The radical of functorially finite subcategories (Q6588207)

The notions of co- and contravariantly finite subcategories were introduced by Auslander and Smaløin order to transfer, via the so-called approximations, informations from the whole category of modules to the specific subcategory. A functorially finite subcategory is one which is both co- and contravariantly finite and, in many senses, it can be viewed as a \textit{miniature} of the whole category of modules. This idea had already been considered by other authors under different names. The specific aim in the primary work of Auslander and Smaløwas, then, the introduction of the preprojective and the preinjective partitions. Soon, it was clear the utility of such concepts and relations with, for instance, tilting theory and, later on, with \(n\)-cluster theory. In the paper under review, the authors study the problem of when a functorially finite subcategory is representation-finite and, for that, they generalize several results concerning the representation-finiteness of an algebra in terms of, for instance, the notion of radical of a category and some usual concepts of the Auslander-Reiten theory.