Flasque resolutions for connected reductive algebraic groups.
The main object of the paper under review is a connected reductive linear algebraic group \(G\) defined over a field \(k\) of characteristic zero. The main result announced in this note [see also \textit{J.-L.~Colliot-Thélène}, Invent. Math. 159, No. 3, 589-606 (2005; Zbl 1080.14012)] says that there exists a short exact sequence of \(k\)-groups \(1\to S\to H\to G\to 1\) where \(H\) is an extension of a quasitrivial torus (= a torus whose character group viewed as \(\text{Gal}(\overline k/k)\)-module is a permutation module) by a semisimple simply connected group, and \(S\) is a flasque torus (= a torus whose character module \(M\) has the property \(H^1({\mathfrak h},\Hom_\mathbb{Z}(M,\mathbb{Z}))=0\) for all open subgroups \({\mathfrak h}\subseteq\text{Gal}(\overline k/k)\)). In applications, this resolution serves as a nice substitution for earlier known methods such as special coverings and \(z\)-extensions. It can be viewed as a very natural generalization of a flasque resolution of an algebraic torus (known since the early 1970's) and allows one to reprove in a more direct way many known results (in particular, recent results by \textit{M.~Borovoi, B.~Kunyavskii}, and \textit{P.~Gille} [J. Algebra 276, No. 1, 292-339 (2004; Zbl 1057.11023)]). It can be applied to computation of some birational invariants of \(G\) (unramified Brauer group, \(R\)-equivalence) and, in the case when \(k\) is a number field, of arithmetical invariants (the Shafarevich-Tate kernel, the defect of weak approximation).
scientific article; zbMATH DE number 2118258
scientific article