Dualité et principe local-global pour les anneaux locaux henséliens de dimension 2

Let k be an algebraically closed field, a finite field or a p-adic field. Let K ₀ = k((x; y)) be the field of Laurent series in two variables over k, and let K be a finite extension of K0. We define Tate-Shafarevich groups of a commutative group scheme over K via cohomology classes locally trivial at each completion of K coming from a codimension 1 point of Spec O _K , where O _K is the integral closure of k[[x; y]] in K. We establish duality theorems between Tate-Shafarevich groups for finite groups schemes and for tori. We apply these results to the study of the obstruction to the local-global principle for K- torsors under a connected linear algebraic group, answering in that way a question of Colliot-Thélène, Parimala and Suresh, and to the weak approximation for tori over K.