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Cyclic length in the tame Brauer group of the function field of a p-adic curve
- American Journal of Mathematics
- Johns Hopkins University Press
- Volume 138, Number 2, April 2016
- pp. 251-286
- 10.1353/ajm.2016.0020
- Article
- Additional Information
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Let $F$ be the function field of a smooth curve over the $p$-adic number
field ${\Bbb Q}_p$. We show that for each prime-to-$p$ number $n$ the
$n$-torsion subgroup ${\rm H}^2(F,\mu_n)={}_n{\rm Br}(F)$ is generated by
${\Bbb Z}/n$-cyclic classes; in fact the ${\Bbb Z}/n$-length is equal to
two. It follows that the Brauer dimension of $F$ is three (first proved by
Saltman), and any $F$-division algebra of period $n$ and index $n^2$ is
decomposable.