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| Main Authors: | , |
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| Format: | Preprint |
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2024
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| Online Access: | https://arxiv.org/abs/2405.12680 |
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| _version_ | 1866918309255970816 |
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| author | Pisolkar, Supriya Samanta, Biswanath |
| author_facet | Pisolkar, Supriya Samanta, Biswanath |
| contents | For a prime $p$ and a commutative ring $R$ with unity, let $W(R)$ denote the group of $p$-typical Witt vectors. The group $W(R)$ is endowed with a Verschiebung operator $V: W(R)\to W(R)$ and a Teichmüller map $\langle \ \rangle: R\rightarrow W(R)$. 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 is a group-theoretic characterisation, in the sense that it does not use the ring structure of $W(R)$. Most constructions of the group of $p$-typical Witt vectors of non-commutative rings do not have a ring structure, and hence the above characterisation is more suitable for generalisation to the non-commutative setup. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2405_12680 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | A universal group-theoretic characterisation of $p$-typical Witt vectors Pisolkar, Supriya Samanta, Biswanath Number Theory Rings and Algebras 13F35, 16W60 For a prime $p$ and a commutative ring $R$ with unity, let $W(R)$ denote the group of $p$-typical Witt vectors. The group $W(R)$ is endowed with a Verschiebung operator $V: W(R)\to W(R)$ and a Teichmüller map $\langle \ \rangle: R\rightarrow W(R)$. 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 is a group-theoretic characterisation, in the sense that it does not use the ring structure of $W(R)$. Most constructions of the group of $p$-typical Witt vectors of non-commutative rings do not have a ring structure, and hence the above characterisation is more suitable for generalisation to the non-commutative setup. |
| title | A universal group-theoretic characterisation of $p$-typical Witt vectors |
| topic | Number Theory Rings and Algebras 13F35, 16W60 |
| url | https://arxiv.org/abs/2405.12680 |