Edge ideals and their asymptotic syzygies
Antonino Ficarra, Ayesha Asloob Qureshi
TL;DR
This work develops the homological shift-ideal framework to study asymptotic syzygies of powers of edge ideals. It proves that edge ideals satisfy the $0$th and $1$st homological strong persistence, and it provides an explicit description of $\textup{HS}_1(I(G)^k)$ as $\textup{Lin}_1(I(G)^k) + I(G)^{\langle k+1\rangle}$, with a module structure $\textup{HS}_1(\mathcal{R}(I(G)))$ over $\mathcal{R}(I(G))$. The paper conjectures that if $I(G)$ has a linear resolution, then $\textup{HS}_i(I(G)^k)$ eventually have linear resolutions, and it confirms this for several classes of graphs: principal Borel edge ideals, initial lexsegment edge ideals, and complete multipartite graphs. It also develops a theoretical and computational program around homological persistence, providing substantial partial results and verifications for graphs with up to seven vertices.
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.