Editing Universal coefficient theorem (section)

{{short description|Establish relationships between homology and cohomology theories}}
In [[algebraic topology]], '''universal coefficient theorems'''   establish relationships between [[homology group]]s (or [[cohomology group]]s) with different coefficients. For instance, for every [[topological space]] {{mvar|X}}, its ''integral homology groups'':

:<math>H_i(X,\Z)</math>

completely determine its ''homology groups with coefficients in'' {{mvar|A}}, for any [[abelian group]] {{mvar|A}}:

:<math>H_i(X,A)</math>

Here <math>H_i</math> 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 complex]]es of [[free abelian group]]s. The form of the result is that other coefficients {{mvar|A}} may be used, at the cost of using a [[Tor functor]].

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