## Journal of Commutative Algebra

### Associated primes and syzygies of linked modules

#### 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

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

#### 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

#### 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.