Study of almost Gorenstein rings

2020 Fiscal Year Final Research Report

• Project/Area Number: 16K05112
• Research Category: Grant-in-Aid for Scientific Research (C)
• Allocation Type: Multi-year Fund
• Section: 一般
• Research Field: Algebra
• Research Institution: Meiji University
• Principal Investigator: Goto Shiro 明治大学, 研究・知財戦略機構(生田), 研究推進員 (50060091)
• Co-Investigator(Kenkyū-buntansha): 居相 真一郎 北海道教育大学, 教育学部, 准教授 (50333125) ; 松岡 直之 明治大学, 理工学部, 専任准教授 (80440155)
• Project Period (FY): 2016-04-01 – 2021-03-31
• Keywords: almost Gorenstein ring / Gorenstein ring / Cohen-Macaulay ring / Rees algebra / integrally closed ideal / contracted ideal

Outline of Final Research Achievements

The research aims at the classification of non-Gorenstein Cohen-Macaulay rings in terms of the distance from Gorenstein rings. The notion of almost Gorenstein ring was given in 2015 by [1] as a higher-dimensional generalization of the notion defined in 1997 by [2] (resp. in 2013 by [3]) for analytically unramified (resp. arbitrary) Cohen-Macaulay local rings of dimension one. The proposed and achieved tasks are the following. (1) Deepening of the theory of almost Gorenstein local/graded rings started by [1]. (2) Analysis of the almost Gorenstein property of Rees algebras of parameter ideals in Cohen-Macaulay local rings, and those of integrally closed/contracted ideals in two-dimensional regular local rings. (3) Permeation of the notion of almost Gorenstein ring into other branches of algebra, for examples, algebraic geometry, combinatorics, invariant theory.
[1] J. Pure and Appl. Algebra, 219 (2015), 2666-2712. [2] J. Algebra, 188 (1997), 418-442. [3] J. Algebra, 379 (2013), 355-381.

Free Research Field

代数学（可換環論）

Academic Significance and Societal Importance of the Research Achievements

多様かつ豊富に存在するCohen-Macaulay環を分類すること，Gorenstein環からどのくらい遠いか，Gorenstein環との違いを指標に階層化することは，可換環論における喫緊の課題の一つである。可換環論に新たな地平をもたらすべく，基礎環の正準加群への埋め込みの様相によって，与えられたCohen-Macaulay環のGorenstein性との乖離状況を記述し，非Gorenstein Cohen-Macaulay環論に大きな発展をもたらすことに成功したと評価される。