Universal coefficient theorem for cohomology
For coefficients in an abelian group
Suppose is an abelian group and is a space with homology of finite type. The universal coefficients theorem relates the cohomology groups for with integral coefficients (i.e., with coefficients in ) to the cohomology groups with coefficients in .
The theorem comes in two parts.
First, it states that there is a natural short exact sequence:
Second, it states that the short exact sequence splits (non-canonically):