Stable Calabi-yau Dimension for Finite Type Selfinjective Algebras

We show that the Calabi-Yau dimension of the stable module category of a selfinjective algebra of finite representation type is determined by the action of the Nakayama and suspension functors on objects. Our arguments are based on recent results of C. Amiot, and hence apply more generally to triangulated categories having only finitely many indecomposable objects. Throughout, k is an algebraically closed field, and all k-algebras are finite dimensional. A selfinjective k-algebra A has a stable module category stabA. This is a triangulated category with suspension Σ given by taking the first syzygy Ω A in an injective resolution. There is also a Serre functor given by S = ΩA ◦ νA where νA := DA ⊗ − is the Nakayama functor, see [6, prop. 1.2(ii)]. By definition, the Calabi-Yau dimension of a triangulated category with Serre functor S is the smallest integer d ≥ 0 such that there is an equivalence of functors S ≃ Σ, or ∞ if no d exists. Calabi-Yau dimensions of stable module categories of selfinjective algebras have been of considerable interest recently, see for instance [2], [3], [5], [6], [7]. In finite representation type, these dimensions occur in connection with u-cluster categories of Dynkin type [9], [10], [11]. According to the definition, for determining the precise value of the Calabi-Yau dimension of the stable module category of a selfinjective algebra A, one needs to find the minimal d ≥ 0 such that there is an equivalence of functors νA ≃ Ω −d−1 A . In some situations it is not too difficult to show that these functors agree on objects (i.e. νA(M) ∼= Ω A (M) for every A-module M). On the other hand, it can be a 2000 Mathematics Subject Classification. Primary: 16G70, 18E30; Secondary: 16D50, 16G10, 16G60.