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Witt vectors as a polynomial functor

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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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Correspondence to D. Kaledin.

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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).

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