Let $\mathcal{T}$ be a triangulated category that is generated by one object, say $A$ in the sense that the smallest triangulated subcategory containing $A$ and closed under coproducts and isomorphisms is $\mathcal{T}$ itself. Denote by $K\_0(\mathcal{T})$ the Grothendieck group of $\mathcal{T}$. Is $K\_0(\mathcal{T})$ generated by the equivalence class of $A$?



Generators of triangulated category and Grothendieck groups
Fernando Muro has already answered this in the comments, but perhaps a reference would help. This is all spelled out in Neeman's book on triangulated categories. In particular, see Definition 4.5.8 and Proposition 4.5.11, and their proofs.


