Homological rigidity of quiver representations over $\mathbb{F}_1$

Changjian Fu; Liang Yang; Zhiyuan Zeng

Homological rigidity of quiver representations over $\mathbb{F}_1$

Representation Theory 2026-07-11 v1 Rings and Algebras

Authors: Changjian Fu , Liang Yang , Zhiyuan Zeng

Abstract

We establish a homological rigidity phenomenon for the category of representations of quivers over the virtual field $\mathbb{F}_1$ , which is inherently non-additive and does not admit classical homological algebra tools. We prove that all higher Yoneda extension groups vanish beyond degree two for arbitrary quivers, including infinite ones. Consequently, the global dimension of the category is universally bounded by 2. Moreover, we obtain a complete classification of quivers according to their homological dimension, which is determined solely by the underlying orientation structure.

Keywords

quiver representations field F1 homological rigidity Yoneda extension groups global dimension homological dimension

Cite

@article{arxiv.2607.10253,
  title  = {Homological rigidity of quiver representations over $\mathbb{F}_1$},
  author = {Changjian Fu and Liang Yang and Zhiyuan Zeng},
  journal= {arXiv preprint arXiv:2607.10253},
  year   = {2026}
}