On the Gille theorem for the relative projective line: I

TL;DR

This paper extends Gille's theorem on torsors over the projective line to a relative setting, establishing weak homotopy invariance for torsors under reductive group schemes over certain semi-local regular domains.

Contribution

It proves a relative version of Gille's theorem and derives weak homotopy invariance results for torsors over semi-local regular domains with infinite residue fields.

Findings

Proves a relative Gille theorem for torsors on the projective line.

Establishes weak homotopy invariance for torsors under reductive group schemes.

Results are applicable to semi-local regular domains with infinite residue fields.

Abstract

We prove a relative version of a theorem on torsors on the projective line due to Philippe Gille. As a consequence we obtain a ``weak homotopy invariance'' result for torsors under reductive group schemes defined over arbitrary semi-local regular domains. Specifically, only regular semi-local domains with infinite residue fields are regarded in this preprint. However, all results of the present preprint are true (after minor modifications) for arbitrary semi-local regular domains. This will be the topic of our next preprint.