Groupes de Brauer alg\'ebriques modulo les constants d'espaces homog\`enes et leurs compactifications

TL;DR

The paper investigates the relationship between the algebraic Brauer group quotient and Galois cohomology for smooth, geometrically integral varieties, showing the inclusion is not always an isomorphism, especially for homogeneous spaces and their compactifications.

Contribution

It demonstrates that the natural inclusion from the algebraic Brauer group quotient to Galois cohomology is not always an isomorphism, even for homogeneous spaces of connected linear algebraic groups.

Findings

The inclusion is not always an isomorphism.

Counterexamples exist for homogeneous spaces.

Similar results hold for smooth compactifications.

Abstract

Let $X$ be a smooth, geometrically integral variety over a field $K$. Then the quotient of the "algebraic" Brauer group of $X$ by $\operatorname{Br} K$ injects into $\textrm{H}^1(K,\textrm{Pic} \bar{X})$. 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. ----- Soit $X$ une vari\'et\'e lisse, g\'eom\'etriquement int\`egre sur un corps $K$. Alors le quotient du groupe Brauer "alg\'ebrique" de $X$ par $\operatorname{Br} K$ s'injecte dans $\textrm{H}^1(K,\operatorname{Pic} \bar{X})$. Nous montrons que cette inclusion n'est pas toujours un isomorphisme m\^eme dans le cas o\`u $X$ est un espace homog\`ene d'un groupe alg\'ebrique lin\'eaire connexe sur $K$. Un r\'esultat similaire pour les compactifications lisses de $X$ est aussi donn\'e.