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

中文翻译：

---

三角范畴的 Auslander-Reiten 三角形和 Grothendieck 群

我们证明，如果 Auslander-Reiten 三角形生成具有（共）生成器的 Hom-finite Krull-Schmidt 三角范畴的 Grothendieck 群的关系，则该范畴只有有限多个不可分解对象的同构类直到翻译。这给出了巴特勒和 Auslander-Reiten 关于格洛腾迪克群关系的定理的三角反演。我们的方法在 Frobenius 类别的背景下有应用。