Betti numbers for cochordal zero-divisor graphs of commutative rings
Fuente:
arXiv
Gespeichert in:
| 1. Verfasser: | |
|---|---|
| Format: | Preprint |
| Veröffentlicht: |
2026
|
| Schlagworte: | |
| Online-Zugang: | |
| Tags: |
Tag hinzufügen
Keine Tags, Fügen Sie den ersten Tag hinzu!
|
| _version_ | 1866911681016233984 |
|---|---|
| author | Rather, Bilal Ahmad |
| author_facet | Rather, Bilal Ahmad |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2605_13622 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Betti numbers for cochordal zero-divisor graphs of commutative rings Rather, Bilal Ahmad Commutative Algebra Discrete Mathematics Combinatorics 13D02, 05C25, 05E40, 13F55, 13A70 F.2.2 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. |
| title | Betti numbers for cochordal zero-divisor graphs of commutative rings |
| topic | Commutative Algebra Discrete Mathematics Combinatorics 13D02, 05C25, 05E40, 13F55, 13A70 F.2.2 |
| url | https://arxiv.org/abs/2605.13622 |