On Milnor $K$-theory in the imperfect residue case and applications to period-index problems

Abstract

Given a $(0,p)$-mixed characteristic complete discrete valued field $\mathcal{K}$ we define a class of finite field extensions called \emph{pseudo-perfect} extensions such that the natural restriction map on the mod-$p$ Milnor $K$-groups is trivial for all $p\neq 2$. This implies that pseudo-perfect extensions split every element in $H^i(\mathcal{K},\mu_p^{\otimes i-1})$ yielding period-index bounds for Brauer classes as well as higher cohomology classes of $\mathcal{K}$. As a corollary, we prove a conjecture of Bhaskhar-Haase that the Brauer $p$-dimension of $\mathcal{K}$ is upper bounded by $n+1$ where $n$ is the $p$-rank of the residue field. When $\mathcal{K}$ is the fraction field of a complete regular ring, we show that any $p$-torsion element in $Br(\mathcal{K})$ that is nicely ramified is split by a pseudo-perfect extension yielding a bound on its index. We then use patching techniques of Harbater, Hartmann and Krashen to show that the Brauer $p$-dimension of semi-global fields of residual characteristic $p$ is at most $n+2$ and also give uniform $p$-bounds for higher cohomologies. These bounds are sharper than previously known in the work of Parimala-Suresh