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

TL;DR

This paper proves a significant case of the Grothendieck-Serre conjecture for principal bundles over semi-local schemes, extending results to arbitrary reductive group schemes and simplifying existing proofs.

Contribution

It establishes the triviality of principal G-bundles over semi-local schemes under certain conditions and generalizes previous results from simple simply-connected groups to all reductive groups.

Findings

Principal G-bundles trivial over generic fibers are trivial over the entire scheme.

The proof of the Grothendieck-Serre conjecture is simplified.

Results extended from simple simply-connected groups to all reductive groups.

Abstract

Let $U$ be a regular connected affine semi-local 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 $W\times_kU$ is trivial if it is trivial over the generic fiber of the projection $W\times_kU\to U$. We also simplify the proof of the Grothendieck-Serre conjecture: let $U$ be a regular connected affine semi-local 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.