### On the K-theory of finite algebras over Witt vectors of perfect fields (with Ib Madsen)

The purpose of this paper is two fold. Firstly, it gives a thorough
introduction to the topological cyclic homology theory, which to a
ring R associates a spectrum TC(R). We determine TC(k) and
TC(k[x]/(x^{2})) where k is a perfect field of positive
characteristic and k[x]/(x^2) its dual numbers, and sets the stage for
further calculations. Secondly, we show that the cyclotomic trace from
Quillen's K(R) to TC(R) becomes a homotopy equivalence after
*p*-adic completion when R is a finite algebra over the Witt
vectors W(k) of a perfect field of characteristic *p*. This
involves a recent relative result of R. McCarthy, the calculation of
TC(k) and Quillen's theorem about K(k), and continuity results for
TC(R) and K(R), the latter basically due to Suslin and coworkers. In
particular, we obtain a calculation of the tangent space of K(k),
i.e. the homotopy fiber of the map from K(k[x]/(x^{2})) to
K(k) induced from the map that takes x to zero.

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