International Journal of Mathematics and Mathematical Sciences
Volume 30 (2002), Issue 11, Pages 667-696
The de Rham theorem for the noncommutative complex of Cenkl and Porter
Department of Mathematics, Northeastern University, 567 Lake Hall, Boston 02115, MA, USA
Received 22 May 2001
Copyright © 2002 Luis Fernando Mejias. 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.
We use noncommutative differential forms (which were first introduced by Connes) to construct a noncommutative version of the complex of Cenkl and Porter for a simplicial set . The algebra is a differential graded algebra with a filtration , such that is a -module, where and for . Then we use noncommutative versions of the Poincaré lemma and Stokes' theorem to prove the noncommutative tame de Rham theorem: if is a simplicial set of finite type, then for each and any -module , integration of forms induces a natural isomorphism of -modules for all . Next, we introduce a complex of noncommutative tame de Rham currents and we prove the noncommutative tame de Rham theorem for homology: if is a simplicial set of finite type, then for each and any -module , there is a natural isomorphism of -modules for all .