A few days ago, there was a similar question in an other context.

Künneth: Consider the tensor product of modules $H_i(X;\mathbb{Z})\otimes A$, where $A$ is an abelian group, $X$ is a topological space and $H_*$ denotes singular homology. The theorem states there is a short exact sequence $$0 \to H_i(X; \mathbf{Z})\otimes A \, \to \, H_i(X;A) \to \operatorname{Tor}(H_{i-1}(X; \mathbf{Z}),A)\to 0.$$

UCT: There is also a short exact sequence in singular cohomology $H^*$ $$0 \to \operatorname{Ext}_\mathbf{Z}^1(\operatorname{H}_{i-1}(X; \mathbf{Z}), A) \to H^i(X; A) \, \to \, \operatorname{Hom}_\mathbf{Z}(H_i(X; \mathbf{Z}), A)\to 0.$$

What I want to know: I know a proof of both theorems with projective and injective resolutions respectively and Eilenberg-Zilber for Künneth.

Nevertheless, is it possible to deduce the theorems from each other using the fact that Hom and Tensor are adjoint functors ? (A paper "Adjoint Functors and Equivalences" brought me to the idea)


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