On Grothendieck–Serre’s conjecture concerning principal -bundles over reductive group schemes: I

Let k be an infinite field. Let R be the semi-local ring of a finite family of closed points on a k -smooth affine irreducible variety, let K be the fraction field of R , and let G be a reductive simple simply connected R -group scheme isotropic over R . Our Theorem 1.1 states that for any Noetherian k -algebra A the kernel of the map eqnarrayH_et¹(R⊗ ₖA,G)→ H_et¹(K⊗ ₖA,G)eqnarray induced by the inclusion of R into K is trivial. Theorem 1.2 for $A=k$ and some other results of the present paper are used significantly in Fedorov and Panin [ A proof of Grothendieck–Serre conjecture on principal bundles over a semilocal regular ring containing an infinite field , Preprint (2013), arXiv:1211.2678v2 ] to prove the Grothendieck–Serre’s conjecture for regular semi-local rings R containing an infinite field.