Asymptotic depth of invariant chains of edge ideals

TL;DR

This paper investigates the long-term behavior of the depth and projective dimension of invariant chains of edge ideals, revealing new combinatorial and topological properties of associated graphs and complexes.

Contribution

It provides a complete characterization of the asymptotic depth and projective dimension for invariant edge ideal chains, along with insights into their combinatorial and topological structures.

Findings

Determined asymptotic depth and projective dimension of invariant edge ideal chains.

Revealed combinatorial properties of associated graphs.

Analyzed homology groups of independence complexes.

Abstract

We completely determine the asymptotic depth, equivalently, the asymptotic projective dimension of a chain of edge ideals that is invariant under the action of the monoid Inc of increasing functions on the positive integers. Our results and their proofs also reveal surprising combinatorial and topological properties of corresponding graphs and their independence complexes. In particular, we are able to determine the asymptotic behavior of all reduced homology groups of these independence complexes.