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  "body": "\n## Summary\n\nAn 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.\nEvery map into $N$ that is not a split epimorphism lifts to $E$.\nDually, every map out of $L$ that is not a split monomorphism extends to $E$.\nOver an Artin algebra, every indecomposable non-projective module $N$ is the end of one, unique up to isomorphism.\nThese sequences are the building blocks of [[Auslander–Reiten theory|2VY5M3M91Z]].\n\n## Explanations\n\n### L1 human\n\nA short exact sequence glues a module $L$ and a module $N$ into a bigger module $E$.\nIf it splits, $E$ is just $L$ and $N$ side by side; if not, the glueing is genuine.\nAn 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.\nIn this sense it is the smallest possible non-trivial extension ending at $N$.\nFor the path algebra of $1 \\to 2$, the sequence $0 \\to S_2 \\to P_1 \\to S_1 \\to 0$ is almost split.\n",
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  "name": "Almost split sequence",
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