The Auslander-Reiten Theory

Abstract

In this chapter the insight into the structure of module category is deepened. One section is devoted to the example of the Kronecker algebra. A series of notions and results named after Auslander and Reiten is presented, all of them constitute the Auslander-Reiten theory: The “Auslander-Reiten translate”, which associates to every indecomposable non-projective module an indecomposable non-injective module and the the “Auslander Reiten sequences”, which are short exact sequences and in a certain sense minimal among the non-split sequences. The Theorem of Auslander-Reiten concerns the structure of module categories and yield a remarkable deep insight. Since it is valid in full generality it can be considered as the crown jewel of representation theory. It makes it possible to calculate combinatorially, via a process called “knitting” studied in the next chapter, important information about certain parts—and in some cases all of it—of the module category over some algebra.