Relative Brauer groups and 𝑚-torsion

Abstract

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