Universal coefficient theorem for cohomology

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):

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

Related facts

 * Universal coefficient theorem for homology
 * Dual universal coefficient theorem
 * Kunneth formula for homology
 * Kunneth formula for cohomology