year and Date: 2008-03-19
• [Presentation] Contravariantly finite resolving subcategories of modules (2008)
  • Author(s): Ryo Takahashi
  • Organizer: CIMPA school and international conference on commutative algebra
  • Place of Presentation: IIT-Bombay,MuMbai,India
  • Year and Date: 2008-01-08
• [Presentation] Resolving subcategories of modules over commutative rings II (2008)
  • Author(s): Ryo Takahashi
  • Organizer: Workshop on Representation Theory of Algebras 2008
  • Place of Presentation: Shizuoka University
• [Presentation] Resolving subcategories of modules over commutative rings I (2008)
  • Author(s): Ryo Takahashi
  • Organizer: Workshop on Representation Theory of Algebras 2008
  • Place of Presentation: Shizuoka University
• [Presentation] On modules in resolving subcategories over commutative rings (2008)
  • Author(s): Ryo Takahashi
  • Organizer: Mini-Symposium on Ring Theory in Connection with Representation Theory and Homological Algebra
  • Place of Presentation: Osaka Prefecture University
• [Presentation] Cohen-Macaulay環の表現論の一般化に向けて (2008)
  • Author(s): 高橋亮
  • Organizer: 名古屋大学特別セミナー
• [Presentation] Approximating modules by resolving subcategories (2008)
  • Author(s): Ryo Takahashi
  • Organizer: International Conference Towards Establishing a Cutting-Edge Platform for Commutative Ring Theory in the 21st Century
  • Place of Presentation: Port Opening Memorial Hall, Yokohama, Japan
• [Presentation] Contravariantly finite resolving subcategories of modules (2008)
  • Author(s): Ryo Takahashi
  • Organizer: CIMPA School and International Conference on Commutative Algebra
  • Place of Presentation: IIT-Bombay, Mumbai, India, January
• [Presentation] Local cohomology with respect to a pair of ideals (2007)
  • Author(s): Ryo Takahashi
  • Organizer: The Third Japan-Vietnam Joint Seminar on Com mutative Algebra
  • Place of Presentation: Institute of Mathematics,Hanoi,Vietnam
  • Year and Date: 2007-12-04
• [Presentation] Contravariantly finite resolving subcategories over a Gorenstein local ring (2007)
  • Author(s): Ryo Takahashi
  • Organizer: 第29回可換環論シンポジウム
  • Place of Presentation: 愛知厚生年金会館ウェルシティなごや
  • Year and Date: 2007-11-20
• [Presentation] On asi-socle ideals in a Gorenstein local ring (2007)
  • Author(s): 後藤 四郎;松岡 直之;高橋 亮
  • Organizer: 日本数学会代数学分科会
  • Place of Presentation: 東北大学
  • Year and Date: 2007-09-23
• [Presentation] 局所コホモロジー加群の一般化とその消滅についてII (2007)
  • Author(s): 高橋 亮;吉野 雄二;吉澤 毅
  • Organizer: 日本数学会代数学分科会
  • Place of Presentation: 東北大学
  • Year and Date: 2007-09-23
• [Presentation] On contravariantly finit subcategories of finitely generated modules (2007)
  • Author(s): Ryo Takahashi
  • Organizer: The Fifth China-Japan-Korea international Sympo sium on Ring Theory
  • Place of Presentation: National Olympics Meom orial Youth Center(NYC),Tokyo,Japan
  • Year and Date: 2007-09-11
• [Presentation] Local cohomology with respect to a pair of ideals (2007)
  • Author(s): Ryo Takahashi
  • Organizer: The Third Japan-Vietnam Joint Seminar on Commutative Algebra
  • Place of Presentation: Institute of Mathematics, Hanoi, Vietnam
• [Presentation] On contravariantly finite subcategories of finitely generated modules (2007)
  • Author(s): Ryo Takahashi
  • Organizer: The Fifth China-Japan-Korea International Symposium on Ring Theory, National Olympics Meomorial Youth Center (NYC)
  • Place of Presentation: Tokyo, Japan
• [Presentation] ゴレンシュタイン環上の加群圏の反変有限部分圏 (2007)
  • Author(s): 高橋亮
  • Organizer: 岡山大学数学コロキウム
• [Remarks]
  • URL: http://math.shinshu-u.ac.jp/~takahasi/
• [Remarks]
  • URL: http://math.shinshu-u.ac.jp/~takahasi/
• [Remarks]
  • URL: http://math.shinshu-u.ac.jp/~takahasi/
• [Remarks]
  • URL: http://math.shinshu-u.ac.jp/~takahasi