Abelianization of the second nonabelian Galois cohomology
type: Item
instance of: scholarly article
title: Abelianization of the second nonabelian Galois cohomology
zbMATH Open document ID: 0849.12011
DOI: 10.1215/S0012-7094-93-07209-2
author: Mikhail Borovoi
published in: Duke Mathematical Journal
publication date: 1996-11-17T00:00:00Z
review text: 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 fo…
Mathematics Subject Classification ID: 12G05
Mathematics Subject Classification ID: 20G10
Mathematics Subject Classification ID: 11R34
zbMATH DE Number: 559000
zbMATH Keywords: \(k\)-kernel
zbMATH Keywords: connected semisimple algebraic groups
zbMATH Keywords: nonabelian Galois cohomology
zbMATH Keywords: abelian Galois cohomology group
zbMATH Keywords: neutral elements
zbMATH Keywords: Hasse principle
reviewed by: Thong Nguyen Quang Do
MaRDI profile type: Publication
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full work available at URL: s0012-7094-93-07209-2
OpenAlex ID: W2071837189
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description: scientific article; zbMATH DE number 559000