R-equivalence on group schemes
Philippe Gille, A Stavrova
TL;DR
This paper develops a broad framework for R-equivalence of points on group schemes over semilocal and regular bases, extending Manin’s notion from fields to integral settings. It shows that for tori and simply connected isotropic groups over equicharacteristic semilocal regular bases, R-equivalence coincides with Karoubi–Villamayor and with certain A^1-equivalence/K1-functors, linking these invariants to retract rationality and the Kneser–Tits problem. The authors establish birational-invariance properties, parabolic-reduction techniques, specialization maps over DVRs and henselian pairs, and a robust patching formalism that connects A^1-equivalence, non-stable K1, and R-equivalence. Collectively, these results provide new tools to analyze rationality questions, local-global principles, and specialization phenomena for reductive group schemes in arithmetic and algebraic geometry contexts. The work has implications for the Serre–Grothendieck conjecture in equicharacteristic settings and for understanding how rationality properties of fibers reflect in the global structure.
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.