Commutative algebra - towards a better understanding of non-Cohen-Macaulay rings

• Project/Area Number: 22540054
• Research Category: Grant-in-Aid for Scientific Research (C)
• Allocation Type: Single-year Grants
• Section: 一般
• Research Field: Algebra
• Research Institution: Meiji University
• Principal Investigator: GOTO Shiro 明治大学, 理工学部, 教授 (50060091)
• Co-Investigator(Kenkyū-buntansha): OZEKI Kazuho 山口大学, 大学院・理学研究科, 専任講師 (70445849) ; IAI Shinichiro 北海道教育大学, 教育学部, 准教授 (50333125)
• Project Period (FY): 2010 – 2012
• Project Status: Completed (Fiscal Year 2012)
• Budget Amount: ¥4,160,000 (Direct Cost: ¥3,200,000、Indirect Cost: ¥960,000); Fiscal Year 2012: ¥1,300,000 (Direct Cost: ¥1,000,000、Indirect Cost: ¥300,000); Fiscal Year 2011: ¥1,300,000 (Direct Cost: ¥1,000,000、Indirect Cost: ¥300,000); Fiscal Year 2010: ¥1,560,000 (Direct Cost: ¥1,200,000、Indirect Cost: ¥360,000)
• Keywords: 環論(含リー環) / 可換環論 / Cohen-Macaualy ring / Ulrich module / Ulrich ideal / Almost Gorenstein ring / 可換環 / Cohen-Macaulay環 / generalized Cohen-Macaulay環 / Hilbert函数とHilbert係数 / almost Gorenstein環 / Ulrichイデアル/加群 / Hilbert函数 / Hilbert係数 / 重複度 / Buchsbaum環

Research Abstract

The final goal of this research is to answer the naive but important question of what non-Cohen-Macaulay rings are. The present targets are the following. (1) The study of quasi-socle ideals Q : mq, where m is the maximal ideal in a given Noetherian local ring and Q a parameter ideal in it. (2) The analysis of the graded rings associated to I = Q : mq. (3) The study of Gorenstein property in the Rees algebras of powers of parameter ideals. The achievements are summarized into 15 papers.

Research Products

• [Journal Article] Hilbert coefficients and sequentiallyCohen-Macaulay modules (2013)
  • Author(s): N. Cuong, S. Goto, and H. L. Truong
  • Journal Title: Journal of Pure and Applied Algebra Volume: 217 Pages: 470-480
• [Journal Article] The equality $I^2 = ￥q I$ in sequentially Cohen-Macaulay rings (2013)
  • Author(s): N. Cuong, S. Goto, and H. L. Truong
  • Journal Title: J. Algebra Volume: 379 Pages: 50-79
• [Journal Article] Almost Gorenstein rings (2013)
  • Author(s): S. Goto, N. Matsuoka, and T. T. Phuong
  • Journal Title: Journal of Algebra Volume: 379 Pages: 355-381
• [Journal Article] The equality I^2 = q I in sequentially Cohen-Macaulay rings (2013)
  • Author(s): N. Cuong, S. Goto, and H. L. Truong
  • Journal Title: Journal of Algebra Volume: 379 Pages: 50-79
  • Peer Reviewed
• [Journal Article] Hilbert coefficients and sequentially Cohen-Macaulay modules (2013)
  • Author(s): N. Cuong, S. Goto, and H. L. Truong
  • Journal Title: Journal of Pure and Applied Algebra Volume: 217 Pages: 470-480
  • Peer Reviewed
• [Journal Article] The homology of parameter ideals (2012)
  • Author(s): S. Goto, J. Hong, and W. V. Vasconcelos
  • Journal Title: Journal of Algebra Volume: 368 Pages: 271-299
• [Journal Article] Negativity of the Chern Number of Parameter Ideals, Algebra and its applications, Recent developments (2012)
  • Author(s): S. Goto, M. Mandal, and J. Verma
  • Journal Title: Narosa (A. Beg and M. Ashraf (Editors)) Volume: 53-68 Pages: 53-68
• [Journal Article] Variation of the first Hilbert coefficients of parameters with a common integral closure (2012)
  • Author(s): L. Ghezzi, S. Goto, K. Ozeki, J. Hong, T.T. Phuong, and W. V. Vasconcelos
  • Journal Title: Journal of Pure and Applied Algebra Volume: 216 Pages: 216-232
• [Journal Article] The leading ideal of a complete intersectionof height two, Part I I I (2012)
  • Author(s): S. Goto, W. Heinzer, and M.-K. Kim
  • Journal Title: Comm. Algebra Volume: 40 Pages: 2523-2539
• [Journal Article] Variation of the first Hilbert coefficients of parameters with a common integral closure (2012)
  • Author(s): L. Ghezzi, S. Goto, K. Ozeki, J. Hong, T.T. Phuong, and W. V. Vasconcelos
  • Journal Title: Journal of Pure and Applied Algebra Volume: 216 Pages: 216-232
  • Peer Reviewed
• [Journal Article] The positivity of the first normalized Hilbert coefficients (2011)
  • Author(s): S. Goto, J. Hong, and M. Mandal
  • Journal Title: The Proceedings of the American Mathematical Society Volume: 139 Pages: 2399-2406
• [Journal Article] Quasi-socle ideals in Buchsbaum rings (2010)
  • Author(s): S. Goto, J. Horiuchi, and H. Sakurai
  • Journal Title: Nagoya Math. J Volume: 200 Pages: 93-106
• [Journal Article] Buchsbaumnessin local rings possessing constant first Hilbert coefficients of parameters (2010)
  • Author(s): S. Goto and K. Ozeki
  • Journal Title: Nagoya Math. J Volume: 199 Pages: 95-105
• [Journal Article] Cohen-Macaulayness versus the vanishing of the first Hilbert coefficient of parameter ideals (2010)
  • Author(s): L. Ghezzi, S. Goto, K. Ozeki, J. Hong, T.T. Phuong, and W. V. Vasconcelos
  • Journal Title: J. London Math. Soc. Volume: 81(2) Pages: 679-695
• [Journal Article] The structure of Sally modules - towards a theory of non-Cohen-Macaulay cases - (2010)
  • Author(s): S. Goto and K. Ozeki
  • Journal Title: J. Algebra Volume: 324 Pages: 2129-2165
• [Journal Article] Quasi-socle ideals and Goto numbers of parameters (2010)
  • Author(s): S. Goto, S. Kimura, T. T. Phuong, andH. L. Truong
  • Journal Title: Journal of Pure andApplied Algebra Volume: 214 Pages: 501-511
• [Journal Article] Sequentially Cohen-Macaulayness versus parametric decomposition of powers of parameter ideals (2010)
  • Author(s): S. Goto, Y. Horiuchi, and H. Sakurai
  • Journal Title: Journal of Commutative Algebra Volume: 2 Pages: 37-54
• [Journal Article] Cohen-Macaulayness versus the vanishing of the first Hilbert coefficient of parameter ideals (2010)
  • Author(s): L.Ghezzi, 後藤四郎, 大関一秀, J.Hong, T.T.Phuong, W.V.Vasconcelos
  • Journal Title: J.London Math.Soc.(2) Volume: 81 Pages: 679-695
  • Peer Reviewed
• [Journal Article] Buchsbaumness in local rings possessing constant first Hilbert coefficients of parameters (2010)
  • Author(s): 後藤四郎, 大関一秀
  • Journal Title: Nagoya Math.J. Volume: 199 Pages: 95-105
  • Peer Reviewed
• [Journal Article] Quasi-socle ideals in Buchsbaum rings (2010)
  • Author(s): 後藤四郎, 堀内淳, 桜井秀人
  • Journal Title: Nagoya Math.J. Volume: 200 Pages: 93-106
  • Peer Reviewed
• [Journal Article] The structure of Sally modules-towards a theory of non-Cohen-Macaulay cases (2010)
  • Author(s): 後藤四郎, 大関一秀
  • Journal Title: J.Algebra Volume: 324 Pages: 2129-2165
  • Peer Reviewed
• [Journal Article] Sequentially Cohen-Macaulayness versus parametric decomposition of powers of parameter ideals (2010)
  • Author(s): 後藤四郎, 堀内陽介, 桜井秀人
  • Journal Title: J.Commut.Algebra Volume: 2 Pages: 37-54
  • Peer Reviewed
• [Journal Article] Quasi-socle ideals and Goto numbers of parameters (2010)
  • Author(s): 後藤四郎, 木村了, T.T.Phuong, L.H.Truong
  • Journal Title: J.Pure Appl.Algebra Volume: 214 Pages: 501-511
  • Peer Reviewed
• [Journal Article] Uniform bounds for Hilbert coefficients of parameters, "Commutative Algebra and its Connections to Geometry (PASI 2009) (2009)
  • Author(s): S. Goto and K. Ozeki
  • Journal Title: Contemp. Math Volume: 555 Pages: 97-118
• [Presentation] Ulrich ideals and modules (2012)
  • Author(s): 後藤四郎
  • Organizer: Commutative Algebra Program of MSRI, Fall 2012 ～ March 2013, the Seminar organized by C. Huneke, C. Polini, and B. Ulrich
  • Place of Presentation: MSRI, Berkeley, USA.
  • Year and Date: 2012-09-17
• [Presentation] 巴系のホモロジー (2012)
  • Author(s): 後藤四郎
  • Organizer: 日本数学会年会・代数学分科会特別講演
  • Year and Date: 2012-03-28
• [Presentation] Ulrich ideals and modules, I (2012)
  • Author(s): 後藤四郎
  • Organizer: The 12-th ALGA Meeting dedicated to Steven Kleiman and Aron Simis on the occasion of their 70-th birthdays
  • Place of Presentation: IMPA, Rio de Janeiro, Brazil
• [Presentation] Almost Gorenstein rings of higher dimension (2012)
  • Organizer: The 3rd Bluegrass Algebra Conference
  • Place of Presentation: University of Kentucky, USA
• [Presentation] Ulrich ideals and modules (2012)
  • Author(s): 後藤四郎
  • Organizer: The 12-th ALGA Meeting dedicated to Steven Kleiman and Aron Simis on the occasion of their 70-th birthdays
  • Place of Presentation: IMPA, Rio de Janeiro, Brazil
  • Invited
• [Presentation] Almost Gorenstein rings-an attempt towards higher dimensional cases-- (2011)
  • Author(s): 後藤四郎
  • Organizer: The 7-th Japan-Vietnam Joint Seminar on Commutative Algebra
  • Place of Presentation: Quy Nhon University (Vietnam)
  • Year and Date: 2011-12-14
• [Presentation] Almost Gorenstein rings (2011)
  • Author(s): 後藤四郎, 松岡直之, Tran Thi Phuong
  • Organizer: 第33回可換環論シンポジウム
  • Place of Presentation: 浜名湖カリアック
  • Year and Date: 2011-11-09
• [Presentation] Almost Gorenstein rings (2011)
  • Author(s): 後藤四郎
  • Organizer: Juergen Herzogを囲む可換代数の集い
  • Place of Presentation: 大阪大学
  • Year and Date: 2011-06-15
• [Presentation] Almost Gorenstein rings (2011)
  • Author(s): 後藤四郎
  • Organizer: Midwest Commutative Algebra and Geometry Conference at PURDUE
  • Place of Presentation: Purdue University (USA)
  • Year and Date: 2011-05-23
• [Presentation] Almost Gorenstein rings -- an attempt towards higher dimensional cases -- (2011)
  • Author(s): 後藤四郎
  • Organizer: The 7-th Japan-Vietnam Joint Seminar on Commutative Algebra
  • Place of Presentation: Quy Nhon (Vietnam)
• [Presentation] Almost Gorenstein rings (2011)
  • Author(s): 後藤四郎
  • Organizer: Juergen Herzog を囲む可換代数の集い
  • Place of Presentation: 大阪大学
• [Presentation] Almost Gorenstein rings (2011)
  • Author(s): 後藤四郎
  • Organizer: Midwest Commutative Algebra and Geometry Conference at PURDUE
  • Place of Presentation: Purdue University (West Lafayette, USA)
• [Presentation] TranThiPhuong,AlmostGorensteinrings (2011)
  • Author(s): 後藤四郎,松岡直之
  • Organizer: 第33回可換環論シンポジウム
  • Place of Presentation: 浜名湖カリアック
• [Presentation] Ulrich加群の一般化 (2011)
  • Author(s): 後藤四郎,高橋亮,大関一秀
  • Organizer: 第33回可換環論シンポジウム
  • Place of Presentation: 浜名湖カリアック
• [Presentation] Study of Hilbert functions and coefficients of parameters I, I I, I I I, IV (2010)
  • Author(s): 後藤四郎
  • Organizer: The 5-th Japan-Vietnam Joint Seminar on Commutative Algebra, Institute ofMathematics Hanoi
  • Place of Presentation: Hanoi (Vietnam)
• [Presentation] Hilbert coefficients of parameters, 5 lectures, ICTP School and Workshop on Local Rings and Local Study of Algebraic Varieties (2010)
  • Author(s): 後藤四郎
  • Organizer: ICTP (The Abdus Salam International Center for Theoretical Physics)
  • Place of Presentation: Trieste, Italy
• [Book] 日本評論社 (2011)
  • Author(s): 後藤四郎,渡辺敬一
  • Publisher: 可換環論
• [Book] 可換環論 (2011)
  • Author(s): 後藤四郎, 渡辺敬一
  • Total Pages: 329
  • Publisher: 日本評論社