We use noncommutative differential forms (which were first introduced by Connes) to construct a noncommutative version of the complex of Cenkl and PorterΩ∗,∗(X)for a simplicial setX. The algebraΩ∗,∗(X)is a differential graded algebra with a filtrationΩ∗,q(X)⊂Ω∗,q+1(X), such thatΩ∗,q(X)is aℚq-module, whereℚ0=ℚ1=ℤandℚq=ℤ[1/2,…,1/q]forq>1. Then we use noncommutative versions of the Poincaré lemma and Stokes' theorem to prove the noncommutative tame de Rham theorem: ifXis a simplicial set of finite type, then for eachq≥1and anyℚq-moduleM, integration of forms induces a natural isomorphism ofℚq-modulesI:Hi(Ω∗,q(X),M)→Hi(X;M)for alli≥0. Next, we introduce a complex of noncommutative tame de Rham currentsΩ∗,∗(X)and we prove the noncommutative tame de Rham theorem for homology: ifXis a simplicial set of finite type, then for eachq≥1and anyℚq-moduleM, there is a natural isomorphism ofℚq-modulesI:Hi(X;M)→Hi(Ω∗,q(X),M)for alli≥0.
Sobolev spaces of differential forms and de Rham-Hodge isomorphism
