Study on Bourbaki sequences and its applications

2023 Fiscal Year Final Research Report

• Project/Area Number: 21K13766
• Research Category: Grant-in-Aid for Early-Career Scientists
• Allocation Type: Multi-year Fund
• Review Section: Basic Section 11010:Algebra-related
• Research Institution: Oyama National College of Technology
• Principal Investigator: Kumashiro Shinya 小山工業高等専門学校, 一般科, 助教 (40896023)
• Project Period (FY): 2021-04-01 – 2024-03-31
• Keywords: Cohen-Macaulay環 / ヒルベルト関数 / 正準イデアル

Outline of Final Research Achievements

Commutative Algebra, to which this research project belongs, aims to investigate the structure of commutative rings. There are two main approaches to commutative ring theory: ideal theory, which investigates the internal structure of rings, and module theory, which investigates the external representation of rings. In this project, the purpose was to study the Bourbaki sequences, and we constructed a filtration of modules, which is a generalization of the Bourbaki sequences. Furthermore, by applying the filtration to exact modules reflecting the Hilbert function, we clarified the behavior of the Hilbert function of ideals of reduction numbers 2 or 3. We also explored canonical ideals.

Free Research Field

可換環論

Academic Significance and Societal Importance of the Research Achievements

ブルバキ完全列は与えられた加群を自由加群とイデアルに分解する短完全列のことである。ブルバキ完全列を通すことで、加群の性質をイデアルに遺伝させることができるため、可換環論の様々な問題において応用がある。本研究では、ブルバキ完全列を一般化させたフィルトレーションの構成を行った。またその構成を応用することで、ヒルベルト関数解析の研究に対し寄与することができた。