Rost cohomological variants with positive characteristic

Let G/F be a semisimple algebraic group defined over a field F with characteristic p greater than or equal to 0. Let us denote by H-3(F,Q(p)/Z(p)(2)) the Galois cohomology group introduced by Kato. If p >0, we show that the p-primary part of Rost's invariant H-1(F,G) --> H-3(F, Q(p)/Z(p)(2)) lifts in characteristic 0. This result allows to deduce properties of the Rost invariant in positive characteristic from known properties in characteristic 0. The case of Merkurjev-Suslin's invariant is specially interesting, i.e. if G/F=SL(D) for a central simple algebra D/F with degree p and class [D] is an element of Br-p(F) H-2(F,Z/pZ(1)), one has H-1(F,SL(D))=F-x /Nrd(D-x) and an element a is an element of F-x is a reduced norm if and only if the 'cup-product' [D] boolean OR (a) is trivial in H-3(F,Z/pZ(2)); one characterizes also in positive characteristic fields with p-dimension less than or equal to 2 by the surjectivity of reduced norms. In a second part, we study Rost invariants when the base field is complete for a discrete valuation. As planned by Serre, invariants are then linked with Bruhat-Tits' theory, this yields a new proof of their nontriviality.