Title: On Milnor $K$-theory in the imperfect residue case and applications to period-index problems Show Chinese title

Title: 关于不完美剩余域情况下的米尔诺$K$理论及其在周期-指标问题中的应用

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 Show Chinese abstract

Abstract: 给定一个$(0,p)$混合特征的完全离散赋值域$\mathcal{K}$，我们定义了一类有限域扩张，称为\emph{伪完美}扩张，使得对于所有$p\neq 2$，模-$p$Milnor$K$-群上的自然限制映射是平凡的。 这表明伪完美扩张将$H^i(\mathcal{K},\mu_p^{\otimes i-1})$中的每个元素分开，从而为布劳尔类以及$\mathcal{K}$的高阶上同调类提供周期-指标界。作为推论，我们证明了 Bhaskhar-Haase 的一个猜想，即$\mathcal{K}$的布劳尔$p$维度被$n+1$上限所限制，其中$n$是剩余域的$p$秩。 当$\mathcal{K}$是一个完备正则环的分式域时，我们证明了任何在$Br(\mathcal{K})$中的$p$-挠元，如果它是良好分裂的，则被一个伪完美扩张所分裂，并由此得到其指标的界。 然后我们使用 Harbater、Hartmann 和 Krashen 的修补技术，证明了剩余特征为$p$的半全局域的布劳尔$p$维数至多为$n+2$，并且还给出了更高上同调的统一$p$界。 这些界比 Parimala-Suresh 的工作中已知的结果更为精确。