Relative Brauer groups and $m$-torsion

Abstract:

Let $K$ be a field and $Br(K)$ 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 $res_{L/K}:Br(K)\rightarrow Br(L)$. A subgroup of $Br(K)$ is called an algebraic relative Brauer group if it is of the form $Br(L/K)$ for some algebraic extension $L/K$. In this paper, we consider the $m$-torsion subgroup $Br_{m}(K)$ consisting of the elements of $Br(K)$ 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 $m=2$.