### Witt vectors of non-commutative rings and topological cyclic homology, by Lars Hesselholt

Classically, one has for every commutative ring A the associated
ring of p-typical Witt vectors W(A). We extend this construction
to a functor which assigns to every assiciative (but not necessarily
commutative or unital) ring A an abelian group W(A). The extended
functor comes equipped with additive Frobenius and Verschiebung
operators. Let K_*(A;Z_p) 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

K_i(A;Z_p)=L_{i+1}W(A)_F,

the left derived functors in the sense of Quillen of the Frobenius
coinvariants of W(A). The proof is by comparison with the topological
cyclic homology TC_*(A) introduced by Bokstedt-Hsiang-Madsen.

Lars Hesselholt <larsh@math.mit.edu>