Edge ideals and their asymptotic syzygies

TL;DR

This paper studies the long-term behavior of syzygies of powers of edge ideals in graphs, introduces the homological strong persistence property, and proves it for certain cases, with conjectures on linear resolutions.

Contribution

It introduces the homological strong persistence property for monomial ideals and proves it for edge ideals, providing explicit descriptions of their homological shift ideals.

Findings

Edge ideals have the 0th and 1st homological strong persistence properties.

Explicit description of the first homological shift algebra of $I(G)$.

Conjecture that linear resolution of $I(G)$ implies linear resolutions for all $ ext{HS}_i(I(G)^k)$ for large $k$.

Abstract

Let $G$ be a finite simple graph, and let $I(G)$ denote its edge ideal. In this paper, we investigate the asymptotic behavior of the syzygies of powers of edge ideals through the lens of homological shift ideals $\text{HS}_i(I(G)^k)$. We introduce the notion of the $i$th homological strong persistence property for monomial ideals $I$, providing an algebraic characterization that ensures the chain of inclusions $\text{Ass}\,\text{HS}_i(I)\subseteq\text{Ass}\,\text{HS}_i(I^2)\subseteq\text{Ass}\,\text{HS}_i(I^3) \subseteq\cdots$. We prove that edge ideals possess both the $0$th and $1$st homological strong persistence properties. To this end, we explicitly describe the first homological shift algebra of $I(G)$ and show that $\text{HS}_1(I(G)^{k+1}) = I(G) \cdot \text{HS}_1(I(G)^k)$ for all $k \ge 1$. Finally, we conjecture that if $I(G)$ has a linear resolution, then $\text{HS}_i(I(G)^k)$ also has a linear resolution for all $k \gg 0$, and we present partial results supporting this conjecture.