Obstruction sets and extensions of groups
Tom 173 / 2016
Streszczenie
Let be a nice variety over a number field k. We characterise in pure “descent-type” terms some inequivalent obstruction sets refining the inclusion X({\mathbb A}_k)^{\textrm{ét,Br}} \subset X({\mathbb A}_k)^{{\rm Br}_1}. In the first part, we apply ideas from the proof of X({\mathbb A}_k)^{\textrm{ét,Br}} = X({\mathbb A}_k)^{\mathcal{L}_k} by Skorobogatov and Demarche to new cases, by proving a comparison theorem for obstruction sets. In the second part, we show that if \mathcal{A} \subset \mathcal{B} \subset \mathcal{L}_k are such that \mathcal{B} \subset \textrm{Ext}(\mathcal{A}, \mathcal{U}_k), then X({\mathbb A}_k)^{\mathcal{A}} = X({\mathbb A}_k)^{\mathcal{B}}. This allows us to conclude, among other things, that X({\mathbb A}_k)^{\textrm{ét,Br}} =X({\mathbb A}_k)^{\mathcal{R}_k} and X({\mathbb A}_k)^ {\rm Sol,Br_1} = X({\mathbb A}_k)^{{\rm Sol}_k}.