A ring theoretic approach to the finite representation type

Abstract

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

We have found some serious errors in the manuscript

References in corpus (1)

• Auslander-Reiten duality for subcategories