Development of the theory of almost Gorenstein rings

2019 Fiscal Year Final Research Report

• Project/Area Number: 17K14176
• Research Category: Grant-in-Aid for Young Scientists (B)
• Allocation Type: Multi-year Fund
• Research Field: Algebra
• Research Institution: Waseda University
• Principal Investigator: Endo Naoki 早稲田大学, グローバルエデュケーションセンター, 講師(任期付) (30782510)
• Project Period (FY): 2017-04-01 – 2020-03-31
• Keywords: Cohen-Macaulay環 / Gorenstein環 / Almost Gorenstein環 / Rees代数 / 行列式環 / Fiber積 / Arf環

Outline of Final Research Achievements

Although the theory of almost Gorenstein rings for higher dimension which was introduced by S. Goto, R. Takahashi, and myself in 2015 is nowadays developing rapidly, there are many unexplored problems, as it is a new concept of Cohen-Macaulay rings. The aim of this research is to explore such problems to enrich the theory of almost Gorenstein rings. More precisely, we investigated the question of when the Rees algebras of ideals, fiber products, and Arf rings are almost Gorenstein, and provided the criterion of them. Besides, we introduce and develop the theory of weakly Arf rings, which is a generalization of Arf rings, initially defined by J. Lipman in 1971.

Free Research Field

代数学

Academic Significance and Societal Importance of the Research Achievements

Cohen-Macaulay環の階層化問題への第一歩として導入されたalmost Gorenstein環論は，研究代表者を含めた日本人研究者を中心とする日本発の新規性の高いオリジナルな研究である。本研究で得られた結果は，国内外における学会・研究集会講演・専門学術誌を通して，世界に広く公表している。これらの成果は，可換環論だけではなく，代数幾何学や特異点論，表現論，組合せ論など関連する諸分野への波及効果も期待できるものである。