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.