Thick subcategories and dimensions of derived categories of commutative rings
| Project/Area Number |
16K05098
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| Research Category |
Grant-in-Aid for Scientific Research (C)
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| Allocation Type | Multi-year Fund |
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| Section | 一般 |
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| Research Field |
Algebra
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| Research Institution | Nagoya University |
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Principal Investigator |
Takahashi Ryo 名古屋大学, 多元数理科学研究科, 准教授 (40447719)
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| Project Period (FY) |
2016-04-01 – 2020-03-31
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| Project Status |
Completed (Fiscal Year 2019)
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| Budget Amount *help |
¥4,550,000 (Direct Cost: ¥3,500,000、Indirect Cost: ¥1,050,000)
Fiscal Year 2018: ¥1,430,000 (Direct Cost: ¥1,100,000、Indirect Cost: ¥330,000)
Fiscal Year 2017: ¥1,430,000 (Direct Cost: ¥1,100,000、Indirect Cost: ¥330,000)
Fiscal Year 2016: ¥1,690,000 (Direct Cost: ¥1,300,000、Indirect Cost: ¥390,000)
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| Keywords | 可換環 / thick部分圏 / 三角圏 / 導来圏 / （Rouquier）次元 / 特異圏 / Cohen-Macaulay / Gorenstein / 局所化部分圏 / 退化 / 深度公式 / 改新交差定理 / Auslander-Reiten双対 / Spanier-Whitehead圏 / Rouquier次元 / レベル / Gorenstein環 / Cohen-Macaulay環 / 極大Cohen-Macaulay加群 / 超曲面 / Ulrich加群 / Auslander-Reiten予想 / 有限表現型 / Noether環 / 局所環 / 有限生成加群 |
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| Outline of Final Research Achievements |
The thick subcategories of the singularity category of a local ring with quasi-decomposable maximal ideal are completely classified. The cocompactly generated thick tensor ideals of the right bounded derived category of finitely generated modules over a commutative noetherian ring are completely classified. For the stable category of Cohen-Macaulay modules over an Iwanaga Gorenstein isolated singularity, being locally finite and having finite representation type turn out to be equivalent. It turns out to hold that if a finitely generated Krull-Schmidt triangulated category is locally finite, then it has dimension zero, and the converse is true if it is Ext-finite. It is found out that the residue field of a local hypersurface of countable representation type has level at most one with respect to each nonzero object.
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| Academic Significance and Societal Importance of the Research Achievements |
表現論は数学全体に跨っている分野ですが、私は（可換）環の表現論を中心に研究しています。この分野の主題は、与えられた環の外部表現（加群や複体）全体のなす圏構造を明らかにすることであり、部分圏の分類を行うこと、次元を評価するということは重要なアプローチになっています。本研究で得られたいくつかの部分圏の分類定理や次元とレベルの評価は、それに寄与するものです。
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Report
(5 results)
Research Products
(120 results)
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[Presentation] Quasi-projective dimension2019
Author(s)
Mohsen Gheibi; David A. Jorgensen; Ryo Takahashi
Organizer
AMS Fall Southeastern Sectional Meeting, Special Session on Homological Methods in Algebra, University of Florida, Gainesville, FL, USA
Related Report
Int'l Joint Research / Invited
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[Presentation] Applications of fiber product rings2018
Author(s)
Saeed Nasseh; Sean Sather-Wagstaff; Ryo Takahashi; Keller VandeBogert
Organizer
AMS Spring Eastern Sectional Meeting, Special Session on Homological Commutative Algebra, Northeastern University, Boston, MA, USA
Related Report
Int'l Joint Research / Invited
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[Presentation] Applications of fiber product rings2018
Author(s)
Saeed Nasseh; Sean Sather-Wagstaff; Ryo Takahashi; Keller VandeBogert
Organizer
AMS Spring Eastern Sectional Meeting, Special Session on Homological Commutative Algebra
Related Report
Int'l Joint Research / Invited
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[Presentation] TBA2018
Author(s)
Ryo Takahashi
Organizer
AMS Sectional Meeting, Special Session: Homological aspects of Commutative Algebra and Representation Theory
Related Report
Int'l Joint Research / Invited
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[Presentation] TBA2017
Author(s)
Ryo Takahashi
Organizer
Third Pacific Rim Mathematical Association (PRIMA) Congress
Place of Presentation
Instituto Tecnologico de Oaxaca (ITO)
Year and Date
2017-08-14
Related Report
Int'l Joint Research / Invited
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[Presentation] Almost Gorenstein fiber product rings2017
Author(s)
Saeed Nasseh; Sean Sather-Wagstaff; Ryo Takahashi; Keller VandeBogert
Organizer
AMS Fall Southeastern Sectional Meeting, Special Session on Commutative Algebra: Interactions with Algebraic Geometry and Algebraic Topology
Related Report
Int'l Joint Research / Invited
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