When stable Cohen-Macaulay Auslander algebra is semisimple

Abstract

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