Betti numbers for cochordal zero-divisor graphs of commutative rings

Abstract

This paper studies the zero-divisor graphs attached to several finite chain-ring families and computes the homological invariants of their edge ideals by using cochordal constructible systems. We begin with a general layered graph $C(q,L)$, whose vertices are arranged according to valuation layers and whose adjacency is governed by the single rule $k+\ell \ge L$, form some integers $k$ and $\ell$. This graph models the zero-divisor structure of a finite chain ring with residue field of order $q$ and nilpotency index $L$. We prove that $C(q,L)$ is cochordal, determine its type sequence, then correct and refine the Betti formula of its edge ideal [Dung and Vu, Cochordal zero divisor graphs and Betti numbers of their edge ideals, Comm. Algebra 54(2) (2026) 736--744]. The results are then specialized to the Gaussian quotient rings ${\mathbb{Z}}_{2^m}[i]$ and to the truncated polynomial rings ${\mathbb{Z}}_p[x]/(x^c)$. We compute projective dimension, regularity, independence number, height, Hilbert series, and Cohen--Macaulay behavior. The computations show that these quotient rings have $2$-linear resolutions, while Cohen--Macaulayness occurs only in the expected degenerate or complete-graph cases.

67 pages, 13 figures