Structure of almost Gorenstein rings and Ulrich modules

• Project/Area Number: 25400051
• 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, 理工学部, 教授 (50060091)
• Co-Investigator(Kenkyū-buntansha): IAI Shin-ichiro 北海道教育大学, 教育学部, 准教授 (50333125)
• Co-Investigator(Renkei-kenkyūsha): TRAN Thi Phuong 明治大学, 研究知財戦略機構, ポストドクター (00649824)
• Project Period (FY): 2013-04-01 – 2016-03-31
• Project Status: Completed (Fiscal Year 2015)
• Budget Amount: ¥5,070,000 (Direct Cost: ¥3,900,000、Indirect Cost: ¥1,170,000); Fiscal Year 2015: ¥1,690,000 (Direct Cost: ¥1,300,000、Indirect Cost: ¥390,000); Fiscal Year 2014: ¥1,690,000 (Direct Cost: ¥1,300,000、Indirect Cost: ¥390,000); Fiscal Year 2013: ¥1,690,000 (Direct Cost: ¥1,300,000、Indirect Cost: ¥390,000)
• Keywords: Cohen-Macaulay ring / Canonical module / Almost Gorenstein ring / Ulrich module / Gorenstein ring / 可換環論 / Ａｌｍｏｓｔ Ｇｏｒｅｎｓｔｅｉｎ環 / canonical module / Cohen-Macaulay環 / 代数学 / almost Gorenstein ring / Ulrich ideal

Outline of Final Research Achievements

The notion of almost Gorenstein local ring given by V. Barucci and R. Froeberg for one-dimensional analytically unramified Noetherian local rings is generalized for arbitrary Noetherian local rings and a basic theory is developed. The higher dimensional definition is safely introduced also for Noetherian local/graded rings. The theory of almost Gorenstein Rees algebras associated to ideals is well developed and there are obtained satisfactory results in the case where the base rings are regular local rings or the case where the ideals are generated by subsystems of parameters in Cohen-Macaulay local rings.

Research Products

• [Int'l Joint Research] VAST/Thai Nguyen University/Ton Duc Thang University(ベトナム)
• [Int'l Joint Research] Southern Connecticut State University/Rutgers University/New York City College of Technology-Cuny(米国)
• [Journal Article] Sectional genera of parameter ideals (2016)
  • Author(s): Shiro Goto and Kazuho Ozeki
  • Journal Title: Journal of Algebra Volume: 446 Pages: 58-76
  • DOI: 10.1016/j.jalgebra.2015.09.005
  • Peer Reviewed
• [Journal Article] Ulrich ideals and almost Gorenstein rings (2016)
  • Author(s): S. Goto, R. Takahashi, and N. Taniguchi
  • Journal Title: Proc. Amer. Math. Soc. Volume: 144 Pages: 2811-2823
  • NAID: 120005844131
  • Peer Reviewed
• [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] 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] 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] The second Hilbert coefficients and the homological torsions of parameters (2015)
  • Author(s): Shiro Goto, Kazuho Ozeki
  • Journal Title: Journal of Pure and Applied Algebra Volume: 219 Issue: 5 Pages: 1685-1703
  • DOI: 10.1016/j.jpaa.2014.07.006
  • Peer Reviewed
• [Journal Article] On the cofiniteness of generalized local cohomology modules (2015)
  • Author(s): Nguyen Tu Cuong, Shiro Goto, and Nguyen Van Hoang
  • Journal Title: Kyoto J. Math. Volume: 55 Pages: 169-185
  • Peer Reviewed / Int'l Joint Research
• [Journal Article] The Chern and Euler coefficients of modules (2015)
  • Author(s): L. Ghezzi, S. Goto, J. Hong, K. Ozeki, T. T. Phuong, and V. W. Vasconcelos
  • Journal Title: Acta Math Vietnam Volume: 40 Pages: 37-60
  • Peer Reviewed / Int'l Joint Research
• [Journal Article] The first Euler characteristics versus the homological degrees (2014)
  • Author(s): S. Goto and K. Ozeki
  • Journal Title: The Proceedings of the 2012 Conference at IMPA，Bull Braz Math Soc Volume: 45 Issue: 4 Pages: 679-709
  • DOI: 10.1007/s00574-014-0069-3
  • Peer Reviewed / Acknowledgement Compliant
• [Journal Article] Ulrich ideals and modules (2014)
  • Author(s): S. Goto, K. Ozeki, R. Takahashi, K. Watanabe, K. Yoshida
  • Journal Title: Math. Proc. Cambridge Philos. Soc. Volume: 156 Issue: 1 Pages: 137-166
  • DOI: 10.1017/s0305004113000571
  • Peer Reviewed
• [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
  • DOI: 10.1016/j.jalgebra.2013.01.025
  • Peer Reviewed
• [Journal Article] J. Sally's question and a conjecture of Y. Shimoda (2013)
  • Author(s): S. Goto, L. O'Carroll, and F. Planas-Vilanova
  • Journal Title: Nagoya Math. J. Volume: 211 Pages: 137-161
  • Peer Reviewed
• [Journal Article] The equality $I^2 = \q I$ in sequentially Cohen-Macaulay rings (2013)
  • Author(s): N. Cuong, S. Goto
  • Journal Title: J. Algebra Volume: 379 Pages: 50-79
  • DOI: 10.1016/j.jalgebra.2012.12.007
  • 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 Issue: 3 Pages: 470-480
  • DOI: 10.1016/j.jpaa.2012.06.026
  • Peer Reviewed
• [Journal Article] Variation of Hilbert Coefficients (2013)
  • Author(s): L. Ghezzi, S. Goto, J. Hong, and W. V. Vasconcelos
  • Journal Title: Proc. Amer. Math. Soc. Volume: 141 Pages: 3037-3048
• [Presentation] When are the Rees algebras of parameter ideals almost Gorenstein graded rings? (2016)
  • Author(s): S. Goto
  • Organizer: Workshop on Commutative Algebra – Part II, Local algebra and Representation theory
  • Place of Presentation: VIASM’s villa, Ha Long, Vietnam
  • Year and Date: 2016-01-03
  • Int'l Joint Research / Invited
• [Presentation] The sectional genera and the second Hilbert coefficients of parameter ideals (2014)
  • Author(s): 後藤四郎，大関一秀
  • Organizer: 第36回可換環論シンポジウム
  • Place of Presentation: IPC 生産性国際交流センター
  • Year and Date: 2014-11-21 – 2014-11-25
• [Presentation] Almost Gorenstein rings – towards a theory of higher dimension (2014)
  • Author(s): 後藤四郎
  • Organizer: 24-th Iranian Algebra Seminar (dedicated to Professor Zakeri)
  • Place of Presentation: Kharazmi University, Faculty of Mathematical Sciences and Computer, Karaj, Iran
  • Year and Date: 2014-11-12 – 2014-11-13
  • Invited
• [Presentation] Huneke-Wiegand conjecture and change of rings (2014)
  • Author(s): 後藤四郎
  • Organizer: 11th Seminar on Commutative Algebra and Related Topics
  • Place of Presentation: Institute for Research in Fundamental Sciences (Iran)
  • Year and Date: 2014-11-10 – 2014-11-11
  • Invited
• [Presentation] Ulrich ideals and modules (2014)
  • Author(s): 後藤四郎
  • Organizer: 11th Seminar on Commutative Algebra and Related Topics
  • Place of Presentation: Institute for Research in Fundamental Sciences (Iran)
  • Year and Date: 2014-11-10 – 2014-11-11
  • Invited
• [Presentation] Almost Gorenstein rings -towards a theory of higher dimension (2014)
  • Author(s): 後藤四郎
  • Organizer: Commutative Algebra and Singularity Theory (dedicated to Kei-ichi Watanabe on occasion of his 70-th birthday)
  • Place of Presentation: 立山国際ホテル
  • Year and Date: 2014-07-28 – 2014-08-01
  • Invited
• [Presentation] Huneke-Wiegand conjecture (2013)
  • Author(s): 後藤四郎
  • Organizer: Commutative algebra and its interactions with algebraic geometry
  • Place of Presentation: CIRM (Luminy，フランス)
  • Invited
• [Presentation] Ulrich ideals and modules (2013)
  • Author(s): 後藤四郎
  • Organizer: 第58回代数学分科会シンポジウム
  • Place of Presentation: 広島大学
  • Invited
• [Presentation] On the Hilbert polynomials of j-transforms (2013)
  • Author(s): 後藤四郎
  • Organizer: International conference on commutative algebra and its interaction to algebraic geometry and combinatorics
  • Place of Presentation: IMVAST (Hanoi, Vietnam)
  • Invited
• [Remarks] 後藤研究室
  • URL: http://www.isc.meiji.ac.jp/~math/labo/goto.html
• [Remarks] 後藤研究室
  • URL: http://www.commalg.jp/~goto/message/20140408/
• [Remarks] 後藤研究室
  • URL: http://www.math.meiji.ac.jp/~goto/
• [Funded Workshop] The 8th Japan-Vietnam Joint Seminar on Commutative Algebra (2016)
  • Place of Presentation: VIASM’s villa, Ha Long, Vietnam
  • Year and Date: 2016-03-21
• [Funded Workshop] School on Commutative Algebra at Thai Nguyen University (2016)
  • Place of Presentation: Thai Nguyen University, Thai Nguyen, Vietnam
  • Year and Date: 2016-03-17
• [Funded Workshop] Workshop on Commutative Algebra – Part II, Local algebra and Representation theory –, VIASM’s villa, Ha Long, Vietnam (2016)
  • Place of Presentation: VIASM’s villa, Ha Long, Vietnam
  • Year and Date: 2016-01-03
• [Funded Workshop] Workshop on Commutative Algebra – Part I, Combinatorial aspects in Commutative Algebra–, VIASM’s villa, Ha Long, Vietnam (2015)
  • Place of Presentation: VIASM’s villa, Ha Long, Vietnam
  • Year and Date: 2015-12-26
• [Funded Workshop] The JV Seminar on Commutative Algebra (2015)
  • Place of Presentation: 明治大学生田校舎（理工学部）第ＩＩ校舎６号館
  • Year and Date: 2015-09-18