Variations on a Theme of Groups Splitting by a Quadratic Extension and Grothendieck-Serre Conjecture for Group Schemes $F_4$ with Trivial $g_3$ Invariant

We study structure properties of reductive group schemes defined over a local ring and splitting over its étale quadratic extension. As an application we prove Serre--Grothendieck conjecture on rationally trivial torsors over a local regular ring containing a field of characteristic 0 for group schemes of type $F_4$ with trivial $g_3$ invariant.

2010 Mathematics Subject Classification: 20G07, 20G10, 20G15, 20G41

Keywords and Phrases: Linear algebraic groups, exceptional groups, torsors, non-abelian cohomology, local regular rings, Grothendieck--Serre conjecture

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