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In a series of papers [Pan0], [Pan1], [Pan2], [Pan3] we give a detailed andbetter structured proof of the Grothendieck--Serre's conjecture for semi-localregular rings containing a finite field. The outline of the proof is the sameas in [P1],[P2],[P3]. If the semi-local regular ring contains an infinitefield, then the conjecture is proved in [FP]. Thus the conjecture is true forregular local rings containing a field.  The present paper is the one [Pan1] in that new series. Theorem 1.1 is one ofthe main result of the paper. It is also one of the key steps in the proof ofthe Grothendieck--Serre's conjecture for semi-local regular rings containing afield (see [Pan3]). The proof of Theorem 1.1 is completely geometric.

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