Non-Stable $K_1$-Functors of Discrete Valuation Rings Containing a Field

Philippe Gille; Anastasia Stavrova

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arXiv · 2512.10681

Non-Stable $K_1$-Functors of Discrete Valuation Rings Containing a Field

Philippe Gille·Anastasia Stavrova

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Abstract

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 $G(D)$.

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BibTeX RIS

Philippe Gille, Anastasia Stavrova. 2025-12-11. Non-Stable $K_1$-Functors of Discrete Valuation Rings Containing a Field. https://arxiv.org/abs/2512.10681

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