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arXiv · 2501.01403

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

Philippe Gille·Diego Izquierdo·Giancarlo Lucchini Arteche
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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.

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Philippe Gille, Diego Izquierdo, Giancarlo Lucchini Arteche. 2025-01-02. Central simple algebras, Milnor $K$-theory and homogeneous spaces over complete discretely valued fields of dimension 2. https://arxiv.org/abs/2501.01403

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