Geodesic
On the Elementary Obstruction to the Existence of Rational Points
Matematičeskie zametki, Tome 81 (2007) no. 1, pp. 112-124

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The differentials of a certain spectral sequence converging to the Brauer–Grothendieck group of an algebraic variety $X$ over an arbitrary field are interpreted as the $\cup$-product with the class of the so-called “elementary obstruction.” This class is closely related to the cohomology class of the first-degree Albanese variety of $X$. If $X$ is a homogeneous space of an algebraic group, then the elementary obstruction can be described explicitly in terms of natural cohomological invariants of $X$. This reduces the calculation of the Brauer–Grothendieck group to the computation of a certain pairing in the Galois cohomology.
Keywords: Brauer–Grothendieck group, algebraic variety over a field, elementary obstruction to the existence of rational points, Albanese variety, Picard variety, Galois cohomology. 
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A. N. Skorobogatov. On the Elementary Obstruction to the Existence of Rational Points. Matematičeskie zametki, Tome 81 (2007) no. 1, pp. 112-124. http://geodesic.mathdoc.fr/item/MZM_2007_81_1_a8/