The de Rham theorem for the noncommutative complex of Cenkl and Porter
In [Trans. Am. Math. Soc. 347, 4277-4299 (1995; Zbl 0852.55009)] \textit{M. Karoubi} started to use noncommutative differential forms to study classical algebraic topology. This paper takes up Karoubi's idea by introducing a noncommutative version of the tame de Rham complex of \textit{B. Cenkl} and \textit{R. Porter} [Adv. Math. 48, 189-204 (1983; Zbl 0526.55013)]. For any simplicial set \(X\) of finite type the author introduces the noncommutative tame de Rham complex \(\Omega^{*,*}(X)\). Given \(q\geq 1\) and a module \(M\) over \(\mathbb Q_q:=\mathbb Z[1/2,1/3,\dots,1/q]\), integration of noncommutative forms is shown to induce an isomorphism between \(H^i(\Omega^{*,q}(X),M)\) and \(H^i(X,M)\). The necessary background on simplicial objects, tame forms, noncommutative forms and integration is reviewed in the paper. Finally, the author also discusses a dual (homological) version of his complex and its relation to noncommutative tame de Rham currents.
scientific article; zbMATH DE number 1779151