Covering theory for linear categories with application to derived categories

Abstract

We extend the Galois covering theory introduced by Bongartz–Gabriel for skeletal linear categories to general linear categories. We show that a Galois covering between Krull–Schmidt categories preserves irreducible morphisms and almost splits sequences. Specializing to derived categories, we study when a Galois covering between locally bounded linear categories induces a Galois covering between the bounded derived categories of finite dimensional modules. As an application, we show that each locally bounded linear category with radical squared zero admits a gradable Galois covering, which induces a Galois covering between the bounded derived categories of finite dimensional modules, and a Galois covering between the Auslander–Reiten quivers of these bounded derived categories. In a future paper, this will enable us to obtain a complete description of the bounded derived category of finite dimensional modules over a finite dimensional algebra with radical squared zero.