Asymptotic regularity of invariant chains of edge ideals

TL;DR

This paper investigates the long-term behavior of the regularity of invariant chains of edge ideals, showing it stabilizes at 2 or 3 and providing explicit conditions for this stability, supporting a broader conjecture.

Contribution

It proves the eventual constancy of regularity in invariant edge ideal chains and characterizes when this occurs, advancing understanding of their asymptotic properties.

Findings

Regularity stabilizes at 2 or 3

Explicit criteria for constancy behavior

Unveils unexpected combinatorial properties

Abstract

We study chains of nonzero edge ideals that are invariant under the action of the monoid $\mathrm{Inc}$ of increasing functions on the positive integers. We prove that the sequence of Castelnuovo--Mumford regularity of ideals in such a chain is eventually constant with limit either 2 or 3, and we determine explicitly when the constancy behaviour sets in. This provides further evidence to a conjecture on the asymptotic linearity of the regularity of $\mathrm{Inc}$-invariant chains of homogeneous ideals. The proofs reveal unexpected combinatorial properties of $\mathrm{Inc}$-invariant chains of edge ideals.