Relative Brauer groups and m-torsion

TY - JOUR

T1 - Relative Brauer groups and m-torsion

AU - Aljadeff, Eli

AU - Sonn, Jack

PY - 2002

Y1 - 2002

N2 - 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.

AB - 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.

UR - http://www.scopus.com/inward/record.url?scp=0035994091&partnerID=8YFLogxK

U2 - 10.1090/S0002-9939-01-06286-4

DO - 10.1090/S0002-9939-01-06286-4

M3 - 文章

AN - SCOPUS:0035994091

SN - 0002-9939

VL - 130

SP - 1333

EP - 1337

JO - Proceedings of the American Mathematical Society

JF - Proceedings of the American Mathematical Society

IS - 5

ER -