A study of noncommutative projective hypersurfaces

2022 Fiscal Year Final Research Report

• Project/Area Number: 18K13381
• Research Category: Grant-in-Aid for Early-Career Scientists
• Allocation Type: Multi-year Fund
• Review Section: Basic Section 11010:Algebra-related
• Research Institution: Hirosaki University
• Principal Investigator: Ueyama Kenta 弘前大学, 教育学部, 准教授 (30746409)
• Project Period (FY): 2018-04-01 – 2023-03-31
• Keywords: 非可換超曲面 / 非可換射影超曲面 / 非可換射影スキーム / 非可換行列因子化 / Cohen-Macaulay加群の安定圏 / Knoerrer周期性

Outline of Final Research Achievements

Various studies related to noncommutative graded hypersurface rings and their associated noncommutative projective schemes (i.e., noncommutative projective hypersurfaces) were carried out. For example, I introduced and studied the notion of noncommutative matrix factorization and gave a noncommutative analogue of Eisenbud's theorem. Applying this result, I also gave a noncommutative graded version of Knoerrer's periodicity theorem. In addition, I proved that the stable category of graded maximal Cohen-Macaulay modules over a (±1)-skew graded (A_1) hypersurface singularity can be explicitly computed using combinatorial methods. Applying this result, I also proved that the derived category of a smooth (±1)-skew quadric hypersurface has a full strong exceptional sequence.

Free Research Field

非可換代数幾何学

Academic Significance and Societal Importance of the Research Achievements

非可換次数付き超曲面環や非可換射影超曲面の研究を今後さらに展開させる上で礎となる結果や手法を多数提供できたことは，非常に意義深いことである．また，今回の研究は非可換代数幾何学，代数幾何学，可換環論，表現論，組合せ論など，様々な分野に関連しており，異なる分野の知見を統一的に理解するための足掛かりを与えたことにも大きな意義がある．