Principal homogeneous spaces under flasque tori: Applications

Rationally trivial hermitian spaces are locally trivial

“…If R is an essentially smooth local k-algebra and G is defined over k (we say that G is constant) Grothendieck's conjecture has been proved in most cases: by Colliot-Thélène and Ojanguren [4] for a perfect infinite field k and then by Raghunathan [19] for any infinite k. One notable open case is that of a finite base field. For a non-constant group G only two cases have been proved: when G is a torus, by Colliot-Thélène and Sansuc [5], and when G is the group SL 1 (D) of norm one elements of an Azumaya R-algebra D, by Panin and Suslin [13].…”

Groupe de Brauer non ramifié d’espaces homogènes de tores

“…Un k-tore Q est dit coflasque (voir [6]) si pour tout sous-groupe ouvert h ⊂ g on a H 1 (h,Q) = 0. Il suffit pour le vérifier de montrer H 1 (H,Q) = 0 pour tout sous-groupe H ⊂ G, où G est le groupe de Galois d'une extension finie déployant Q.…”

Cohomological invariants of algebraic tori

“…The following corollary is essentially equivalent to [12,Prop. 9.5] in the case when F is of zero characteristic.…”