Classical groups and the Hasse principle.
type: Item
instance of: scholarly article
title: Classical groups and the Hasse principle.
zbMATH Open document ID: 0909.20029
DOI: 10.2307/120961
published in: Annals of Mathematics. Second Series
publication date: 1999-02-01T00:00:00Z
review text: The paper adds some important notes to the solution of the ``Hasse principle conjectures'' on the structure of the cohomology \(H^1(k,G)\) of the algebraic group \(G\), defined over the field \(k\). In 1962, Serre conjectured the following: -- Let \(k\) be a perfect field of cohomological dimension \(\leq 1\). Let \(G\) be a connected linear algebraic group defined over \(k\). Then \(H^1(k,G)=0\). -- Let \(k\) be a perfect field of cohomological dimension \(\leq 2\). Let \(G\) be a semi-simple simply connected linear algebraic group defined over \(k\). Then \(H^1(k,G)=0\). The first was proved by Steinberg in 1965, the second by the authors in 1995, for groups of classical type. Of course, the structure of \(H^1(k,G)\) is strongly related to the ground field \(k\). In this paper the authors generalize the conjecture by proving, amongst other things, that -- If \(k\) is a perfect field such that the cohomological dimension of \(k(\sqrt{-1})\) is \(\leq 2\), and \(G\) a semi-simple, simply connected group, which is a product of groups of type \(A_n\), \(B_n\), \(C_n\), \(D_n\), the case \(D_4\) being excluded, or \(G\) is of type \(G_2\) or \(F_4\), then the natural map \(H^1(k,G)\to\prod_vH^1(k_v,G)\) is injective, where \(v\) runs over the orderings of \(k\) and \(k_v\) denotes the real closure of \(k\) at \(v\). This was proved, for number fields, by Kneser, Springer, Harder and Chernousov, and restricts, for \(k\) having no real places, to the case \(H^1(k,G)=0\). Added in 2007: From the text of the correction: Lemma 2.4, on page 654, is not correct as stated. Indeed, \(I^3(k)\) torsion-free does not imply \(I^3(k\sqrt{-1})=0\). Hence, throughout the paper the assumption \(I^3(k)\) torsion-free should be replaced by \(I^3(k\sqrt{-1})=0\).
reviewed by: G. Falcone
Mathematics Subject Classification ID: 20G10
Mathematics Subject Classification ID: 11E72
Mathematics Subject Classification ID: 11E57
Mathematics Subject Classification ID: 12G05
zbMATH DE Number: 1196882
zbMATH Keywords: connected linear algebraic groups
zbMATH Keywords: semisimple simply connected algebraic groups
zbMATH Keywords: cohomological dimension
author: Eva Bayer-Fluckiger
author: R. Parimala
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.2307/120961
OpenAlex ID: W1994642986
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description: scientific article; zbMATH DE number 1196882