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In noncommutative geometry and related branches of mathematics, cyclic homology and cyclic cohomology are certain (co)homology theories for associative algebras which generalize the de Rham (co)homology of manifolds. These notions were independently introduced by Boris Tsygan (homology)〔Boris L. Tsygan. Homology of matrix Lie algebras over rings and the Hochschild homology. Uspekhi Mat. Nauk, 38(2(230)):217–218, 1983. Translation in Russ. Math. Survey 38(2) (1983), 198–199. 〕 and Alain Connes (cohomology)〔Alain Connes. Noncommutative differential geometry. Inst. Hautes Études Sci. Publ. Math., 62:257–360, 1985. 〕 in 1980s. These invariants have many interesting relationships with several older branches of mathematics, including de Rham theory, Hochschild (co)homology, group cohomology, and the K-theory. The principal contributors to the development of theory include Max Karoubi, Yuri L. Daletskii, Boris Feigin, Jean-Luc Brylinski, Mariusz Wodzicki, , Victor Nistor, Daniel Quillen, Joachim Cuntz, Ryszard Nest, Ralf Meyer, Michael Puschnigg, and many others. == Hints about definition == The first definition of the cyclic homology of a ring ''A'' over a field of characteristic zero, denoted :''HC''''n''(''A'') or ''H''''n''λ(''A''), proceeded by the means of an explicit chain complex related to the Hochschild homology complex of ''A''. Connes later found a more categorical approach to cyclic homology using a notion of cyclic object in an abelian category, which is analogous to the notion of simplicial object. In this way, cyclic homology (and cohomology) may be interpreted as a derived functor, which can be explicitly computed by the means of the (''b'', ''B'')-bicomplex. One of the striking features of cyclic homology is the existence of a long exact sequence connecting Hochschild and cyclic homology. This long exact sequence is referred to as the periodicity 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「Cyclic homology」の詳細全文を読む スポンサード リンク
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