R-equivalence on group schemes

TL;DR

This paper introduces R-equivalence for group schemes over semilocal rings, explores its relation to rational properties, and investigates specific cases like tori and isotropic semisimple groups, connecting it to known equivalences and the Kneser-Tits problem.

Contribution

It defines R-equivalence in a new setting and relates it to existing concepts, providing insights into its behavior for certain classes of group schemes.

Findings

R-equivalence coincides with Karoubi-Villamayor equivalence in specific cases

Connections established between R-equivalence and the Kneser-Tits problem

Constructs specialization maps for R-equivalence over regular algebras containing a field

Abstract

We define R-equivalence for group schemes over a semilocal ring and relate this with rational properties. Two main cases are investigated: tori and isotropic semisimple simply connected group schemes where we show in certain cases that R-equivalence coincide with Karoubi-Villamayor equivalence and is also related to the Kneser-Tits problem in this setting. Finally we construct specialization maps for R-equivalence in the case of regular algebras containing a field.