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.


Thomas Geisser <geisser@math.usc.edu>
Lars Hesselholt <larsh@math.mit.edu>