In algebraic topology, universal coefficient theorems establish relationships between homology groups (or cohomology groups) with different coefficients. For instance, for every topological space X, its integral homology groups:

completely determine its homology groups with coefficients in A, for any abelian group A:

Here ${\displaystyle H_{i}}$ might be the simplicial homology, or more generally the singular homology. The usual proof of this result is a pure piece of homological algebra about chain complexes of free abelian groups. The form of the result is that other coefficients A may be used, at the cost of using a Tor functor.

For example, it is common to take ${\displaystyle A}$ to be ${\displaystyle \mathbb {Z} /2\mathbb {Z} }$, so that coefficients are modulo 2. This becomes straightforward in the absence of 2-torsion in the homology. Quite generally, the result indicates the relationship that holds between the Betti numbers ${\displaystyle b_{i}}$ of ${\displaystyle X}$ and the Betti numbers ${\displaystyle b_{i,F}}$ with coefficients in a field ${\displaystyle F}$. These can differ, but only when the characteristic of ${\displaystyle F}$ is a prime number ${\displaystyle p}$ for which there is some ${\displaystyle p}$-torsion in the homology.

Consider the tensor product of modules ${\displaystyle H_{i}(X,\mathbb {Z} )\otimes A}$. The theorem states there is a short exact sequence involving the Tor functor

Furthermore, this sequence splits, though not naturally. Here ${\displaystyle \mu }$ is the map induced by the bilinear map ${\displaystyle H_{i}(X,\mathbb {Z} )\times A\to H_{i}(X,A)}$.

If the coefficient ring ${\displaystyle A}$ is ${\displaystyle \mathbb {Z} /p\mathbb {Z} }$, this is a special case of the Bockstein spectral sequence.

Let ${\displaystyle G}$ be a module over a principal ideal domain ${\displaystyle R}$ (for example ${\displaystyle \mathbb {Z} }$, or any field.)