Cyclic length in the tame Brauer group of the function field of a \(p\)-adic curve (Q2804435)

Let \(F\) be a field and \(n\) prime to \(\mathrm{char}(F)\). A central simple \(F\)-algebra is said to be \(\mathbb{Z}/n\)-cyclic if it contains a \(\mathbb{Z}/n\)-cyclic Galois maximal etale subalgebra. One says that \(\mathrm{H}^2(F,\mu_n) = {}_n\mathrm{Br}(F)\) is generated by \(\mathbb{Z}/n\)-cyclic classes if the cup product \(\mathrm{H}^1(F,\mu_n) \otimes_{\mathbb{Z}} \mathrm{H}^1(F,\mathbb{Z}/n) \to \mathrm{H}^2(F,\mu_n)\) is surjective. When \(\mathrm{H}^2(F,\mu_n)\) is generated by \(\mathbb{Z}/n\)-cyclic classes. we define the \(\mathbb{Z}/n\)-length \({}_n\mathrm{L}(F)\) to be the smallest \(N \in \mathbb{N} \cup \{\infty\}\) such that any class in \(\mathrm{H}^2(F,\mu_n)\) can be expressed as a sum of \(N\) \(\mathbb{Z}/n\)-cyclic classes, and the \(n\)-Brauer dimension \({}_n\mathrm{Br.dim}(F)\) to be the smallest \(d \in \mathbb{N} \cup \{\infty\}\) such that any class in \(\mathrm{H}^2(F,\mu_n)\) has index dividing \(n^{d-1}\). One always has \({}_n\mathrm{Br.dim}(F) \leq {}_n\mathrm{L}(F) + 1\).NEWLINENEWLINEIt is an open problem if \(\mathrm{H}^2(F,\mu_n)\) is always generated by \(\mathbb{Z}/n\)-cyclic classes.NEWLINENEWLINEThe main result is that for the function field \(F\) of a smooth \(p\)-adic curve and \(n\) prime to \(p\), \(\mathrm{H}^2(F,\mu_n)\) is generated by \(\mathbb{Z}/n\)-cyclic classes, and \({}_n\mathrm{L}(F) = 2\), in accordance with the \(n\)-Brauer dimension. It follows that \({}_n\mathrm{Br.dim}(F) = 3\).