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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?

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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.

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  • $\begingroup$ 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. $\endgroup$
    – Smart Yao
    Feb 22, 2020 at 4:14

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