Periodic resolutions and self-injective algebras of finite type

By Alex Dugas. 2008

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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.

Subfields: Algebra and Number Theory, Geometry and Topology, Statistical and Nonlinear Physics

View Paper: https://www.semanticscholar.org/paper/b3eeeb3c6c070ddacee6269017fda7ce13d689be