Studies of the structure of module categories, derived categories and singularity categories of commutative rings

2015 Fiscal Year Final Research Report

• Project/Area Number: 25400038
• 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): 2013-04-01 – 2016-03-31
• Keywords: 可換環の表現論 / 加群圏 / 導来圏 / 特異圏 / 分解部分圏 / 特異点 / Ext関手 / Rouquier次元

Outline of Final Research Achievements

I did several researches in the representation theory of commutative rings. More precisely, for a given commutative noetherian ring, I investigated the structure of the module category (i.e., the category of finitely generated modules), the (bounded) derived category and the singularity category. As main achievements, I obtained complete classification of the resolving subcategories of the module category of a complete intersection ring. Also, using the dimension and radius of a subcategory of the module category, I explored certain properties of singularities. Furthermore, I studied annihilation of the Ext functor, and related it to generation of the derived and singularity categories, especially to the Rouquier dimension.

Free Research Field

可換環論