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):

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

Related facts

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