On Frobenius (Completed) Orbit Categories

Abstract

Abstract. Let E be a Frobenius category, P its subcategory of projective objects and F : E → E an exact automorphism. We prove that there is a fully faithful functor from the orbit category E/F into gpr(P/F ), the category of finitely-generated Gorensteinprojective modules over P/F . We give sufficient conditions to ensure that the essential image of E/F is an extension-closed subcategory of gpr(P/F ). If E is in addition KrullSchmidt, we give sufficient conditions to ensure that the completed orbit category E /F is a Krull-Schmidt Frobenius category. Finally, we apply our results on completed orbit categories to the context of Nakajima categories associated to Dynkin quivers and sketch applications to cluster algebras.