  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. We prove that for any Noetherian k-algebra A, the map
of etale cohomology sets H^1(A\otimes_k R,G)-> H^1(A\otimes_ k K,G), induced by
the inclusion of R into K, has trivial kernel. This implies the
Serre-Grothendieck conjecture for such groups G.
  The main theorem for A=k and some other results of the present paper are used
significantly in arXiv:1211.2678 to prove the Serre-Grothendieck conjecture for
all reductive groups over a regular semi-local ring containing an infinite
field.
