Universal Group-theoretic Characterisation of Witt Vectors

Abstract

Let Ab, CRings, Rings respectively denote the category of abelian groups, unital commutative rings and unital associative rings. For a prime p, we have the classical construction of a p-typical Witt vector functor W : CRings → CRings given by E. Witt. There are multiple constructions of the group of p-typical Witt vectors of associative (possibly non-commutative) rings. It is known that all these constructions match with classical Witt functor W, when restricted to the CRings. One of the natural questions we have tried to answer in this thesis is - Is there a universal Witt functor on the Rings?

The first part of the thesis is devoted to the commutative set-up. Note that for a commutative ring R, the group W(R) is endowed with a Verschiebung operator V: W(R)→ W(R) and a Teichmüller map < >: R → W(R). One of the properties satisfied by V, < > is that the map R → W(R) seding x to V<x^p>-p<x> is an additive map. In this thesis, we show that for an odd prime p, this property essentially characterises the functor W. Unlike other known characterisations, this is a group-theoretic characterisation, in the sense that it does not use the ring structure of W(R). This is important because most of the constructions of a Witt functor defined on non-commutative rings do not have a ring structure. Hence, we can use our group theoretic characterisation of W to answer the above question.

The second part of the thesis is devoted to the associative rings. We first define a notion of a pre-Witt functor which abstracts the above group theoretic property. We give a construction of a pre-Witt functor E : Rings → Ab adapting the construction of the Witt functor given by Cuntz-Deninger for the commutative rings. We prove that E when restricted to CRings matches with the functor W. We then define a Witt functor \hat{E}: Rings → Ab and give a universal group theoretic characterisation of \hat{E} modulo an explicit conjecture about non-commutative polynomials. We prove that \hat{E} admits a natural surjection to the Hesselholt’s Witt functor W_H, without using the conjecture. We also suspect that the Witt functor W_H is the universal Morita invariant Witt functor.