Grothendieck--Serre for constant reductive group schemes

Abstract

The Grothendieck-Serre conjecture predicts that on a regular local ring, no nontrivial reductive torsor becomes trivial over the fraction field. While this conjecture has been proven in the equicharacteristic case, it remains open in the mixed characteristic case. This article establishes a generalized version of the conjecture over Pr\"ufer bases for constant reductive group schemes. In particular, the Noetherian restriction of our main result settles the constant, unramified case of the Grothendieck-Serre conjecture. Along the way, inspired by the recent preprint of Cesnavicius, we also prove several versions of the Nisnevich conjecture in our context.