Obstruction sets and extensions of groups
Tom 173 / 2016
Streszczenie
Let $X$ 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 $X({\mathbb A}_k)^ {\rm Sol,Br_1} = X({\mathbb A}_k)^{{\rm Sol}_k}$.
