Exercise 10, § 2

Let $u : C \to C'$ be a morphism of A-complexes, P a projective A-module, $n$ an integer $\geqslant 0$,

an A-homomorphism. Now prove that there exists a complex $\tilde{C}$, an exact sequence of complexes

and a morphism $\tilde{u} : \tilde{C} \to C'$ such that $\tilde{u} \circ i = u$ and such that the morphism $C(i) : \mathrm{Con}\,(u) \to \mathrm{Con}\,(\tilde{u})$ induced by $i$ satisfies the following properties:

$\alpha)$ The mapping $H_p(C(i)) : H_p(\mathrm{Con}\,(u)) \to H_p(\mathrm{Con}\,(\tilde{u}))$ is bijective for $p \neq n, n+1$; $\beta)$ There exists an exact sequence