  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.
