Abstract
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.
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Original Russian Text © A. N. Skorobogatov, 2007, published in Matematicheskie Zametki, 2007, Vol. 81, No. 1, pp. 112–124.
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Skorobogatov, A.N. On the elementary obstruction to the existence of rational points. Math Notes 81, 97–107 (2007). https://doi.org/10.1134/S0001434607010099
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DOI: https://doi.org/10.1134/S0001434607010099