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  "body": "\n## Definition\n\nLet $k$ be an algebraically closed field and let $A$ be a finite-dimensional [[associative algebra|3CTW1P4HBP]] over $k$.\nLet $M$ and $N$ be finitely generated left $A$-modules.\nLet $\\operatorname{rad}(M, N)$ be the space of morphisms $f \\colon M \\to N$ such that $h f g$ is never an isomorphism, for $X$ [[indecomposable|E2P29GCD9T]], $g \\colon X \\to M$ and $h \\colon N \\to X$.\nLet $\\operatorname{rad}^2(M, N)$ be the space spanned by the composites $f_2 f_1$ with $f_1 \\in \\operatorname{rad}(M, Y)$ and $f_2 \\in \\operatorname{rad}(Y, N)$ for some module $Y$.\nThe *Auslander–Reiten quiver* of $A$ is the [[quiver|RE9YFFS3TS]] $\\Gamma_A$ such that:\n\n1. its vertices are the isomorphism classes $[M]$ of indecomposable finitely generated $A$-modules;\n2. the number of arrows $[M] \\to [N]$ is $\\dim_k \\operatorname{rad}(M, N) / \\operatorname{rad}^2(M, N)$.\n\nIt comes with the partial bijection $\\tau$ on vertices, the Auslander–Reiten translation, defined on the classes of non-projective modules.\n\n## Remarks\n\nFor indecomposable $M$ and $N$, the nonzero elements of $\\operatorname{rad}(M, N) \\setminus \\operatorname{rad}^2(M, N)$ are exactly the irreducible morphisms.\nLet $0 \\to \\tau N \\to E \\to N \\to 0$ be an [[almost split sequence|M5D95FN7G3]], with $E = \\bigoplus_r E_r^{m_r}$ and $E_r$ indecomposable.\nThen the arrows ending at $[N]$ start exactly at the classes $[E_r]$, and the arrows starting at $[\\tau N]$ end exactly there.\nExample: for the path algebra of $1 \\to 2$, the indecomposable modules are $S_2 = P_2$, $P_1$ and $S_1$.\nThe quiver is $[S_2] \\to [P_1] \\to [S_1]$, with $\\tau [S_1] = [S_2]$, coming from the almost split sequence $0 \\to S_2 \\to P_1 \\to S_1 \\to 0$.\nNon-example: there is no arrow $[S_2] \\to [S_1]$, since $\\Hom(S_2, S_1) = 0$.\n",
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