### The de Rham-Witt complex and p-adic vanishing cycles (with Thomas
Geisser)

We determine the structure of the de Rham-Witt complex of a smooth
scheme X over a discrete valuation ring of mixed characteristic with
log-poles along the special fiber Y and show that the sub-sheaf fixed
by the Frobenius is isomorphic to the sheaf of *p*-adic vanishing
cycles. We use this result to evaluate the algebraic K-theory
with coefficients of the quotient field K of the henselian local
ring of X at a generic point of Y. The result affirms the
Lichtenbaum-Quillen conjecture for the field K.

Appendix B of the paper, written by Viorel Costeanu, proves that the
Steinberg relation holds in the de Rham-Witt complex of a log-ring
(R,M) such that R is a **Z**_{(p)}-algebra with *p* odd.

Lars Hesselholt <larsh@math.nagoya-u.ac.jp>