Witt vectors as a polynomial functor

Abstract

For every commutative ring A, one has a functorial commutative ring W(A) of p-typical Witt vectors of A, an iterated extension of A by itself. If A is not commutative, it has been known since the pioneering work of L. Hesselholt that W(A) is only an abelian group, not a ring, and it is an iterated extension of the Hochschild homology group $HH_0(A)$ by itself. It is natural to expect that this construction generalizes to higher degrees and arbitrary coefficients, so that one can define “Hochschild–Witt homology” $WHH_*(A,M)$ for any bimodule M over an associative algebra A over a field k. Moreover, if one want the resulting theory to be a trace theory, then it suffices to define it for $A=k$. This is what we do in this paper, for a perfect field k of positive characteristic p. Namely, we construct a sequence of polynomial functors $W_m$, $m \ge 1$ from k-vector spaces to abelian groups, related by restriction maps, we prove their basic properties such as the existence of Frobenius and Verschiebung maps, and we show that $W_m$ are trace functors. The construction is very simple, and it only depends on elementary properties of finite cyclic groups.