Groupe de Brauer non ramifi\'e d'espaces homog\`enes de tores

Abstract

Let k be a field, X a smooth, projective k-variety. If X is geometrically rational, there is an injective map from the quotient of Brauer groups Br(X)/Br(k) into the first Galois cohomology group of the lattice given by the geometric Picard group. In this note, where the main attention is on smooth compactifications of homogeneous spaces of algebraic k-tori, we show how under some hypotheses the map is onto, and how one may in some special case exhibit concrete generators in Br(X). This is applied to the analysis of counterexamples to the local-global principle for norms in biquadratic extensions of number fields.