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

TL;DR

This paper investigates the arithmetic properties of complete discretely valued fields of dimension 2, establishing key results on central simple algebras, Milnor K-theory, and Galois cohomology, including the period-index equality and Serre's Conjecture II.

Contribution

It proves the period equals index property, symbol representation in Milnor K-theory, and verifies Serre's Conjecture II for such fields, advancing understanding of their algebraic structure.

Findings

Period equals index for central simple algebras over K

Every Milnor K-theory class mod p is represented by a symbol

Serre's Conjecture II holds for the field K

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.