Study of almost Gorenstein rings
| Project/Area Number |
16K05112
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| Research Category |
Grant-in-Aid for Scientific Research (C)
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| Allocation Type | Multi-year Fund |
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| Section | 一般 |
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| Research Field |
Algebra
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| Research Institution | Meiji University |
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Principal Investigator |
Goto Shiro 明治大学, 研究・知財戦略機構(生田), 研究推進員 (50060091)
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| 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)
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| Project Period (FY) |
2016-04-01 – 2021-03-31
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| Project Status |
Completed (Fiscal Year 2020)
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| Budget Amount *help |
¥4,550,000 (Direct Cost: ¥3,500,000、Indirect Cost: ¥1,050,000)
Fiscal Year 2020: ¥910,000 (Direct Cost: ¥700,000、Indirect Cost: ¥210,000)
Fiscal Year 2019: ¥910,000 (Direct Cost: ¥700,000、Indirect Cost: ¥210,000)
Fiscal Year 2018: ¥910,000 (Direct Cost: ¥700,000、Indirect Cost: ¥210,000)
Fiscal Year 2017: ¥910,000 (Direct Cost: ¥700,000、Indirect Cost: ¥210,000)
Fiscal Year 2016: ¥910,000 (Direct Cost: ¥700,000、Indirect Cost: ¥210,000)
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| Keywords | almost Gorenstein ring / Gorenstein ring / Cohen-Macaulay ring / Rees algebra / integrally closed ideal / contracted ideal / Commutative algebra / 可換環論 / Rees代数 / Cohen-Macaulay環 / Gorenstein環 / Almost Gorenstein環 / Generalized Gorenstein環 / Arf環 / Almost Gorenstein ring / 2-AGL ring / Sally module / canonical module / 整閉イデアル / Contracted ideal / 2-almost Gorenstein ring / Ulrich module / 数学 / 代数学 |
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| 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.
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| 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。
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Report
(6 results)
Research Products
(53 results)
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[Journal Article] Hilbert polynomials of j-transforms2016
Author(s)
Shiro Goto, Jooyoun Hong, Wolmer V. Vasconcelos
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Journal Title
Mathematical Proceedings of the Cambridge Philosophical Society
Volume: 161
Issue: 2
Pages: 305-337
DOI
Related Report
Peer Reviewed / Int'l Joint Research / Acknowledgement Compliant
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[Presentation] Almost Gorenstein rings2017
Author(s)
Shiro Goto
Organizer
Special session on “Commutative Algebra”, the AMS Sectional Meeting, Hunter College, City University of New York, May 6~7, 2017.
Related Report
Int'l Joint Research / Invited
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