On the linearity defect of the residue field

TL;DR

This paper investigates whether a finite linearity defect of the residue field in a local ring necessarily implies a linear resolution, exploring special cases and various interpretations of this question.

Contribution

It provides partial answers and discusses different perspectives on the relationship between finite and zero linearity defect of the residue field.

Findings

Finite linearity defect does not always imply a linear resolution.

Special cases where the implication holds are identified.

The paper offers multiple interpretations and refinements of the core question.

Abstract

Given a commutative Noetherian local ring $R$, the linearity defect of a finitely generated $R$-module $M$, denoted $\ld_R(M)$, is an invariant that measures how far $M$ and its syzygies are from having a linear resolution. Motivated by a positive known answer in the graded case, we study the question of whether $\ld_R(k)<\infty$ implies $\ld_R(k)=0$. We give answers in special cases, and we discuss several interpretations and refinements of the question.