Grothendieck-Serre conjecture for groups of type F_4 with trivial f_3 invariant

TL;DR

This paper proves that for certain algebraic groups of type F_4 over semi-local regular rings, rational triviality of torsors implies their triviality over the ring, advancing the understanding of the Grothendieck-Serre conjecture.

Contribution

It establishes the Grothendieck-Serre conjecture for groups of type F_4 with trivial f_3 invariant over semi-local regular rings containing an infinite perfect field.

Findings

Rationally trivial H-torsors are trivial over R.

Extends previous results on the Grothendieck-Serre conjecture for F_4 groups.

Complements existing work on invariants of algebraic groups.

Abstract

Let R be a semi-local regular ring containing an infinite perfect field, and let K be the field of fractions of R. Let H be a simple algebraic group of type F_4 over R such that H_K is the automorphism group of a 27-dimensional Jordan algebra which is a first Tits construction. If char K is not 2, this means precisely that the f_3 invariant of H_K is trivial. We prove that if an H-torsor is rationally trivial, then it is trivial over R. This result is a particular case of the Grothendieck-Serre conjecture. It continues the recent series of papers by I. Panin, N.Vavilov and the authors, and complements the result of V. Chernousov on the Grothendieck-Serre conjecture for groups of type F_4 with trivial g_3 invariant.