Universal Coefficient Theorem

A theorem relating homology or cohomology with arbitrary coefficients to homology or cohomology with integer coefficients via short exact sequences involving Tor and Ext, allowing the computation of (co)homology groups with new coefficients from integral (co)homology data.

A family of exact sequences relating (co)homology groups with arbitrary coefficients to homology or cohomology with base coefficients via Hom, Ext, and Tor functors; prototype statements express H^n(X;G) in terms of Hom(H_n(X),G) and Ext^1(H_{n-1}(X),G), and similarly for homology with tensor and Tor terms.

A theorem relating homology (or cohomology) with arbitrary coefficients to homology (or cohomology) with base coefficients via short exact sequences involving Tor and Ext (or Hom) terms that measure extension and torsion phenomena.