Depth and regularity of monomial ideals via polarization and combinatorial optimization

TL;DR

This paper investigates how polarization and combinatorial optimization techniques can analyze the depth and regularity of monomial ideals, especially edge ideals of clutters, revealing monotonic behaviors under certain conditions.

Contribution

It introduces new methods using polarization and combinatorial optimization to study depth and regularity of monomial ideals and their powers, with specific results for edge ideals of clutters.

Findings

Powers of edge ideals of unmixed clutters with the max-flow min-cut property have non-increasing depth.

Regularity of these powers is non-decreasing.

Symbolic powers of ideals of covers of clique clutters of strongly perfect graphs have non-increasing depth.

Abstract

In this paper we use polarization to study the behavior of the depth and regularity of a monomial ideal $I$, locally at a variable $x_i$, when we lower the degree of all the highest powers of the variable $x_i$ occurring in the minimal generating set of $I$, and examine the depth and regularity of powers of edge ideals of clutters using combinatorial optimization techniques. If $I$ is the edge ideal of an unmixed clutter with the max-flow min-cut property, we show that the powers of $I$ have non-increasing depth and non-decreasing regularity. In particular edge ideals of unmixed bipartite graphs have non-decreasing regularity. We are able to show that the symbolic powers of the ideal of covers of the clique clutter of a strongly perfect graph have non-increasing depth. A similar result holds for the ideal of covers of a uniform ideal clutter.