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The universal coefficient theorem relates homology or cohomology with different coefficient groups through tensor and Tor, or Hom and Ext, respectively. The universal coefficient theorem for homology is its tensor-Tor sequence; the universal coefficient theorem for cohomology is its Hom-Ext sequence.

This is the cohomological form of the universal coefficient theorem, with Ext and Hom terms rather than the homological tensor and Tor terms.

For an abelian coefficient group $G$, cohomology fits into a split short exact sequence

For an abelian coefficient group $G$, there is a split short exact sequenceFor $G=\mathbb{Q}$, the Tor term vanishes.

This is the homology case of the universal coefficient theorem; its cohomology case uses Hom and Ext rather than tensor product and Tor.