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.

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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