G-semisimple algebras (Q6597177)

Let \(\Lambda\) be an Artin algebra, and \(S(\text{Gprj-}\Lambda)\) the monomorphism category with terms in \(\text{Gprj-}\Lambda \) whose cokernels lie in \(\text{Gprj-}\Lambda\), where \(\text{Gprj-}\Lambda\) is the category of finitely generated Gorenstein projective \(\Lambda\)-modules, known to be a resolving subcategory of mod-\(\Lambda\). Hafezi, in unpublished work, introduced \(G\)-semisimple algebras as Artin algebras where the functor category mod-\((\underline{\text{Gprj-}}\Lambda)\) forms a semisimple abelian category, achieved when the global projective dimension of mod-\((\underline{\text{Gprj-}}\Lambda)\) is zero.\N\NIn this article the authors build on prior unpublished work, enhancing and extending results to study the monomorphism category \(S(\text{Gprj-}\Lambda)\) in relation to mod-\((\text{Gprj-}\Lambda)\), particularly under CM-finite conditions. They demonstrate that a \(G\)-semisimple algebra is CM-finite, meaning there are only finitely many isomorphism classes of indecomposable Gorenstein projective modules. For a \(G\)-semisimple algebra, the stable category \(S(\text{Gprj-}\Lambda)\), which describes monomorphisms with terms in \(\text{Gprj-}Λ\), is of finite representation type. Moreover, the class of \(G\)-semisimple algebras contains important subclasses such as gentle algebras and quadratic monomial algebras, which are well-studied in representation theory. Finally, they establish that the path algebra \(\Lambda Q\) of a \(G\)-semisimple algebra \(\Lambda\) is Cohen-Macaulay (CM)-finite if and only if the quiver \(Q\) is of Dynkin type.