A classification of torsors over Laurent polynomial rings

TL;DR

This paper classifies torsors over Laurent polynomial rings by showing unramified torsors extend uniquely, enabling a comprehensive classification of G-torsors over these rings.

Contribution

It provides a novel classification of G-torsors over Laurent polynomial rings by establishing extension properties of unramified torsors.

Findings

Unramified K_n-torsors extend uniquely to R_n-torsors.

Complete classification of G-torsors over R_n achieved.

Extension results facilitate understanding of torsor structures.

Abstract

Let R\_n be the ring of Laurent polynomials in n variables over a field k of characteristic zero and let K\_n be its fraction field.Given a linear algebraic k-group $G$, we show that a K\_n-torsor under G which is unramified with respect to X=Spec(R\_n)extends to a unique toral R\_n-torsor under G. This result, in turn, allows us to classify all G-torsors over R\_n.