On higher extensions of quiver representations over $\mathbb{F}_1$

Abstract

We show that higher extension spaces between finite-dimensional nilpotent $\mathbb{F}_1$-representations maybe infinite-dimensional, thereby clarifying a misconception in the literature. Our examples arise from cyclic quivers. In particular, for a cyclic quiver $Δ_n$, we show that $\operatorname{Ext}^3(-,-)$ vanishes for any pair of finite-dimensional nilpotent $\mathbb{F}_1$-representations of $Δ_n$, while $\operatorname{Ext}^2(-,-)$ is infinite-dimensional for any pair of simple representations.