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

2025/01/02 by Philippe Gille, Gille, Philippe, Diego Munguía Izquierdo +3
Mathematics · #11E72 #12G05 #12G10 #16K50 #19F15 #20G10 #Algebraic Geometry (math.AG) #Algebraic Geometry and Number Theory #Algebraic structures and combinatorial models #FOS: Mathematics #Homotopy and Cohomology in Algebraic Topology #K-Theory and Homology (math.KT) #Rings and Algebras (math.RA)

paper · pdf · doi:10.48550/arxiv.2501.01403

openalex publication_date 2025/01/02 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

Let K be a complete discretely valued field with residue field 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 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 H¹(K,G) is trivial.

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