Relations for the Grothendieck groups of triangulated categories (Q1858242)

The authors study the relations of the Grothendieck groups of triangulated categories of finite type by using the Auslander-Reiten triangle as well as the Auslander-Reiten quivers of triangulated categories of finite type. It is proved that the relations of the Grothendieck group of triangulated categories of finite type are generated by all Auslander-Reiten triangles. Also, it is shown that if there is no loop in the Auslander-Reiten quiver of a triangulated category of finite type then it can be described in terms of Dynkin diagrams. Some special triangulated categories without loops in its Auslander-Reiten quivers are presented.