On the Lefschetz property for quotients by monomial ideals containing squares of variables

Abstract

Let Δ be an (abstract) simplicial complex on n vertices. One can define the Artinian monomial algebra A ( Δ ) = k [ x 1 , … , x n ] / 〈 x 1 2 , … , x n 2 , I Δ 〉 , where k is a field of characteristic 0 and I Δ is the Stanley-Reisner ideal associated to Δ . In this paper, we aim to characterize the Weak Lefschetz Property (WLP) of A ( Δ ) in terms of the simplicial complex Δ . We are able to completely analyze when the WLP holds in degree 1, complementing work by Migliore, Nagel and Schenck on the WLP for quotients by quadratic monomials. We give a complete characterization of all 2-dimensional pseudomanifolds Δ such that A ( Δ ) satisfies the WLP. We also construct Artinian Gorenstein algebras that fail the WLP by combining our results and the standard technique of Nagata idealization.