Indecomposable and noncrossed product division algebras over curves over complete discrete valuation rings

Let T be a complete discrete valuation ring and a smooth projective curve over S=spec(T) with closed fibre X. Denote by F the function field of and by the completion of F with respect to the discrete valuation defined by X, the closed fibre. In this paper, we construct indecomposable and noncrossed product division algebras over F. This is done by defining an index preserving group homomorphism , and using it to lift indecomposable and noncrossed product division algebras over .