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>