Example 3

Modules of type $\mathscr{C}_{\leq 0}$ are the modules of dimension $\leq 0$, that is to say the modules of finite length (No. 4, Remark 1). We have $\mathrm{long}_A(M) = \varepsilon(z_0(M))$ for every finite-length A-module M, where $\varepsilon : Z_0 \to \mathbf{Z}$ associates to the linear combination $\sum_m n_m[V(m)]$ the integer $\sum_m n_m$ (IV, § 2, No. 5, corollary to Prop. 8).