The künneth formula in periodic cyclic homology

Ioannis Emmanouil

doi:10.1007/bf00536612

Topological K-theory of complex noncommutative spaces

Compositio Mathematica ◽

10.1112/s0010437x15007617 ◽

2015 ◽

Vol 152 (3) ◽

pp. 489-555 ◽

Cited By ~ 7

Author(s):

Anthony Blanc

Keyword(s):

Chern Character ◽

Cyclic Homology ◽

Finite Dimensional ◽

Perfect Complexes ◽

Topological Realization ◽

Kunneth Formula ◽

K Theory ◽

Simplicial Presheaves ◽

Definition Of ◽

Dg Algebra

The purpose of this work is to give a definition of a topological K-theory for dg-categories over$\mathbb{C}$and to prove that the Chern character map from algebraic K-theory to periodic cyclic homology descends naturally to this new invariant. This topological Chern map provides a natural candidate for the existence of a rational structure on the periodic cyclic homology of a smooth proper dg-algebra, within the theory of noncommutative Hodge structures. The definition of topological K-theory consists in two steps: taking the topological realization of algebraic K-theory and inverting the Bott element. The topological realization is the left Kan extension of the functor ‘space of complex points’ to all simplicial presheaves over complex algebraic varieties. Our first main result states that the topological K-theory of the unit dg-category is the spectrum$\mathbf{BU}$. For this we are led to prove a homotopical generalization of Deligne’s cohomological proper descent, using Lurie’s proper descent. The fact that the Chern character descends to topological K-theory is established by using Kassel’s Künneth formula for periodic cyclic homology and the proper descent. In the case of a dg-category of perfect complexes on a separated scheme of finite type, we show that we recover the usual topological K-theory of complex points. We show as well that the Chern map tensorized with$\mathbb{C}$is an equivalence in the case of a finite-dimensional associative algebra – providing a formula for the periodic homology groups in terms of the stack of finite-dimensional modules.

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Cyclic-Homology Chern–Weil Theory for Families of Principal Coactions

Communications in Mathematical Physics ◽

10.1007/s00220-020-03804-2 ◽

2020 ◽

Author(s):

Piotr M. Hajac ◽

Tomasz Maszczyk

Keyword(s):

Cyclic Homology ◽

Quantum Deformation ◽

Homotopy Formula ◽

Standard Quantum ◽

Unital Algebra ◽

Hopf Fibrations ◽

Chain Homotopy ◽

Cartan Model ◽

Extension Algebra ◽

Hochschild Complex

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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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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