Generic initial ideals, graded Betti numbers and $k$-Lefschetz properties

TL;DR

This paper introduces generalized Lefschetz properties for graded Artinian algebras, characterizes their generic initial ideals as almost revlex ideals under certain conditions, and establishes bounds on their graded Betti numbers.

Contribution

It generalizes Lefschetz properties to the $k$-SLP and $k$-WLP, characterizes generic initial ideals as almost revlex ideals, and provides bounds on Betti numbers for these algebras.

Findings

Generic initial ideal of an algebra with SLP is the unique almost revlex ideal.

For $n$-SLP, the generic initial ideal is the unique almost revlex ideal if Hilbert function differences are quasi-symmetric.

Established sharp upper bounds on graded Betti numbers for algebras with $k$-WLP.

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 $R=K[x_1, x_2, x_3]$ 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 $R=K[x_1, x_2, ..., x_n]$ 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.