可換環論：Ｕｌｒｉｃｈ加群とＵｌｒｉｃｈイデアルの構造解析

• Project/Area Number: 14J00126
• Research Category: Grant-in-Aid for JSPS Fellows
• Allocation Type: Single-year Grants
• Section: 国内
• Research Field: Algebra
• Research Institution: Meiji University
• Principal Investigator: 谷口 直樹 明治大学, 理工学部, 特別研究員(DC1)
• Project Period (FY): 2014-04-25 – 2017-03-31
• Project Status: Declined (Fiscal Year 2016)
• Budget Amount: ¥3,200,000 (Direct Cost: ¥3,200,000); Fiscal Year 2016: ¥1,000,000 (Direct Cost: ¥1,000,000); Fiscal Year 2015: ¥1,000,000 (Direct Cost: ¥1,000,000); Fiscal Year 2014: ¥1,200,000 (Direct Cost: ¥1,200,000)
• Keywords: Almost Gorenstien環 / Rees代数 / Ulrich ideal / Almost Gorenstein環 / 系列的Cohen-Macaulay加群

Outline of Annual Research Achievements

平成27年度は，almost Gorenstein環論のさらなる進化・発展を目指して，下記2課題に着手した。
課題① Rees代数のalmost Gorenstein性解析
課題② Almost Gorenstein環内におけるUlrich idealsの挙動解析
Rees代数のCohen-Macaulay性解析は，1979年の後藤四郎教授と下田保博教授の共同研究まで遡る。現在までに数多くの優れた研究者により理論の精密化が行われているが，中でもGorenstein Rees代数はかなり稀な存在であることが知られている。従って，Rees代数のalmost Gorenstein性解析はRees代数の環構造論としても新たな地平を開くと期待される。課題①は，後藤四郎教授，吉田健一教授，松岡直之専任講師との共同研究として実施され，得られた成果は2本の論文に纏められており，J. Algebra，J. Pure Appl. Algebraから発表されている。
Almost Gorenstein環の理論には，B. Ulrich教授が創始し，後藤四郎教授を中心に一般化されたUlrich加群とUlrich idealsの理論が不可分に関連している。既存の理論ではUlrich idealsの挙動を制御するためには，環のGorenstein性が不可欠であった。課題②はGorensteinとは限らないが，Gorenstein 環の次に優れた構造を持つalmost Gorenstein環内におけるUlrich idealsの挙動解析に挑戦するものである。課題②は，後藤四郎教授，高橋亮准教授との共同研究として実施され，得られた成果は論文に纏め，Proc. Amer. Math. Soc. から発表されている。

Research Progress Status

翌年度、交付申請を辞退するため、記入しない。

Strategy for Future Research Activity

翌年度、交付申請を辞退するため、記入しない。

Research Products

• [Journal Article] Sequentially Cohen-Macaulay Rees algebras (2016)
  • Author(s): Naoki Taniguchi, Tran Thi Phuong, Nguyen Thi Dung, and Tran Nguyen An
  • Journal Title: Journal of the Mathematical Society of Japan
  • Peer Reviewed / Int'l Joint Research / Acknowledgement Compliant
• [Journal Article] Topics on sequentially Cohen-Macaulay modules (2016)
  • Author(s): Naoki Taniguchi, Tran Thi Phuong, Nguyen Thi Dung, and Tran Nguyen An
  • Journal Title: Journal of Commutative Algebra
  • Peer Reviewed / Int'l Joint Research / Acknowledgement Compliant
• [Journal Article] The almost Gorenstein Rees algebras of parameters (2016)
  • 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
  • Peer Reviewed / Open Access / Int'l Joint Research / Acknowledgement Compliant
• [Journal Article] The almost Gorenstein Rees algebras over two-dimensional regular local rings (2016)
  • 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
  • Peer Reviewed / Open Access / Acknowledgement Compliant
• [Journal Article] Ulrich ideals and almost Gorenstein rings (2016)
  • 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
  • Peer Reviewed / Acknowledgement Compliant
• [Journal Article] Bounds for the first Hilbert coefficients of m-primary ideals (2015)
  • Author(s): Asuki Koura and Naoki Taniguchi
  • Journal Title: Tokyo Journal of Mathematics Volume: 38 Pages: 125-133
  • Peer Reviewed / Acknowledgement Compliant
• [Journal Article] Bounds for the first Hilbert coefficients of m-primary ideals (2015)
  • Author(s): Asuki Koura and Naoki Taniguchi
  • Journal Title: Tokyo journal of mathematics Volume: 38
  • Peer Reviewed / Acknowledgement Compliant
• [Journal Article] Huneke-Wiegand conjecture and change of rings (2015)
  • Author(s): Shiro Goto, Ryo Takahashi, Naoki Taniguchi and Hoang Le Truong
  • Journal Title: Journal of Algebra Volume: 422 Pages: 33-52
  • DOI: 10.1016/j.jalgebra.2014.09.006
  • Peer Reviewed / Acknowledgement Compliant
• [Journal Article] Almost Gorenstein rings -towards a theory of higher dimension (2015)
  • Author(s): Shiro Goto, Ryo Takahashi and Naoki Taniguchi
  • Journal Title: Journal of Pure and Applied Algebra Volume: 219 Issue: 7 Pages: 2666-2712
  • DOI: 10.1016/j.jpaa.2014.09.022
  • Peer Reviewed / Acknowledgement Compliant
• [Journal Article] Sequentially Cohen-Macaulay Rees modules (2015)
  • Author(s): Naoki Taniguchi
  • Journal Title: The Proceedings of the 36th Symposium on Commutative Algebra Pages: 10-21
  • Acknowledgement Compliant
• [Journal Article] The almost Gorenstein property of associated graded rings (2014)
  • Author(s): Naoki Taniguchi
  • Journal Title: 第26回可換環論セミナー報告集 Pages: 44-52
• [Presentation] Almost Gorenstein rings (2016)
  • Author(s): 谷口直樹
  • Organizer: 第61回代数学シンポジウム
  • Place of Presentation: 佐賀大学
  • Year and Date: 2016-09-07
  • Invited
• [Presentation] The almost Gorenstein Rees algebras of contracted ideals (2016)
  • Author(s): 後藤四郎, 松岡直之, 谷口直樹, 吉田健一
  • Organizer: 日本数学会2016年度年会
  • Place of Presentation: 筑波大学
  • Year and Date: 2016-03-16
• [Presentation] Ulrich ideals and almost Gorenstein rings (2016)
  • Author(s): 谷口直樹
  • Organizer: 第28回可換環論セミナー
  • Place of Presentation: 岡山理科大学
  • Year and Date: 2016-01-24
• [Presentation] The almost Gorenstein Rees algebras of contracted ideals (2016)
  • Author(s): 後藤四郎, 松岡直之, 谷口直樹, 吉田健一
  • Organizer: 第28回可換環論セミナー
  • Place of Presentation: 岡山理科大学
  • Year and Date: 2016-01-24
• [Presentation] Ulrich ideals and almost Gorenstein rings (2016)
  • Author(s): Naoki Taniguchi
  • Organizer: Workshop on Commutative Algebra -Part II, Local algebra and Representation theory-
  • Place of Presentation: VIASM Ha Long Bay (Vietnam)
  • Year and Date: 2016-01-03
  • Int'l Joint Research
• [Presentation] Almost Gorenstein homogeneous rings (2015)
  • Author(s): 谷口直樹
  • Organizer: 名古屋大学環論表現論セミナー
  • Place of Presentation: 名古屋大学
  • Year and Date: 2015-12-14
  • Invited
• [Presentation] Almost Gorenstein rings (2015)
  • Author(s): 谷口直樹
  • Organizer: 第2回グレブナーセミナー
  • Place of Presentation: 関西学院大学
  • Year and Date: 2015-11-24
  • Invited
• [Presentation] The almost Gorenstein Rees algebras of socle ideals (2015)
  • Author(s): Shiro Goto, Naoyuki Matsuoka, Naoki Taniguchi, and Ken-ichi Yoshida
  • Organizer: The 37th Japan Symposium on Commutative Algebra
  • Place of Presentation: Kurashiki Seaside Hotel
  • Year and Date: 2015-11-18
  • Int'l Joint Research
• [Presentation] Almost Gorenstein Rees algebras of some integrally closed ideals (2015)
  • Author(s): Ken-ichi Yoshida, Shiro Goto, Naoki Taniguchi, and Naoyuki Matsuoka
  • Organizer: The 37th Japan Symposium on Commutative Algebra
  • Place of Presentation: Kurashiki Seaside Hotel
  • Year and Date: 2015-11-18
  • Int'l Joint Research
• [Presentation] Determinantal ideals and the straightening law (2015)
  • Author(s): 谷口直樹
  • Organizer: 第12回可換環論サマースクール
  • Place of Presentation: 関西学院大学
  • Year and Date: 2015-09-24
  • Invited
• [Presentation] The almost Gorenstein Rees algebras of parameters (2015)
  • Author(s): Naoki Taniguchi
  • Organizer: The JV seminar on Commutative Algebra -by and for young mathematicians-
  • Place of Presentation: Meiji University
  • Year and Date: 2015-09-18
  • Int'l Joint Research
• [Presentation] ２次元正規特異点の p_g-idealのRees代数のAlmost Gorenstein性 (2015)
  • Author(s): 吉田健一，後藤四郎，谷口直樹
  • Organizer: 日本数学会2015年度秋季総合分科会
  • Place of Presentation: 京都産業大学
  • Year and Date: 2015-09-13
• [Presentation] The almost Gorenstein Rees algebras of socle ideals (2015)
  • Author(s): 後藤四郎，松岡直之，谷口直樹，吉田健一
  • Organizer: 日本数学会2015年度秋季総合分科会
  • Place of Presentation: 京都産業大学
  • Year and Date: 2015-09-13
• [Presentation] Almost Gorenstein Rees algebras (2015)
  • Author(s): 吉田健一，後藤四郎，谷口直樹，松岡直之
  • Organizer: 第48回環論および表現論シンポジウム
  • Place of Presentation: 名古屋大学
  • Year and Date: 2015-09-07
  • Int'l Joint Research
• [Presentation] The almost Gorenstein Rees algebras of socle ideals (2015)
  • Author(s): 谷口直樹
  • Organizer: 第10回岡山可換代数表現セミナー
  • Place of Presentation: 岡山大学
  • Year and Date: 2015-08-03
  • Invited
• [Presentation] Sequentially Cohen-Macaulay Rees modules (2015)
  • Author(s): 谷口直樹
  • Organizer: 日本数学会2015年度年会
  • Place of Presentation: 明治大学
  • Year and Date: 2015-03-21 – 2015-03-24
• [Presentation] Almost Gorenstein rings of higher dimension (2015)
  • Author(s): Naoki Taniguchi
  • Organizer: AMS Special Session on Algebra and Representation theory
  • Place of Presentation: Georgetown University
  • Year and Date: 2015-03-07 – 2015-03-08
  • Invited
• [Presentation] Sequentially Cohen-Macaulay Rees modules (2015)
  • Author(s): 谷口直樹
  • Organizer: 第27回可換環論セミナー
  • Place of Presentation: 静岡大学
  • Year and Date: 2015-01-25 – 2015-01-28
• [Presentation] Sequentially Cohen-Macaulay Rees modules (2014)
  • Author(s): Naoki Taniguchi
  • Organizer: The 36th Symposium on Commutative Algebra
  • Place of Presentation: International Productivity Center
  • Year and Date: 2014-11-21 – 2014-11-25
• [Presentation] Matlis双対性と局所双対性 (2014)
  • Author(s): 谷口直樹
  • Organizer: 第11回可換環論サマースクール
  • Place of Presentation: 奈良工業高等専門学校
  • Year and Date: 2014-09-04 – 2014-09-07
• [Presentation] Sequentially Cohen-Macaulay Rees modules (2014)
  • Author(s): Naoki Taniguchi
  • Organizer: Commutative Algebra and Singularity Theory 2014
  • Place of Presentation: Tateyama Kokurai Hotel
  • Year and Date: 2014-07-28 – 2014-08-01
• [Remarks] Naoki Taniguchiのweb page
  • URL: http://www.isc.meiji.ac.jp/~taniguchi/
• [Remarks] Naoki TANIGUCHIのweb page
  • URL: http://www.isc.meiji.ac.jp/~taniguci/