Proposition 12

Let $0 \to C' \to C \to C'' \to 0$ be an exact sequence of complexes.

a) If $H(C), H(C')$ and $H(C'')$ are bounded of type $\mathscr{C}$, one has

b) If $\mathscr{C}$ is stable, and if two of the graded modules $H(C), H(C')$ and $H(C'')$ are bounded of type $\mathscr{C}$, the same holds for the third.

Part a) follows from Lemma 4 applied to the complex with zero homology defined by the homology exact sequence associated with the given exact sequence. Part b) follows, by considering this homology exact sequence, from the following lemma: