A question of notation in cochains groups and cohomology

Question

The (singular) homology groups are derived from (singular) chain groups $C_n$, formed by linear combinations of mappings $\sigma:\triangle^n \to X$, for some space $X$. These linear combinations can be expressed $\sum_in_i\sigma_i$, where our coefficients $n_i$ come from some abelian group, typically $\mathbb{Z}$. $C_n(X;G)$ denotes the group of (singular) $n$-chains in a space $X$ with coefficient group $G$.

When we move to cohomology, we now have cochain groups $C^n$ defined $C^n(X;G)=Hom(C_n(X),G)$. However, this notation does not specify what coefficient group we are using for $C_n$. Is it just assumed that, in this case, $C_n(X)$ denotes $C_n(X;G)$? Is there a way to construct cohomology groups using a different coefficient group for $C^n$ than for $C_n$? I would appreciate some guidance on this question.

Answer 1

Note:You'd better forget about my deleted comment if you did see.

No, the chain complex used to define the cochain complex $C^n(X;G)$ must be $C_n(X)=C_n(X;\mathbb{Z})$. And if you use two independent coefficient groups $H$ and $G$ for the chain complex and the cochain complex respectively so that you have the cochain complex $\mathrm{Hom}(C_n(X)\otimes H,G)$, then you can use the adjunction between tensor product and $\mathrm{Hom}$ functor to have $\mathrm{Hom}(C_n(X)\otimes H,G)=\mathrm{Hom}(C_n(X),\mathrm{Hom}(H,G))$, so the homology group of this cochain complex is just the cohomology group of $X$ with coefficient grop $\mathrm{Hm}(H,G)$.