On the topological cyclic homology of the algebraic closure of a local field

Let V be a complete discrete valuation ring with quotient field K of characteristic 0 and perfect residue field k of odd characteristic p. In joint work with Ib Madsen, we previously constructed a map from the localization sequence in K-theory to a similar cofibration sequence of topological Hochschild spectra
      T(k) → T(V) → T(V|K) → T(k)[-1].
We also determined, up to pro-isomorphism, the structure of the homotopy groups with Z/pvZ-coefficients of the fixed point spectra
      TRn(V|K;p) = T(V|K)Cpn-1
under the assumption that the field K contains a primitive pvth root of unity. This was used to evaluate the p-adic K-groups of the field K. In the present paper, we completely determine the structure of the homotopy groups with Z/pvZ-coefficients, for all v, of the colimit
          _ _
      TRn(V|K;p) = colim TRn(VL|L;p)
over all finite extensions L of K contained in an algebraic closure of K. Based on this result, we formulate a conjecture for the value of the homotopy group with Z/pvZ-coefficients, for all v, of the spectrum TRn(V|K;p).


Lars Hesselholt <larsh@math.mit.edu>