A hasse principle for function fields over PAC fields

Let K be a perfect pseudo-algebraically closed field and let F be an extension of K of relative transcendence degree 1. It is shown that the restriction map Res: Br(F) → ∏_p Br(F_p^h) is injective, where p ranges over all non-trivial K-places of F, and F_p^h is the corresponding henselization. Conversely, the validity of this Hasse principle for all such extensions F implies a weaker version of pseudo-algebraic closedness. As an application we determine the finitely generated pro-p closed subgroups of the absolute Galois group of K(t).