Groupes de Brauer algébriques modulo les constantes d’espaces homogènes et leurs compactifications

Abstract

Let $X$ be a smooth, geometrically integral variety without non-constant invertible functions over a field $K$. Then the quotient of the “algebraic” Brauer group of $X$ by $\mathrm{Br}\,K$ injects into $\mathrm{H}^1(K,\mathrm{Pic}{\overline{X}})$. We show that this inclusion is not always an isomorphism, even in the case where $X$ is a homogeneous space of a connected linear algebraic group over $K$. A similar result for the smooth compactifications of $X$ is also given.