The Cyclotomic Trace in Algebraic K-Theory

Abstract

This paper reviews the relations between algebraic K-theory and topological cyclic homology given by cyclotomic trace. If one, very superficially, views algebraic K-theory as classifying invertible matrices, then the cyclotomic trace records the trace of all powers of matrices. In a more relevant formulation, the topological cyclic homology has the same relationship to Bökstedt’s topological Hochschild homology as Connes’ cyclic homology has to Hochschild homology, and the cyclotomic trace is a topological cyclic version of the Dennis trace map.KeywordsSpectral SequenceHomotopy GroupFinite SubgroupCyclic HomologyDual NumberThese keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.