When stable Cohen-Macaulay Auslander algebra is semisimple

Abstract

Let $\text{Gprj}\mbox{-}Λ$ denote the category of Gorenstein projective modules over an Artin algebra $Λ$ and the category $\text{mod}\mbox{-} (\underline{\text{Gprj}}\mbox{-}Λ)$ of finitely presented functors over the stable category $\underline{\text{Gprj}}\mbox{-}Λ$. In this paper, we study those algebras $Λ$ with $\text{mod}\mbox{-} (\underline{\text{Gprj}}\mbox{-}Λ)$ to be a semisimple abelian category, and called $Ω_{\mathcal{G}}$-algebras. The class of $Ω_{\mathcal{G}}$-algebras contains important classes of algebras, including gentle algebras. Over an $Ω_{\mathcal{G}}$-algebra $Λ$, the structure of the almost split sequences in the morphism categories $\text{H}(\text{Gprj}\mbox{-}Λ)$ and the monomorphism categories $\mathcal{S}(\text{Gprj}\mbox{-}Λ)$ of $\text{Gprj}\mbox{-}Λ$ is investigated. Among other applications, we provide some results for the Cohen-Macaulay Auslander algebras of $Ω_{\mathcal{G}}$-algebras.