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

Abstract

Let A = ⊕ i = 0 c A i be a graded Artinian K-algebra, where A c ≠ ( 0 ) and char K = 0 . ( The grading may not necessarily be standard.) Then A has the strong Lefschetz property if there exists an element g ∈ A 1 such that the multiplication × g c − 2 i : A i → A c − i 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 ∈ A 1 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.