New development of theory of integral closure and its applications to local rings

2023 Fiscal Year Final Research Report

• Project/Area Number: 20K03535
• Research Category: Grant-in-Aid for Scientific Research (C)
• Allocation Type: Multi-year Fund
• Section: 一般
• Review Section: Basic Section 11010:Algebra-related
• Research Institution: Okayama University
• Principal Investigator: Hayasaka Futoshi 岡山大学, 環境生命自然科学学域, 教授 (20409460)
• Project Period (FY): 2020-04-01 – 2024-03-31
• Keywords: 整閉イデアル / 2次元正則局所環 / 整閉加群 / 直既約加群 / 行列式イデアル

Outline of Final Research Achievements

We studied the existence of indecomposable integrally closed modules over two-dimensional regular local rings and the associated determinantal ideals. We obtained a method for constructing indecomposable integrally closed modules associated with integrally closed monomial ideals, and gave a large class of indecomposable integrally closed modules of arbitrary rank. Furthermore, we obtained a characterization of integrally closed ideals which arise as the determinantal ideals of indecomposable integrally closed modules of rank 2 and 3.

Free Research Field

可換環論

Academic Significance and Societal Importance of the Research Achievements

古典的な整閉イデアルの理論を高階数化した2次元正則局所環上の整閉加群の理論では、直既約整閉加群がどれくらい存在するか？という問いが理論の非自明性を示す上で重要である。得られた成果は、直既約整閉加群とそれに付随する行列式イデアルが予想より大量かつ多様に存在することを示すもので、理論の非自明性を強化するばかりでなく、整閉加群の分類可能性を示唆するものである。