Journal of Commutative Algebra

Associated primes and syzygies of linked modules

Olgur Celikbas, Mohammad T. Dibaei, Mohsen Gheibi, Arash Sadeghi, and Ryo Takahashi

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

Motivated by the notion of geometrically linked ideals, we show that over a Gorenstein local ring $R$, if a Cohen-Macaulay $R$-module $M$ of grade $g$ is linked to an $R$-module $N$ by a Gorenstein ideal $c$, such that $\mathrm {Ass}_R(M)$ and $\mathrm {Ass}_R(N)$ are disjoint, then $M\otimes _RN$ is isomorphic to direct sum of copies of $R/\mathfrak {a}$, where $\mathfrak {a}$ is a Gorenstein ideal of $R$ of grade $g+1$. We give a criterion for the depth of a local ring $(R,\mathfrak {m},k)$ in terms of the homological dimensions of the modules linked to the syzygies of the residue field $k$. As a result we characterize a local ring $(R,\mathfrak {m},k)$ in terms of the homological dimensions of the modules linked to the syzygies of $k$.

Article information

Source
J. Commut. Algebra, Volume 11, Number 3 (2019), 301-323.

Dates
First available in Project Euclid: 3 December 2019

Permanent link to this document
https://projecteuclid.org/euclid.jca/1575363814

Digital Object Identifier
doi:10.1216/JCA-2019-11-3-301

Mathematical Reviews number (MathSciNet)
MR4038052

Zentralblatt MATH identifier
07140749

Subjects
Primary: 13D07: Homological functors on modules (Tor, Ext, etc.)
Secondary: 13D02: Syzygies, resolutions, complexes 13C40: Linkage, complete intersections and determinantal ideals [See also 14M06, 14M10, 14M12] 13D05: Homological dimension

Keywords
Linkage of modules geometrically linked ideals Huneke-Wiegand conjecture torsion-free module vanishing of $\mathrm {Tor}$ homological dimension.

Citation

Celikbas, Olgur; Dibaei, Mohammad T.; Gheibi, Mohsen; Sadeghi, Arash; Takahashi, Ryo. Associated primes and syzygies of linked modules. J. Commut. Algebra 11 (2019), no. 3, 301--323. doi:10.1216/JCA-2019-11-3-301. https://projecteuclid.org/euclid.jca/1575363814


Export citation

References

  • F.W. Anderson and K.R. Fuller, Rings and categories of modules, Second edition, Springer, 1992.
  • M. Auslander and M. Bridger, Stable module theory, Mem. Amer. Math. Soc. 94 (1969).
  • L.L. Avramov, Infinite free resolution, in Six lectures on commutative algebra (Bellaterra, 1996), 1–118, Progr. Math. 166, Birkhäuser, Basel, 1998.
  • L.L. Avramov and H.-B. Foxby, Locally Gorenstein homomorphisms, Amer. J. Math. \bbb114 (1992), 1007–1047.
  • L.L. Avramov, V.N. Gasharov and I.V. Peeva, Complete intersection dimension, Publ. Math. I.H.E.S. \bbb86 (1997), 67–114.
  • W. Bruns and J. Herzog, Cohen-Macaulay rings, Cambridge Studies in Advanced Mathematics \bf39, Cambridge University Press, 1993.
  • O. Celikbas, A. Sadeghi and R. Takahashi, Bounds on depth of tensor products of modules, J. Pure Appl. Algebra \bbb219 (2015), no. 5, 1670–1684.
  • L.W. Christensen, Gorenstein dimensions, Lecture Notes in Mathematics \bf1747, Springer, Berlin, 2000.
  • M.T. Dibaei, M. Gheibi, S.H. Hassanzadeh and A. Sadeghi, Linkage of modules over Cohen-Macaulay rings, J. Algebra, \bbb335 (2011), 177–187.
  • M.T. Dibaei and A. Sadeghi, Linkage of finite Gorenstein dimension modules, J. Algebra \bbb376 (2013), 261–278.
  • M.T. Dibaei and A. Sadeghi, Linkage of modules and the Serre conditions, J. Pure Appl. Algebra \bbb219 (2015), 4458–4478.
  • S.P. Dutta, Syzygies and homological conjectures, in Commutative algebra (Berkeley, 1987), 139–156, Math. Sci. Res. Inst. Publ. \bbb15, Springer, New York, 1989.
  • E.G. Evans and P. Griffith, Syzygies, London Math. Soc. Lecture Note Series 106, Cambridge University Press, 1985.
  • H.B. Foxby, Gorenstein modules and related modules, Math. Scand. \bbb31 (1972), 267–284 (1973).
  • P.A. Garía-Sánchez and M.J. Leamer, Huneke-Wiegand conjecture for complete intersection numerical semigroup rings, J. Algebra \bbb391 (2013), 114–124.
  • E.S. Golod, G-dimension and generalized perfect ideals, Trudy Mat. Inst. Steklov. \bbb165 (1984), 62–66; English transl. in Proc. Steklov Inst. Math. \bbb165 (1985).
  • S. Goto, R. Takahashi, N. Taniguchi and H.L. Troung, Huneke-Wiegand conjecture and change of rings, J. Algebra, \bbb422 (2015), 33–52.
  • H. Holm, Rings with finite Gorenstein injective dimension, Proc. Amer. Math. Soc. \bbb132 (2004), no. 5, 1279–1283.
  • C. Huneke, Almost Complete Intersections and Factorial Rings, J. Algebra \bbb71 (1981), 179–188.
  • C. Huneke, Linkage and the Koszul homology of ideals, Amer.J. Math. \bbb104 (1982), no. 5, 1043–1062.
  • C. Huneke and R. Wiegand, Tensor products of modules and the rigidity of Tor, Math. Ann. \bbb299 (1994), 449–476.
  • K.I. Iima and R. Takahashi, Perfect linkage of Cohen-Macaulay modules over Cohen-Macaulay rings, J. Algebra \bbb458 (2016), 134–155.
  • M.R. Johnson, Linkage and sums of ideals, Trans. Amer. Math. Soc. \bbb350 (1998), no. 5, 1913–1930.
  • G.J. Leuschke and R. Wiegand, Cohen-Macaulay Representations, Mathematical Surveys and Monographs \bbb181, Amer. Math. Soc., 2012.
  • A. Martsinkovsky and J.R. Strooker, Linkage of modules, J. Algebra \bbb271 (2004), 587–626.
  • U. Nagel, Liaison classes of modules, J. Algebra \bbb284 (2005), no. 1, 236–272.
  • C. Peskine and L. Szpiro, Liasion des variétés algébriques I, Inv. Math. \bbb26 (1974), 271–302.
  • P. Schenzel, Notes on liaison and duality , J. Math. Kyoto Univ. \bbb22 (1982/83), no. 3, 485–498.
  • R. Takahashi, Syzygy modules with semidualizing or G-projective summands J. Algebra \bbb295 (2006), 179–194.
  • B. Ulrich, Sums of linked ideals, Trans. Amer. Math. Soc. \bbb318 (1990), no. 1, 1–42.
  • Y. Yoshino and S. Isogawa, Linkage of Cohen-Macaulay modules over a Gorenstein ring, J. Pure Appl. Algebra \bbb149 (2000), no. 3, 305–318.
  • M. Zargar, O. Celikbas, M. Gheibi and A. Sadeghi, Homological dimensions of rigid modules, Kyoto J. Math. \bf58 (2018), no. 3, 639–669.