Edge ideals and their asymptotic syzygies

Official URL: https://dx.doi.org/10.1016/j.jpaa.2025.108079

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 HSi(I(G)k). We introduce the notion of the ith homological strong persistence property for monomial ideals I, providing an algebraic characterization that ensures the chain of inclusions AssHSi(I)⊆AssHSi(I2)⊆AssHSi(I3)⊆⋯. We prove that edge ideals possess both the 0th and 1st homological strong persistence properties. To this end, we explicitly describe the first homological shift algebra of I(G) and show that HS1(I(G)k+1)=I(G)⋅HS1(I(G)k) for all k≥1. Finally, we conjecture that if I(G) has a linear resolution, then HSi(I(G)k) also has a linear resolution for all k≫0, and we present partial results supporting this conjecture.