See Ostvaer for an introduction to maps from algebraic K-theory to Hochschild homology (Dennis trace map), topological Hochschild homology, and topological cyclic homology (cyclotomic trace map). These fit into a commutative diagram with maps from TC to THH and from THH to HH. Here we “evaluate” the theories at any functor with smash product. See Madsen: Algebraic K-theory and traces, for a better overview.
Hesselholt and Geisser: Topological cyclic homology of schemes
Bokstedt, Hsiang, Madsen: The cyclotomic trace and algebraic K-theory of spaces. (1993)
Geisser in K-theory handbook (includes applications to arithmetic geometry and relations to etale K-theory). See memo notes!
Hesselholt in K-theory handbook.
arXiv: Experimental full text search
AG (Algebraic geometry), NCG (Algebra and noncommutative geometry)?
On the descent proplem for topological cyclic homology and algebraic K-theory, by Stavros Tsalidis: http://www.math.uiuc.edu/K-theory/0329
Geisser and Hesselholt: K-theory and topological cyclic homology of smooth schemes over discrete valuation rings
McCarthy: Relative algebraic K-theory and topological cyclic homology (1997)
Preprint in progress of Rognes: Homotopy operations in TC(; p)
Preprint in progress of Rognes: Algebraic models for topological cyclic homology and Whitehead spectra of simply connected spaces
Preprint in progress of Rognes: Algebraic K-theory of group rings and topological cyclic homology
arXiv:1003.2810 Cyclotomic complexes from arXiv Front: math.AT by D. Kaledin We construct a triangulated category of cyclotomic complexes, a homological counterpart of cyclotomic spectra of Bokstedt and Madsen. We also construct a version of the Topological Cyclic Homology functor TC for cyclotomic complexes, and an equivariant homology functor from cycloctomic spectra to cyclotomic complexes which commutes with TC. Then on the other hand, we prove that the category of cyclotomic complexes is essentially a twisted 2-periodic derived category of the category of filtered Dieudonne modules of Fontaine and Lafaille. We also show that under some mild conditions, the functor TC on cyclotomic complexes is the syntomic cohomology functor.
nLab page on Topological cyclic homology