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