# Universal coefficient theorem for cohomology

## Statement

### For coefficients in an abelian group

Suppose $M$ is an abelian group and $X$ is a space with homology of finite type. The universal coefficients theorem relates the cohomology groups for $X$ with integral coefficients (i.e., with coefficients in $\mathbb{Z}$) to the cohomology groups with coefficients in $M$.

The theorem comes in two parts.

First, it states that there is a natural short exact sequence:

$0 \to H^n(X;\mathbb{Z}) \otimes M \to H^n(X;M) \to \operatorname{Tor}(H^{n+1}(X;\mathbb{Z}),M) \to 0$

Second, it states that the short exact sequence splits (non-canonically):

$H^n(X;M) \cong (H^n(X;\mathbb{Z}) \otimes M) \oplus \operatorname{Tor}(H^{n+1}(X;\mathbb{Z}),M)$