I have a (maybe dumb) question about the relation between the Künneth theorem and the Theorem-universal coefficient theorem (UCT for short) in $KK$-theory (for the setting see "The Künneth theorem and the universal coefficient theorem for Kasparov’s generalized K-functor"- Rosenberg and Schochet). 


Let $N$ be the bootstrap class of $C^\*$-algebras (i.e. see here [commutative diagram with $K\_{i+1}(A)\to K\_i(A\rtimes\_{\rho} \mathbb{R})$ (for $C^\*$-algebras)](https://mathoverflow.net/questions/241884/commutative-diagram-with-k-i1a-to-k-ia-rtimes-rho-mathbbr-for-c) for the definition). 


**UCT**: Let $A\in N$ and $B$ be $\sigma$-unital. Then there is a short exact sequence $$0\to Ext\_{\mathbb{Z}}^1(K\_\*(A), K\_\*(B))\to KK\_\*(A,B)\to Hom(K\_\*(A), K\_\*(B))\to 0.$$


**Künneth**: Let $A\in N$, $B$ be $\sigma$-unital and $K\_\*(B)$ finitely generated. Then there is a short exact sequence $$0\to K^\*(A)\otimes K\_\*(B)\to KK\_\*(A,B)\to Tor\_1^{\mathbb{Z}}(K^\*(A),K\_\*(B))\to 0.$$



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Edit: In the paper the theorems are proved independently. 


By definition it is $Ext\_{\mathbb{Z}}^1(K\_\*(A), K\_\*(B))=H^1(Hom\_{\mathbb{Z}}(P\_\*,K\_\*(B)) [\cong H^1(Hom\_{\mathbb{Z}}(K\_\*(A),I^\*))\enspace ]$, where $P\_\*$ is a projective resolution of $K\_\*(A)$ and $I^\*$ an injective resolution of $K\_\*(B)$. $H^1$ denotes the first cohomology of the cochain complex $Hom\_{\mathbb{Z}}(P\_\*,K\_\*(B))$.


Then it is $Tor\_1^{\mathbb{Z}}(K^\*(A),K\_\*(B))=H\_1(P\_\*\otimes K\_\*(B)) [\cong H\_1(K^\*(A)\otimes D\_\*))]$, where $P\_\*$ is a projective resolution of $K^\*(A)$ and $D\_\*$ is a projective resolution of $K\_\*(B)$. $H\_1$ is the first homology of the chain complex $P\_\*\otimes K\_\*(B)$. 


I'm not sure if in this case we can use that homology and cohomology commute with exact functors (since the tensor product is exact for $P\_\*$ projective and Hom is exact for $P\_\*$ projective and $I^\*$ injective respectively). .


Edit (very simple example): If you take $A=\mathbb{C}$ and $B=K(H)$ for a separable finite dimensional complex Hilbert space $H$, it is $Ext\_{\mathbb{Z}}^1(K\_\*(A), K\_\*(B))=0=Tor\_1^{\mathbb{Z}}(K^\*(A),K\_\*(B))$ and $KK\_\*(A,B)\cong Hom(K\_\*(A), K\_\*(B))\cong K^\*(A)\otimes K\_\*(B)\cong \mathbb{Z} $.



Relation between the UCT and Künneth ($C^*$-algebras)
It is absolutely true that these two theorems are conceptually related. But instead of one being a consequence of the other, both rather follow from the same machinery being applied to slightly different setups: left deriving the covariant hom functor vs right deriving the contravariant hom functor. This is clarified in Example 4.5 in <http://arxiv.org/pdf/math/0702146.pdf>


