Motivic torsors

The torsor P_σ = Hom ^⊗ (H_DR, H_σ) under the motivic Galois group G_σ = Aut^⊗ H_σ of the Tannakian category M_k generated by one-motives related by absolute Hodge cycles over a field k with an embedding σ:k (rightwards arrow with hook) ℂ is shown to be determined by its projection P_σ → P_σ/G⁰_σ to a Gal(k̄/k)- torsor, and by its localizations P_σ ⊗_k k_ξ at a dense subset of orderings ξ of the field k, provided k has virtual cohomological dimension (vcd) one. This result is an application of a recent local-global principle for connected linear algebraic groups over a field k of vcd ≤ 1.