Notes on the linearity defect and applications

TL;DR

This paper investigates the linearity defect, a measure of the complexity of minimal free resolutions, exploring its behavior in short exact sequences, generalizing Koszul filtrations, and applying these results to properties of local rings and modules.

Contribution

It introduces new insights into the behavior of linearity defect in short exact sequences and extends Koszul filtrations to local rings, with applications to Koszul modules and algebra specializations.

Findings

Linearity defect behaves well along certain short exact sequences.

Generalization of Koszul filtrations from graded to local rings.

Confirmation that specializations of absolutely Koszul algebras remain absolutely Koszul.

Abstract

The linearity defect, introduced by Herzog and Iyengar, is a numerical measure for the complexity of minimal free resolutions. Employing a characterization of the linearity defect due to \c{S}ega, we study the behavior of linearity defect along short exact sequences. We point out two classes of short exact sequences involving Koszul modules, along which linearity defect behaves nicely. We also generalize the notion of Koszul filtrations from the graded case to the local setting. Among the applications, we prove that if $R\to S$ is a surjection of noetherian local rings such that $S$ is a Koszul $R$-module, and $N$ is a finitely generated $S$-module, then the linearity defect of $N$ as an $R$-module is the same as its linearity defect as an $S$-module. In particular, we confirm that specializations of absolutely Koszul algebras are again absolutely Koszul, answering positively a question due to Conca, Iyengar, Nguyen and R\"omer.