text: Here  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  may be used, at the cost of using a Tor functor.
  For example it is common to take  to be , 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  of  and the Betti numbers  with coefficients in a field . These can differ, but only when the characteristic of  is a prime number  for which there is some -torsion in the homology.
  Statement of the homology case
  Consider the tensor product of modules . The theorem states there is a short exact sequence involving the Tor functor
  Furthermore, this sequence splits, though not naturally. Here  is the map induced by the bilinear map .
  If the coefficient ring  is , this is a special case of the Bockstein spectral sequence.
  Universal coefficient theorem for cohomology
  Let  be a module over a principal ideal domain  (e.g.,  or a field.)
  There is also a universal coefficient theorem for cohomology involving the Ext functor, which asserts that there is a natural short exact sequence
  As in the homology case, the sequence splits, though not naturally.
  In fact, suppose
  and define:
  Then  above is the canonical map: