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The periodic cyclic homology of crossed products of finite type algebras

The periodic cyclic homology of crossed products of finite type algebras
The periodic cyclic homology of crossed products of finite type algebras
We study the periodic cyclic homology groups of the cross-product of a finite type algebra A by a discrete group. In case A is commutative and the group is finite, our results are complete and given in terms of the singular cohomology of the sets of fixed points. These groups identify our cyclic homology groups with the `orbifold cohomology'of the underlying (algebraic) orbifold. The proof is based on a careful study of localization at fixed points and of the resulting Koszul complexes. This is achieved by extending to our class of noncommutative algebras the concept of an `infinitesimal neighborhood' that plays such an important role in commutative algebra. We provide examples of Azumaya algebras for which this identification is, however, no longer valid. As an example, we discuss some affine Weyl groups.
0001-8708
494-523
Brodzki, Jacek
b1fe25fd-5451-4fd0-b24b-c59b75710543
Dave, Shantanu
61d050c3-7410-48b5-93ed-1ac017702e1f
Nistor, Victor
c9f95327-cc97-4d9a-a8fb-3c89e3d84bdf
Brodzki, Jacek
b1fe25fd-5451-4fd0-b24b-c59b75710543
Dave, Shantanu
61d050c3-7410-48b5-93ed-1ac017702e1f
Nistor, Victor
c9f95327-cc97-4d9a-a8fb-3c89e3d84bdf

Brodzki, Jacek, Dave, Shantanu and Nistor, Victor (2017) The periodic cyclic homology of crossed products of finite type algebras. Advances in Mathematics, 306, 494-523. (doi:10.1016/j.aim.2016.10.025).

Record type: Article

Abstract

We study the periodic cyclic homology groups of the cross-product of a finite type algebra A by a discrete group. In case A is commutative and the group is finite, our results are complete and given in terms of the singular cohomology of the sets of fixed points. These groups identify our cyclic homology groups with the `orbifold cohomology'of the underlying (algebraic) orbifold. The proof is based on a careful study of localization at fixed points and of the resulting Koszul complexes. This is achieved by extending to our class of noncommutative algebras the concept of an `infinitesimal neighborhood' that plays such an important role in commutative algebra. We provide examples of Azumaya algebras for which this identification is, however, no longer valid. As an example, we discuss some affine Weyl groups.

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Accepted/In Press date: 20 October 2016
e-pub ahead of print date: 4 November 2016
Published date: 14 January 2017
Organisations: Pure Mathematics

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Local EPrints ID: 401819
URI: https://eprints.soton.ac.uk/id/eprint/401819
ISSN: 0001-8708
PURE UUID: 5e4a90e1-2ca0-475b-8cfb-8a786ca237b0

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Date deposited: 25 Oct 2016 10:16
Last modified: 22 Jan 2018 05:01

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