Corollary 3 (universal coefficient formula)

Suppose the ring $A$ is principal. If the complexes $C$ and $C'$ are free, the sequences of $A$-modules (11) are exact and split; if the complex $C$ is free, the sequences of $A$-modules (13) are exact and split for every $A$-module $N$. Indeed, $B(C), Z(C)$ and $B(C')$ are submodules of the free modules $C, C, C'$, so are free (VII, § 3, cor. 2 to th. 1), and we apply cor. 2.