Abstract:
This talk is about our joint work with Roman Fedorov.
Assume that $U$ is a regular scheme, $G$ is a reductive $U$-group scheme,
and $\mathcal{G}$ is a principal $G$-bundle.
It is well known that such a bundle is trivial locally in étale topology
but in general not in Zariski topology.
A. Grothendieck and J.-P. Serre conjectured that $\mathcal{G}$
is trivial locally in Zariski topology,
if it is trivial at all the generic points.
We proved this conjecture for regular local rings $R$,
containing infinite fields. Our proof was inspired by the theory
of affine Grassmannians.
It is also based significantly on the geometric
part of a paper of the second author with A. Stavrova and N. Vavilov.
