Artin algebras of finite type and finite categories of $\Delta$-good modules

TL;DR

This paper provides an alternative proof that Artin algebras with a nilpotent infinite radical are of finite type, and characterizes finite subcategories of -good modules in quasi-hereditary algebras using morphism depth.

Contribution

It introduces a new proof technique using postprojective and preinjective partitions and characterizes finite subcategories of -good modules.

Findings

Proves that if the square of the infinite radical is zero, the algebra is finite type.

Characterizes finite subcategories of -good modules via morphism depth.

Provides a new approach to understanding the structure of Artin algebras.

Abstract

We give an alternative proof to the fact that if the square of the infinite radical of the module category of an Artin algebra is equal to zero then the algebra is of finite type by making use of the theory of postprojective and preinjective partitions. Further, we use this new approach in order to get a characterization of finite subcategories of $\Delta$-good modules of a quasi-hereditary algebra in terms of depth of morphisms similar to a recently obtained characterization of Artin algebras of finite type.