[Math] Confusion about cohomology and universal coefficients theorem.

algebraic-topologyhomological-algebrahomology-cohomology

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:

  1. $h^n(X,R) \cong H^n(X,R)$? Which one is the 'correct' definition of
    cohomology?
  2. Is there a split exact sequence of $R$-modules $$0\to
    \text{Ext}_\mathbb{Z}(H_{n-1}(X),R)\to h^n(X,R)\to
    \hom_\mathbb{Z}(H_n(X),R)\to 0?$$

Best Answer

EDIT: The version of (2) using homology with coefficients in R is not true. Consider $X = \Bbb{RP}^2$, and $R = \Bbb{Z}/4$.

The homology groups are $$ \Bbb Z, \Bbb Z/2, 0, \ldots $$ The universal coefficient theorem tells you that the homology and cohomology with mod-4 coefficients is $$ \Bbb Z/4, \Bbb Z/2, \Bbb Z/2, 0, \ldots $$ but your formula predicts that the cohomology with mod-4 coefficients is $$ \Bbb Z/4, \Bbb Z/2, \Bbb Z/2^2, \Bbb Z/2, 0, \ldots $$

However, (2) is true as you've stated it. Sorry; I got mixed up as to the two displayed equations.

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