Associative rings and algebras Definition

Auslander–Reiten quiver

Builds on Quiver Associative algebra Indecomposable module Almost split sequence

Definition

Let $k$ be an algebraically closed field and let $A$ be a finite-dimensional associative algebra over $k$ . Let $M$ and $N$ be finitely generated left $A$ -modules. Let $\operatorname{rad}(M, N)$ be the space of morphisms $f \colon M \to N$ such that $h f g$ is never an isomorphism, for $X$ indecomposable, $g \colon X \to M$ and $h \colon N \to X$ . Let $\operatorname{rad}^2(M, N)$ 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 $Y$ . The Auslander–Reiten quiver of $A$ is the quiver $\Gamma_A$ such that:

its vertices are the isomorphism classes $[M]$ of indecomposable finitely generated $A$ -modules;
the number of arrows $[M] \to [N]$ is $\dim_k \operatorname{rad}(M, N) / \operatorname{rad}^2(M, N)$ .

It comes with the partial bijection $\tau$ on vertices, the Auslander–Reiten translation, defined on the classes of non-projective modules.

Remarks

For indecomposable $M$ and $N$ , the nonzero elements of $\operatorname{rad}(M, N) \setminus \operatorname{rad}^2(M, N)$ are exactly the irreducible morphisms. Let $0 \to \tau N \to E \to N \to 0$ be an almost split sequence, with $E = \bigoplus_r E_r^{m_r}$ and $E_r$ indecomposable. Then the arrows ending at $[N]$ start exactly at the classes $[E_r]$ , and the arrows starting at $[\tau N]$ end exactly there. Example: for the path algebra of $1 \to 2$ , the indecomposable modules are $S_2 = P_2$ , $P_1$ and $S_1$ . The quiver is $[S_2] \to [P_1] \to [S_1]$ , with $\tau [S_1] = [S_2]$ , coming from the almost split sequence $0 \to S_2 \to P_1 \to S_1 \to 0$ . Non-example: there is no arrow $[S_2] \to [S_1]$ , since $\Hom(S_2, S_1) = 0$ .

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Tag	`V1X38GS4BX`
URI	`m2://V1X38GS4BX`
Kind	definition
Writing	W0 stub
MSC	16G70
Defines	Auslander–Reiten quiver
Uses	Quiver, Associative algebra, Indecomposable module, Almost split sequence
Changed	2 November 2026

Auslander–Reiten quiver

Definition

Remarks

Reading path

Further reading

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