Motivic torsors
  The torsor P_s=Hom(H_{\DR},H_s) under the motivic Galois group G_s=Aut H_s of the Tannakian category M_k generated by one-motives related by absolute Hodge cycles over a field k with an embedding s into the complex numbers is shown to be determined by its global projection [P_s\to (P_s)/(G_s)^0] to a Gal(\ov k/k)-torsor, and by its localizations (P_s) x_k (k_\xi) at a dense subset of orderings \xi 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. 
Yuval Z. Flicker