Study of weakly Arf rings

2023 Fiscal Year Final Research Report

• Project/Area Number: 21K13767
• Research Category: Grant-in-Aid for Early-Career Scientists
• Allocation Type: Multi-year Fund
• Review Section: Basic Section 11010:Algebra-related
• Research Institution: National Institute of Technology(KOSEN), Oshima College (2022-2023); Chiba University (2021)
• Principal Investigator: Isobe Ryotaro 大島商船高等専門学校, 一般科目, 助教 (50897882)
• Project Period (FY): 2021-04-01 – 2024-03-31
• Keywords: Cohen-Macaulay環 / Arf環 / Weakly Arf環 / Strictly closed環 / 整閉イデアル / 反射的加群

Outline of Final Research Achievements

The notion of weakly Arf rings was introduced by weakening the defining conditions of Arf rings, to extend the theory of Arf rings over arbitrary commutative rings. It is known that Arf rings have some good structure next to integrally closed rings from their definition and characterization. Therefore, it is expected that weakly Arf rings also have similar properties.
In this research project, we have been engaged in analyzing the structure of ideals and modules over weakly Arf rings. We determined the structure of integrally closed ideals in Arf rings, the structure of reflexive modules over Arf rings, and provided a specific method for constructing weakly Arf closure and strict closure of rings.

Free Research Field

可換環論

Academic Significance and Societal Importance of the Research Achievements

本研究の目的は，weakly Arf環の構造解析を通して「整閉環に準ずる環構造とは如何なる存在であるか」を考察し，その構造を明らかにすることである。整閉環論と並行する新たな非整閉環の環構造論を確立することは，可換環論のみならず，整閉環論と親和性の高い代数幾何学・表現論・不変式論・組合せ論といった周辺分野の発展にも大きく貢献することが期待される。
本研究ではArf環やweakly Arf環，環のstrict closureに関する基礎的な構造を明らかにし，weakly Arf環論の基盤を構成している。新たな非整閉環論の原型を確立したことは，当該分野の発展に貢献したと考えられる。