Resolutions of mesh algebras: periodicity and Calabi–Yau dimensions

Abstract

A triangulated category $${(\mathcal{T},\Sigma)}$$ is said to be Calabi–Yau of dimension d if Σ d is a Serre functor. We determine which stable module categories of self-injective algebras Λ of finite type are Calabi–Yau and compute their Calabi–Yau dimensions, correcting errors in previous work. We first show that the Calabi–Yau property of mod-Λ can be detected in the minimal projective resolution of the stable Auslander algebra Γ of Λ, over its enveloping algebra. We then describe the beginning of such a minimal resolution for any mesh algebra of a stable translation quiver and apply covering theory to relate these minimal resolutions to those of the (generalized) preprojective algebras of Dynkin graphs. For representation-finite self-injective algebras of torsion order t = 1, we obtain a complete description of their stable Calabi–Yau properties, but only partial results for those algebras of torsion order t = 2. We also obtain some new information about the periods of the representation-finite self-injective algebras of torsion order t > 1. Finally, we describe how these questions can also be approached by realizing the stable categories of representation-finite self-injective algebras as orbit categories of the bounded derived categories of hereditary algebras, and illustrate this technique with several explicit computations that our previous methods left unsettled.