Universal coefficient theorem allows us to calculate $H^*(X,M)$ from $H_*(X,Z)$. Do we have a "dual" universal coefficient theorem that allows us to calculate $H_*(X,M)$ from $H^*(X,Z)$?
Here $Z$ is the set of integers.
Universal coefficient theorem allows us to calculate $H^*(X,M)$ from $H_*(X,Z)$. Do we have a "dual" universal coefficient theorem that allows us to calculate $H_*(X,M)$ from $H^*(X,Z)$? Here $Z$ is the set of integers. 


Yes, there is such a universal coefficient theorem. $$0 \to Ext(H^{q+1}(X,R), G) \to H_q(X, G) \to Hom(H^q(X, R), G) \to 0$$ see Theorem 6.5.12 in Spanier's textbook "Algebraic Topology". It's on page 248. 


The proof of Spanier's 6.5.12 starts from a free chain complex with homology of finite type and replaces it by a quasiisomorphic free chain complex of finite type. Then the conclusion reduces directly to application of the usual universal coefficient theorem for computing cohomology of chain complexes from homology. The reference to EKMM should be to p. 82, which states "A reference to Adams [1] is mandatory''; [1] is "Lectures on generalized cohomology". Springer Lecture Notes 99, 1969, pp 1138. As recalled on EKMM p. 82, Adams shows how to deduce further spectral sequences from those listed on that page, by duality. Adams' (UCT 4), page 5 op cit, gives the spectral sequence that generalizes the result cited from Spanier to generalized homology/cohomology theories. 

