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Study of almost Gorenstein rings

Research Project

Project/Area Number 16K05112
Research Category

Grant-in-Aid for Scientific Research (C)

Allocation TypeMulti-year Fund
Section一般
Research Field Algebra
Research InstitutionMeiji 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
Project Status Completed (Fiscal Year 2020)
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)
Keywordsalmost 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
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.

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。

Report

(6 results)
  • 2020 Annual Research Report   Final Research Report ( PDF )
  • 2019 Research-status Report
  • 2018 Research-status Report
  • 2017 Research-status Report
  • 2016 Research-status Report
  • Research Products

    (53 results)

All 2021 2020 2019 2018 2017 2016 Other

All Int'l Joint Research (10 results) Journal Article (29 results) (of which Int'l Joint Research: 13 results,  Peer Reviewed: 27 results,  Open Access: 12 results,  Acknowledgement Compliant: 6 results) Presentation (12 results) (of which Int'l Joint Research: 7 results,  Invited: 7 results) Book (1 results) Funded Workshop (1 results)

  • [Int'l Joint Research] West Virginia University/North Dakota State University(米国)

    • Related Report
      2020 Annual Research Report
  • [Int'l Joint Research] West Virginia University(米国)

    • Related Report
      2019 Research-status Report
  • [Int'l Joint Research] Thai Nguyen University of Education/Vinh University(ベトナム)

    • Related Report
      2019 Research-status Report
  • [Int'l Joint Research] Institute of Mathematics, Hanoi, VAST/Thai Nguyen University(ベトナム)

    • Related Report
      2018 Research-status Report
  • [Int'l Joint Research] West Virginia University(米国)

    • Related Report
      2018 Research-status Report
  • [Int'l Joint Research] West Virginia University(米国)

    • Related Report
      2017 Research-status Report
  • [Int'l Joint Research] Thai Nguyen University(ベトナム)

    • Related Report
      2017 Research-status Report
  • [Int'l Joint Research] Thai Nguyen University/Institute of Mathematics, VAST(ベトナム)

    • Related Report
      2016 Research-status Report
  • [Int'l Joint Research] Southern Connecticut State University(米国)

    • Related Report
      2016 Research-status Report
  • [Int'l Joint Research] Kharazmi University(Iran)

    • Related Report
      2016 Research-status Report
  • [Journal Article] Construction of strictly closed rings2021

    • Author(s)
      Shiro Goto, Naoki Endo
    • Journal Title

      Proc. Amer. Math. Soc.

      Volume: -

    • Related Report
      2020 Annual Research Report
    • Peer Reviewed
  • [Journal Article] Almost Gorenstein rings arising from fiber products2021

    • Author(s)
      Naoki Endo, Shiro Goto, Ryotaro Isobe
    • Journal Title

      Canadian Math. Bull.

      Volume: -

    • Related Report
      2020 Annual Research Report
    • Peer Reviewed
  • [Journal Article] Efficient generation of ideals in core subalgebras of the polynomial ring $k[t]$ over a field $k$2020

    • Author(s)
      Endo Naoki、Goto Shiro、Matsuoka Naoyuki、Yamamoto Yuki
    • Journal Title

      Proceedings of the American Mathematical Society

      Volume: 148 Issue: 8 Pages: 3283-3292

    • DOI

      10.1090/proc/15032

    • Related Report
      2020 Annual Research Report
    • Peer Reviewed
  • [Journal Article] Ulrich ideals and 2-AGL rings2020

    • Author(s)
      Shiro Goto, Ryotaro Isobe, and Naoki Taniguchi
    • Journal Title

      Journal of Algebra

      Volume: 555 Pages: 96-130

    • DOI

      10.1016/j.jalgebra.2020.01.028

    • Related Report
      2020 Annual Research Report 2019 Research-status Report
    • Peer Reviewed
  • [Journal Article] Correspondence between trace ideals and birational extensions with application to the analysis of the Gorenstein property of rings2020

    • Author(s)
      Goto Shiro、Isobe Ryotaro、Kumashiro Shinya
    • Journal Title

      Journal of Pure and Applied Algebra

      Volume: 224 Issue: 2 Pages: 747-767

    • DOI

      10.1016/j.jpaa.2019.06.008

    • Related Report
      2020 Annual Research Report 2019 Research-status Report
    • Peer Reviewed
  • [Journal Article] Generalized Gorenstein Arf rings2019

    • Author(s)
      Ela Celikbas, Olgur Celikbas, Shiro Goto, and Naoki Taniguchi
    • Journal Title

      Ark. Mat.

      Volume: 57 Issue: 1 Pages: 35-53

    • DOI

      10.4310/arkiv.2019.v57.n1.a3

    • Related Report
      2019 Research-status Report
    • Peer Reviewed / Int'l Joint Research
  • [Journal Article] The structure of Ulrich ideals in Cohen-Macaulay local rings of dimension one2019

    • Author(s)
      Shiro Goto, Ryotaro Isobe, and Shinya Kumashiro
    • Journal Title

      Acta Math. Vietnam

      Volume: 44 Issue: 1 Pages: 65-82

    • DOI

      10.1007/s40306-018-0283-y

    • Related Report
      2019 Research-status Report 2018 Research-status Report
    • Peer Reviewed
  • [Journal Article] On the ideal case of a conjecture of Huneke and Wiegand2019

    • Author(s)
      Olgur Celikbas, Shiro Goto, Ryo Takahashi, Naoki Taniguchi
    • Journal Title

      Proc. Edinburgh Math. Soc.

      Volume: 62 Pages: 847-859

    • Related Report
      2019 Research-status Report 2018 Research-status Report
    • Peer Reviewed / Int'l Joint Research
  • [Journal Article] Sally modules of canonical ideals in dimension one and 2-AGL rings2019

    • Author(s)
      Chau Tran Do Minh、Goto Shiro、Kumashiro Shinya、Matsuoka Naoyuki
    • Journal Title

      Journal of Algebra

      Volume: 521 Pages: 299-330

    • DOI

      10.1016/j.jalgebra.2018.11.023

    • Related Report
      2019 Research-status Report 2018 Research-status Report
    • Peer Reviewed / Int'l Joint Research
  • [Journal Article] On the almost Gorenstein property in Rees algebras of contracted ideals2019

    • Author(s)
      Shiro Goto, Naoyuki Matsuoka, Naoki Taniguchi, Ken-ichi Yoshida
    • Journal Title

      Kyoto J. Math.

      Volume: 59 Issue: 4 Pages: 769-785

    • DOI

      10.1215/21562261-2018-0001

    • Related Report
      2019 Research-status Report
    • Peer Reviewed
  • [Journal Article] Residually faithful modules and the Cohen–Macaulay type of idealizations2019

    • Author(s)
      Shiro Goto and Nguyen Thi Hong Loan
    • Journal Title

      Journal of the Mathematical Society of Japan

      Volume: 71 Issue: 4 Pages: 1269-1291

    • DOI

      10.2969/jmsj/80398039

    • NAID

      130007733379

    • ISSN
      0025-5645, 1881-1167, 1881-2333
    • Related Report
      2019 Research-status Report
    • Peer Reviewed / Int'l Joint Research
  • [Journal Article] Residually faithful modules and the Cohen-Macaulay type of idealizations2019

    • Author(s)
      Shiro Goto,Shinya Kumashiro, and Nguyen Thi Hong Loan
    • Journal Title

      J. Math. Soc. Japan

      Volume: 印刷中

    • NAID

      130007733379

    • Related Report
      2018 Research-status Report
    • Peer Reviewed / Int'l Joint Research
  • [Journal Article] Correspondence between trace ideals and birational extensions with application to the analysis of the Gorenstein property of rings2019

    • Author(s)
      Shiro Goto, Ryotaro Isobe, and Shinya Kumashiro
    • Journal Title

      J. Pure and Appl. Algebra

      Volume: 印刷中

    • Related Report
      2018 Research-status Report
    • Peer Reviewed / Open Access
  • [Journal Article] Almost Gorenstein rings arising from fiber products2019

    • Author(s)
      Noki Endo, Shiro Goto, and Ryotaro Isobe
    • Journal Title

      arXiv:1904.07051, 2019

      Volume: -

    • Related Report
      2018 Research-status Report
    • Open Access
  • [Journal Article] Ulrich ideals and $2$-AGL rings2019

    • Author(s)
      Shiro Goto and Naoki Taniguchi
    • Journal Title

      arXiv:1704.01278, 2017

      Volume: -

    • Related Report
      2018 Research-status Report
    • Open Access
  • [Journal Article] Pseudo-Frobenius numbers versus generation of defining ideals in numerical semigroup rings2018

    • Author(s)
      Shiro Goto, Do Van Kien, Naoyuki Matsuoka, Hoang Le Truong
    • Journal Title

      J. Algebra

      Volume: 508 Pages: 1-15

    • DOI

      10.1016/j.jalgebra.2018.04.025

    • Related Report
      2018 Research-status Report 2017 Research-status Report
    • Peer Reviewed / Int'l Joint Research
  • [Journal Article] Invariants of Cohen-Macaulay rings associated to their canonical ideals2018

    • Author(s)
      Laura Ghezzi, Shiro Goto, Jooyoun Hong, and Wolmer V. Vasconcelos
    • Journal Title

      J. Algebra

      Volume: 印刷中

    • Related Report
      2017 Research-status Report
    • Peer Reviewed / Open Access / Int'l Joint Research
  • [Journal Article] On the almost Gorenstein property in Rees algebras of contracted ideals, Kyoto J. Math.2018

    • Author(s)
      Shiro Goto, Naoyuki Matsuoka, Naoki Taniguchi, Ken-ichi Yoshida
    • Journal Title

      Kyoto J. Math.

      Volume: 印刷中

    • Related Report
      2017 Research-status Report
    • Peer Reviewed / Open Access
  • [Journal Article] The almost Gorenstein Rees algebras of pg-ideals, good ideals, and powers of the maximal ideals2018

    • Author(s)
      Shiro Goto, Naoyuki Matsuoka, Naoki Taniguchi, Ken-ichi Yoshida
    • Journal Title

      Michigan Math. J.

      Volume: 67 Issue: 1 Pages: 159-174

    • DOI

      10.1307/mmj/1516330972

    • Related Report
      2017 Research-status Report
    • Peer Reviewed / Open Access / Int'l Joint Research
  • [Journal Article] On the sequential polynomial type of modules2018

    • Author(s)
      Shiro Goto and Le Thanh Nhan
    • Journal Title

      Journal of the Mathematical Society of Japan

      Volume: 70 Issue: 1 Pages: 365-385

    • DOI

      10.2969/jmsj/07017535

    • NAID

      130006334147

    • ISSN
      0025-5645, 1881-1167, 1881-2333
    • Related Report
      2017 Research-status Report
    • Peer Reviewed / Open Access / Int'l Joint Research
  • [Journal Article] When is $R \ltimes I$ an almost Gorenstein local ring?2018

    • Author(s)
      Shiro Goto and Shinya Kumashiro
    • Journal Title

      Proc. Amer. Math. Soc.

      Volume: 146 Pages: 1431-1437

    • Related Report
      2017 Research-status Report
    • Peer Reviewed / Open Access
  • [Journal Article] When are the Rees algebras of parameter ideals almost Gorenstein graded rings?2017

    • Author(s)
      Shiro Goto, Mehran Rahimi, Naoki Taniguchi, and Hoang Le Truong
    • Journal Title

      Kyoto J. Math.

      Volume: 57 Issue: 3 Pages: 655-666

    • DOI

      10.1215/21562261-2017-0010

    • Related Report
      2017 Research-status Report
    • Peer Reviewed / Open Access / Int'l Joint Research
  • [Journal Article] On the sequential polynomial type of modules2017

    • Author(s)
      Shiro Goto, Le Thanh Nhan
    • Journal Title

      Journal of the Mathematical Society of Japan

      Volume: 印刷中

    • NAID

      130006334147

    • Related Report
      2016 Research-status Report
    • Peer Reviewed / Int'l Joint Research / Acknowledgement Compliant
  • [Journal Article] Hilbert polynomials of j-transforms2016

    • Author(s)
      Shiro Goto, Jooyoun Hong, Wolmer V. Vasconcelos
    • Journal Title

      Mathematical Proceedings of the Cambridge Philosophical Society

      Volume: 161 Issue: 2 Pages: 305-337

    • DOI

      10.1017/s0305004116000268

    • Related Report
      2016 Research-status Report
    • Peer Reviewed / Int'l Joint Research / Acknowledgement Compliant
  • [Journal Article] Ulrich ideals and modules of two-dimensional rational singularities2016

    • Author(s)
      Shiro Goto; Kazuho Ozeki; Ryo Takahashi; Kei-ichi Watanabe; Ken-ichi Yoshida
    • Journal Title

      Nagoya Math. J.

      Volume: 221 Issue: 1 Pages: 69-110

    • DOI

      10.1017/nmj.2015.1

    • Related Report
      2016 Research-status Report
    • Peer Reviewed / Open Access / Acknowledgement Compliant
  • [Journal Article] Extension closedness of syzygies and local Gorensteinness of commutative rings2016

    • Author(s)
      Shiro Goto, Ryo Takahashi
    • Journal Title

      Algebras and Representation Theory

      Volume: 19 Issue: 3 Pages: 511-521

    • DOI

      10.1007/s10468-015-9585-0

    • NAID

      120005889992

    • Related Report
      2016 Research-status Report
    • Peer Reviewed
  • [Journal Article] The almost Gorenstein Rees algebras over two-dimensional regular local rings2016

    • Author(s)
      Shiro Goto, Naoyuki Matsuoka, Naoki Taniguchi, and Ken-ichi Yoshida
    • Journal Title

      Journal of Pure and Applied Algebra

      Volume: 220 Issue: 10 Pages: 3425-3436

    • DOI

      10.1016/j.jpaa.2016.04.007

    • Related Report
      2016 Research-status Report
    • Peer Reviewed / Open Access / Acknowledgement Compliant
  • [Journal Article] Ulrich ideals and almost Gorenstein rings2016

    • Author(s)
      Shiro Goto, Ryo Takahashi, and Naoki Taniguchi
    • Journal Title

      Proceedings of the American Mathematical Society

      Volume: 144 Issue: 7 Pages: 2811-2823

    • DOI

      10.1090/proc/12970

    • Related Report
      2016 Research-status Report
    • Peer Reviewed / Acknowledgement Compliant
  • [Journal Article] The almost Gorenstein Rees algebras of parameters2016

    • Author(s)
      Shiro Goto, Naoyuki Matsuoka, Naoki Taniguchi, and Ken-ichi Yoshida
    • Journal Title

      Journal of Algebra

      Volume: 452 Pages: 263-278

    • DOI

      10.1016/j.jalgebra.2015.12.022

    • Related Report
      2016 Research-status Report
    • Peer Reviewed / Open Access / Int'l Joint Research / Acknowledgement Compliant
  • [Presentation] Efficient generation of ideals in core subalgebras of the polynomial ring $k[t]$ over a field $k$2019

    • Author(s)
      後藤四郎
    • Organizer
      第32回可換環論セミナー
    • Related Report
      2019 Research-status Report
    • Invited
  • [Presentation] One-dimensional 2-AGL rings - towards a stratification of Cohen-Macaulay rings2018

    • Author(s)
      後藤四郎
    • Organizer
      第40回可換環論シンポジウム
    • Related Report
      2018 Research-status Report
    • Int'l Joint Research / Invited
  • [Presentation] トレース加群からみたGorenstein性の解析2018

    • Author(s)
      後藤四郎,磯部遼太郎,神代真也
    • Organizer
      日本数学会2018年度秋季総合分科会
    • Related Report
      2018 Research-status Report
  • [Presentation] The structure of chains of Ulrich ideals in Cohen-Macaulay local rings of dimension one2018

    • Author(s)
      後藤四郎,磯部遼太郎,神代真也
    • Organizer
      日本数学会2018年度秋季総合分科会
    • Related Report
      2018 Research-status Report
  • [Presentation] Residually faithful modules and the Cohen-Macaulay type of idealizations II2018

    • Author(s)
      Shiro Goto
    • Organizer
      Mini-workshop on Commutative Algebra by and for young mathematicians, March 26~28, 2018, Sapporo Ekimae Satellite, Hokkaido University of Education.
    • Related Report
      2017 Research-status Report
    • Int'l Joint Research / Invited
  • [Presentation] イデアル化R × I のalmost Gorenstein 性2018

    • Author(s)
      後藤四郎,神代真也〇
    • Organizer
      日本数学会年会,東京大学 , 2018年3月20日
    • Related Report
      2017 Research-status Report
  • [Presentation] Characterization of generalized Gorenstein rings2018

    • Author(s)
      後藤四郎,谷口直樹,神代真也,磯部遼太郎〇
    • Organizer
      日本数学会年会,東京大学, 2018年3月20日
    • Related Report
      2017 Research-status Report
  • [Presentation] Almost Gorenstein rings - towards a stratification of Cohen-Macaulay rings2017

    • Author(s)
      Shiro Goto
    • Organizer
      The Prospects for Commutative Algebra, Osaka, July 10~14, 2017
    • Related Report
      2017 Research-status Report
    • Int'l Joint Research / Invited
  • [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
      2017 Research-status Report
    • Int'l Joint Research / Invited
  • [Presentation] Ulrich ideals and G-regular rings2017

    • Author(s)
      Shiro Goto
    • Organizer
      RIMS 共同研究(公開型),『第39回可換環論シンポジウム』,2017年 11 月 13 日 17 日.
    • Related Report
      2017 Research-status Report
    • Int'l Joint Research / Invited
  • [Presentation] On the almost Gorenstein property of Rees algebras of powers of parameter ideals2016

    • Author(s)
      Shiro Goto
    • Organizer
      第38回可換環論シンポジウム
    • Place of Presentation
      生産性国際交流センター
    • Year and Date
      2016-11-18
    • Related Report
      2016 Research-status Report
    • Int'l Joint Research
  • [Presentation] On the almost Gorenstein property of Rees algebras of powers of ideals2016

    • Author(s)
      Shiro Goto
    • Organizer
      Japan-Vietnam Workshop on Commutative Algebra, 2016
    • Place of Presentation
      Meiji University
    • Year and Date
      2016-09-19
    • Related Report
      2016 Research-status Report
    • Int'l Joint Research / Invited
  • [Book] 可換環論の勘どころ2017

    • Author(s)
      後藤四郎
    • Total Pages
      228
    • Publisher
      共立出版
    • ISBN
      9784320110731
    • Related Report
      2017 Research-status Report
  • [Funded Workshop] International workshop on commutative algebra (Thai Nguyen University of Education)2018

    • Related Report
      2017 Research-status Report

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Published: 2016-04-21   Modified: 2022-02-22  

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