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. 1. A has the Weak Lefschetz Property (resp. Strong Lefschetz Property) if and only if Gr ( z ) ( A ) has the Weak Lefschetz Property (resp. Strong Lefschetz Property) for some linear form z of A . 2. 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.