# universal coefficient theorem

The Universal Coefficient Theorem for homology expresses the homology groups with coefficients in an arbitrary abelian group $G$ in terms of the homology groups with coefficients in $\mathbb{Z}$.

Theorem (Universal Coefficients for Homology)
Let $K$ be a chain complex of free abelian groups, and let $G$ any abelian group. Then there exists a split short exact sequence