Journal of Commutative Algebra

Regularity and linearity defect of modules over local rings

Rasoul Ahangari Maleki and Maria Evelina Rossi

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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

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

First available in Project Euclid: 5 January 2015

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

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

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


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.

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