# 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