Adaptions I have to make to go from integer coefficients to coefficients in $R$

Question

Let $R$ be a unital ring. I can prove that $\partial \circ \partial$ for the boundary map between singular homology groups with integer coefficients. Now I want to generalise to coefficients in $R$.

My question is: do I have to make any adaptions at all? It seems to me that in the integer case the chain groups are taken to be free abelian groups and if I replace this chain group definition with one using a free $R$-module (generated by the $p$-simplices) then the rest of the proof stays the same.

Answer 1

You have two options:

• either do as you say and notice that the proof stays the same (which is true: none of the formulas depend on the fact that the coefficients are integers);
• or remark that $C_*(X; R) = C_*(X; \mathbb{Z}) \otimes_\mathbb{Z} R$, and that the map induced by $\partial$ over the integers is still $\partial$ over $R$. It follows immediately that $\partial \circ \partial = 0$ for the chains with $R$ coefficients.