Cyclic length in the tame Brauer group of the function field of a p -adic curve

Abstract

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.