On the asymptotic behavior of the linearity defect

TL;DR

This paper investigates the asymptotic behavior of the linearity defect of modules over noetherian local rings, proving that certain sequences related to ideals and modules stabilize eventually.

Contribution

It establishes the eventual constancy of the linearity defect sequences for powers of ideals and their quotients over regular local rings, extending previous understanding.

Findings

Sequences of linearity defect for $I^nM$ and $M/I^nM$ are eventually constant.

The result applies to finitely generated modules over regular local rings.

Generalizes to graded modules over standard graded algebras.

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.