The universal coefficient theorem for cohomology over a PID

Under the stated Choice hypothesis, $B_{n-1}C$ is free and projective. Choose a section $s_n:B_{n-1}C\to C_n$ of the corestricted differential and put ${\pi }_n=1-s_n{d}_n:C_n\to Z_{n}C$. For any $h:H_{n}C\to G$, the cochain $h\circ q_n\circ {\pi }_n$ restricts on $Z_{n}C$ to $h\circ q_n$, vanishes on $B_{n}C$, and hence is a cocycle. Its evaluation is $h$, proving surjectivity.

If a cocycle $f:C_n\to G$ has zero evaluation, then $f{\mathrm{\mid }}_{Z_{n}C}=0$: its restriction factors through $H_{n}C$, and that induced map is zero. Since $d_n:C_n\twoheadrightarrow B_{n-1}C$ has kernel $Z_{n}C$, there is a unique $\psi :B_{n-1}C\to G$ with $f=\psi d_n$. Conversely every such $\psi d_n$ is a cocycle because $d_n{d}_{n+1}=0$.

The cochain $\psi d_n$ is a coboundary exactly when $\psi$ extends to $Z_{n-1}C$. Indeed, if $\psi d_n=\phi d_n$ for a cochain $\phi :C_{n-1}\to G$, surjectivity of $d_n$ onto $B_{n-1}C$ gives $\psi =\phi {\mathrm{\mid }}_{B_{n-1}C}$. Conversely, if $g:Z_{n-1}C\to G$ extends $\psi$, Choice and projectivity of $B_{n-2}C$ give a projection ${\pi }_{n-1}:C_{n-1}\to Z_{n-1}C$; then $\psi d_n=(g{\pi }_{n-1})d_n$ is a coboundary. Thus $\ker {\mathrm{ev}}_n$ is ${\mathrm{Hom}}_R(B_{n-1}C,G)\mathrm{/}\mathrm{im}({\mathrm{Hom}}_R(Z_{n-1}C,G))$.

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 ${\mathrm{Ext}}_R^1(H_{n-1}C,G)$ by Ext via a projective resolution of the first variable. Define the extension map on a representative $\psi$ by $[\psi ]\mapsto [\psi d_n]$; step 2.1 proves it is well defined and injective, while step 1.2 identifies its image with the evaluation kernel.

Naturality requires no chosen projection: for a chain map $u:C\to D$, pullback of $\psi d_n^D$ is $(\psi u_{n-1}{\mathrm{\mid }}_{B_{n-1}C})d_n^C$, exactly the map induced on the free presentations; a map $G\to G^{\mathrm{\prime }}$ commutes by postcomposition. Evaluation also commutes with these maps. The chosen projections in steps 1.1 and 2.1 establish existence and exactness only. Thus the displayed sequence is natural.