Symmetry, Unimodality, and Lefschetz Properties for Graded Modules

TL;DR

This paper explores the Weak Lefschetz Property in graded modules with specific resolutions, analyzing Betti numbers, Hilbert functions, and the non-Lefschetz locus, revealing new connections with Gorenstein and Artin level modules.

Contribution

It generalizes previous results on the non-Lefschetz locus and studies symmetry, unimodality, and Lefschetz properties in a broader module setting.

Findings

Characterization of the non-Lefschetz locus for finite length modules

Connections between Gorenstein and Artin level modules

Conditions for symmetry and unimodality in Hilbert functions

Abstract

We investigate the Weak Lefschetz Properties for modules whose minimal free resolutions are given by generalized Kosuzl complexes in dimension three through a careful study of their Betti numbers and the symmetry and unimodality of their Hilbert functions. We also study the non-Lefschetz locus for finite length modules in arbitrary dimension, and are able to generalize several previous results on the non-Lefschetz locus in this setting. Along the way, we find several connections with a Gorenstein analogue for finite length modules and Artin level modules that are both interesting and useful throughout this paper.