Lefschetz elements of Artinian Gorenstein algebras and Hessians of homogeneous polynomials

Abstract

We give a characterization of the Lefschetz elements in Artinian Gorenstein rings over a field of characteristic zero in terms of the higher Hessians. As an application, we give new examples of Artinian Gorenstein rings which do not have the strong Lefschetz property.

The Jordan type of graded Artinian Gorenstein algebras

“…The following result is Theorem [GZ2, Theorem 2.4] that generalizes [Wa1,Theorem 4] and [MW,Theorem 3.1].…”

“…For standard graded Artinian Gorenstein K algebras A, the Strong Lefschetz property (SLP) means that there is l ∈ A 1 such that the multiplication maps µ l d−2k : A k → A d−k is an isomorphism for all k. The Weak Lefschetz property (WLP) means that there is l ∈ A 1 such that the multiplication maps µ l : A k → A k+1 have maximal rank for all k. In the algebraic context the Lefschetz properties is also a very important theme of research the last decades (see, for instance, [BI,BL,BMMNZ,MN1,MN2,St,St2,HMNW,MW,Go,GZ]).…”

The weak Lefschetz property for Artinian Gorenstein algebras of codimension three

“…3 An important tool needed to study whether a Gorenstein algebra has the WLP is the Macaulay inverse system, and especially the higher Hessians. We give now some definitions and results taken from a paper by Maeno and Watanabe [16] and from a recent paper by Gondim and Zappalá [9]. The general facts on the Macaulay's inverse system can be seen in [8].…”

Complete intersection Jordan types in height two

“…There have been many studies of graded Artinian Gorenstein algebras satisfying the strong or weak Lefschetz property (see [7] and the references cited there). Recently, there have been studies of more general questions about the Jordan type of pairs ( , A) (see [4,5,7,10,16] and references cited.) By a result of F.H.S.…”

“…We denote by R = k[x, y] the polynomial ring in two variables over k. We will consider Artinian Gorenstein (so by F.H.S. Macaulay's result complete intersection) algebras A = R/ Ann F , where F ∈ E = k[X, Y ] is the Macaulay dual generator of A. T. Maeno and J. Watanabe in 2009 introduced a method of using higher Hessians to determine the strong or weak Lefschetz properties of a graded Artinian algebra [16]; this was further developed and used by T. Maeno and Y. Numata [15] and by R. Gondim and colleagues [4,5,3]. The Hilbert function of a graded CI quotient A of R satisfies H(A) = T , a symmetric sequence of the form T = (1 0 , 2 1 , .…”