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

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

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.