---
tag: M5D95FN7G3
kind: concept
name: Almost split sequence
msc: [16G70]
specializes: [JB7ZBFBA9C]
sources:
  - url: https://en.wikipedia.org/wiki/Auslander%E2%80%93Reiten_theory
---

## Summary

An almost split sequence, or Auslander–Reiten sequence, is a non-split short [[exact sequence|JB7ZBFBA9C]] $0 \to L \to E \to N \to 0$ of modules that is as close to splitting as possible.
Every map into $N$ that is not a split epimorphism lifts to $E$.
Dually, every map out of $L$ that is not a split monomorphism extends to $E$.
Over an Artin algebra, every indecomposable non-projective module $N$ is the end of one, unique up to isomorphism.
These sequences are the building blocks of [[Auslander–Reiten theory|2VY5M3M91Z]].

## Explanations

### L1 human

A short exact sequence glues a module $L$ and a module $N$ into a bigger module $E$.
If it splits, $E$ is just $L$ and $N$ side by side; if not, the glueing is genuine.
An almost split sequence is a genuine glueing that splits as soon as it is pulled back along any map into $N$ that is not a split epimorphism.
In this sense it is the smallest possible non-trivial extension ending at $N$.
For the path algebra of $1 \to 2$, the sequence $0 \to S_2 \to P_1 \to S_1 \to 0$ is almost split.