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Ulrich ideals and almost Gorenstein rings


Authors: Shiro Goto, Ryo Takahashi and Naoki Taniguchi
Journal: Proc. Amer. Math. Soc. 144 (2016), 2811-2823
MSC (2010): Primary 13H10, 13H15, 13D07
DOI: https://doi.org/10.1090/proc/12970
Published electronically: December 3, 2015
MathSciNet review: 3487216
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Abstract: The structure of the complex $ \mathrm {{\bf R}Hom}_R(R/I,R)$ is explored for an Ulrich ideal $ I$ in a Cohen-Macaulay local ring $ R$. As a consequence, it is proved that in a one-dimensional almost Gorenstein but non-Gorenstein local ring, the only possible Ulrich ideal is the maximal ideal. It is also studied when Ulrich ideals have the same minimal number of generators.


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  • [1] T. Aihara; R. Takahashi, Generators and dimensions of derived categories, Comm. Algebra 43 (2015), no 11, 5003-5029, DOI: 10.1080/00927872.2014.957384.
  • [2] Luchezar L. Avramov and Alex Martsinkovsky, Absolute, relative, and Tate cohomology of modules of finite Gorenstein dimension, Proc. London Math. Soc. (3) 85 (2002), no. 2, 393-440. MR 1912056 (2003g:16009), https://doi.org/10.1112/S0024611502013527
  • [3] Valentina Barucci and Ralf Fröberg, One-dimensional almost Gorenstein rings, J. Algebra 188 (1997), no. 2, 418-442. MR 1435367 (98a:13033), https://doi.org/10.1006/jabr.1996.6837
  • [4] Winfried Bruns and Jürgen Herzog, Cohen-Macaulay rings, Cambridge Studies in Advanced Mathematics, vol. 39, Cambridge University Press, Cambridge, 1993. MR 1251956 (95h:13020)
  • [5] Lars Winther Christensen, Gorenstein dimensions, Lecture Notes in Mathematics, vol. 1747, Springer-Verlag, Berlin, 2000. MR 1799866 (2002e:13032)
  • [6] Shiro Goto, Naoyuki Matsuoka, and Tran Thi Phuong, Almost Gorenstein rings, J. Algebra 379 (2013), 355-381. MR 3019262, https://doi.org/10.1016/j.jalgebra.2013.01.025
  • [7] Shiro Goto, Kazuho Ozeki, Ryo Takahashi, Kei-Ichi Watanabe, and Ken-Ichi Yoshida, Ulrich ideals and modules, Math. Proc. Cambridge Philos. Soc. 156 (2014), no. 1, 137-166. MR 3144215, https://doi.org/10.1017/S0305004113000571
  • [8] S. Goto, K. Ozeki, R. Takahashi, K.-i. Watanabe, and K.-i. Yoshida, Ulrich ideals and modules over two-dimensional rational singularities, Nagoya Math. J. (to appear).
  • [9] Shiro Goto, Ryo Takahashi, and Naoki Taniguchi, Almost Gorenstein rings--towards a theory of higher dimension, J. Pure Appl. Algebra 219 (2015), no. 7, 2666-2712. MR 3313502, https://doi.org/10.1016/j.jpaa.2014.09.022
  • [10] S. Goto, N. Matsuoka, N. Taniguchi, and K.-i. Yoshida, The almost Gorenstein Rees algebras over two-dimensional regular local rings, Preprint (2015).
  • [11] Jürgen Herzog and Ernst Kunz, Der kanonische Modul eines Cohen-Macaulay-Rings, Lecture Notes in Mathematics, Vol. 238, Springer-Verlag, Berlin-New York, 1971. Seminar über die lokale Kohomologietheorie von Grothendieck, Universität Regensburg, Wintersemester 1970/1971. MR 0412177 (54 #304)
  • [12] R. Takahashi, Some characterizations of Gorenstein local rings in terms of G-dimension, Acta Math. Hungar. 104 (2004), no. 4, 315-322. MR 2082781 (2005f:13015), https://doi.org/10.1023/B:AMHU.0000036291.12706.46
  • [13] Ryo Takahashi, On $ G$-regular local rings, Comm. Algebra 36 (2008), no. 12, 4472-4491. MR 2473342 (2009i:13040), https://doi.org/10.1080/00927870802179602
  • [14] Yuji Yoshino, Modules of G-dimension zero over local rings with the cube of maximal ideal being zero, Commutative algebra, singularities and computer algebra (Sinaia, 2002), NATO Sci. Ser. II Math. Phys. Chem., vol. 115, Kluwer Acad. Publ., Dordrecht, 2003, pp. 255-273. MR 2030276 (2004m:13039)

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Additional Information

Shiro Goto
Affiliation: Department of Mathematics, School of Science and Technology, Meiji University, 1-1-1 Higashi-mita, Tama-ku, Kawasaki 214-8571, Japan
Email: goto@math.meiji.ac.jp

Ryo Takahashi
Affiliation: Graduate School of Mathematics, Nagoya University, Furocho, Chikusa-ku, Nagoya 464-8602, Japan
Email: takahashi@math.nagoya-u.ac.jp

Naoki Taniguchi
Affiliation: Department of Mathematics, School of Science and Technology, Meiji University, 1-1-1 Higashi-mita, Tama-ku, Kawasaki 214-8571, Japan
Email: taniguti@math.meiji.ac.jp

DOI: https://doi.org/10.1090/proc/12970
Keywords: Almost Gorenstein ring, Cohen--Macaulay ring, Ulrich ideal
Received by editor(s): July 16, 2015
Received by editor(s) in revised form: September 3, 2015
Published electronically: December 3, 2015
Additional Notes: The first author was partially supported by JSPS Grant-in-Aid for Scientific Research 25400051
The second author was partially supported by JSPS Grant-in-Aid for Scientific Research 25400038
The third author was partially supported by Grant-in-Aid for JSPS Fellows 26-126 and by JSPS Research Fellow
Communicated by: Irena Peeva
Article copyright: © Copyright 2015 American Mathematical Society

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