On the elementary obstruction to the existence of rational points

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 ∪-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.