The existence of covering functors for orders and consequences for stable components with oriented cycles

Abstract

The starting point for the covering theory for algebras of finite type over algebraically closed fields was Riedtmann’s paper, “Algebren, Darstellungskocher, uberlagerungen und zurtick” [ 14 1. Her main result, from today’s point of view, is the existence of covering functors, i.e., the existence of a bijection between the morphisms between indecomposable modules and purely combinatorial defined vector spaces generated by paths in a translation quiver (cf. (1.1) [ 141). This bijection has recently turned out to be more and more important in the representation theory of algebras. It was the first step leading to the use of Auslander-Reiten quivers as a tool for the classification of algebras of linite type [ 15, 81. The main purpose of this paper is to initiate an analog for classical orders over complete Dedekind domains. We shall formulate our results and proofs only for orders, but it is always obvious that everything carries immediately over to the algebra situation. Let R be a complete Dedekind domain with quotient field K, and let n be an R-order in a separable K-algebra A [ 161: n is a subring of A with the same identity, and moreover A is a full R-sublattice, which means that /i is an R-lattice and contains a K-basis of A. We denote by r the Auslander-Reiten quiver of/i, i.e., vertices of r are the isomorphism classes of indecomposable n-lattices, and two vertices are joined by an arrow provided there exists an irreducible /i-map between the corresponding A-lattices. Moreover, we denote by f the universal cover of I[8], in particular, the vertices of r” are homotopy classes of walks in I-, with covering morphism F: F-P r [8]. For each arrow x + y in F we define $,, as the space of irreducible maps Irr(Fx, Fy) [4, 171 from Fx to Fy. Moreover, for indecomposable /i-lattices IV, N let r’(M, N) c Hom,(M, N) be the ith functorial radical, i.e., r’(M, N) = ri(-, N)(M), where ri(-, N) is the ith power of the radical of the presentable fun&or (-, N) = Hom,((, N) [ 21. 292 0021-8693/85 $3.00