Classically, one has for every commutative ring A the associated
ring Wn(A) of p-typical Witt vectors of length
n in A. We extend this construction to a functor which
assigns to every assiciative (but not necessarily commutative or
unital) ring A an abelian group Wn(A). The extended
functors comes equipped with additive restriction, Frobenius,
and Verschiebung operators. We write W(A)F for the quotient
group of coinvariant for the Frobenius operator. Let K*(A;
Zp) denote the p-adic K-groups of A, that
is, the homotopy groups of the p-completion of the spectrum
K(A). We prove that, if A is a finite dimensional associative algebra
over a perfect field k of positive characteristic p, then
there is a canonical isomorphism
Kq(A;Zp) = (Lq+1W(-)F)(A),
where the left-hand side is the q+1th left derived functor in the
sense of Quillen of the functor W(-)F. The proof is by
comparison with the topological cyclic homology TC*(A;p)
introduced by Bokstedt-Hsiang-Madsen.
The original paper contained two mistakes. These mistakes, however, do
not affect the main conclusions of the paper. Please see the erratum
below for a correction.
Lars Hesselholt <larsh@math.nagoya-u.ac.jp>