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D on a compact smooth manifold defines a class in K homology. One invariant of this class is the analytic index of the operator. This is seen as the pairing of the class , with the element 1 in HC(C(M)). Cyclic cohomology can be seen as a way to get higher invariants of elliptic differential
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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
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44:(cohomology) in the 1980s. These invariants have many interesting relationships with several older branches of mathematics, including de Rham theory, Hochschild (co)homology, group cohomology, and the
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229:
393:{\displaystyle {\begin{aligned}t_{n}:A^{\otimes n}\to A^{\otimes n},\quad a_{1}\otimes \dots \otimes a_{n}\mapsto (-1)^{n-1}a_{n}\otimes a_{1}\otimes \dots \otimes a_{n-1}.\end{aligned}}}
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is the restriction of the
Hochschild differential to this quotient. One can check that the Hochschild differential does indeed factor through to this space of coinvariants.
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953:, analytic cyclic homology due to Ralf Meyer or asymptotic and local cyclic homology due to Michael Puschnigg. The last one is very close to
1402:
Boris L. Fegin and Boris L. Tsygan. Additive K-theory and crystalline cohomology. Funktsional. Anal. i
Prilozhen., 19(2):52–62, 96, 1985.
1102:
In some situations, this map can be used to compute K-theory by means of this map. A pioneering result in this direction is a theorem of
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1575:
578:
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Alain Connes and Henri
Moscovici. The local index formula in noncommutative geometry. Geom. Funct. Anal., 5(2):174–243, 1995.
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Michael
Puschnigg. Diffeotopy functors of ind-algebras and local cyclic cohomology. Doc. Math., 8:143–245 (electronic), 2003.
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While
Goodwillie's result holds for arbitrary rings, a quick reduction shows that it is in essence only a statement about
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1521:
Clausen, Dustin; Mathew, Akhil; Morrow, Matthew (2018), "K-theory and topological cyclic homology of henselian pairs",
1279:, cyclic homology must be replaced by topological cyclic homology in order to keep a close connection to K-theory. (If
1638:
974:
906:. Cyclic cohomology is in fact endowed with a pairing with K-theory, and one hopes this pairing to be non-degenerate.
886:
This formula suggests a way to define de Rham cohomology for a 'noncommutative spectrum' of a noncommutative algebra
876:{\displaystyle HC_{n}(A)\simeq \Omega ^{n}\!A/d\Omega ^{n-1}\!A\oplus \bigoplus _{i\geq 1}H_{\text{dR}}^{n-2i}(V).}
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There has been defined a number of variants whose purpose is to fit better with algebras with topology, such as
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connecting
Hochschild and cyclic homology. This long exact sequence is referred to as the periodicity sequence.
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Alain Connes. Noncommutative differential geometry. Inst. Hautes Études Sci. Publ. Math., 62:257–360, 1985.
1315:, the relative K-theory is isomorphic to relative topological cyclic homology (without tensoring both with
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contains the rational numbers, the definition in terms of the Connes complex calculates the same homology.
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1375:. Uspekhi Mat. Nauk, 38(2(230)):217–218, 1983. Translation in Russ. Math. Survey 38(2) (1983), 198–199.
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proved a far-reaching generalization of
Goodwillie's result, stating that for a commutative ring
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to the relative cyclic homology (measuring the difference between K-theory or cyclic homology of
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734:
57:
33:
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Ryszard Nest and Boris Tsygan. Algebraic index theorem. Comm. Math. Phys., 172(2):223–262, 1995.
940:-algebras, etc. The reason is that K-theory behaves much better on topological algebras such as
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One of the applications of cyclic homology is to find new proofs and generalizations of the
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Jean-Louis Loday. Cyclic
Homology. Vol. 301. Springer Science & Business Media, 1997.
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1514:-theory, commutative algebra, and algebraic geometry (Santa Margherita Ligure, 1989)
1323:, asserting that in this situation the relative K-theory spectrum modulo an integer
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Connes later found a more categorical approach to cyclic homology using a notion of
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is less well-behaved than topological
Hochschild homology for rings not containing
1197:{\displaystyle K_{n}(A,I)\otimes \mathbf {Q} \to HC_{n-1}(A,I)\otimes \mathbf {Q} }
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977:. Among these generalizations are index theorems based on spectral triples and
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Ralf Meyer. Analytic cyclic cohomology. PhD thesis, Universität Münster, 1999
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1487:, Grundlehren der mathematischen Wissenschaften, vol. 301, Springer,
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899:
670:. In this way, cyclic homology (and cohomology) may be interpreted as a
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568:{\displaystyle C_{n}^{\lambda }(A):=HC_{n}(A)/\langle 1-t_{n+1}\rangle }
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One motivation of cyclic homology was the need for an approximation of
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One of the striking features of cyclic homology is the existence of a
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Boris L. Tsygan. Homology of matrix Lie algebras over rings and the
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1452:
Jardine, J. F. (1993), "The K-theory of finite fields, revisited",
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operators not only for smooth manifolds, but also for foliations,
997:, and singular spaces that appear in noncommutative geometry.
648:{\displaystyle d:C_{n}^{\lambda }(A)\to C_{n-1}^{\lambda }(A)}
36:
of manifolds. These notions were independently introduced by
476:. Then the components of the Connes complex are defined as
1287:, then cyclic homology and topological cyclic homology of
48:. Contributors to the development of the theory include
674:, which can be explicitly computed by the means of the (
1508:-theory of Henselian local rings and Henselian pairs",
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agree.) This is in line with the fact that (classical)
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that is defined, unlike K-theory, as the homology of a
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The first definition of the cyclic homology of a ring
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1516:, Contemp. Math., vol. 126, AMS, pp. 59–70
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to reprove
Quillen's computation of the K-theory of
1268:{\displaystyle A\otimes _{\mathbf {Z} }\mathbf {Q} }
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713:of characteristic zero can be computed in terms of
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1535:Goodwillie, Thomas G. (1986), "Relative algebraic
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1651:A personal note on Hochschild and Cyclic homology
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125:proceeded by the means of the following explicit
1319:). Their result also encompasses a theorem of
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701:Cyclic cohomology of the commutative algebra
403:Recall that the Hochschild complex groups of
175:which generates the natural cyclic action of
8:
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1092:{\displaystyle tr:K_{n}(A)\to HC_{n-1}(A).}
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465:{\displaystyle HC_{n}(A):=A^{\otimes n+1}}
203:{\displaystyle \mathbb {Z} /n\mathbb {Z} }
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28:are certain (co)homology theories for
20:and related branches of mathematics,
7:
1588:, vol. 147, Berlin, New York:
1301:Clausen, Mathew & Morrow (2018)
1001:Computations of algebraic K-theory
957:as it is endowed with a bivariant
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1207:between the relative K-theory of
1311:holds with respect to the ideal
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729:is smooth, cyclic cohomology of
721:. In particular, if the variety
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1586:Graduate Texts in Mathematics
1017:, say), to cyclic homology:
411:itself are given by setting
1639:Encyclopedia of Mathematics
1275:. For rings not containing
975:Atiyah-Singer index theorem
894:Variants of cyclic homology
705:of regular functions on an
131:Hochschild homology complex
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1106:: it asserts that the map
682:)-bicomplex. If the field
1335:used Gabber's result and
719:algebraic de Rham complex
697:Case of commutative rings
1231:) is an isomorphism for
979:deformation quantization
707:affine algebraic variety
1353:Noncommutative geometry
1327:which is invertible in
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144:For any natural number
18:noncommutative geometry
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80:Hints about definition
1541:Annals of Mathematics
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933:{\displaystyle C^{*}}
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168:{\displaystyle t_{n}}
52:, Yuri L. Daletskii,
32:which generalize the
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1007:cyclotomic trace map
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34:de Rham (co)homology
30:associative algebras
1666:Homological algebra
1634:"Cyclic cohomology"
1576:Rosenberg, Jonathan
1373:Hochschild homology
1293:Hochschild homology
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691:long exact sequence
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1466:10.1007/BF00961219
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1211:with respect to a
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735:de Rham cohomology
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58:Jean-Luc Brylinski
1599:978-0-387-94248-3
1543:, Second Series,
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1481:Loday, Jean-Louis
1104:Goodwillie (1986)
990:elliptic operator
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668:simplicial object
68:, Victor Nistor,
26:cyclic cohomology
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83:
54:Boris Feigin
42:Alain Connes
38:Boris Tsygan
25:
21:
15:
1013:(of a ring
946:C*-algebras
50:Max Karoubi
1616:0801.19001
1528:1803.10897
1510:Algebraic
1446:References
1331:vanishes.
1235:≥1.
1644:EMS Press
1504:(1992), "
1250:⊗
1213:nilpotent
1187:⊗
1164:−
1150:→
1142:⊗
1070:−
1056:→
995:orbifolds
963:KK-theory
926:∗
851:−
830:≥
823:⨁
819:⊕
807:−
800:Ω
778:Ω
774:≃
632:λ
624:−
613:→
599:λ
563:⟩
544:−
538:⟨
494:λ
449:⊗
376:−
365:⊗
362:⋯
359:⊗
346:⊗
328:−
314:−
308:↦
295:⊗
292:⋯
289:⊗
267:⊗
259:→
251:⊗
1660:Category
1578:(1994),
1483:(1998),
1454:K-Theory
1347:See also
955:K-theory
900:K-theory
472:for all
46:K-theory
1646:, 2001
1608:1282290
1569:0855300
1561:1971283
1474:1268594
1223:and of
210:on the
1621:Errata
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1606:
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1567:
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1491:
1472:
725:=Spec
662:in an
1557:JSTOR
1523:arXiv
1359:Notes
961:from
474:n ≥ 0
146:n ≥ 0
108:) or
1594:ISBN
1489:ISBN
1005:The
24:and
1619:.
1612:Zbl
1549:doi
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988:An
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737:of
717:'s
133:of
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1038:K
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104:(
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86:A
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