Splittings of monomial ideals

TL;DR

This paper introduces new conditions for splitting monomial ideals to facilitate Betti number computations, applies these to graph edge and cover ideals, and explores the prevalence of such splittings and their characteristic dependence.

Contribution

It extends the splitting approach for monomial ideals, providing new criteria and methods for calculating Betti numbers of specific classes of ideals.

Findings

Edge ideals of graphs are splittable.

An iterative method for Betti numbers of cover ideals of Cohen-Macaulay bipartite graphs.

Discussion on the frequency and characteristic dependence of splittings.

Abstract

We provide some new conditions under which the graded Betti numbers of a monomial ideal can be computed in terms of the graded Betti numbers of smaller ideals, thus complementing Eliahou and Kervaire's splitting approach. As applications, we show that edge ideals of graphs are splittable, and we provide an iterative method for computing the Betti numbers of the cover ideals of Cohen-Macaulay bipartite graphs. Finally, we consider the frequency with which one can find particular splittings of monomial ideals and raise questions about ideals whose resolutions are characteristic-dependent.