Obstruction sets and extensions of groups

Francesca Balestrieri¹

¹ Mathematical Institute Oxford, OX2 6HD, United Kingdom

Acta Arithmetica, Tome 173 (2016) no. 2, pp. 151-181

Résumé

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 $X({\mathbb A}_k)^ {\rm Sol,Br_1} = X({\mathbb A}_k)^{{\rm Sol}_k}$.

Détail

DOI : 10.4064/aa8154-12-2015

Keywords: nice variety number field characterise pure descent type terms inequivalent obstruction sets refining inclusion mathbb textrm subset mathbb first part apply ideas proof mathbb textrm mathbb mathcal skorobogatov demarche cases proving comparison theorem obstruction sets second part mathcal subset mathcal subset mathcal mathcal subset textrm ext mathcal mathcal mathbb mathcal mathbb mathcal allows conclude among other things mathbb textrm mathbb mathcal mathbb sol mathbb sol