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In homological algebra, the hyperhomology or hypercohomology of a complex of objects of an abelian category is an extension of the usual homology of an object to complexes. It is a sort of cross between the derived functor cohomology of an object and the homology of a chain complex. Hyperhomology is no longer used much: since about 1970 it has been largely replaced by the roughly equivalent concept of a derived functor between derived categories. ==Definition== We give the definition for hypercohomology as this is more common. As usual, hypercohomology and hyperhomology are essentially the same: one converts from one to the other by dualizing, i.e. by changing the direction of all arrows, replacing injective objects with projective ones, and so on. Suppose that ''A'' is an abelian category with enough injectives and ''F'' a left exact functor to another abelian category ''B''. If ''C'' is a complex of objects of ''A'' bounded on the left, the hypercohomology :H''i''(''C'') of ''C'' (for an integer ''i'') is calculated as follows: # Take a quasi-isomorphism ''Φ'' : ''C'' → ''I'', here ''I'' is a complex of injective elements of ''A''. # The hypercohomology H''i''(''C'') of ''C'' is then the cohomology ''H''''i''(''F''(''I'')) of the complex ''F''(''I''). The hypercohomology of ''C'' is independent of the choice of the quasi-isomorphism, up to unique isomorphisms. The hypercohomology can also be defined using derived categories: the hypercohomology of ''C'' is just the cohomology of ''F''(''C'') considered as an element of the derived category of ''B''. 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「hyperhomology」の詳細全文を読む スポンサード リンク
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