Let $\mathcal A$ be an essentially small abelian length category, so that its isomorphism classes of objects form a set, and let $\{S_\lambda\}_{\lambda \in \Lambda}$ be a set of representatives for the isomorphism classes of simple objects of $\mathcal A$. For each object $M$ of $\mathcal A$, let $m_\lambda(M)$ denote the multiplicity of $S_\lambda$ among the composition factors of $M$, which is independent of the chosen composition series by the Jordan-Hölder theorem.

Then there is an isomorphism of abelian groups

\begin{align*} K_0(\mathcal A) \cong \bigoplus_{\lambda \in \Lambda} \mathbb Z e_\lambda, \end{align*}

where $e_\lambda$ denotes the basis element corresponding to $S_\lambda$. Under the identification of $e_\lambda$ with the class $[S_\lambda]$ in $K_0(\mathcal A)$, every object $M$ satisfies

\begin{align*} [M] = \sum_{\lambda \in \Lambda} m_\lambda(M)[S_\lambda]. \end{align*}