The universal coefficient theorem for cohomology over a PID

By the boundary-and-cycle lemma under Choice, $B_{n-1}C$ is free and therefore projective. Hence $0\to Z_{n}C\to C_n\stackrel{d_n}{\to }{B}_{n-1}C\to 0$ splits. Every map $H_{n}C\to G$ pulls back to a map $Z_{n}C\to G$ vanishing on $B_{n}C$ and extends across a chosen projection $C_n\to Z_{n}C$ to a cocycle. Thus ${\mathrm{ev}}_n$ is surjective.

A cocycle lies in $\ker {\mathrm{ev}}_n$ exactly when its restriction to $Z_{n}C$ vanishes modulo $B_{n}C$. Subtracting a representative that is zero on $Z_{n}C$ shows that the kernel is ${\mathrm{Hom}}_R(B_{n-1}C,G)/\mathrm{im}({\mathrm{Hom}}_R(Z_{n-1}C,G))\mathrm{.}$ Applying ${\mathrm{Hom}}_R(-,G)$ to the free presentation $0\to B_{n-1}C\to Z_{n-1}C\to H_{n-1}C\to 0$ identifies this quotient with ${\mathrm{Ext}}_R^1(H_{n-1}C,G)$. The inclusion is the extension map of the preceding lemma, and all constructions before the optional splitting are natural, proving the exact sequence.