## 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.

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To Sasha Beilinson, on his birthday.

Partially supported by the Russian Academic Excellence Project ‘5-100’.

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Kaledin, D. Witt vectors as a polynomial functor.
*Sel. Math. New Ser.* **24**, 359–402 (2018). https://doi.org/10.1007/s00029-017-0365-z

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DOI: https://doi.org/10.1007/s00029-017-0365-z