Unramified division algebras do not always contain Azumaya maximal orders

TL;DR

This paper demonstrates that certain division algebras over high-dimensional schemes lack Azumaya maximal orders, revealing limitations in the algebraic structures related to Brauer classes and their topological properties.

Contribution

It shows the existence of Brauer classes on high-dimensional schemes where associated division algebras lack Azumaya maximal orders, combining algebraic and topological methods.

Findings

Existence of such Brauer classes on schemes of dimension ≥6

Division algebras over generic points without Azumaya maximal orders

Topological methods used to establish algebraic counterexamples

Abstract

We show that, in general, over a regular integral noetherian affine scheme X of dimension at least 6, there exist Brauer classes on X for which the associated division algebras over the generic point have no Azumaya maximal orders over X. Despite the algebraic nature of the result, our proof relies on the topology of classifying spaces of algebraic groups.