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  "body": "\n## Definition\n\nLet $R$ be a [[ring|0B1261B5ED]] and let $0 \\to L \\xrightarrow{f} E \\xrightarrow{g} N \\to 0$ be a short [[exact sequence|JB7ZBFBA9C]] of left $R$-[[modules|0NZ8JA25WK]].\nThe sequence is *almost split* if:\n\n1. it does not split;\n2. every morphism $u \\colon L \\to Y$ that is not a split monomorphism factors as $u = u' f$ for some $u' \\colon E \\to Y$;\n3. every morphism $h \\colon X \\to N$ that is not a split epimorphism factors as $h = g h'$ for some $h' \\colon X \\to E$.\n\n## Remarks\n\nIn an almost split sequence, $L$ and $N$ are [[indecomposable|E2P29GCD9T]] with local endomorphism rings.\nExample: let $A$ be the path algebra of $1 \\to 2$ over a field $k$, with modules $S_2 = (0 \\to k)$, $P_1 = (k \\xrightarrow{\\id} k)$, $S_1 = (k \\to 0)$.\nThe sequence $0 \\to S_2 \\to P_1 \\to S_1 \\to 0$ does not split, since $P_1$ is indecomposable.\nFor clause 3 it suffices to test indecomposable $X$: maps from $S_1$ are $0$ or isomorphisms, $\\Hom(S_2, S_1) = 0$, and maps from the projective $P_1$ lift.\nClause 2 is checked in the same way.\nNon-example: the split sequence $0 \\to S_2 \\to S_2 \\oplus S_1 \\to S_1 \\to 0$ satisfies clauses 2 and 3, since $f$ is a split monomorphism and $g$ a split epimorphism, but it fails clause 1.\nFor a finite-dimensional algebra, the left end of the almost split sequence ending at $N$ is $\\tau N$, the Auslander–Reiten translate.\n",
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