It is, it seems, rather well-known that there's no universal coefficient theorem for cohomology with local coefficients. However in Spanier's book Algebraic topology, there is an exercise that asks the reader to do something that at least goes by that name! Specifically (from page 283),
4 If $\Gamma$ is a local system of $R$-modules on $X$ and $G$ is an $R$-module, there is a local system $\mathrm{Hom}(\Gamma,G)$ of $R$-modules on $X$ which assigns to $x\in X$ the module $\mathrm{Hom}(\Gamma(x),G)$. Prove that
$$
\Delta^*(X,A;\mathrm{Hom}(\Gamma,G)) \approx \mathrm{Hom}(\Delta(X,A;\Gamma),G)
$$
Deduce a universal-coefficient formula for cohomology with local coefficients.
It does not appear that he is assuming $R$ is a PID or anything special. A 2018 paper (Local Coefficients Revisited) says "there is a version [of the local coefficient UCT] in [Spanier], p. 283, though its application is limited".
Though there's no actual statement in Spanier, as is apparent from the quote above!
What should the statement be, and why is it of limited application? (I'm not particularly interested in the proof at present, just what the theorem is meant to say!)
Best Answer
I think the identity Spanier proposes is true, and should not be difficult to prove. The question is what piece of homological algebra one should then apply to it.
If $R$ is left hereditary (eg a PID) and either $G$ is an injective $R$-module (unlikely) or else $\Delta(X, A;\Gamma)$ is a complex of projective $R$-modules (which holds iff the $\Gamma(x)$ are projective $R$-modules), then page 114 of Cartan-Eilenberg gives a standard-looking UCT, of the form $$0 \to Ext^1_R(H_{i-1}(X,A;\Gamma), G) \to H^i(X,A; Hom_R(\Gamma, G)) \to Hom_R(H_i(X,A;\Gamma), G) \to 0.$$