A presentation of the symmetric Grothendieck-Witt group of local rings over $\mathbb{F}_2$
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866908642507227136 |
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| author | Nicolas, Marcus |
| author_facet | Nicolas, Marcus |
| contents | Let $R$ be a commutative local ring. We provide an explicit presentation of the symmetric Grothendieck-Witt ring $\mathrm{GW}^{\mathrm{s}}(R)$ of $R$ as an abelian group when $R$ has residue field $\mathbb{F}_2$. This completes a recent work by Rogers and Schlichting, where an explicit presentation of $\mathrm{GW}^{\mathrm{s}}(R)$ is given when the residue field is different from $\mathbb{F}_2$. We then use this result to compute the symmetric Grothendieck-Witt rings for the sequences of local rings $\mathbb{Z}/2^n\mathbb{Z}$ and $\mathbb{F}_2[x]/(x^n)$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2504_18909 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | A presentation of the symmetric Grothendieck-Witt group of local rings over $\mathbb{F}_2$ Nicolas, Marcus K-Theory and Homology Rings and Algebras Let $R$ be a commutative local ring. We provide an explicit presentation of the symmetric Grothendieck-Witt ring $\mathrm{GW}^{\mathrm{s}}(R)$ of $R$ as an abelian group when $R$ has residue field $\mathbb{F}_2$. This completes a recent work by Rogers and Schlichting, where an explicit presentation of $\mathrm{GW}^{\mathrm{s}}(R)$ is given when the residue field is different from $\mathbb{F}_2$. We then use this result to compute the symmetric Grothendieck-Witt rings for the sequences of local rings $\mathbb{Z}/2^n\mathbb{Z}$ and $\mathbb{F}_2[x]/(x^n)$. |
| title | A presentation of the symmetric Grothendieck-Witt group of local rings over $\mathbb{F}_2$ |
| topic | K-Theory and Homology Rings and Algebras |
| url | https://arxiv.org/abs/2504.18909 |