Local-global principle for over semiglobal fields

TL;DR

This paper investigates local-global principles for torsors under reductive groups over semiglobal fields, establishing conditions under which these principles hold for certain valuations.

Contribution

It proves that the local-global principle holds for completions at divisorial valuations when the group is a retract rational variety over the semiglobal field.

Findings

Local-global principle holds for divisorial valuations

Applicable to reductive groups that are retract rational varieties

Enhances understanding of torsor behavior over semiglobal fields

Abstract

We compare different local-global principles for torsors under a reductive group G defined over a semiglobal field F. In particular if the F-group G s a retract rational F-variety, we prove that the local global principle holds for the completions with respect to divisorial valuations of F.