Luis Fernando Mejias

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.

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 ℚq-module, where ℚ0=ℚ1=ℤ and ℚq=ℤ[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 q≥1 and any ℚq-module M, integration of forms induces a natural isomorphism of ℚq-modules I:Hi(Ω∗,q(X),M)→Hi(X;M) for all i≥0. 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 q≥1 and any ℚq-module M, there is a natural isomorphism of ℚq-modules I:Hi(X;M)→Hi(Ω∗,q(X),M) for all i≥0.