Generic Initial Ideals, Graded Betti Numbers, and k-Lefschetz Properties

Abstract

We introduce the k-strong Lefschetz property and the k-weak Lefschetz property for graded Artinian K-algebras, which are generalizations of the Lefschetz properties. The main results are: 1. Let I be an ideal of R = K[x 1, x 2,…, x n ] whose quotient ring R/I has the n-SLP. Suppose that all kth 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. 2. We give a sharp upper bound on the graded Betti numbers of Artinian K-algebras with the k-WLP and a fixed Hilbert function.