The de Rham-Witt complex and 𝑝-adic vanishing cycles

We determine the structure of the reduction modulo p p of the absolute de Rham-Witt complex of a smooth scheme over a discrete valuation ring of mixed characteristic ( 0 , p ) (0,p) with log-poles along the special fiber and show that the sub-sheaf fixed by the Frobenius map is isomorphic to the sheaf of p p -adic vanishing cycles. We use this result together with the main results of op. cit. to evaluate the algebraic K K -theory with finite coefficients of the quotient field of the henselian local ring at a generic point of the special fiber. The result affirms the Lichtenbaum-Quillen conjecture for this field.