I would like to know whether the universal coefficients theorem (for homology) holds when the coefficient ring is $\mathbb{Z}[t,t^{-1}]$. This ring is not a PID (so the classical theorem cannot be applied) but at least it is a gcd domain.
More concretely, I am interested in knowing whether the following isomorphism holds, $$H_1(X; \mathbb{Z}[t,t^{-1}]) \otimes_{\mathbb{Z}[t,t^{-1}]} \mathbb{Q}(t) \cong H_1(X; \mathbb{Q}(t)), $$ where $\mathbb{Q}(t)$ is the field of fractions of $\mathbb{Z}[t,t^{-1}]$. If the UCT held, then we would be done, since the field of fractions of a ring is always flat and $\mathrm{Tor}$ vanishes.
Best Answer
You are interested in the Universal coefficient spectral sequence. It involved higher Tor's for homology and higher Ext's for cohomology. For $H_1$ your isomorphism holds for any ring and any module because $H_0$ is always free and $H_{-n}$ are always trivial. For a flat module, the isomorphism will always hold in the homological case regardless of what homological degree you are in because all nonzero Tor's are trivial.