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International Journal of Mathematics and Mathematical Sciences
Volume 30 (2002), Issue 11, Pages 667-696
http://dx.doi.org/10.1155/S016117120200769X

The de Rham theorem for the noncommutative complex of Cenkl and Porter

Department of Mathematics, Northeastern University, 567 Lake Hall, Boston, MA 02115, USA

Received 22 May 2001

Copyright © 2002 Hindawi Publishing Corporation. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

We use noncommutative differential forms (which were first introduced by Connes) to construct a noncommutative version of the complex of Cenkl and Porter Ω,(X) for a simplicial set X. The algebra Ω,(X) is a differential graded algebra with a filtration Ω,q(X)Ω,q+1(X), such that Ω,q(X) is a q-module, where 0=1= and q=[1/2,,1/q] for q>1. Then we use noncommutative versions of the Poincaré lemma and Stokes' theorem to prove the noncommutative tame de Rham theorem: if X is a simplicial set of finite type, then for each q1 and any q-module M, integration of forms induces a natural isomorphism of q-modules I:Hi(Ω,q(X),M)Hi(X;M) for all i0. Next, we introduce a complex of noncommutative tame de Rham currents Ω,(X) and we prove the noncommutative tame de Rham theorem for homology: if X is a simplicial set of finite type, then for each q1 and any q-module M, there is a natural isomorphism of q-modules I:Hi(X;M)Hi(Ω,q(X),M) for all i0.