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.