Periodic resolutions and self-injective algebras of finite type

We say that an algebra AA is periodic if it has a periodic projective resolution as an (A,A)(A,A)-bimodule. We show that any self-injective algebra of finite representation type is periodic. To prove this, we first apply the theory of smash products to show that for a finite Galois covering B→AB→A, BB is periodic if and only if AA is. In addition, when AA has finite representation type, we build upon results of Buchweitz to show that periodicity passes between AA and its stable Auslander algebra. Finally, we use Asashiba’s classification of the derived equivalence classes of self-injective algebras of finite type to compute bounds for the periods of these algebras, and give an application to stable Calabi–Yau dimensions.