## Journal of Commutative Algebra

### Regularity and linearity defect of modules over local rings

#### 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

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

#### 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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