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International conference "Geometry of Algebraic Varieties" dedicated to the memory of V. A. Iskovskikh
October 24, 2013 15:00–16:00, Moscow, Steklov Mathematical Institute of RAS

A proof of the geometric case of a conjecture of Grothendieck and Serre concerning principal bundles

I. A. Panin
 Video records: Flash Video 464.6 Mb Flash Video 2,784.1 Mb MP4 464.6 Mb

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.