Homological shifts of a complementary edge ideal

TL;DR

This paper investigates the homological shifts and projective dimension of complementary edge ideals I_c(G) and their powers, focusing on trees and cycles. It establishes a general relation for HS_i(I) when I has a linear resolution, and leverages this to compute explicit HS descriptions and generation degrees for powers of I_c(G). For trees, it yields a precise closed form for HS_i(I^s) and proves HS_i(R(I)) is generated in degree i, with a Veronese-type realization for HS^i(I^i). For cycles, it derives detailed formulas and generation bounds that depend on cycle parity, and shows strict monotonic behavior of pd(I_c(G)^s) in many bipartite cases. Overall, the work connects homological shift algebras to Veronese-type ideals and provides exact pd and HS descriptions that enhance understanding of the syzygies of complementary edge ideals.

Abstract

The homological shift algebra and the projective dimension function of complementary edge ideals are investigated. Let $G$ be a connected graph, and let $I$ be its complementary edge ideal. For bipartite graphs $G$, we show that the projective dimension of $I^s$ increases strictly with $s$ until reaching its maximum value. For trees and cycles, explicit expressions for the projective dimension of $I^s$ are provided, along with detailed descriptions of their homological shift algebras. In particular, it is shown that the $i$-th homological shift algebra of such ideals is generated in degree at most $i$. Additionally, we prove that if $G$ is a tree, then the homological shift ideal $\mathrm{HS}_i(I^i)$, when divided by a suitable monomial, becomes a Veronese-type ideal, and every Veronese-type ideal arises in this manner.

Figures (1)

• Figure 1: Caterpillar tree $T$

Theorems & Definitions (67)

• Theorem 1
• Proposition 2
• Theorem 3
• Theorem 4
• Theorem 5
• Theorem 6
• Theorem 7
• Definition 1.1
• Lemma 1.2
• Lemma 1.3