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

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 $W{\times }_{k}U$ 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 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.

Final version to be published in the Journal of Algebra and Number Theory. We slightly change the isotropy condition as well as the terminology: see Definition 1.1 of strongly locally isotropic reductive groups. We remove the assumption that U is geometrically regular in Theorem 1 (regular is enough). Other minor improvements

References in corpus (1)

• Nice triples and Grothendieck--Serre's conjecture concerning principal G-bundles over reductive group schemes

Cited by in corpus (1)

• The Grothendieck-Serre Conjecture over Semilocal Dedekind Rings