Principal bundles admitting a rational section
The main result of the paper is the following theorem. Let \(k\) be an infinite field and \(G\) a smooth reductive group-scheme over \(k\). Let \(\mathbb{X}\) be an irreducible smooth scheme of finite type over \(k\). Let \((P, \pi, m)\) be a principal \(G\)-bundle on \(\mathbb{X}\) which is trivial over a nonempty Zariski open subset of \(\mathbb{X}\). Then for each \(x \in \mathbb{X}\) there is a Zariski open neighbourhood \(\mathbb{X} (x)\) of \(x\) in \(\mathbb{X}\) such that \((P, \pi, m)\) is trivial over \(\mathbb{X} (x)\). This result is an affirmation of the conjecture of Grothendieck and Serre in the case of an infinite field and \(\mathbb{X}\) is of finite type over \(k\). It is new only when \(k\) is not perfect and \(G\) has an anisotropic factor over \(k\); when \(k\) is infinite perfect or \(G\) has no anisotropic factors, the theorem is due to \textit{J.-L. Colliot-Thélène} and \textit{M. Ojanguren} [see Publ. Math., Inst. Hautes Étud. Sci. 75, 97-122 (1992; Zbl 0795.14029)].
scientific article; zbMATH DE number 554023