Thursday, December 2, 2021 - 11:00

### Abstract or Additional Information

Let R be a regular semilocal integral domain. Let G be a reductive

group scheme over R. A famous conjecture of Grothendieck and Serre

predicts that a G-torsor (a.k.a. a principal G-bundle) over R is

trivial, provided it is rationally trivial. After briefly discussing

the conjecture and its state, I'll talk about the following

generatlization: let R and G be as above, let f be an element of R

that does not belong to the square of any maximal ideal of R

(equivalently, the hypersurface {f=0} is regular). Then a G-torsor

over the localization R_f is trivial, provided that it is rationally

trivial. I will only assume basic knowledge of algebraic geometry and minimal

familiarity with torsors.