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 $\mathbb{A}\Omega$.