On higher extensions of quiver representations over F1

We show that higher extension spaces between finite-dimensional nilpotent F1-representations may be infinite-dimensional, thereby clarifying a misconception in the literature. Our examples arise from cyclic quivers. In particular, for a cyclic quiver Δn, we show that Ext3(−,−) vanishes for any pair of finite-dimensional nilpotent F1-representations of Δn, while Ext2(−,−) is infinite-dimensional for any pair of nilpotent simple representations.