Arbeitsgemeinschaft: Topological Cyclic Homology

Abstract

Introduced by Bökstedt-Hsiang-Madsen in the nineties, topological cyclic homology is a manifestation of the dual visions of Connes-Tsygan and Waldhausen to extend de Rham cohomology to a noncommutative setting and to replace algebra by higher algebra. The cohomology theory that ensues receives a denominator-free Chern character from algebraic K-theory, used by Hesselholt-Madsen to evaluate the p-adic K-theory of p-adic fields. More recently, Bhatt-Morrow-Scholze have defined a “motivic” filtration of topological cyclic homology and its variants, the filtration quotients of which give rise to their denominator-free p-adic Hodge theory AΩ.