Covering functors without groups

Abstract

Coverings in the representation theory of algebras were introduced for the Auslander-Reiten quiver of a representation-finite algebra in [Ch. Riedtmann, Algebren, Darstellungsköcher, Überlagerungen und zurüch, Comment. Math. Helv. 55 (1980) 199-224] and later for finite-dimensional algebras in [K. Bongartz, P. Gabriel, Covering spaces in representation theory, Invent. Math. 65 (3) (1982) 331-378; P. Gabriel, The universal cover of a representation-finite algebra, in: Proc. Representation Theory I, Puebla, 1980, in: Lecture Notes in Math., vol. 903, Springer, 1981, pp. 68-105; R. Martínez-Villa, J.A. de la Peña, The universal cover of a quiver with relations, J. Pure Appl. Algebra 30 (3) (1983) 277-292]. The best understood class of covering functors is that of Galois covering functorsF : A → B determined by the action of a group of automorphisms of A. In this work we introduce the balanced covering functors which include the Galois class and for which classical Galois covering-type results still hold. For instance, if F : A → B is a balanced covering functor, where A and B are linear categories over an algebraically closed field, and B is tame, then A is tame.