Relative Brauer groups and \(m\)-torsion. (English) Zbl 1099.11066

Summary: Let \(K\) be a field and \(\text{Br}(K)\) its Brauer group. If \(L/K\) is a field extension, then the relative Brauer group \(Br(L/K)\) is the kernel of the restriction map \(res_{L/K}:\text{Br}(K)\rightarrow \text{Br}(L)\). A subgroup of \(\text{Br}(K)\) is called an algebraic relative Brauer group if it is of the form \(\text{Br}(L/K)\) for some algebraic extension \(L/K\). We consider the \(m\)-torsion subgroup \(\text{Br}_{m}(K)\) consisting of the elements of \(\text{Br}(K)\) killed by \(m\), where \(m\) is a positive integer, and ask whether it is an algebraic relative Brauer group. The case \(K=\mathbb{Q} \) is already interesting: the answer is yes for \(m\) squarefree, and we do not know the answer for \(m\) arbitrary. A counterexample is given with a two-dimensional local field \(K=k((t))\) and \(m=2\).


11R34 Galois cohomology
11S25 Galois cohomology
12F05 Algebraic field extensions
12G05 Galois cohomology
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