Galois cohomology of the classical groups over imperfect fields
type: Item
instance of: scholarly article
title: Galois cohomology of the classical groups over imperfect fields
zbMATH Open document ID: 1121.11035
author: Grégory Berhuy
author: Christoph Frings
author: Jean-Pierre Tignol
published in: Journal of Pure and Applied Algebra
publication date: 2007-09-03T00:00:00Z
review text: Serre's Conjecture II states that if \(G\) is a simply connected semi-simple linear algebraic group over a perfect field \(F\) of cohomological dimension at most \(2\), then the Galois cohomology set \(H^1(F,G)\) is trivial. Using Shapiro's Lemma, this problem can be reduced to the case where \(G\) is simply connected absolutely simple. This conjecture has been proved for groups of type \(^1A_n\) (Merkurjev and Suslin) and \(^2A_n\), \(B_n\), \(C_n\), \(D_n\) (except for trialitarian \(D_4\)), \(F_4\) and \(G_2\) (Bayer-Fluckiger and Parimala). Serre himself proposed a stronger version of the conjecture in the case of nonperfect fields, stating that \(H^1(F,G)\) is trivial for any simply connected absolutely simple \(G\) provided the \(p\)-separable dimension [in the sense of \textit{Ph.~Gille}, \(K\)-Theory 21, No. 1, 57--100 (2000; Zbl 0993.20031)] is at most \(2\) for all torsion primes \(p\) of the root system of \(G\). In the present paper, the authors prove this stronger conjecture for groups of type \(^2A_n\), \(B_n\), \(C_n\) and \(D_n\) (except trialitarian \(D_4\)). Similar to Bayer-Fluckiger and Parimala's approach, the idea is to reduce the question to certain classification results about quadratic and hermitian forms, involutions, and quadratic pairs which roughly say that these objects are classified by their ``classical'' invariants provided the separable dimension of the base field is at most \(2\). The crucial contribution in the present paper to these classification results is the characteristic \(2\) case. A further new aspect is the emphasis on involutions and quadratic pairs on central simple algebras rather than hermitian forms as in Bayer-Fluckiger and Parimala's work, thus avoiding the use of Morita equivalence and, as an additional bonus, making the proofs characteristic-free. In particular, the paper contains a detailed study, valid in all characteristics, of the hyperbolicity behavior of involutions and quadratic pairs under quadratic and …
Mathematics Subject Classification ID: 11E72
Mathematics Subject Classification ID: 11E04
Mathematics Subject Classification ID: 11E39
Mathematics Subject Classification ID: 11E81
Mathematics Subject Classification ID: 12G05
Mathematics Subject Classification ID: 12G10
Mathematics Subject Classification ID: 16K20
Mathematics Subject Classification ID: 16W10
zbMATH DE Number: 5186628
zbMATH Keywords: linear algebraic group
zbMATH Keywords: Serre's Conjecture II
zbMATH Keywords: Galois cohomology
zbMATH Keywords: hermitian form
zbMATH Keywords: quadratic form
zbMATH Keywords: central simple algebra
zbMATH Keywords: involution
zbMATH Keywords: quadratic pair
zbMATH Keywords: cohomological dimension
zbMATH Keywords: separable dimension
zbMATH Keywords: hyperbolic involution
zbMATH Keywords: hyperbolic quadratic pair
reviewed by: Detlev Hoffmann
MaRDI profile type: Publication
full work available at URL: j.jpaa.2007.01.001
OpenAlex ID: W2031341889
cites work: Invarianten hermitescher Formen über Schiefkörpern. (Invariant Hermitean forms over skew fields)
cites work: Forms in Odd Degree Extensions and Self-Dual Normal Bases
cites work: Galois cohomology of the classical groups over fields of cohomological dimension \(\leq 2\)
cites work: Hyperbolic involutions
cites work: Galois cohomology of Henselian discrete valuation fields, after K. Kato and S. Bloch
cites work: Somme orthogonal d'algebres a involution et algebra de clifford
cites work: Witt equivalence of central simple algebras with involution
cites work: Quadratic pairs in characteristic \(2\) and the Witt cancellation theorem.
cites work: Classification of quadratic forms over skew fields of characteristic 2
cites work: Rost's cohomological invariants in positive characteristic
cites work: Galois cohomology of quasi-split groups over fields of cohomological dimension \(\leqslant 2\)
cites work: Q4742893
cites work: Q4210545
cites work: Classification theorems for central simple algebras with involution (with an appendix by R. Parimala)
cites work: A local-global principle for algebras with involution and Hermitian forms.
cites work: Fields of characteristic 2 with prescribed u-invariants
cites work: The multipliers of similitudes and the Brauer group of homogeneous varieties.
cites work: Symmetric bilinear forms and quadratic forms
cites work: Q3708857
cites work: Galois Cohomology
cites work: Cohomology of artinian group schemes over local fields
cites work: Q3257832
cites work: Q4166831
cites work: Q3884236
DOI: 10.1016/J.JPAA.2007.01.001
description: scientific article; zbMATH DE number 5186628