Non-Stable $K_1$-Functors of Discrete Valuation Rings Containing a Field
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| Format: | Preprint |
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2025
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| _version_ | 1866911314619662336 |
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| author | Gille, Philippe Stavrova, Anastasia |
| author_facet | Gille, Philippe Stavrova, Anastasia |
| contents | Let $k$ be a field, and let $G$ be a simply connected semisimple k-group which is isotropic and contains a strictly proper parabolic $k$-subgroup $P$. Let $D$ be a discrete valuation ring which is a local ring of a smooth algebraic curve over $k$. Let $K$ be the fraction field of $D$. We show that the corresponding non-stable $K_1$-functor (for $G$ and $P$, also called the Whitehead group of $G$) coincide over $D$ and $K$. As a consequence, $K^G_1 (D)$ coincides with the (generalized) Manin's $R$-equivalence class group of $G(D)$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2512_10681 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Non-Stable $K_1$-Functors of Discrete Valuation Rings Containing a Field Gille, Philippe Stavrova, Anastasia Algebraic Geometry Let $k$ be a field, and let $G$ be a simply connected semisimple k-group which is isotropic and contains a strictly proper parabolic $k$-subgroup $P$. Let $D$ be a discrete valuation ring which is a local ring of a smooth algebraic curve over $k$. Let $K$ be the fraction field of $D$. We show that the corresponding non-stable $K_1$-functor (for $G$ and $P$, also called the Whitehead group of $G$) coincide over $D$ and $K$. As a consequence, $K^G_1 (D)$ coincides with the (generalized) Manin's $R$-equivalence class group of $G(D)$. |
| title | Non-Stable $K_1$-Functors of Discrete Valuation Rings Containing a Field |
| topic | Algebraic Geometry |
| url | https://arxiv.org/abs/2512.10681 |