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 in the sense of arXiv:1308.3743, 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 $W_m$, $m \geq 1$ 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 in the sense of arXiv:1308.3743. The construction is very simple, and it only depends on elementary properties of finite cyclic groups.