Abelianization of the second nonabelian Galois cohomology
Let \(k\) be a field of characteristic 0, \(\overline k\) an algebraic closure of \(k, \overline G\) an algebraic group over \(\overline k\). T. A. Springer has defined the notion of a \(k\)-kernel \(L\) attached to \(\overline G\), as well as its nonabelian Galois cohomology set \(H^2 (k,L)\). A \(k\)-kernel is a pair \(L = (\overline G, \kappa)\), where \(\kappa\) is a homomorphism from \(\Gamma = \text{Gal} (\overline k/k)\) into the group \(SOut G\) of semi-algebraic outer automorphisms of \(\overline G\) over \(k\), such that: i) \(\kappa\) is a splitting of the exact sequence \(1 \to Out \overline G \to SOut \overline G \to \Gamma\); ii) \(\kappa \) can be lifted to a continuous map from \(\Gamma\) to the group \(SAut \overline G\) of semi-algebraic automorphisms of \(\overline G\) over \(k\). Obstructions to certain constructions over \(k\) lie in \(H^2 (k,L)\), such a construction being possible if and only if the obstruction is a so-called ``neutral element'' in \(H^2 (k,L)\). In this paper, for \(\overline G\) connected, the author defines an abelian Galois cohomology group \(H^2_{ab} (k,L)\), as well as an abelianization map \(ab^2\): \(H^2 (k,L) \to H^2_{ab} (k,L)\) which takes the neutral elements to zero. If \(k\) is a local or a number field, he shows that the converse holds: an element \(\eta\) is neutral if and only if \(ab^2 (\eta) = 0\). He then applies this criterion to derive a Hasse principle for neutral elements. In particular, if \(\overline G\) is connected semisimple and \(k\) is any number field, he shows that an element \(\eta \in H^2 (k,L)\) is neutral if and only if all its archimedean localizations are neutral. He also recovers an old result of \textit{J.-C. Douai} [C. R. Acad. Sci., Paris. Sér. A 280, 321-323 (1975; Zbl 0328.20036)] that, for \(\overline G\) connected semisimple and \(k\) any nonarchimedean local field or totally imaginary number field, all elements of \(H^2 (k,L)\) are neutral. Finally, he uses his Hasse principle for \(H^2 (k,L)\) to give new proofs of various former results on the Hasse principle for homogeneous spaces due to \textit{G. Harder} [Jahresber. Deutsch. Math.-Ver. 70, 182-216 (1968; Zbl 0194.05701)], \textit{A. S. Rapinchuk} [Dokl. Akad. Nauk BSSR 31, 773-776 (1987; Zbl 0679.14027)] or himself [J. Reine Angew. Math. 426, 179-192 (1992; Zbl 0739.14030)].
scientific article; zbMATH DE number 559000
scientific article