On homological properties of the category of $\mathbb{F}_1$-representations over a linear quiver of type $\mathbb{A}_n

Let $Q$ be a quiver of type $\mathbb{A}_n$ with linear orientation and $\operatorname{rep}(Q,\mathbb{F}_1)$ the category of representations of $Q$ over the virtual field $\mathbb{F}_1$.It is proved that $\operatorname{rep}(Q,\mathbb{F}_1)$ has global dimension $2$ whenever $n\geq 3$ and it is hereditary if $n\leq 2$. As a consequence, the Euler form $\langle L, M\rangle=\sum_{i=0}^\infty (-1)^i\operatorname{dim} \operatorname{Ext}^i(L,M)$ is well-defined. However, it does not descend to the Grothendieck group of $\operatorname{rep}(Q,\mathbb{F}_1)$. This yields negative answers to questions raised by Szczesny in [IMRN, Vol. 2012, No. 10, pp. 237-2404].