Studies of maximal Cohen-Macaulay modules over graded hypersurfaces

• Project/Area Number: 26400056
• Research Category: Grant-in-Aid for Scientific Research (C)
• Allocation Type: Multi-year Fund
• Section: 一般
• Research Field: Algebra
• Research Institution: Okayama University of Science
• Principal Investigator: Araya Tokuji 岡山理科大学, 理学部, 准教授 (70613222)
• Co-Investigator(Renkei-kenkyūsha): YOSHINO Yuji 岡山大学, 理学部, 教授 (00135302)
• Project Period (FY): 2014-04-01 – 2017-03-31
• Project Status: Completed (Fiscal Year 2016)
• Budget Amount: ¥4,810,000 (Direct Cost: ¥3,700,000、Indirect Cost: ¥1,110,000); Fiscal Year 2016: ¥1,560,000 (Direct Cost: ¥1,200,000、Indirect Cost: ¥360,000); Fiscal Year 2015: ¥1,430,000 (Direct Cost: ¥1,100,000、Indirect Cost: ¥330,000); Fiscal Year 2014: ¥1,820,000 (Direct Cost: ¥1,400,000、Indirect Cost: ¥420,000)
• Keywords: 超曲面 / 極大コーエン・マコーレー加群 / ゴーレンシュタイン環 / 次数付き環 / 三角圏 / 安定圏 / Gorenstein 環 / 次数付環 / weighted projective line

Outline of Final Research Achievements

The aim this research is to study the category of maximal Cohen-Macaulay modules over graded hypersurfaces. It is well-known that the stable category of maximal Cohen-Macaulay modules over Gorenstein rings has a structure of triangulated category and that hypersurface is a typical example of Gorenstein.
The main result of this research is to give a classification of the thick subcategories of the steble category over the simple hypersurface singularity of type A and D. I gave a talk about this result at the international conference which held at Syracuse university.

Research Products

• [Journal Article] Remarks on torsionfreeness and its applications (2017)
  • Author(s): Tokuji Araya and Kei-ichiro Iima
  • Journal Title: Communications in Algebra Volume: 印刷中 Issue: 1 Pages: 191-200
  • DOI: 10.1080/00927872.2017.1324865
  • Peer Reviewed / Acknowledgement Compliant
• [Journal Article] Remarks on Reflexive Subcategories (2016)
  • Author(s): Araya, Tokuji and Kageyama, Masaru
  • Journal Title: Communications in Algebra Volume: 44 Issue: 7 Pages: 3038-3052
  • DOI: 10.1080/00927872.2015.1065868
  • Peer Reviewed / Acknowledgement Compliant
• [Journal Article] Dimensions of triangulated categories with respect to subcategories (2014)
  • Author(s): T. Aihara, T. Araya, O. Iyama, R. Takahashi, M. Yoshiwaki
  • Journal Title: J. Algebra Volume: 399 Pages: 205-219
  • DOI: 10.1016/j.jalgebra.2013.09.035
  • Peer Reviewed
• [Presentation] Thick subcategories over graded simple singularities of type A (2016)
  • Author(s): Tokuji Araya
  • Organizer: Workshop on Weighted Projective Spaces and Representation Theory
  • Place of Presentation: University of Science and Technology of China
  • Year and Date: 2016-03-05
  • Int'l Joint Research / Invited
• [Presentation] Thick subcategories over graded simple singularities of type D (2016)
  • Author(s): 荒谷 督司
  • Organizer: Ring Theory and Representation Theory Seminar
  • Place of Presentation: 名古屋大学
  • Year and Date: 2016-01-18
  • Invited
• [Presentation] Thick subcategories over graded simple hypersurface singularities (2016)
  • Author(s): Tokuji Araya
  • Organizer: Workshop and International Conference on Representations of Algebras (ICRA 2016)
  • Place of Presentation: Syracuse University
  • Int'l Joint Research
• [Presentation] An introduction to reducible homological dimension (2016)
  • Author(s): Tokuji Araya
  • Organizer: 第38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] Remarks on some homological conjectures (2015)
  • Author(s): Tokuji Araya
  • Organizer: 第37回可換環論シンポジウム
  • Place of Presentation: 倉敷シーサイドホテル
  • Year and Date: 2015-11-18
  • Int'l Joint Research
• [Presentation] On the locally Gorensteinness and Serre's (Sn)-condition of modules (2015)
  • Author(s): 飯間圭一郎，荒谷督司
  • Organizer: 2015年度日本数学会年会
  • Place of Presentation: 明治大学駿河台キャンパス
  • Year and Date: 2015-03-21
• [Presentation] Locally-Gorensteinness of Cohen-Macaulay rings (2015)
  • Author(s): Tokuji Araya
  • Organizer: 第27回可換環論セミナー
  • Place of Presentation: 静岡大学
  • Year and Date: 2015-01-28
• [Presentation] Locally-Gorensteinness of Cohen-Macaulay rings (2014)
  • Author(s): Tokuji Araya
  • Organizer: 第36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: 2014-11-24
• [Presentation] Gorensteinness on the punctured spectrum (2014)
  • 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: 大阪市立大学
  • Year and Date: 2014-09-14
• [Presentation] Gorensteinenss on the punctured spectrum (2014)
  • Author(s): Tokuji Araya
  • Organizer: The 16th International Conference on Representations of Algebras and Workshop
  • Place of Presentation: singhua Sanya International Mathematics Forum
  • Year and Date: 2014-08-26
• [Funded Workshop] 第37回可換環論シンポジウム (2015)
  • Place of Presentation: 倉敷シーサイドホテル
  • Year and Date: 2015-11-18