La descente sur les variétés rationnelles. II. (The descent on rational varieties. II) (Q1112126)

[For part I see Journées de géometrie algébrique, Angers/France 1979, 223-237 (1980; Zbl 0451.14018).] The article under review gives a general framework for the study of rational points on varieties X over number fields k. For such a variety the Hasse-principle holds if it has a k-rational point, provided that it has rational points over all local fields \(k_ v\) arising from completions). Yu. I. Manin has found an obstruction to the Hasse- principle, involving Brauer-groups. The authors observe that one can give a geometric interpretation by forming torsors \(E\to X\) (under tori), such that the Manin-obstruction vanishes for E. Any rational point on X lifts to one such torsor E, and it is hoped that E satisfies the Hasse- principle. This is true for many important examples. However these assertions have been shown elsewhere. The bulk of this paper consists of general and very detailed cohomological constructions and computations.