Vol. 16, No. 2, 2022

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

Roman Fedorov

Vol. 16 (2022), No. 2, 447–465
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 ×kU is trivial if it is trivial over the generic fiber of the projection W ×kU 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