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A proof of Grothendieck--Serre conjecture on principal bundles over a semi-local regular ring containing an infinite field is given in [FP] recently. That proof is based significantly on Theorem 1.0.1 stated below in the Introduction and proven in the present preprint. Theorem 1.0.1 itself is a consequence of two purity theorems (Theorems A and 10.0.30) proven below in the present preprint. The geometric part of a new preprint [PSV] and the main result of an article [C-T-S] are used significantly in proofs of those two purity theorems. One of that purity result looks as follows. Let O be a semi-local ring of finitely many closed points on a k-smooth irreducible affine scheme, where k is an infinite field. Given a smooth O-group scheme morphism mu G to C of reductive O-group schemes, with a torus C one can form a functor from O-algebras to abelian groups, which takes an O-algebra S to the quotient group F(S)=C(S) modulo mu(G(S)). Assuming additionally that the kernel of mu is a reductive O-group scheme, we prove that this functor satisfies a purity theorem for the k-algebra O. Examples to mentioned purity results are considered at the very end of the preprint.

On Grothendieck-Serre's conjecture concerning principalG-bundles over reductive group schemes: II