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

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 ${\mathrm{\Delta }}_n$, we show that ${\mathrm{Ext}}^3(-,-)$ vanishes for any pair of finite-dimensional nilpotent ${\mathbb{F}}_1$-representations of ${\mathrm{\Delta }}_n$, while ${\mathrm{Ext}}^2(-,-)$ is infinite-dimensional for any pair of simple representations.