Studies of the structure of triangulated categories associated with noncommutative graded isolated singularities

2017 Fiscal Year Final Research Report

• Project/Area Number: 15K17503
• Research Category: Grant-in-Aid for Young Scientists (B)
• Allocation Type: Multi-year Fund
• Research Field: Algebra
• Research Institution: Hirosaki University
• Principal Investigator: Ueyama Kenta 弘前大学, 教育学部, 講師 (30746409)
• Project Period (FY): 2015-04-01 – 2018-03-31
• Keywords: 非可換次数付き孤立特異点 / 三角圏 / 安定圏 / 非可換射影スキーム / 非可換射影空間

Outline of Final Research Achievements

Triangulated categories are increasingly important in many areas of mathematics including algebraic geometry and representation theory of algebras. In particular, triangulated categories associated with isolated singularities have made rapid progress. In this research, I studied noncommutative graded isolated singularities, and abelian categories and triangulated categories associated with them. As main achievements, I proved that the stable category of graded maximal Cohen-Macaulay modules over an AS-Gorenstein isolated quotient singularity has a tilting object and therefore it is triangle equivalent to the derived category of a finite dimensional algebra. Also I gave conditions for the noncommutative projective scheme associated with a noncommutative graded isolated singularity to be realized as a noncommutative projective space.

Free Research Field

非可換代数幾何学