The strong Lefschetz property for Artinian algebras with non-standard grading

Let be a graded Artinian K-algebra, where Ac≠(0) and charK=0. (The grading may not necessarily be standard.) Then A has the strong Lefschetz property if there exists an element g∈A1 such that the multiplication is bijective for every i=0,1,…,[c/2]. The main results obtained in this paper are as follows:1.A has the strong Lefschetz property if and only if there is a linear form z∈A1 such that Gr(z)(A) has the strong Lefschetz property.2.If A is Gorenstein, then A has the strong Lefschetz property if and only if there is a linear form z∈A such that all central simple modules of (A,z) have the strong Lefschetz property.3.A finite free extension of an Artinian K-algebra with the strong Lefschetz property has the strong Lefschetz property if the fiber does.4.The complete intersection defined by power sums of consecutive degrees has the strong Lefschetz property.