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 $\mathrm{rep}(Q,{\mathbb{F}}_1)$ the category of representations of $Q$ over the virtual field ${\mathbb{F}}_1$.It is proved that $\mathrm{rep}(Q,{\mathbb{F}}_1)$ has global dimension $2$ whenever $n\ge 3$ and it is hereditary if $n\le 2$. As a consequence, the Euler form $⟨L,M⟩={\sum }_{i=0}^{\mathrm{\infty }}(-1{)}^i\dim {\mathrm{Ext}}^i(L,M)$ is well-defined. However, it does not descend to the Grothendieck group of $\mathrm{rep}(Q,{\mathbb{F}}_1)$. This yields negative answers to questions raised by Szczesny in [IMRN, Vol. 2012, No. 10, pp. 237-2404].