# Coefficient dependences of cohomology group of spaces

In Hatcher's textbook, the cohomology groups of spaces are defined by the homology of the cochain complex Hom$$_R(C_i(X;R),G)$$, dual of the original chain complex $$C_i(X;R)$$ where $$R$$ is a principal ideal domain and the abelian group $$G$$ is an $$R$$-module.

Clearly, we have two coefficient dependences: $$G$$ and $$R$$. However, when we write down the notation of the cohomology group, we use $$\begin{eqnarray} H^i(X;G), \end{eqnarray}$$ which does not specify the $$R$$-dependence. In my understanding, $$H^i(X;G)$$ is used in the case when $$R=\mathbb{Z}$$, but, later in Hatcher's book, when the coefficient ring $$R$$ of cochains is different from $$\mathbb{Z}$$, the notation $$H^i(X;G)$$ is still used.

My question is whether this notation will bring about any ambiguity. If so, why not specify explicitly the $$R$$-ring dependence of the chain complex?

It makes no difference, since the two versions are naturally isomorphic. Indeed, the version that goes through $$R$$ is obtained by applying the functor $$\operatorname{Hom}_R(R\otimes -,G)$$ to the integral chain complex, and the version that does not go through $$R$$ is obtained by applying the functor $$\operatorname{Hom}_{\mathbb{Z}}(-,G)$$ to the integral chain complex. These two functors are naturally isomorphic. Very explicitly, in the case of (say) singular homology, in both cases an $$n$$-cochain is just a function from the set of singular $$n$$-simplices to $$G$$, since $$C_n(X)$$ is the $$\mathbb{Z}$$-module on the set of singular $$n$$-simplices and $$C_n(X;R)$$ is the free $$R$$-module on the set of singular $$n$$-simplices.
• Thanks for the clear answer! It means, for a fixed $H^i(X,G)$, we can choose various applicable P.I.D. $R$ to get several splitting of it by the universal coefficient theorem. Feb 22, 2020 at 4:14