On noncommutative differential forms

Abstract

We review the topic of noncommutative differential forms, following the works of Karoubi, Cuntz–Quillen, Cortiñas, Ginzburg–Schedler, and Waikit Yeung. In particular we give a new proof of the theorem of Ginzburg and Schedler that compares extended noncommutative De Rham cohomology to cyclic homology. This theorem is a stronger version of a theorem of Karoubi. We also describe an algebraic structure, namely a category in DG cocategories, that noncommutative forms and other versions of noncommutative calculus are particular cases of.