🤖 AI Summary
This study investigates homological invariants of zero-divisor graphs over finite chain rings, with a focus on the Betti numbers and algebraic properties of their edge ideals. By constructing a layered graph \( C(q,L) \) that encodes the zero-divisor structure, the authors prove that this graph is cochordal and thereby determine its type sequence, leading to a corrected and refined formula for the Betti numbers of the associated edge ideal. Employing techniques from combinatorial commutative algebra, cochordal graph theory, and homological algebra, they compute the projective dimension and Castelnuovo–Mumford regularity of Gaussian quotient rings \( \mathbb{Z}_{2^m}[i] \) and truncated polynomial rings \( \mathbb{Z}_p[x]/(x^c) \), establishing that these rings admit 2-linear resolutions. Moreover, they show that such rings are Cohen–Macaulay only in degenerate or complete graph cases.
📝 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.