Lemma 4

Let $f : M' \to M$ be a homomorphism of $A$-modules of type $\mathscr{C}$, and $p : P \to M$ a bounded left resolution of $P$ of type $\mathscr{C}_0$. There exist a bounded left resolution $p' : P' \to M'$ of type $\mathscr{C}_0$ and a morphism of complexes $u : P' \to P$ such that $p \circ u = f \circ p'$.

Let us argue by induction on the length $n$ of $P$, the assertion being trivial when the latter is $< 0$. Consider the mapping $g : M' \times P_0 \to M$ such that

for $x \in M', r' \in P_0$,

and its kernel $K$; the $A$-module $K$ is of type $\mathscr{C}$ since $g$ is surjective and $M' \times P_0$ and $M$ are of type $\mathscr{C}$. Let $h : P'_0 \to K$ be a surjective homomorphism, where $P'_0$ is of type $\mathscr{C}_0$; let us denote by $p'_0 : P'_0 \to M'$ (resp. $u_0 : P'_0 \to P_0$) the homomorphism composed of $h$ and the projection $K \to M$ (resp. $K \to P_0$); the homomorphism $p'_0$ is surjective and we have a commutative diagram

It is then enough to apply the induction hypothesis to the homomorphism

deduced from $u_0$.