Noetherianity and Gelfand–Kirillov dimension of components

The category of all additive functors Mod(modΛ) 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 modΛ 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.