Corollary

Let $M$ be an $A$-module, $P$ and $P'$ two projective resolutions of $M$; suppose that $P$ is minimal. Let $f:P\to P'$ and $g:P'\to P$ be two morphisms of resolutions. Then $f$ is injective, $g$ is surjective, and $P'$ is the direct sum of the subcomplexes $\operatorname{Im}f$ and $\operatorname{Ker}g$. Moreover $\operatorname{Ker}g$ has zero homology.

In fact, $\alpha=g\circ f$ is an automorphism of $P$ (Prop. 8). Put $\widetilde f=f\circ\alpha^{-1}$.

We have

which shows that $P'=\operatorname{Im}\widetilde f\oplus\operatorname{Ker}g$. Since the sequence

is exact and $g$ is a homomorphism, $\operatorname{Ker}g$ has zero homology.