On a universal characterisation of \(p\)-typical Witt vectors

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 \(W(R)\xrightarrow{V}W(R)\) and a Teichm\"{u}ller map \(R\xrightarrow{\langle \ \rangle}W(R)\). One of the properties...

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Bibliographic Details
Published in:arXiv.org 2024-05
Main Authors: Pisolkar, Supriya, Samanta, Biswanath
Format: Article
Language:English
Subjects:
Commutativity
Operators (mathematics)
Rings (mathematics)
Online Access:Get full text
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Summary: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 \(W(R)\xrightarrow{V}W(R)\) and a Teichm\"{u}ller map \(R\xrightarrow{\langle \ \rangle}W(R)\). One of the properties satisfied by \(V, \langle \ \rangle\) is that the map \(R \to W(R)\) given by \(x\mapsto V\langle x^p\rangle - p\langle x \rangle\) is an additive map. In this paper we show that for \(p\neq 2\), 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 \(p\neq 2\). Moreover we provide evidence for this conjecture.
ISSN:2331-8422