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

TL;DR

This paper provides a geometric proof of a key step in confirming Grothendieck--Serre's conjecture for principal G-bundles over semi-local regular rings containing a finite field, advancing the understanding of algebraic group schemes.

Contribution

It offers a new geometric proof of a crucial theorem that supports the broader proof of Grothendieck--Serre's conjecture for certain regular rings.

Findings

Proves Theorem 1.1 as a key step in the conjecture's proof.

Provides a geometric approach to the problem.

Advances the proof for rings containing a finite field.

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. 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.