### On the K-theory of finite algebras over Witt vectors of perfect fields, by Lars Hesselholt and 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>0. 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 which maps x
to zero.

Lars Hesselholt <larsh@math.mit.edu>

Ib Madsen <imadsen@mi.aau.dk>