| Universal Coefficient Theorem |
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| CATEGORIES ABOUT UNIVERSAL COEFFICIENT THEOREM | |
| algebraic topology | |
| homological algebra | |
| mathematical theorems | |
H do in a certain, definite sense determine the groups H Here ''H''∗ might be the about Chain Complex es of Free Abelian Group s. 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 ''A'' to be Z/2Z, 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 Number s ''b''''i'' of ''X'' and the Betti numbers ''b''''i'',''F'' with coefficients in a Field ''F''. These can differ, but only when the Characteristic of ''F'' is a Prime Number ''p'' for which there is some ''p''-torsion in the homology. The statement of the universal coefficient theorem runs as follows. Consider : where ''H''''i'' means ''H''''i''(''X'',Z). Then there is an Injective Group Homomorphism ι from it to ''H''''i''(''X'',''A''). The theorem describes the Cokernel of ι as :Tor(''H''''i'' − 1,''A''). This Tor group can therefore be described as the obstruction to ι being an isomorphism, which could be thought of as the 'expected' result. This can be summarized saying there is a Natural Short Exact Sequence : Furthermore, this is a Split Sequence (but the splitting is not natural). There is also a universal coefficient theorem for cohomology involving the Ext Functor , stating that there is a natural short exact sequence : As in the homological case, the sequence splits, though not naturally. REFERENCES
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