Auslander–Reiten–Smalø conjecture on components of Auslander–Reiten quivers

Let $A$ be an Artin algebra, and let $\Gamma_A$ be its associated Auslander–Reiten quiver. An Artin algebra is of infinite representation type if it has infinitely many pairwise nonisomorphic indecomposable finitely generated modules.

Auslander–Reiten–Smalø conjecture. If $A$ is of infinite representation type, then $\Gamma_A$ has infinitely many connected components.

This conjecture concerns the global structure of Auslander–Reiten quivers and was posed for Artin algebras. The paper establishes it for finite-dimensional algebras over perfect fields, while the formulation above is stated in the broader Artin-algebra setting.