Invariants Cohomologiques de Rost En Caractéristique PositiveRost's Cohomological Invariants in Positive Characteristic

Let G/F be a semisimple algebraic group defined over a field F with characteristic p> 0. Let us denote byH(F,Qp/Zp(2)) the Galois cohomology group introduced by Kato. If p > 0, we show that thep-primary part of Rost’s invariant H1(F,G) → H(F,Qp/Zp(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 degreep and class [D] ∈p Br(F) ≈ H2(F,Z/pZ(1)), one hasH1(F, SL(D)) = F×/Nrd(D×) and an element a ∈ F× is a reduced norm if and only if the ‘cup-product’ [ D] ∪ (a) is trivial in H3(F,Z/pZ(2)); one characterizes also in positive characteristic fields with p-dimension6 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. Mathematics Subject Classifications ( 2000):11E72, 20G10.

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