Generic initial ideals, graded Betti numbers and k-Lefschetz properties
Abstract
We introduce the k-strong Lefschetz property (k-SLP) and the k-weak Lefschetz property (k-WLP) for graded Artinian K-algebras, which are generalizations of the Lefschetz properties. The main results obtained in this paper are as follows: 1. Let I be a graded ideal of R=K[x1, x2, x3] whose quotient ring R/I has the SLP. Then the generic initial ideal of I is the unique almost revlex ideal with the same Hilbert function as R/I. 2. Let I be a graded ideal of R=K[x1, x2, ..., xn] whose quotient ring R/I has the n-SLP. Suppose that all k-th differences of the Hilbert function of R/I are quasi-symmetric. Then the generic initial ideal of I is the unique almost revlex ideal with the same Hilbert function as R/I. 3. We give a sharp upper bound on the graded Betti numbers of Artinian K-algebras with the k-WLP and a fixed Hilbert function.