Theorem 1

Let $\mathscr{C}_0$ and $\mathscr{C}$ be two additive and left exact sets of class of $A$-modules such that $\mathscr{C}_0 \subset \mathscr{C}$ and every $A$-module of type $\mathscr{C}$ possesses a bounded left resolution of type $\mathscr{C}_0$. Then the homomorphism $\alpha : K(\mathscr{C}_0) \to K(\mathscr{C})$ deduced from the inclusion of $\mathscr{C}_0$ in $\mathscr{C}$ is bijective; if $M$ is an $A$-module of type $\mathscr{C}$ and $P$ a bounded left resolution of $M$ of type $\mathscr{C}_0$, one has $\alpha^{-1}([M]_{\mathscr{C}}) = \chi_{\mathscr{C}_0}(P)$ (X, p. 41, example 6).