Proposition 11

Let C be a bounded complex of type $\mathscr{C}$. a) If $\mathscr{C}$ is stable, $H(C)$ is of type $\mathscr{C}$. b) If $H(C)$ is of type $\mathscr{C}$, it is analogously the case for $B(C)$ and $Z(C)$, and one has $\chi(H(C)) = \chi(C)$.

a) If $\mathscr{C}$ is stable, for every $n$ the module $Z_n(C)$ is of type $\mathscr{C}$ as kernel of $d_n : C_n \to C_{n-1}$, and $H_n(C)$ is of type $\mathscr{C}$ as cokernel of $C_{n+1} \to Z_n$. On the other hand, $H_n(C) = 0$ as soon as $C_n = 0$.

b) Suppose $H(C)$ of type $\mathscr{C}$. The canonical exact sequences:

show by induction on $n$ starting from the right bound of $C$ that $Z_n(C)$ and $B_n(C)$ are of type $\mathscr{C}$ for all $n$. One then has