Proposition 1

Let $(P, d_P)$ and $(E, d_E)$ be two complexes of $A$-modules and $r$ an integer.

a) Let $(u_i : P_i \to E_i)_{i \leq r}$ be a family of homomorphisms such that $d_E \circ u_i = u_{i-1} \circ d_P$ for $i \leq r$. Suppose that $P_i$ is projective for $i > r$ and that $H_i(E) = 0$ for $i \geq r$. Then the family of the $u_i$ extends to a morphism of complexes from $P$ into $E$; two such extensions are homotopic.

b) Let $(u^i : P^i \to E^i)_{i \leq r}$ be a family of homomorphisms such that $u^i \circ d_P = d_E \circ u^{i-1}$ for $i \leq r$. Suppose that $E^i$ is injective for $i > r$ and that $H^i(P) = 0$ for $i \geq r$. Then the family of the $u^i$ extends to a morphism of complexes from $P$ into $E$; two such extensions are homotopic.

Let us prove a). The existence of an extension $v$ of the family $(u_i)_{i \leq r}$ follows at once from Lemma 2 by induction. Let $v'$ be another extension; put $f = v' - v$, and construct by induction on the integer $n$ a homomorphism $k_n : P_n \to E_{n+1}$ such that $f_n = d_E \circ k_n + k_{n-1} \circ d_P$. For $i \leq r$, take $k_i = 0$. Let $n \geq r$ and suppose that the $k_i$ have been constructed for $i \leq n$. Consider then the diagram

The hypotheses of Lemma 1 are satisfied; there therefore exists an $A$-homomorphism $k_{n+1} : P_{n+1} \to E_{n+2}$ such that $f_{n+1} = d_E \circ k_{n+1} + k_n \circ d_P$, whence a).

The proof of b) is analogous, via Lemmas 1 bis and 2 bis.