Two purity theorems and the Grothendieck--Serre's conjecture concerning principal G-bundles

Abstract

In a series of papers [Pan0], [Pan1], [Pan2], [Pan3] we give a detailed and better structured proof of the Grothendieck--Serre's conjecture for semi-local regular rings containing a finite field. The outline of the proof is the same as in [P1],[P2],[P3]. If the semi-local regular ring contains an infinite field, then the conjecture is proved in [FP]. Thus the conjecture is true for regular local rings containing a field. A proof of Grothendieck--Serre conjecture on principal bundles over a semi-local regular ring containing an arbitrary field is given in [Pan3]. That proof is heavily based on Theorem 1.3 stated below in the Introduction and proven in the present paper.