Witt vectors as a polynomial functor

• Published in: Selecta mathematica (Basel, Switzerland) Switzerland), 2018-03, Vol.24 (1), p.359-402
• Main Author: Kaledin, D.
• Format: Article
• Language: English
• Subjects: Group theory; Homology; Mathematical analysis; Mathematics; Mathematics and Statistics; Polynomials; Vector spaces; Vectors (mathematics)
• ISSN: 1022-1824; 1420-9020
• DOI: 10.1007/s00029-017-0365-z

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 H H 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” W H H ∗ ( 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 ≥ 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.