Nice triples and Grothendieck--Serre's conjecture concerning principal G-bundles over reductive group schemes

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. The present paper is the one [Pan1] in that new series. Theorem 1.1 is one of the main result of the paper. It is also one of the key steps in the proof of the Grothendieck--Serre's conjecture for semi-local regular rings containing a field (see [Pan3]). The proof of Theorem 1.1 is completely geometric.