Year: 2026
Type: preprint
Abstract: In 2006, Katzman showed that the first six Betti numbers of the Stanley--Reisner ring of a flag complex are field independent. He also found flag complexes on eleven vertices whose eighth Betti number depends on the field, and asked whether the seventh is always field independent. We answer this affirmatively by proving a stronger, purely topological result. Let $τ(d)$ be the least number of vertices of a flag complex whose $d$-th reduced integral homology has torsion. We prove that $τ(d)\geq d+10$ for every $d\ge0$. This bound yields the field independence of the seventh Betti number. Equivalently, combining our result with Katzman's, for every finite simple graph $G$, the first seven Betti numbers of the edge ideal $I(G)$ are field independent.
Source: arXiv (Cornell University)
Author: Javadekar, Omkar
Language: English
Subfield: Algebra and Number Theory
Field: Mathematics
Domain: Physical Sciences
Open Access status: green
Funder: Infosys Foundation
OpenAlex ID: W7214243833