Generation problems in module categories and derived categories of commutative noetherian rings

2022 Fiscal Year Final Research Report

• Project/Area Number: 19K03443
• Research Category: Grant-in-Aid for Scientific Research (C)
• Allocation Type: Multi-year Fund
• Section: 一般
• Review Section: Basic Section 11010:Algebra-related
• Research Institution: Nagoya University
• Principal Investigator: Takahashi Ryo 名古屋大学, 多元数理科学研究科, 教授 (40447719)
• Project Period (FY): 2019-04-01 – 2023-03-31
• Keywords: 可換環 / thick部分圏 / 三角圏 / 導来圏 / 特異圏 / 加群圏 / 分解部分圏 / 支配的局所環

Outline of Final Research Achievements

I introduced a new class of commutative noetherian local rings which are called dominant local rings. I investigated basic properties of dominant local rings, and compare them with other classes of commutative noetherian local rings. Also, I gave various methods to get another one from a given dominant local ring. Moreover, under certain assumptions on the dominance of localizations, I classified suitable resolving subcategories of finitely generated modules, suitable thick subcategories of the bounded derived category of finitely generated modules, and the whole thick subcategories of the singularity category. This classification theorem recovers all the existing classification theorems in the same context.

Free Research Field

可換環論

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きました。この分野の主題は、与えられた環の外部表現（加群や複体）全体のなす構造を明らかにすることであり、環に付随する各種のアーベル圏や三角圏の適切な充満部分圏の分類を行うことは、そのための重要なアプローチの一つになっています。本研究で得られた部分圏の分類定理は、環の表現論の進展に寄与するものであると言えます。