Koszul property and finite linearity defect over $g$-stretched local rings

TL;DR

This paper proves that Cohen-Macaulay local rings of almost minimal multiplicity with residue field of characteristic zero are Koszul if their residue field has finite linearity defect, advancing understanding of the linearity defect and Koszul property.

Contribution

It provides a positive answer to Herzog and Iyengar's open question for a specific class of local rings and characterizes g-stretched local rings, including a complete classification of one-dimensional cases.

Findings

Proves Cohen-Macaulay local rings of almost minimal multiplicity are Koszul under certain conditions.

Characterizes g-stretched local rings and their relation to Koszul property.

Provides a numerical characterization of g-stretched Koszul rings.

Abstract

The linearity defect is a measure for the non-linearity of minimal free resolutions of modules over noetherian local rings. A tantalizing open question due to Herzog and Iyengar asks whether a noetherian local ring $(R,\mathfrak{m})$ is Koszul if its residue field $R/\mathfrak{m}$ has a finite linearity defect. We provide a positive answer to this question when $R$ is a Cohen-Macaulay local ring of almost minimal multiplicity with the residue field of characteristic zero. The proof depends on the study of noetherian local rings $(R,\mathfrak{m})$ such that $\mathfrak{m}^2$ is a principal ideal, which we call $g$-$stretched$ local rings. The class of $g$-stretched local rings subsumes stretched artinian local rings studied by Sally, and generic artinian reductions of Cohen-Macaulay local rings of almost minimal multiplicity. An essential part in the proof of our main result is a complete characterization of one-dimensional complete $g$-stretched local rings. Beside partial progress on Herzog-Iyengar's question, another consequence of our study is a numerical characterization of all $g$-stretched Koszul rings, strengthening previous work of Avramov, Iyengar, and \c{S}ega.