Homological Invariants of Edge Ideals Associated with Cozero-Divisor Graphs of Zn

Cozero-divisor graphs Γ′(Zn) have been studied from structural, combinatorial and spectral points of view. The edge ideals they define appear to have received much less attention, and the purpose of this paper is to begin their homological study. We first show that two vertices of Γ′(Zn) are adjacent exactly when the corresponding divisors of n are incomparable, so that the graph is a blow-up of the incomparability graph of the divisor poset of n. This reduction is then combined with Fröberg’s theorem and with Hochster’s formula. The small cases become transparent: Γ′(Zpa) has no edges, Γ′(Zpq) is the complete bipartite graph Kq−1,p−1, and Γ′(Zp2q) splits into four divisor classes joined bipartitely. We prove that Γ′(Zn) is cochordal if and only if n=pa, n=paq or n=pqb for distinct primes p,q, which settles when I(Γ′(Zn)) has a linear resolution, and we obtain closed formulas for the graded Betti numbers when n=pq and n=p2q, together with the projective dimension, depth, regularity, Hilbert series and Cohen–Macaulay defect of the quotient rings. Every Betti table reported here has been confirmed by independent computation. In these families the arithmetic of n alone determines the homological behaviour of I(Γ′(Zn)), so that the divisor lattice of n is tied directly to a minimal free resolution. The cochordality classification is a statement about the complement of Γ′(Zn) and about resolutions, and is therefore of a different kind from the structural and spectral descriptions already available in the literature.