I want to check that my understanding is correct about cohomology.
Let $X$ be a topological space $G$ be an abelian group. The universal coefficients theorem, as stated in hatcher, says that the following sequence of $\mathbb{Z}$-modules is split exact $$0\to \text{Ext}(H_{n-1}(X),G)\to H^n(X,G)\to \hom(H_n(X),G)\to 0,$$ where $H_i(X)$ is the singular homology group with $\mathbb{Z}$-coefficients.
Later in a comment, Hatcher mentioned that we take the chain complex of free $R$-modules $$\cdots\to C_{n-1}(X,R)\to C_n(X,R)\to C_{n+1}(X,R)\to \cdots$$ and take its homology modules $H_n(X,R)$, as well as the homology modules after applying $\hom_R(-,R)$, denoted $H^n(X,R)$. Then the universal coefficients theorem says that the following is a split short exact sequence of $R$-modules $$0\to \text{Ext}_R(H_{n-1}(X,R),R)\to H^n(X,R)\to \hom_R(H_n(X,R),R)\to 0.$$
However I feel that Hatcher did not give the more general case, so the following is my speculation:
Since $\mathbb{Z}$ is initial in the category of unital rings, let $R$ be any unital ring, and $\varphi:\mathbb{Z}\to R$ be the unique homomorphism, making $R$ a $\mathbb{Z}-R$ left bimodule. Consider the chain complex with $\mathbb{Z}$ coefficients $$\cdots\to C_{n-1}(X)\to C_n(X)\to C_{n+1}(X)\to \cdots$$ and apply $\hom_\mathbb{Z}(-,R)$ to obtain a complex of $R$-modules $$\cdots\to \hom_\mathbb{Z}(C_{n+1}(X),R)\to \hom_\mathbb{Z}(C_{n}(X),R)\to \hom_\mathbb{Z}(C_{n-1}(X),R)\to \cdots,$$ whose homology yields $h^n(X,R)$.
My two questions are:
- $h^n(X,R) \cong H^n(X,R)$? Which one is the 'correct' definition of cohomology?
- Is there a split exact sequence of $R$-modules $$0\to \text{Ext}_\mathbb{Z}(H_{n-1}(X),R)\to h^n(X,R)\to \hom_\mathbb{Z}(H_n(X),R)\to 0?$$