On Frobenius (completed) orbit categories

Let ? be a Frobenius category, \({\mathcal P}\) its subcategory of projective objects and F : ? → ? an exact automorphism. We prove that there is a fully faithful functor from the orbit category ?/F into \(\operatorname {gpr}({\mathcal P}/F)\), the category of finitely-generated Gorenstein-projective modules over \({\mathcal P}/F\). We give sufficient conditions to ensure that the essential image of ?/F is an extension-closed subcategory of \(\operatorname {gpr}({\mathcal P}/F)\). If ? is in addition Krull-Schmidt, we give sufficient conditions to ensure that the completed orbit category \({\mathcal E} \ \widehat {\!\! /} F\) 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.