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 $\cup$-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.