Formes antihermitiennes devenant hyperboliques sur un corps de déploiement

Abstract

Let H=( a, b) F be a division quaternion algebra over a field F of characteristic not 2. Denote by τ the canonical involution on H and by K a splitting field of H. If h is a skew-hermitian form over ( H, τ) then, by extension of scalars to K and by Morita equivalence, we obtain a quadratic form h K over K. This gives a map of Witt groups ρ:W −1( H, τ)→W( K) induced by ρ( h)= h K . When K is a generic splitting field of H we prove in this note that the map ρ is injective.