Noetherianity and Gelfand–Kirillov dimension of components

Abstract

The category of all additive functors Mod ( mod Λ ) for a finite dimensional algebra Λ were shown to be left Noetherian if and only if Λ is of finite representation type by M. Auslander. Here we consider the category of all additive graded functors from the category of associated graded category of mod Λ to graded vector spaces. This category decomposes into subcategories corresponding to the components of the Auslander–Reiten quiver. For a regular component we show that the corresponding graded functor category is left Noetherian if and only if the section of the component is extended Dynkin or infinite Dynkin.