  We study local-global questions for Galois cohomology over the function field
of a curve defined over a p-adic field (a field of cohomological dimension 3).
We define Tate-Shafarevich groups of a commutative group scheme via cohomology
classes locally trivial at each completion of the base field coming from a
closed point of the curve. In the case of a torus we establish a perfect
duality between the first Tate-Shafarevich group of the torus and the second
Tate-Shafarevich group of the dual torus. As an application, we show that the
failure of the local-global principle for rational points on principal
homogeneous spaces under tori is controlled by a certain subquotient of a third
etale cohomology group. We also prove a generalization to principal homogeneous
spaces of certain reductive group schemes in the case when the base curve has
good reduction.
