On a theorem of Harder

TL;DR

This paper proves that principal G-bundles over Dedekind domains are trivial if they are Zariski-locally trivial, extending Harder's 1967 result for quasi-split groups to all simply connected isotropic reductive groups.

Contribution

It generalizes Harder's theorem by establishing triviality of principal bundles for a broader class of groups over Dedekind domains.

Findings

Principal G-bundles over Dedekind domains are trivial if Zariski-locally trivial.

Extension of Harder's 1967 result to all simply connected isotropic reductive groups.

Provides a significant advancement in the theory of algebraic groups and principal bundles.

Abstract

We prove that for any simply connected isotropic reductive group G over a Dedekind domain D, any Zariski-locally trivial principal G-bundle over D is trivial. The corresponding result for quasi-split groups was proved in 1967 by G. Harder.