On the K-theory of local fields (with Ib Madsen)

Let K be a complete discrete valuation field of char. 0 with perfect residue field of odd characteristic p. We establish a connection between the K-theory of K and the de Rham-Witt complex of the valuation ring OK with logarithmic poles at the maximal ideal. We use this to show that for s greater than or equal to 1,

K2s(K, Z/pv Z) = H0(K, Z/pv Z(s)) + H2(K, Z/pv Z(s+1)),

K2s+1(K, Z/pv Z) = H1(K, Z/pv Z(s+1)).

This confirms the Licthenbaum-Quillen conjecture for the field K.

Lars Hesselholt <larsh@math.mit.edu>
Ib Madsen <imadsen@imf.au.dk>