Grothendieck group of an essentially small triangulated category

Let $\mathcal{T}$ be an essentially small triangulated category (Triangulated category), with translation $[1]$ and class of distinguished triangles. Essential smallness supplies a set $\mathrm{Iso}(\mathcal{T})$ of isomorphism classes of objects, so the free abelian group $\mathbb{Z}[\mathrm{Iso}(\mathcal{T})]$ on that set exists (Free abelian group on a set). Write $[X]$ for the image of the generator belonging to the isomorphism class of $X$. The triangulated Grothendieck group of $\mathcal{T}$ is

$K_0^{\mathrm{t}\mathrm{r}\mathrm{i}}(\mathcal{T}):=\mathbb{Z}[\mathrm{Iso}(\mathcal{T})]/⟨[Y]-[X]-[Z]:X\to Y\to Z\to X[1]\text{ is a distinguished triangle}⟩\mathrm{.}$

Thus the defining relation is $[Y]=[X]+[Z]$ for every distinguished triangle $X\to Y\to Z\to X[1]$ of $\mathcal{T}$, with the cochain orientation fixed by the translation (Distinguished triangle). The subgroup generated by these elements is a subgroup of an abelian group, so the quotient is an abelian group; the presentation uses triangle relations and not merely the biproduct relations that define a split Grothendieck group.

The definition applies in particular to the strictly full subcategory $D_{\mathrm{p}\mathrm{e}\mathrm{r}\mathrm{f}}(A)$ of perfect objects inside $D(A\text{-}\mathrm{M}\mathrm{o}\mathrm{d})$, introduced in the preceding definition, and to $D^b(\mathcal{C})$ for an essentially small abelian category $\mathcal{C}$ under the standing derived-localization size convention of Derived category of an abelian category; in both cases the ambient category is triangulated and the subcategory or localization is essentially small, so $\mathrm{Iso}$ is a set.

Shift and functoriality properties of $K_0^{\mathrm{t}\mathrm{r}\mathrm{i}}$ are not assumed here: they are proved in the following lemma.