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

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, $K_1^G(D)$ coincides with the (generalized) Manin's $R$-equivalence class group of $G(D)$.