Azumaya algebras over unramified extensions of function fields

2024/08/28 by Mohammed Moutand, Moutand, Mohammed
Mathematics · #Advanced Topics in Algebra #Rings, Modules, and Algebras #Holomorphic and Operator Theory

paper · pdf · doi:10.48550/arxiv.2408.15893

Abstract

Let X be a smooth variety over a field K with function field K(X). Using the interpretation of the torsion part of the étale cohomology group H_ét²(K(X), \mathbbG_m) in terms of Milnor-Quillen algebraic K-group K₂(K(X)), we prove that under mild conditions on the norm maps along unramified extensions of K(X) over X, there exist cohomological Brauer classes in H_ét²(X, \mathbbG_m) that are representable by Azumaya algebras on X. Theses conditions are almost satisfied in the case of number fields, providing then, a partial answer on a question of Grothendieck.

Related

• Brauer groups and quotient stacks
• Norm tori of etale algebras and unramified Brauer groups
• Galois-Azumaya Extensions and the Brauer-Galois Group of a Commutative Ring
• Moduli of étale subalgebras in an Azumaya algebra
• Cohomology of varieties over the maximal Kummer extension of a number field