On the purity conjecture of Nisnevich for torsors under reductive group schemes

Let R be a regular semilocal integral domain containing a field k. Assume that k is infinite or there is a homomorphism R k left inverse to the inclusion. Let f R be an element such that for all maximal ideals m of R we have f m². Let G be a reductive group scheme over R. Under an isotropy assumption on G we show that a G-torsor over Rf is trivial, provided it is rationally trivial. We show that it is not true without the isotropy assumption. The first statement is derived from its abstract version concerning Nisnevich sheaves satisfying some properties. The counterexample is constructed by providing a torsor over a local family of affine lines that cannot be extended to the projective lines. The latter is accomplished using the technique of affine Grassmannians.