Journal of Commutative Algebra

Characterizations of regular local rings via syzygy modules of the residue field

Dipankar Ghosh, Anjan Gupta, and Tony J. Puthenpurakal

Full-text: Access denied (no subscription detected)

We're sorry, but we are unable to provide you with the full text of this article because we are not able to identify you as a subscriber. If you have a personal subscription to this journal, then please login. If you are already logged in, then you may need to update your profile to register your subscription. Read more about accessing full-text

Abstract

Let $R$ be a commutative Noetherian local ring with residue field $k$. We show that, if a finite direct sum of syzygy modules of $k$ maps onto `a semidualizing module' or `a non-zero maximal Cohen-Macaulay module of finite injective dimension,' then $R$ is regular. We also prove that $R$ is regular if and only if some syzygy module of $k$ has a non-zero direct summand of finite injective dimension.

Article information

Source
J. Commut. Algebra, Volume 10, Number 3 (2018), 327-337.

Dates
First available in Project Euclid: 9 November 2018

Permanent link to this document
https://projecteuclid.org/euclid.jca/1541754162

Digital Object Identifier
doi:10.1216/JCA-2018-10-3-327

Mathematical Reviews number (MathSciNet)
MR3874655

Zentralblatt MATH identifier
06976318

Subjects
Primary: 13D02: Syzygies, resolutions, complexes
Secondary: 13D05: Homological dimension 13H05: Regular local rings

Keywords
Regular local rings syzygy and cosyzygy modules semi-dualizing modules injective dimension

Citation

Ghosh, Dipankar; Gupta, Anjan; Puthenpurakal, Tony J. Characterizations of regular local rings via syzygy modules of the residue field. J. Commut. Algebra 10 (2018), no. 3, 327--337. doi:10.1216/JCA-2018-10-3-327. https://projecteuclid.org/euclid.jca/1541754162


Export citation

References

  • L.L. Avramov, Modules with extremal resolutions, Math. Res. Lett. 3 (1996), 319–328.
  • W. Bruns and J. Herzog, Cohen-Macaulay rings, Cambr. Stud. Adv. Math. 39 (1998).
  • S.P. Dutta, Syzygies and homological conjectures, in Commutative algebra, Springer, Berkeley, CA, 1987.
  • E.S. Golod, $G$-dimension and generalized perfect ideals, in Algebraic geometry and its applications, Trudy Mat. Inst. Steklov. 165 (1984), 62–66.
  • T.Y. Lam, A first course in noncommutative rings, Springer-Verlag, New York, 2001.
  • G. Levin and W.V. Vasconcelos, Homological dimensions and Macaulay rings, Pacific J. Math 25 (1968), 315–323.
  • A. Martsinkovsky, A remarkable property of the (co) syzygy modules of the residue field of a nonregular local ring, J. Pure Appl. Alg. 110 (1996), 9–13.
  • P. Roberts, Le théorème d'intersection, C.R. Acad. Sci. Paris 304 (1987), 177–180.
  • R. Takahashi, Syzygy modules with semidualizing or $G$-projective summands, J. Algebra 295 (2006), 179–194.