On a universal characterisation of p-typical Witt vectors

2024/05/21 by Pisolkar, Supriya, Samanta, Biswanath
#13F35 #16W60 #FOS: Mathematics #Number Theory (math.NT) #Rings and Algebras (math.RA)

paper · doi:10.48550/arxiv.2405.12680

Abstract

For a prime p and a commutative ring R with unity, let W(R) denote the ring of p-typical Witt vectors. The ring W(R) is endowed with a Verschiebung operator W(R)\xrightarrowVW(R) and a Teichmüller map R\xrightarrow⟨ ⟩W(R). One of the properties satisfied by V, ⟨ ⟩ is that the map R → W(R) given by x↦ V⟨ x^p⟩ - p⟨ x ⟩ is an additive map. In this paper we show that for p≠ 2, this property essentially characterises the functor W. Unlike other characterisations, this only uses the group structure on W(R) and hence is suitable for generalising to the non-commutative setup. We give a conjectural characterisation of Hesselholt's functor of p-typical Witt vectors using a universal property for p≠ 2. Moreover we provide evidence for this conjecture.

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