  In this paper we study local-global principles for tori over semi-global
fields, which are one variable function fields over complete discretely valued
fields. In particular, we show that for principal homogeneous spaces for tori
over the underlying discrete valuation ring, the obstruction to a local-global
principle with respect to discrete valuations can be computed using methods
coming from patching. We give a sufficient condition for the vanishing of the
obstruction, as well as examples were the obstruction is nontrivial or even
infinite. A major tool is the notion of a flasque resolution of a torus.
