Edge ideals and their asymptotic syzygies

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 Ass HSi(I) subset of Ass HSi(I-2) subset of Ass HSi(I-3) subset of . 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. (c) 2025 Elsevier B.V. All rights are reserved, including those for text and data mining, AI training, and similar technologies.