symmetric monoidal (∞,1)-category of spectra
Topological Hochschild homology (resp. topological cyclic homology) survey is in (May) is a refinement of Hochschild homology/cyclic homology from commutative rings/algebras to the higher algebra of ring spectra/E-∞ rings/E-∞ algebras.
In particular, given a ring , then there is a natural morphism of spectra
from the algebraic K-theory spectrum to the topological Hochschild homology spectrum, the Dennis trace map. Since Hochschild homology spectra are naturally cyclotomic spectra, that factors through the topological cyclic homology spectrum via a map called the cyclotomic trace which much like a Chern character map for algebraic K-theory.
The spectra and are typically easier to analyze than . Moreover, the difference between them and is “locally constant” (Dundas-Goodwillie-McCarthy13) and often otherwise bounded in complexity. Accordingly, and are in practice computationally useful approximations to .
There are various generalizations:
Just as for basic Hochschild homology, there is higher topological Hochschild homology (Carlsson-Douglas-Dundas 08) given not just by derived loop spaces but by derived mapping spaces out of higher dimensional tori.
The original references are
Marcel Bökstedt, Topological Hochschild homology, Bielefeld, 1985, 1988
and a further generalization is defined in
Higher topological Hochschild Homology is discussed in
Review and exposition includes
Anatoly Preygel, Hochschild homology notes, juvitop seminar, pdf
Thomas Geisser, Motivic Cohomology, K-Theory and Topological Cyclic Homology, Handbook of K-theory II.1, pdf
Ib Madsen, Algebraic K-theory and traces, pdf
Abstract characterization of the Dennis trace and cyclotomic trace is discussed in
Thomas Geisser, Lars Hesselhoft, Topological cyclic homology of schemes, in: Algebraic -theory (Seattle, WA, 1997), 41–87, Proc. Sympos. Pure Math. 67, Amer. Math. Soc. 1999, MR2001g:19003; K-theory archive
R. McCarthy, Relative algebraic K-theory and topological cyclic homology, Acta Math. 179 (1997), 197-222.
J. McClure, R. Staffeldt, On the topological Hochschild homology of , I, pdf
Daniel Joseph Vera, Topological Hochschild homology of twisted group algebra, MIT Ph. D. thesis 2006, pdf
V. Angeltveit, A. Blumberg, T. Gerhardt, M. Hill, T. Lawson, M. Mandell, Topological cyclic homology via the norm (arXiv:1401.5001)
and of tmf in