On the K-theory of local fields

Abstract

Introduction 1. Topological Hochschild homology and localization 2. The homotopy groups of T (A|K) 3. The de Rham-Witt complex and TR • * (A|K; p) 4. Tate cohomology and the Tate spectrum 5. The Tate spectral sequence for T (A|K) 6. The pro-system TR • * (A|K; p, Z/p v ) Appendix A. Truncated polynomial algebras References * The first named author was supported in part by NSF Grant and the Alfred P. Sloan Foundation. The second named author was supported in part by The American Institute of Mathematics. i ! −→ TC(C b z (P A ); p) j −→ TC(C b q (P A ); p) ∂ −→ Σ TC(C b z (P A ) q ; p), *

Localization theorems in topological Hochschild homology and topological cyclic homology

“…We need functorial point-set versions of these maps to construct TC as a functor on small spectral categories. In [21], the connectivity and convergence hypotheses used there imply that THH.C/ is an equivariant -spectrum relative to the family of finite subsets of S 1 ; the point-set maps F; R in [21] are then constructed using the point-set fixed point spectra as models for the derived fixed point spectra. In our context, we need to use an -spectrum replacement functor in the category of cyclotomic spectra; see Definition 4.5 and Theorem 4.7 in the next section.…”

Hopf algebra structure on topological Hochschild homology

“…When R = HO E is the Eilenberg-Mac Lane spectrum of the valuation ring in a local number field, systematic computations of the topological cyclic homology of R were made in [HM03], thereby verifying the Lichtenbaum-Quillen conjectures for the algebraic K -theory of these fields. Particular computations for other commutative S -algebras, like connective complex K -theory ku and its p-local Adams summand ℓ, have revealed a more general pattern of how algebraic K -theory creates a "red-shift" in chromatic filtration and satisfies a Galois descent property [AR02], [Au].…”

Homology of Enring spectra and iterated THH

“…Waldhausen originally conjectured the theory THH and used an ad hoc version of it to split the algebraic K -theory of spaces into stable homotopy theory and stable pseudo-isotopy theory [26]. The theory TC provides the key tool in the proof of the K -theoretic Novikov conjecture by Bökstedt, Hsiang, and Madsen [4], and in the algebraic K -theory computations pioneered by Hesselholt and Madsen [12;13;14]. K -theory computations in TC also led to Ausoni and Rognes' mysterious chromatic red shift phenomenon [1].…”