Non-Stable K₁-Functors of Discrete Valuation Rings Containing a Field

2025/12/11 by Gille, Philippe, Stavrova, Anastasia
Mathematics · #Advanced Algebra and Geometry #Algebraic Geometry (math.AG) #Algebraic Geometry and Number Theory #FOS: Mathematics #Homotopy and Cohomology in Algebraic Topology

paper · doi:10.48550/arxiv.2512.10681

openalex publication_date 2025/12/11 · openalex created_date 2025/12/13 · openalex updated_date 2026/07/28

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 K₁-functor (for G and P, also called the Whitehead group of G) coincide over D and K. As a consequence, K^G₁ (D) coincides with the (generalized) Manin's R-equivalence class group of G(D).

Citations

• Group schemes over LG-rings and applications to cancellation theorems and Azumaya algebras
• Unramified Grothendieck-Serre for isotropic groups
• A¹-invariance of non-stable K₁-functors in the equicharacteristic case
• Homotopy invariance of non-stable K₁-functors

Related

• Homotopy invariance of non-stable K₁-functors
• A¹-invariance of non-stable K₁-functors in the equicharacteristic case
• Transfers for non-stable K₁-functors of classical type
• Non-stable K₁-functors of multiloop groups
• On the product functor on inner forms of the general linear group over a non-Archimedean local field