Criterion for surjectivity of localization in Galois cohomology of a reductive group over a number field

Let G be a connected reductive group over a number field F, and let S be a set (finite or infinite) of places of F. We give a necessary and sufficient condition for the surjectivity of the localization map from H¹(F,G) to the “direct sum” of the sets H¹(Fv,G) where v runs over S. In the appendices, we give a new construction of the abelian Galois cohomology of a reductive group over a field of arbitrary characteristic.