Torsors on the affine line
Let \(G\) be a connected algebraic reductive group over an arbitrary field \(k\). The classification of \(G\)-torsors over the projective line was given by \textit{A. Grothendieck} [Am. J. Math. 79, 121--138 (1957; Zbl 0079.17001)] for the case when \(k\) is an algebraically closed field of characteristic zero and by \textit{G. Harder} [Invent. Math. 6, 107--149 (1968; Zbl 0186.25902)] for a general field. Torsors over an affine line for a general field have been studied by \textit{M. S. Raghunathan} and \textit{A. Ramanathan} [Proc. Indian Acad. Sci., Math. Sci. 93, 137--145 (1984; Zbl 0587.14007)]. In this article, the author gives a new proof of a result of Raghunathan and Ramanathan stating that any \(G\)-torsor on the affine line which is trivial over the separable closure of \(k\) comes from a \(G\)-torsor over \(\text{Spec}\,k\).
scientific article; zbMATH DE number 1885650