On the asymptotic behavior of the linearity defect

Abstract

This work concerns the linearity defect of a module $M$ over a noetherian local ring $R$, introduced by Herzog and Iyengar in 2005, and denoted by $\text{ld}_R M$. Roughly speaking, $\text{ld}_R M$ is the homological degree beyond which the minimal free resolution of $M$ is linear. In the paper, it is proved that for any ideal $I$ in a regular local ring $R$ and for any finitely generated $R$-module $M$, each of the sequences $(\text{ld}_R (I^nM))_n$ and $(\text{ld}_R (M/I^nM))_n$ is eventually constant. The first statement follows from a more general result about the eventual constancy of the sequence $(\text{ld}_R C_n)_n$ where $C$ is a finitely generated graded module over a standard graded algebra over $R$. The second statement follows from the first together with a result of Avramov on small homomorphisms.