Exercise in Spanier to get universal coefficient theorem for cohomology with local coefficients

Solution 1:

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 $$0 \to Ext^1_R(H_{i-1}(X,A;\Gamma), G) \to H^i(X,A; Hom_R(\Gamma, G)) \to Hom_R(H_i(X,A;\Gamma), G) \to 0.$$

Solution 2:

After discussion with the OP, here's a little note that I wrote offering a modern point of view on this theorem - viewing it as a special case of the algebraic UCT, using the point of view that local coefficient systems are functors $X\to D(R)$, and that their co/homology can be interpreted as the homology of a limit/colimit of that functor.