Linearity defect of the residue field of short local rings

TL;DR

This paper investigates the linearity defect of the residue field in short local rings, proving that finite linearity defect implies it is zero when the maximal ideal's fourth power vanishes.

Contribution

It provides a positive answer to whether finite linearity defect of the residue field implies it is zero in rings with =0, extending understanding of linear resolutions in local rings.

Findings

If =0, then \u2113_R(k)< implies _R(k)=0

Finite linearity defect of the residue field is zero in rings with =0

Supports Herzog and Iyengar's question in specific cases

Abstract

Let $(R,\m,k)$ be a Noetherian local ring with maximal ideal $\m$ and residue field $k$. The linearity defect of a finitely generated $R$-module $M$, which is denoted $\ld_R(M)$, is a numerical measure of how far $M$ is from having linear resolution. We study the linearity defect of the residue field. We give a positive answer to the question raised by Herzog and Iyengar of whether $\ld_R(k)<\infty$ implies $\ld_R(k)=0$, in the case when $\m^4=0$.