The De Rham Theorem for the Noncommutative Complex of Cenkl and Porter

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 anyQq-moduleM , 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 Qqmodules I :Hi(X;M)→Hi(Ω∗,q(X),M) for all i≥ 0.