Two purity theorems and the Grothendieck--Serre's conjecture concerning principal G-bundles

TL;DR

This paper advances the proof of Grothendieck--Serre's conjecture for principal G-bundles over semi-local regular rings, especially those containing finite or arbitrary fields, by establishing key purity theorems.

Contribution

It provides a detailed proof of the conjecture for semi-local regular rings with finite or arbitrary fields, based on new purity theorems and structural insights.

Findings

Proof of Grothendieck--Serre conjecture for rings containing finite fields

Extension of the conjecture to rings with arbitrary fields

Establishment of two purity theorems related to principal G-bundles

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. A proof of Grothendieck--Serre conjecture on principal bundles over a semi-local regular ring containing an arbitrary field is given in [Pan3]. That proof is heavily based on Theorem 1.3 stated below in the Introduction and proven in the present paper.