Auslander-reiten Theory for Modules of Finite Complexity over Selfinjective Algebras

In this paper we describe the shapes of the stable components containing modules with finite complexity over a selfinjective finite dimensional algebra over an algebraically closed field. We prove that the associated orbit graph of each such component, is either a Dynkin diagram (finite or infinite), or an extended Dynkin diagram. Peter Webb has described in [16] the Auslander-Reiten quiver of a group algebra kG of a finite group over a field. Specifically, if Cs is a stable component of the Auslander-Reiten quiver, then its tree class is a Dynkin diagram (finite or infinite), or an extended Dynkin diagram. The purpose of this paper is to extend Webb’s theorem to a more general setting, namely to components of selfinjective algebras containing modules of finite complexity. The complexity of a finitely generated module measures the growth of the terms in a minimal projective resolution of the module. To be more precise, let Λ be a finite dimensional algebra over a field k and let P • : · · · −→ P 2 δ 2 −→ P 1 δ 1 −→ P 0 δ 0 −→M → 0 be a minimal projective resolution of a finitely generated Λ-module M . The i-th Betti number of M , βi(M), is defined as the number of indecomposable summands of P . The complexity of M is defined as cxM = inf{n ∈ N|βi(M) ≤ cin−1 for some positive c ∈ Q and all i ≥ 0} If no such n exists, then we say that the complexity ofM is infinite. It is well-known that, if Λ is a selfinjective algebra and C is a component of the Auslander-Reiten quiver of Λ, then all nonprojective modules in C have the same complexity [7, 2.2]. Note that over the group algebra kG of a finite group, every module has finite complexity by the Alperin-Venkov theorem. Let Λ be a selfinjective algebra, and let τ denote the Auslander-Reiten translate, see [2]. An indecomposable non projective Λ-module M is called τ -periodic, if τM ∼= M for some m > 0. If an Auslander-Reiten component C contains a τ -periodic module, then its stable part Cs, is a τ -periodic component, that is, every module in Cs is τ -periodic.The tree class of Cs is either a finite Dynkin diagram, or is of type A∞, which means that Cs is a tube [9]. The complexity of nonprojective 2000 Mathematics Subject Classification. Primary 16G70. Secondary 16D50, 16E05. Most of the results of this paper were obtained while both authors were visiting the University of Bielefeld in 2009 as part of the SFB’s “Topologische und spektrale Strukturen in der Darstellungstheorie” program. We both thank Claus Michael Ringel, the Faculty of Mathematics in Bielefeld and the SFB for inviting us and for making our stay possible. The second author is supported by a grant from NSA.