Arbeitsgemeinschaft: Topological Cyclic Homology

  • Lars Hesselholt

    University of Copenhagen, Denmark and Nagoya University, Japan
  • Peter Scholze

    Universität Bonn, Germany

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 -theory, used by Hesselholt–Madsen to evaluate the -adic -theory of -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 -adic Hodge theory .

Cite this article

Lars Hesselholt, Peter Scholze, Arbeitsgemeinschaft: Topological Cyclic Homology. Oberwolfach Rep. 15 (2018), no. 2, pp. 805–940

DOI 10.4171/OWR/2018/15