Arnab Kundu: Torsors on Valuation Rings

Abstract

A conjecture of Grothendieck and Serre states that a torsor under a reductive group over a Noetherian regular scheme X is Zariski locally trivial if it is generically trivial. Recently, this conjecture has seen progress through the work of Fedorov, Panin and Česnavičius. We shall see the historical background of this conjecture, followed by the techniques that go into the proof of the quasi-split case in the analogous situation when X is a smooth scheme over a valuation ring of rank one.