Finite dimensional objects in distinguished triangles

Auslander-Reiten triangles and Grothendieck groups of triangulated categories

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.

Auslander-Reiten 三角存在性的推论及其在自内射代数完美复形中的应用

Consequences of the existence of Auslander-Reiten triangles with applications to perfect complexes for self-injective algebras

本文利用 Auslander-Reiten 三角与 Grothendieck 群上的双线性型刻画自内射代数完美复形在 AR 箭图中的位置，揭示同调分布规律，并在对称代数情形获得 Hermite 型计算方法。