Definition 8

Let M be a bounded graded A-module of type $\mathscr C$ and $(M_n)$ its grading. We call the $\varphi$-characteristic of M and denote $\chi_\varphi(M)$ or simply $\chi(M)$ the element $\sum (-1)^n \varphi([M_n])$ of G.

This definition applies in particular when M is the graded module underlying a complex of A-modules.