Azumaya algebras over unramified extensions of function fields

TL;DR

This paper investigates conditions under which certain cohomological Brauer classes on smooth varieties over fields can be represented by Azumaya algebras, extending understanding in algebraic geometry and number theory.

Contribution

It establishes new conditions related to unramified extensions that ensure the existence of Azumaya algebra representatives for Brauer classes.

Findings

Existence of Azumaya algebra representatives under mild unramified extension conditions.

Partial resolution of Grothendieck's question for number fields.

Connection between étale cohomology, K-theory, and Azumaya algebras.

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 \'etale cohomology group $H_{\text{\'et}}^2(K(X), \mathbb{G}_m)$ in terms of Milnor-Quillen algebraic $K$-group $K_2(K(X))$, we prove that under mild conditions on the norm maps along unramified extensions of $K(X)$ over $X$, there exist cohomological Brauer classes in $H_{\text{\'et}}^2(X, \mathbb{G}_m)$ 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.