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Working paper

Witt vectors as a polynomial functor

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 HH0(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 Wm, m≥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 Wm 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.