#### 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 $W\phantom{\rule{-0.17em}{0ex}}$, a principal $G$-bundle over $W{×}_{k}U$ is trivial if it is trivial over the generic fiber of the projection $W{×}_{k}U\to U\phantom{\rule{-0.17em}{0ex}}$.

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 $U\phantom{\rule{-0.17em}{0ex}}$.

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
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