Journal of Commutative Algebra

Regularity and linearity defect of modules over local rings

Rasoul Ahangari Maleki and Maria Evelina Rossi

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Abstract

Given a finitely generated module $M$ over a commutative local ring (or a standard graded $k$-algebra) $(R,\m,k)$, we detect its complexity in terms of numerical invariants coming from suitable $\m$-stable filtrations $\mathbb{M}$ on $M.$ We study the Castelnuovo-Mumford regularity of $gr_{\mathbb{M}}(M) $ and the linearity defect of $M, $ denoted $\ld_R(M), $ through a deep investigation based on the theory of standard bases. If $M$ is a graded $R$-module, then $\reg_R(gr_{\mathbb{M}}(M)) \lt \infty $ implies $\reg_R(M)\lt \infty$ and the converse holds provided $M$ is of homogenous type. An analogous result can be proved in the local case in terms of the linearity defect. Motivated by a positive answer in the graded case, we present for local rings a partial answer to a question raised by Herzog and Iyengar of whether $\ld_R(k)\lt \infty$ implies $R$ is Koszul.

Article information

Source
J. Commut. Algebra, Volume 6, Number 4 (2014), 485-504.

Dates
First available in Project Euclid: 5 January 2015

Permanent link to this document
https://projecteuclid.org/euclid.jca/1420466341

Digital Object Identifier
doi:10.1216/JCA-2014-6-4-485

Mathematical Reviews number (MathSciNet)
MR3294859

Zentralblatt MATH identifier
1321.13004

Subjects
Primary: 16W50: Graded rings and modules 13D07: Homological functors on modules (Tor, Ext, etc.)
Secondary: 16W70: Filtered rings; filtrational and graded techniques 16S37: Quadratic and Koszul algebras

Keywords
Regularity linearity defect minimal free resolutions standard basis associated graded module filtered modules Koszul algebras

Citation

Maleki, Rasoul Ahangari; Rossi, Maria Evelina. Regularity and linearity defect of modules over local rings. J. Commut. Algebra 6 (2014), no. 4, 485--504. doi:10.1216/JCA-2014-6-4-485. https://projecteuclid.org/euclid.jca/1420466341


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References

  • R. Ahangari Maleki, Koszul rings and modules and Castelnuovo-Mumford regularity, Ph.D. thesis, Kharazmi University, Iran, 2013.
  • L.L. Avramov, Infinite free resolutions, six lectures on commutative algebra (Bellaterra, 1996). Progr. Math. 166, Birkhauser, Basel, 1998, 1–118.
  • L.L. Avramov and D. Eisenbud, Regularity of modules over a Koszul algebra, J. Alg. 153 (1992), 85–90.
  • L.L. Avramov and I. Peeva, Finite regularity and Koszul algebras, Amer. J. Math. 123 (2001), 275–281.
  • A. Conca, E. De Negri and M.E. Rossi, Koszul algebras and regularity, Irena Peeva ed., Commutative algebra: Expository papers dedicated to David Eisenbud, Springer, New York,
  • J. Herzog and S. Iyengar, Koszul modules, J. Pure Appl. Alg. 201 (2005), 154–188.
  • J. Herzog, M.E. Rossi and G. Valla, On the depth of the symmetric algebra, Trans. Amer. Math. Soc. 296 (1986), 577-–606.
  • S. Iyengar and T. Römer, Linearity defects of modules over commutative rings, J. Alg. 322 (2009), 3212–3237.
  • L. Robbiano and G. Valla, Free resolutions for special tangent cones, Comm. Alg. Lect. Notes Pure Appl. Math. 84, Dekker, New York, 1983.
  • M.E. Rossi and L. Sharifan, Minimal free resolution of a finitely generated module over a regular local ring, J. Alg. 322 (2009), 3693–-3712.
  • ––––, Consecutive cancellation in Betti numbers of local rings, Proc. Amer. Math. Soc. 138 (2010), 61–73.
  • M.E. Rossi and G. Valla, Hilbert functions of filtered modules, Lect. Notes Un. Matem. Ital. 9, Springer-Verlag, Berlin; UMI, Bologna, 2010.
  • L.M. Şega, On the linearity defect of the residue field, arXiv:1303.4680v1 [math.AC], 2013.
  • ––––, Homological properties of powers of the maximal ideal of a local ring, J. Alg. 241 (2001), 827–858.
  • T. Shibuta, Cohen–-Macaulyness of almost complete intersection tangent cones, J. Alg. 319 (2008), 3222–-3243.