---
tag: 2VY5M3M91Z
kind: concept
name: Auslander–Reiten theory
msc: [16G70, 16G10]
sources:
  - url: https://en.wikipedia.org/wiki/Auslander%E2%80%93Reiten_theory
same_as:
  wikidata: Q4822775
  wikipedia: 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|M5D95FN7G3]].
Its main tools are the Auslander–Reiten translation $\tau$ and the [[Auslander–Reiten quiver|125Z033VCQ]].
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.

## Explanations

### L1 human

A module category has too many maps to look at all of them.
Auslander–Reiten theory keeps only the indecomposable modules and the maps that cannot be split into simpler maps, the irreducible ones.
These form a quiver, a map of the whole category.
The translation $\tau$ moves along this map, taking a module to the left end of the almost split sequence that ends at it.
For the path algebra of $1 \to 2$, the map has three modules and two irreducible maps between them.