Periodic resolutions and self-injective algebras of finite type

TL;DR

This paper proves that all self-injective algebras of finite representation type are periodic, using Galois coverings, stable Auslander algebras, and classification results, with implications for Calabi-Yau dimensions.

Contribution

It establishes the periodicity of self-injective finite type algebras and provides bounds on their periods using advanced algebraic techniques.

Findings

Self-injective finite type algebras are periodic.

Periodicity passes through Galois coverings and stable Auslander algebras.

Bounds for periods and applications to Calabi-Yau dimensions are given.

Abstract

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.