Thick subcategories and dimensions of derived categories of commutative rings

2019 Fiscal Year Final Research Report

• Project/Area Number: 16K05098
• Research Category: Grant-in-Aid for Scientific Research (C)
• Allocation Type: Multi-year Fund
• Section: 一般
• Research Field: Algebra
• Research Institution: Nagoya University
• Principal Investigator: Takahashi Ryo 名古屋大学, 多元数理科学研究科, 准教授 (40447719)
• Project Period (FY): 2016-04-01 – 2020-03-31
• Keywords: 可換環 / thick部分圏 / 三角圏 / 導来圏 / （Rouquier）次元 / 特異圏 / Cohen-Macaulay / Gorenstein

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.

Free Research Field

可換環論，表現論

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。