Azumaya algebras over unramified extensions of function fields

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.