Auslander–Reiten theory

Summary

Auslander–Reiten theory studies the category of finitely generated modules over an Artin algebra, such as a finite-dimensional algebra, through irreducible morphisms and almost split sequences. Its main tools are the Auslander–Reiten translation $\tau$ and the Auslander–Reiten quiver. Maurice Auslander and Idun Reiten introduced it in the 1970s. It is the standard method to describe module categories of algebras of finite representation type and of many tame algebras.