Proof of Grothendieck--Serre conjecture on principal G-bundles over regular local rings containing a finite field

TL;DR

This paper proves the Grothendieck--Serre conjecture for principal G-bundles over regular local rings containing a finite field, establishing their triviality when trivial over the fraction field, thus confirming a key case of the conjecture.

Contribution

The paper establishes the conjecture for regular local rings containing a finite field, filling a gap in the known cases of the Grothendieck--Serre conjecture.

Findings

Principal G-bundles over R are trivial if trivial over the fraction field K.

The induced map H^1_{et}(R,G) to H^1_{et}(K,G) has a trivial kernel.

The result complements previous work for rings containing infinite or characteristic zero fields.

Abstract

Let R be a regular local ring, containing a finite field. Let G be a reductive group scheme over R. We prove that a principal G-bundle over R is trivial, if it is trivial over the fraction field of R. In other words, if K is the fraction field of R, then the map of pointed sets H^1_{et}(R,G) \to H^1_{et}(K,G), induced by the inclusion of R into K, has a trivial kernel. Certain arguments used in the present preprint do not work if the ring R contains a characteristic zero field. In that case and, more generally, in the case when the regular local ring R contains an infinite field this result is proved in joint work due to R.Fedorov and I.Panin (see [FP]). Thus the Grothendieck--Serre conjecture holds for regular local rings containing a field.