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Let k be an infinite field. Let R be the semi-local ring of a finite familyof closed points on a k-smooth affine irreducible variety, let K be thefraction field of R, and let G be a reductive simple simply connected R-groupscheme isotropic over R. We prove that for any Noetherian k-algebra A, the mapof etale cohomology sets H^1(A\otimes_k R,G)-> H^1(A\otimes_ k K,G), induced bythe inclusion of R into K, has trivial kernel. This implies theSerre-Grothendieck conjecture for such groups G.  The main theorem for A=k and some other results of the present paper are usedsignificantly in arXiv:1211.2678 to prove the Serre-Grothendieck conjecture forall reductive groups over a regular semi-local ring containing an infinitefield.

On Grothendieck--Serre's conjecture concerning principal G-bundles over reductive group schemes:I