Covering theory, (mono)morphism categories and stable Auslander algebras

Let $\mathcal{A}$ be a locally bounded $k$-category and $G$ a torsion-free group of $k$-linear automorphisms of $\mathcal{A}$ acting freely on the objects of $\mathcal{A},$ and $F:\mathcal{A}\to \mathcal{B}$ is a Galois functor. We extend naturally the push-down functor $F_{\lambda }$ to the functor $\rm{H}\rm{F}{\lambda}:\rm{H}(\rm{mod}\mbox{-} $\mathcal{A}$)\rightarrow \rm{H}(\rm{mod}\mbox{-} $\mathcal{B}$)$,resp\mathrm{.}$$\mathcal{S}$ \rm{F}{\lambda}:$\mathcal{S}$(\rm{mod}\mbox{-} $\mathcal{A}$)\rightarrow $\mathcal{S}$(\rm{mod}\mbox{-} $\mathcal{B}$)$, between the corresponding morphism categories, resp. monomorphism categories, of $\mathrm{m}\mathrm{o}\mathrm{d}-\mathcal{A}$ and $\mathrm{m}\mathrm{o}\mathrm{d}-\mathcal{B}$. Under some additional conditions, we show that $\mathrm{H}(\mathrm{m}\mathrm{o}\mathrm{d}-\mathcal{A})$, resp. $$\mathcal{S}$( \rm{mod}\mbox{-}$\mathcal{A}$)$,islocallyboundedifandonlyif$\rm{H}(\rm{mod}\mbox{-} $\mathcal{B}$)$,resp\mathrm{.}$$\mathcal{S}$(\rm{mod}\mbox{-}$\mathcal{B}$)$, is of finite representation type. As an application, we show that the stable Auslander algebra of a representation-finite selfinjective algebra $\mathrm{\Lambda }$ is again representation-finite if and only if $\mathrm{\Lambda }$ is of Dynkin type ${\mathbb{A}}_n$ with $n⩽4$.