Operations on Hochschild and Cyclic Homology

AbstractViewing the space of cotraces in the structural coalgebra of a principal coaction as a noncommutative counterpart of the classical Cartan model, we construct the cyclic-homology Chern–Weil homomorphism. To realize the thus constructed Chern–Weil homomorphism as a Cartan model of the homomorphism tautologically induced by the classifying map on cohomology, we replace the unital subalgebra of coaction-invariants by its natural H-unital nilpotent extension (row extension). Although the row-extension algebra provides a drastically different model of the cyclic object, we prove that, for any row extension of any unital algebra over a commutative ring, the row-extension Hochschild complex and the usual Hochschild complex are chain homotopy equivalent. It is the discovery of an explicit homotopy formula that allows us to improve the homological quasi-isomorphism arguments of Loday and Wodzicki. We work with families of principal coactions, and instantiate our noncommutative Chern–Weil theory by computing the cotrace space and analyzing a dimension-drop-like effect in the spirit of Feng and Tsygan for the quantum-deformation family of the standard quantum Hopf fibrations.

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Relative algebraic K-theory and topological cyclic homology

Acta Mathematica ◽

10.1007/bf02392743 ◽

1997 ◽

Vol 179 (2) ◽

pp. 197-222 ◽

Cited By ~ 37

Author(s):

Randy McCarthy

Keyword(s):

Cyclic Homology ◽

Topological Cyclic Homology ◽

K Theory

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The künneth formula in periodic cyclic homology

K-Theory ◽

10.1007/bf00536612 ◽

1996 ◽

Vol 10 (2) ◽

pp. 197-214 ◽

Cited By ~ 2

Author(s):

Ioannis Emmanouil

Keyword(s):

Cyclic Homology ◽

Kunneth Formula

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A new description of equivariant cohomology for totally disconnected groups

Journal of K-theory K-theory and its Applications to Algebra Geometry and Topology ◽

10.1017/is007011019jkt020 ◽

2008 ◽

Vol 1 (3) ◽

pp. 431-472 ◽

Cited By ~ 1

Author(s):

Christian Voigt

Keyword(s):

Equivariant Cohomology ◽

Simplicial Complexes ◽

Cyclic Homology ◽

Cohomology Theory ◽

Cohomology Groups ◽

Natural Class ◽

New Approach ◽

Totally Disconnected

AbstractWe consider smooth actions of totally disconnected groups on simplicial complexes and compare different equivariant cohomology groups associated to such actions. Our main result is that the bivariant equivariant cohomology theory introduced by Baum and Schneider can be described using equivariant periodic cyclic homology. This provides a new approach to the construction of Baum and Schneider as well as a computation of equivariant periodic cyclic homology for a natural class of examples. In addition we discuss the relation between cosheaf homology and equivariant Bredon homology. Since the theory of Baum and Schneider generalizes cosheaf homology we finally see that all these approaches to equivariant cohomology for totally disconnected groups are closely related.

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Twisted Homology of Quantum SL(2) - Part II

Journal of K-theory K-theory and its Applications to Algebra Geometry and Topology ◽

10.1017/is009009022jkt091 ◽

2009 ◽

Vol 6 (1) ◽

pp. 69-98 ◽

Cited By ~ 5

Author(s):

Tom Hadfield ◽

Ulrich Krähmer

Keyword(s):

Noncommutative Geometry ◽

Hochschild Cohomology ◽

Cyclic Homology ◽

Coordinate Ring ◽

Volume Form ◽

Twisted Homology

AbstractWe complete the calculation of the twisted cyclic homology of the quantised coordinate ring = ℂq [SL(2)] of SL(2) that we began in [14]. In particular, a nontrivial cyclic 3-cocycle is constructed which also has a nontrivial class in Hochschild cohomology and thus should be viewed as a noncommutative geometry analogue of a volume form.

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