Non-stable $K_1$-functors of discrete valuation rings containing a field

P. Gille^ab, A. Stavrova<sup>c

a</sup> UMR 5208, Institut Camille Jordan, Université Claude Bernard Lyon 1, 43 boulevard du 11 novembre 1918, 69622 Villeurbanne cedex, France
^b Institute of Mathematics “Simion Stoilow” of the Romanian Academy, 21 Calea Grivitei Street, 010702 Bucharest, Romania
^c St. Petersburg Department of Steklov Mathematical Institute, Fontanka 27, 191023 St. Petersburg, Russia

Abstract: Let $k$ be a field, and let $G$ be a simply connected semisimple $k$-group which is isotropic and contains a stricty proper parabolic $k$-subgroup $P$. Let $D$ be a discrete valuation ring which is a local ring of a smooth algebraic curve over $k$. We show that $K_1^G(D)=K_1^G(K)$, where $K$ is the fraction field of $D$ and $K_1^G(-)=G(-)/E_P(-)$ is the corresponding non-stable $K_1$-functor, also called the Whitehead group of $G$. As a consequence, $K_1^G(D)$ coincides with the (generalized) Manin's $R$-equivalence class group of $G(D)$.

Key words and phrases: non-stable $K_1$-functor, $R$-equivalence, reductive group, parabolic subgroup.

UDC: 512.743

Received: 08.12.2025

Language: English