On Frobenius (completed) orbit categories

Abstract

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

arXiv comment19 pages