Central simple algebras, Milnor $K$-theory and homogeneous spaces over complete discretely valued fields of dimension 2

TL;DR

The paper analyzes complete discretely valued fields $K$ of dimension $2$ (with residue field dimension $1$, possibly imperfect) and proves three central results: the period equals index for central simple $K$-algebras, that Milnor $K$-theory modulo $p$ is generated by symbols for all primes $p$ in characteristic $0$, and Serre's Conjecture II holds for $K$ across all semisimple simply connected groups. It extends Kato’s $K$-theory descriptions to imperfect residue fields by constructing generalized residue maps and absorption lemmas that reduce Milnor $K$-theory classes to symbols, and it leverages these results to deduce arithmetic properties of homogeneous spaces, including extensions of torsors via parahoric group schemes. The work unifies and extends known results from perfect-residue cases, providing new tools for analysis of central simple algebras, $K$-theory, and torsors over low-dimensional fields, with broad implications for arithmetic geometry and the theory of algebraic groups. These results yield concrete consequences for Serre II over $K$, Cyclicity of CSAs in suitable settings, and torsor behavior on curves and for $E_8$-type groups in mixed and equal characteristic regimes.

Abstract

Let $K$ be a complete discretely valued field with residue field $\bar K$ of dimension $1$ (not necessarily perfect). This occurs if and only if $K$ has dimension $2$. We prove the following statements on the arithmetic of such fields: - The "period equals index" property holds for central simple $K$-algebras. - For every prime $p$, every class in the Milnor $\mathrm{K}$-theory modulo $p$ is represented by a symbol. - Serre's Conjecture II holds for the field $K$. That is, for every semisimple and simply connected $K$-group $G$, the set $H^1(K,G)$ is trivial.

Theorems & Definitions (38)

• Theorem 1.1: Theorem \ref{['thm period index']}
• Conjecture 1.2: Serre's conjecture II
• Theorem 1.3: Theorem \ref{['thm Serre II']}
• Definition 2.1
• Lemma 2.2
• proof
• Example 2.3
• Proposition 3.1
• proof
• Proposition 3.2