On Grothendieck-Serre conjecture concerning principal G-bundles over regular semi-local domains containing a finite field: II

TL;DR

This paper advances the proof of Grothendieck-Serre's conjecture for principal G-bundles over regular semi-local domains containing a finite field, reducing the problem to semi-simple simply-connected groups and confirming the conjecture in this case.

Contribution

It reduces the Grothendieck-Serre conjecture to semi-simple simply-connected groups over finite fields and completes the proof for regular semi-local domains containing such fields.

Findings

The conjecture holds for regular semi-local domains over finite fields.

Reduction to semi-simple simply-connected groups is achieved.

Proof of the conjecture for these groups is established.

Abstract

In three preprints [Pan1], [Pan3] and the present one we prove Grothendieck-Serre's conjecture concerning principal G-bundles over regular semi-local domains R containing a finite field (here $G$ is a reductive group scheme). The preprint [Pan1] contains main geometric presentation theorems which are necessary for that. The present preprint contains reduction of the Grothendieck--Serre's conjecture to the case of semi-simple simply-connected group schemes (see Theorem 1.0.1). The preprint [Pan3] contains a proof of that conjecture for regular semi-local domains R containing a finite field. The Grothendieck--Serre conjecture for the case of regular semi-local domains containing an infinite field is proven in joint work due to R.Fedorov and I.Panin (see [FP]). Thus the conjecture holds for regular semi-local domains containing a field. The reduction is based on two purity results (Theorem 1.0.2 and Theorem 10.0.29).