# Kunneth Theorem for free chain complexes with free cycles and boundaries, over a graded commutative ring

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).

• This is in Hatcher, end of Chapter 3. – Sheel Stueber Jun 5 '18 at 2:34
• No, Hatcher assumes that R is a PID. – Ben Jun 5 '18 at 2:41
• Ah, my fault then – Sheel Stueber Jun 5 '18 at 19:34
• The proof in Spanier around page 228 can be adapted to your hypothesis. – Pedro Tamaroff Jun 6 '18 at 13:15