A ring theoretic approach to the finite representation type

Abstract

An Artin algebra $\Lambda$ is said to be of finite Cohen-Macaulay type, $\rm{CM}$-finite for short, if the full subcategory $\rm{Gprj}\mbox{-} \Lambda$ of finitely generated Gorenstein projective $\Lambda$-modules is of finite representation type. If $\Lambda$ is a $\rm{CM}$-finite algebra, then we denote by $\rm{Aus}(\underline{\rm{Gprj}}\mbox{-} \Lambda)$ the stable Cohen-Macaulay Auslander algebra, i.e. $\rm{\underline{End}}_{\Lambda}(G)$, where $G $ is a basic representation generator of $\rm{Gprj}\mbox{-}\Lambda$. In this paper, we will explain how by defining an equivalence relation on the elements of algebra $\rm{Aus}(\underline{\rm{Gprj}}\mbox{-} \Lambda)$ can be used to give a characterization for $\rm{Aus}(\underline{\rm{Gprj}}\mbox{-} \Lambda)$ to be of finite representation type, or equivalently, the $\rm{CM}$-finiteness of the algebra of $2 \times 2$ lower triangular matrices over $\Lambda,$ where $\Lambda$ is a $\rm{CM}$-finite Artin algebra over an algebraic closed filed. Then, by presenting some examples we will show how our results work.