nLab Dennis trace

Idea

The Dennis trace is a natural morphism from algebraic K-theory of rings to Hochschild homology

$$K(-) \longrightarrow THH(-) \,.$$

While various versions of Chern character are maps from K-theory to some (co)homology of a space, algebra, scheme, the Dennis trace map underlies this phenomenon at the more fundamental level of K-theory space.

The Dennis trace lifts to to ring spectra and as such factors through the map $TC(-) \longrightarrow THH(-)$ ]] from topological cyclic homology, this lift is called the cyclotomic trace.

Literature

• Ib Madsen, Algebraic K-theory and traces, Current Developments in Mathematics, 1995.

• John Rognes, after Marcel Bökstedt, Trace maps from the algebraic K-theory of the integers, K-theory archive

• M. R. Kantorovitz, Adams operations and the Dennis trace map, JPAA 144, 1 (Dec 1999), 21-27 [doi]

• M. R. Kantorovitz, C. Miller, An explicit description of the Dennis trace map, claudia-revised.ps

• Andrew Blumberg, David Gepner, Goncalo Tabuada, Uniqueness of the multiplicative cyclotomic trace (arXiv:1103.3923)

• Bjørn Dundas, Thomas Goodwillie, Randy McCarthy, The local structure of algebraic K-theory, Springer 2013

Relation to bicategorical trace:

• Jonathan Campbell, John Lind, Cary Malkiewich, Kate Ponto, Inna Zakharevich, K-theory of endomorphisms, the TR-trace, and zeta functions [arXiv:2005.04334]