I'm looking for a reference for the following version of the Kunneth theorem.
Let $R$ be a commutative ring, graded by a monoid $M$. Let $A_*$ and $B_*$ be chain complexes, where $A_n$ and $B_n$ are free $M$-graded $R$-modules for $n \geq 0$. Assume that, for all $n \geq 0$, the submodule of cycles $\ker(\partial^A : A_n \rightarrow A_{n-1})$ is free and the submodule of boundaries im$(\partial^A : A_{n+1} \rightarrow A_{n})$ is free (and similarly for boundaries and cycles in $B_*$). Then, for all $n \geq 0$ there is a short exact sequence $$0 \rightarrow \bigoplus_{p+q = n} H_p(A_*) \otimes_R H_q(B_*) \rightarrow H_n(A_* \otimes B_*) \rightarrow \bigoplus_{p+q=n-1} Tor_1^R(H_p(A_*),H_q(B_*)) \rightarrow 0$$ which is natural in the arguments $A_*$ and $B_*$, and which splits (but not necessarily naturally).