On the Grothendieck-Serre conjecture about principal bundles and its generalizations

Fedorov, Roman

doi:10.2140/ant.2022.16.447

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On the Grothendieck-Serre conjecture about principal bundles and its generalizations

Roman Fedorov

Vol. 16 (2022), No. 2, 447–465

DOI: 10.2140/ant.2022.16.447

Abstract

Let $U$ be a regular connected affine semilocal scheme over a field $k$ . Let $G$ be a reductive group scheme over $U$ . Assuming that $G$ has an appropriate parabolic subgroup scheme, we prove the following statement. Given an affine $k$ -scheme $W$ , a principal $G$ -bundle over $W \times_{k} U$ is trivial if it is trivial over the generic fiber of the projection $W \times_{k} U \to U$ .

We also simplify the proof of the Grothendieck–Serre conjecture: let $U$ be a regular connected affine semilocal scheme over a field $k$ . Let $G$ be a reductive group scheme over $U$ . A principal $G$ -bundle over $U$ is trivial if it is trivial over the generic point of $U$ .

We generalize some other related results from the simple simply connected case to the case of arbitrary reductive group schemes.

Keywords

algebraic groups, principal bundles, local schemes, Grothendieck–Serre conjecture, affine Grassmannians

Mathematical Subject Classification

Primary: 14L15

Milestones

Received: 30 August 2020

Revised: 3 February 2022

Accepted: 13 June 2021

Published: 27 April 2022

Authors

Roman Fedorov
University of Pittsburgh Pittsburgh, PA United States

Vol. 16, No. 2, 2022