Abstract: The Grothendieck–Serre conjecture predicts that every torsor under a reductive group scheme G over a regular local ring A is trivial if it trivializes over the fraction field Frac A. When A contains a field (the equicharacteristic case), the conjecture was settled in the affirmative by the works of Fedorov-Panin and Panin, but it remains open in the mixed characteristic case. In this talk, we consider the case when A is (essentially) smooth over a valuation ring R (for instance, a DVR) and G is the base change of a reductive R-group scheme. The proof rests on our recent work on the purity of reductive torsors on schemes smooth over valuation rings, a mixed-characteristic Lindel type geometric lemma, and a section theorem for the triviality of torsors over the relative affine line (whose proof depends crucially on the geometry of affine Grassmannian and the rigidity of torsors over the projective line). If time permits, we will also discuss the closed related Nisnevich conjecture and the application to the Bass-Quillen conjecture for torsors.
This is a joint work with Ning Guo.