Auslander–Reiten Triangles and Grothendieck Groups of Triangulated Categories

Abstract We prove that if the Auslander–Reiten triangles generate relations for Grothendieck group of a Hom-finite Krull–Schmidt triangulated category with (co)generator, then has only finitely many isomorphism classes indecomposable objects up to translation. This gives converse theorem Butler and on groups. Our approach applications in context Frobenius categories.