Proposition 7

Let $\mathscr{C}$ be a hereditary set of classes of modules such that every module of type $\mathscr{C}$ has finite length. Let $\mathscr{S}$ be the set of classes of simple modules belonging to $\mathscr{C}$. Then the family $([S]_{\mathscr{C}})_{S\in\mathscr{S}}$ is a basis of the $\mathbf{Z}$-module $K(\mathscr{C})$, and we have

for every module E of type $\mathscr{C}$.

Formula (8) follows from Proposition 3 applied to a Jordan–Hölder series of E. By Lemma 1, for every element S of $\mathscr{S}$, there exists a $\mathbf{Z}$-linear mapping $\varphi_S$ from $K(\mathscr{C})$ to $\mathbf{Z}$ such that $\varphi_S([E]_{\mathscr{C}}) =\ell_S(E)$ for every module E of type $\mathscr{C}$. In particular, we have $\varphi_S([S]_{\mathscr{C}}) = 1$ and $\varphi_S([S']_{\mathscr{C}}) = 0$ for every $S'\not= S$ in $\mathscr{S}$. It follows that the elements of the form $[S]_{\mathscr{C}}$ (for $S\in \mathscr{S}$) are linearly independent over $\mathbf{Z}$; these elements generate $K(\mathscr{C})$ by formula (8).