On Milnor $K$-theory in the imperfect residue case and applications to period-index problems
Srinivasan Srimathy
TL;DR
The paper addresses period-index bounds for Brauer classes and higher cohomology in mixed characteristic by introducing pseudo-perfect extensions, which mimic stages of forming perfect hulls. The main technical achievement is a vanishing theorem: for a complete discrete valued field $\\mathcal{K}$ with residue characteristic $p \neq 2$, the restriction map $Res_{\\mathcal{L}/\\mathcal{K}}^i: K_i(\\mathcal{K})/p \rightarrow K_i(\\mathcal{L})/p$ vanishes for all $i\ge 2$ along any pseudo-perfect extension $\\mathcal{L}$, and, under suitable hypotheses, the corresponding cohomology restriction $H^i(\\mathcal{K},\\mu_p^{\otimes i-1}) \to H^i(\\mathcal{L},\\mu_p^{\otimes i-1})$ vanishes as well. Building on Bloch–Kato–Kato filtrations and Kato’s results, the authors derive explicit bounds: the Brauer $p$-dimension satisfies $Br_pdim(\\mathcal{K}) \le \mathscr{R}_{ps}(\\mathcal{K})$, and generalized stable $i$-splitting dimensions obey $gssd^i_p(\\mathcal{K}) \le \mathscr{R}_{ps}(\\mathcal{K})$, with strengthened bounds for semi-global fields via patching. They further provide concrete period-index bounds for complete regular local rings and uniform bounds for semi-global fields, improving on prior results such as Parimala–Suresh. The findings unify equicharacteristic and mixed-characteristic cases and have practical impact on explicit splitting fields and uniform cohomological bounds in arithmetic geometry. Overall, the work sharpens our understanding of how residue-field $p$-rank and pseudo-rank govern splitting and index phenomena in higher cohomology and Brauer groups, with concrete applications to semi-global fields and patching methods.
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 $H^i(\mathcal{K},μ_p^{\otimes i-1})$ 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