Abstract: The Grothendieck--Serre conjecture predicts that for a reductive group scheme G over a regular local ring R, there is no nontrivial G-torsor over R trivializes over Frac(R). In this talk, we consider the case when R is instead assumed to be a valuation ring. This result is predicted by the original Grothendieck--Serre and the resolution of singularities. By using flasque resolution of tori, we prove for groups of multiplicative type. Subsequently, taking advantage of techniques in algebraization, we obtain the passage to the Henselian rank one case. Finally, we induct on Levi subgroups and use the integrality of rational points of anisotropic groups to reduce to the semisimple anisotropic case, in which we appeal to properties of parahoric subgroups in Bruhat–Tits theory to conclude. In this talk, time permitting, we focus on techniques in algebraization (such as Gabber--Ramero triples) and their applications to lifting torsors and group schemes from certain closed subschemes.
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