## 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 $$m=2$$.

### MSC:

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

### Keywords:

torsion subgroup; algebraic relative Brauer group
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### References:

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