# Universal coefficient theorem for cohomology

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## Statement

### 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):