Almost split sequence

Summary

An almost split sequence, or Auslander–Reiten sequence, is a non-split short exact sequence $0 \to L \to E \to N \to 0$ 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.