Example 2

We may take as $\mathscr C$ the set of classes of modules of finite length and as $\varphi: K(\mathscr C) \to \mathbf Z$ the homomorphism defined by $\varphi([M]) = \operatorname{long}_A(M)$.

We say that a graded A-module M is of type $\mathscr C$ if $M_n$ is of type $\mathscr C$ for all n (for this, it is necessary when M is bounded and sufficient when $\mathscr C$ is stable that the module M be of type $\mathscr C$).