R-equivalence on group schemes

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.