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| In [[noncommutative geometry]] and related branches of mathematics, '''cyclic homology''' and '''cyclic cohomology''' are certain (co)homology theories for [[associative algebra]]s which generalize the [[de Rham cohomology|de Rham (co)homology]] of manifolds. These notions were independently introduced by [[Alain Connes]] (cohomology)<ref>Alain Connes. Noncommutative differential geometry. Inst. Hautes Études Sci. Publ. Math., 62:257–360, 1985.
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| </ref> and Boris Tsygan (homology)<ref>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.
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| </ref> around 1980. 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, {{Link-interwiki|en=Jean-Louis Loday|lang=fr}}, Victor Nistor, [[Daniel Quillen]], Joachim Cuntz, Ryszard Nest, Ralf Meyer, Michael Puschnigg, and many others.
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| == Hints about definition ==
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| The first definition of the cyclic homology of a ring ''A'' over a field of [[characteristic (algebra)|characteristic]] zero, denoted | |
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| :''HC''<sub>''n''</sub>(''A'') or ''H''<sub>''n''</sub><sup>λ</sup>(''A''),
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| proceeded by the means of an explicit [[chain complex]] related to the [[Hochschild homology|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.
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| One of the striking features of cyclic homology is the existence of a long exact sequence connecting
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| Hochschild and cyclic homology. This long exact sequence is referred to as the periodicity sequence.
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| == Case of commutative rings ==
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| Cyclic cohomology of the commutative algebra ''A'' of regular functions on an [[affine algebraic variety]] over a field ''k'' of characteristic zero can be computed in terms of [[Grothendieck]]'s [[crystalline cohomology|algebraic de Rham complex]].<ref>Boris L. Fegin and Boris L. Tsygan. Additive K-theory and crystalline cohomology. Funktsional. Anal. i Prilozhen., 19(2):52–62, 96, 1985.</ref> In particular, if the variety ''V''=Spec ''A'' is smooth, cyclic cohomology of ''A'' are expressed in terms of the [[de Rham cohomology]] of ''V'' as follows:
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| :<math> HC_n(A)\simeq \Omega^n\!A/d\Omega^{n-1}\!A\oplus \bigoplus_{i\geq 1}H^{n-2i}_{DR}(V).</math>
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| This formula suggests a way to define de Rham cohomology for a 'noncommutative spectrum' of a noncommutative algebra ''A'', which was extensively developed by Connes.
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| == Variants of cyclic homology ==
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| One motivation of cyclic homology was the need for an approximation of [[K-theory]] that be defined, unlike K-theory, as the homology of a [[chain complex]]. Cyclic cohomology is in fact endowed with a pairing with K-theory, and one hopes this pairing to be non-degenerate. | |
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| There has been defined a number of variants whose purpose is to fit better with algebras with topology, such as [[Fréchet algebra]]s, <math>C^*</math>-algebras, etc. The reason is that K-theory behaves much better on topological algebras such as [[Banach algebra]]s or [[C*-algebras]] than on algebras without additional structure. Since, on the other hand, cyclic homology degenerates on C*-algebras, there came up the need to define modified theories. Among them are entire cyclic homology due to [[Alain Connes]], analytic cyclic homology due to Ralf Meyer<ref>
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| Ralf Meyer. Analytic cyclic cohomology. PhD thesis, Universität Münster, 1999</ref> or asymptotic and local cyclic homology due to Michael Puschnigg.<ref>Michael Puschnigg. Diffeotopy functors of ind-algebras and local cyclic cohomology. Doc.
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| Math., 8:143–245 (electronic), 2003.</ref> The last one is very near to [[K-theory]] as it is endowed with a bivariant [[Chern character]] from [[KK-theory]].
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| ==Applications==
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| One of the applications of cyclic homology is to find new proofs and generalizations of the [[Atiyah-Singer index theorem]]. Among these generalizations are index theorems based on spectral triples<ref>Alain Connes and Henri Moscovici. The local index formula in noncommutative geometry. Geom. Funct. Anal., 5(2):174–243, 1995.</ref> and [[deformation quantization]] of Poisson structures.<ref>Ryszard Nest and Boris Tsygan. Algebraic index theorem. Comm. Math. Phys., 172(2):223–262, 1995.</ref><!-- needs to be a good representative of the theory, with enough context and relevance. The index theorem for quantum tori is linked to the [[quantum Hall effect]],<ref>http://citeseer.ist.psu.edu/old/404503.html</ref> and the index theorem for deformation quantization to the study of band energy redistribution in the [[Born-Oppenheimer approximation]] in molecular physics.<ref>http://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.29.3618</ref> -->
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| ==See also==
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| *[[Hochschild homology]]
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| *[[Noncommutative geometry]]
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| *[[Homology (mathematics)|Homology]]
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| *[[Homology theory]]
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| == References ==
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| <references/>
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| * Jean-Louis Loday, ''Cyclic Homology'', Grundlehren der mathematischen Wissenschaften Vol. 301, Springer (1998) ISBN 3-540-63074-0
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| ==External links==
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| *[http://mathsci.kaist.ac.kr/~jinhyun/note/cyclic/cyclic.pdf A personal note on Hochschild and Cyclic homology]
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| {{DEFAULTSORT:Cyclic Homology}}
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| [[Category:Homological algebra]]
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