A classification of torsors over Laurent polynomial rings

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 $R_n$.