Betti numbers for cochordal zero-divisor graphs of commutative rings

TL;DR

This paper investigates the homological invariants of zero-divisor graphs of finite chain rings using cochordal systems, providing formulas and computations for various algebraic properties.

Contribution

It introduces a general layered graph model for zero-divisor structures, proves cochordality, and refines Betti number formulas for edge ideals of these graphs.

Findings

C(q,L) is cochordal and its type sequence is determined

Betti formulas for edge ideals are refined and corrected

Various algebraic invariants like projective dimension and Cohen--Macaulayness are computed

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.