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| Format: | Preprint |
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2026
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| Online Access: | https://arxiv.org/abs/2601.20536 |
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| _version_ | 1866914286510538752 |
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| author | Pisolkar, Supriya Samanta, Biswanath |
| author_facet | Pisolkar, Supriya Samanta, Biswanath |
| contents | For a prime $p$ and an associative ring $R$ with unity, there are various constructions of $p$-typical Witt vectors of $R$, all of which specialize to the classical $p$-typical Witt vectors when $R$ is commutative. These constructions are endowed with a Verschiebung operator $V$ and a Teichmüller map $\langle \cdot \rangle$, and they satisfy the property that the map $x \mapsto V\langle x^p\rangle - p\langle x \rangle$ is additive. In this paper, we adapt the group-theoretic universal characterization of classical $p$-typical Witt vectors proposed in arXiv:2405.12680 to the non-commutative setting. Our main result is that this approach yields a construction of Witt vectors for associative rings, denoted $E$, which specializes correctly to the classical Witt functor in the commutative case. The construction of $E$ is inspired by the Witt functor of Cuntz--Deninger, and we show that $E$ is a universal pre-Witt functor, subject to an explicit conjecture concerning non-commutative polynomials. We further introduce the notion of a Witt functor and construct a universal Witt functor $\hat{E}$, which is closely related to Hesselholt's Witt functor $W_H$. We suspect that $W_H$ is, in fact, the universal Morita-invariant Witt functor. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2601_20536 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | A universal construction of $p$-typical Witt vectors of associative rings Pisolkar, Supriya Samanta, Biswanath Number Theory 13F35, 16W60 For a prime $p$ and an associative ring $R$ with unity, there are various constructions of $p$-typical Witt vectors of $R$, all of which specialize to the classical $p$-typical Witt vectors when $R$ is commutative. These constructions are endowed with a Verschiebung operator $V$ and a Teichmüller map $\langle \cdot \rangle$, and they satisfy the property that the map $x \mapsto V\langle x^p\rangle - p\langle x \rangle$ is additive. In this paper, we adapt the group-theoretic universal characterization of classical $p$-typical Witt vectors proposed in arXiv:2405.12680 to the non-commutative setting. Our main result is that this approach yields a construction of Witt vectors for associative rings, denoted $E$, which specializes correctly to the classical Witt functor in the commutative case. The construction of $E$ is inspired by the Witt functor of Cuntz--Deninger, and we show that $E$ is a universal pre-Witt functor, subject to an explicit conjecture concerning non-commutative polynomials. We further introduce the notion of a Witt functor and construct a universal Witt functor $\hat{E}$, which is closely related to Hesselholt's Witt functor $W_H$. We suspect that $W_H$ is, in fact, the universal Morita-invariant Witt functor. |
| title | A universal construction of $p$-typical Witt vectors of associative rings |
| topic | Number Theory 13F35, 16W60 |
| url | https://arxiv.org/abs/2601.20536 |