Almost split sequence

Remarks

In an almost split sequence, $L$ and $N$ are indecomposable with local endomorphism rings. Example: let $A$ be the path algebra of $1 \to 2$ over a field $k$, with modules $S_2 = (0 \to k)$, $P_1 = (k \xrightarrow{\id} k)$, $S_1 = (k \to 0)$. The sequence $0 \to S_2 \to P_1 \to S_1 \to 0$ does not split, since $P_1$ is indecomposable. For 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. Clause 2 is checked in the same way. Non-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. For a finite-dimensional algebra, the left end of the almost split sequence ending at $N$ is $\tau N$, the Auslander–Reiten translate.