Indecomposable and noncrossed product division algebras over function fields of smooth p-adic curves

We construct indecomposable and noncrossed product division algebras over function fields of connected smooth curves X over Zp. This is done by defining an index preserving morphism s:Br(K(ˆX))′→Br(K(X))′ which splits res:Br(K(X))→Br(K(ˆX)), where K(ˆX) is the completion of K(X) at the special fiber, and using it to lift indecomposable and noncrossed product division algebras over K(ˆX).