---
tag: 40F8KNF8Q1
kind: definition
name: Auslander–Reiten theory
msc: [16G70, 16G10]
defines: 2VY5M3M91Z
uses: [3CTW1P4HBP, 0NZ8JA25WK, E2P29GCD9T, BFFA03GE3V, M5D95FN7G3, 125Z033VCQ]
sources:
  - url: https://en.wikipedia.org/wiki/Auslander%E2%80%93Reiten_theory
---

## Definition

Let $k$ be a field and let $A$ be a finite-dimensional [[associative algebra|3CTW1P4HBP]] over $k$.
Let $\operatorname{mod} A$ be the category of finitely generated left $A$-[[modules|0NZ8JA25WK]].

1. A morphism $f \colon M \to N$ between [[indecomposable|E2P29GCD9T]] modules is *irreducible* if it is neither a split monomorphism nor a split epimorphism, and the following holds.
In every factorization $f = g h$, either $h$ is a split monomorphism or $g$ is a split epimorphism.
2. For $M$ in $\operatorname{mod} A$ with minimal [[projective|BFFA03GE3V]] presentation $P_1 \xrightarrow{p} P_0 \to M \to 0$, the *transpose* $\operatorname{Tr} M$ is the cokernel of $\Hom_A(p, A) \colon \Hom_A(P_0, A) \to \Hom_A(P_1, A)$, a right $A$-module.
3. The *Auslander–Reiten translation* is $\tau M = D \operatorname{Tr} M$, where $D = \Hom_k(\cdot, k)$.

*Auslander–Reiten theory* is the study of $\operatorname{mod} A$ through irreducible morphisms, the translation $\tau$, [[almost split sequences|M5D95FN7G3]] and the [[Auslander–Reiten quiver|125Z033VCQ]].

## Remarks

The basic existence theorem: if $M$ is indecomposable and not projective, there is an almost split sequence $0 \to \tau M \to E \to M \to 0$.
Dually, for $N$ indecomposable and not injective there is an almost split sequence starting at $N$, and it ends at $\tau^{-1} N = \operatorname{Tr} D N$.
Example: for the path algebra of $1 \to 2$, with simple modules $S_1, S_2$ and the projective cover $P_1$ of $S_1$, one has $\tau S_1 = S_2$.
The almost split sequence is $0 \to S_2 \to P_1 \to S_1 \to 0$.
The translation $\tau$ sends projective modules to $0$.
The theory works more generally for Artin algebras over a commutative Artinian ring $R$, with $D = \Hom_R(\cdot, E)$ for the injective envelope $E$ of $R/\operatorname{rad} R$.