A theory of nice triples and a theorem due to O.Gabber

TL;DR

This paper provides a geometric proof of a key case of the Grothendieck--Serre conjecture for semi-local regular rings over finite fields, extending the theory of nice triples inspired by Voevodsky.

Contribution

It introduces a geometric approach using nice triples to prove the conjecture for certain reductive groups over finite fields, building on previous work and theories.

Findings

Proves the Grothendieck--Serre conjecture for semi-local regular rings over finite fields with simply-connected reductive groups.

Develops the theory of nice triples inspired by Voevodsky's standard triples.

Establishes a geometric proof technique applicable to the conjecture.

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 [Pan0] in that series. Theorem 1.2 is one of the main result of the paper. The proof of the latter theorem is completely geometric. It is based on a theory of nice triples from [PSV] and on its extension from [P]. The theory of nice triples is inspired by the Voevodsky theory of standart triples [V]. Theorem 1.2 yields an unpublished result due to O.Gabber (see Theorem 1.1=Theorem 3.1). It states that the Grothendieck--Serre's conjecture for semi-local regular rings containing a finite field is true providing that the group is simply-connected reductive and is extended from the base field.