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**/p^{v}**Z**-coefficients of
the fixed point spectra
TR^{n}(V|K;p) = T(V|K)^{Cpn-1}

under the assumption that the field K contains a primitive
*p*^{v}th 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**/p^{v}**Z**-coefficients, for all v, of the colimit
_ _
TR^{n}(V|K;p) = colim TR^{n}(V_{L}|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**/p^{v}**Z**-coefficients,
for all v, of the spectrum TR^{n}(V|K;p).