Example 3

If M is bounded of type $\mathscr{C}$, it is analogously the case for $M(p)$ for every $p \in \mathbf{Z}$, and one has $\chi(M(p)) = (-1)^p \chi(M)$.

4) Let $0 \to M' \to M \to M'' \to 0$ be an exact sequence of graded A-modules and graded homomorphisms of degree 0. If M, M' and M'' are bounded of type $\mathscr{C}$, one has

If M and M'' are bounded of type $\mathscr{C}$, it is analogously the case for M'; if $\mathscr{C}$ is stable and if two of the three modules are bounded of type $\mathscr{C}$, it is analogously the case for the third.

5) Let $u : C' \to C$ be a morphism of bounded complexes of type $\mathscr{C}$. Then Con ($u$) is bounded of type $\mathscr{C}$, and one has:

6) One may take for G the group $K(\mathscr{C})$ itself, and for $\varphi$ the identity; in this case one denotes by $\chi_{\mathscr{C}}(M)$ the element $\chi_\varphi(M) = \sum (-1)^n[M_n]$ of $K(\mathscr{C})$.