The central simple modules of Artinian Gorenstein algebras

Abstract

Let $A$ be a standard graded Artinian $K$-algebra, with char $K=0$. We prove the following.

• $A$ has the Weak Lefschetz Property (resp. Strong Lefschetz Property) if and only if ${\mathrm{Gr}}_{\left(z\right)}\left(A\right)$ has the Weak Lefschetz Property (resp. Strong Lefschetz Property) for some linear form $z$ of $A$.

• If $A$ is Gorenstein, then $A$ has the Strong Lefschetz Property if and only if there exists a linear form $z$ of $A$ such that all central simple modules of $(A,z)$ have the Strong Lefschetz Property.

As an application of these theorems, we give some new classes of Artinian complete intersections with the Strong Lefschetz Property.