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In algebraic topology, universal coefficient theorems establish relationships between homology and cohomology theories. For instance, the ''integral homology theory'' of a topological space , and its ''homology with coefficients'' in any abelian group are related as follows: the integral homology groups : completely determine the groups : Here might be the simplicial homology or more general singular homology theory: the result itself 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 : Furthermore, this sequence splits, though not naturally. Here is a map induced by the bilinear map . If the coefficient ring is , this is a special case of the Bockstein spectral sequence. 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「Universal coefficient theorem」の詳細全文を読む スポンサード リンク
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