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 $({\text{ld}}_R(I^{n}M){)}_n$ and $({\text{ld}}_R(M/I^{n}M){)}_n$ 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.

Following the referee's suggestion, the original paper "Linearity defects of powers are eventually constant" is splitted in two parts. This is the first part (with a new title), and the second part will contain the results of Sections 4 and 5. In this version, we strengthen Theorem 1.1 of the previous version to the effect stated in the abstract. The presentation has been thoroughly revised