Norm tori of étale algebras and unramified Brauer groups (Q6584649)

Let \(k\) be a field, \(L\) am étale \(k\)-algebra of finite rank, and \(N_{L/k} :L\to k\) be the norm map. For \(a\in K^{\times}\), let \(X_a\) be the affine \(k\)-variety defined by \(N_{L/k}(t)=a\), and \(X_a^c\) its smooth compactification.\N\NThe goal of this article is to give a combinatorial description of the unramifield Brauer group \(\mathrm{Br}(X_a^c)\mathrm{Im} \mathrm{Br}(k))\) assuming that \(L\) has at least one factor that is a cyclic field extension of \(k\).\N\NTheorem 1: Let \(p\) be prime and \(n\geq 1\) be an integer. Assume that \(char(k)\neq p\) and let \(F\) be a Galois extension of \(k\) with Galois group \((\mathbb{Z}/p^n\mathbb{Z})^2\). Suppose that \(L\) is a product of \(r\) linearly disjoint cyclic subfields of \(F\) of degree \(p^n\). Then \(\mathrm{Br}(X_a^c)/\mathrm{Im}\mathrm{Br}(k))\) is isomorphic to \((\mathbb{Z}/p^n \mathbb{Z})^{r-2}\).\N\NExplicit generators of this group is given.\N\NLet \(K\) be one of the cyclic subfields of degree \(p^n\) of \(F\), and let \(\chi: \mathrm{Gal}(K/k)\to \mathbf{Q}/\mathbf{Z}\) be an injective morphism. Write \(L=K\times K^{\prime}\) with \(K^{\prime}=\prod_{i\in I} K_i\) where \(K_i\) is a cyclic subfield of \(F\) of degree \(p^n\) for all \(i\in I\). Assume that \(K\) and \(K_i\) are linearly disjoint in \(F\). Assume \(\# I=r-1\). For all \(i\in I\), set \(N_i=N_{K_i/k}(y)\in k(X_a)^{\times}\). Let \(I^{\prime}\) be a subset of \(I\) with \(\#I^{\prime}=r-2\). Now let \((N,\chi)\) denote the class of cyclic algebra over \(k(X_a)\) associated to \(\chi\) and the element \(N_i\in k(X_a)^{\times}\).\N\NTheorem 2: The group \(\mathrm{Br}(X_a^c)/\mathrm{Im} \mathrm{Br}(k))\) is generated by the elements \((N,\chi)\) for \(i\in I^{\prime}\).\N\NProof rests on combinatorial description for the group \(\mathrm{Br}(X_a^c)/\mathrm{Im}\mathrm{Br}(k))\).