Does tensor product commute with homology?

My question is if homology commutes with taking tensor product. I believe in general it is not true but when the tensor product is with a projective module it is. I would like to take a look at a proof but I havent found one yet.

I was trying to proof the naturality of Kunneth exact sequence when I thought the second part of my question. If indeed homology commutes up to isomorphism, with taking the tensor product with a projective module, is this isomorphism natural?

This is: If $C'$ is a chain complex and $C_i$ is the chain complex which is $C_i$ in dimension, $i$ and zero in every other dimension and $C_i$ is a projective module, is this isomorphism natural?

$\;H_n ( C_i\bigotimes C')\cong C_i\otimes H_{n-i}( C').$

It's not true for an arbitrary module; the map $H_n(C) \otimes A \to H_n(C\otimes A)$ is injective but in general has a nontrivial cokernel. If $A$ is projective, then $\operatorname{Tor}(*, A) = 0$ (more or less by definition, as it has an obvious projective resolution), and the universal coefficient theorem does give an isomorphism.
• Does the same happen if A is an arbitrary chain complex and $C$ is the one that is projetive? – allizdog Oct 25 '17 at 20:37
• @allizdog: Just getting back to this. I don't know offhand, but I suspect the result still holds by the symmetry of $\operatorname{Tor}$. – anomaly Nov 9 '17 at 17:13