nLab cyclotomic trace

Idea

Given a ring $R$, then there is a natural morphism of spectra

$$\array{ && \mathbf{TC}(R) \\ & \nearrow & \downarrow \\ \mathbf{K}(R) &\longrightarrow& \mathbf{THH}(R) }$$

from the algebraic K-theory spectrum to the topological Hochschild homology spectrum and factoring through the topological cyclic homology spectrum (a cyclotomic spectrum) called the cyclotomic trace which much like a Chern character map for algebraic K-theory.

This is a refinement to spectra of the Dennis trace.

Related concepts

• Beilinson regulator

References

• Marcel Bökstedt, W.C. Hsiang, Ib Madsen, The cyclotomic trace and algebraic K-theory of spaces, Invent. Math. 111 (1993), 463-539, MR94g:55011, doi

• Ib Madsen, The cyclotomic trace in algebraic K-theory, In: Joseph, A., Mignot, F., Murat, F., Prum, B., Rentschler, R. (eds) First European Congress of Mathematics Paris, July 6–10, 1992. Progress in Mathematics 120, Birkhäuser 1994 doi

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

• David Ayala, Aaron Mazel-Gee, Nick Rozenblyum, The geometry of the cyclotomic trace, (arXiv:1710.06409)

For a very brief, very elementary introduction, see

• Aaron Mazel-Gee, The geometry of the cyclotomic trace, (pdf)

For more see the references at topological Hochschild homology.