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 resL/K: Br(K) → 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 Brm(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 = 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.
|Number of pages||5|
|Journal||Proceedings of the American Mathematical Society|
|State||Published - 2002|