  We say that an algebra A is periodic if it has a periodic projective
resolution as an (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 --> A, B
is periodic if and only if A is. In addition, when A has finite representation
type, we build upon results of Buchweitz to show that periodicity passes
between A 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.
