\(R\)-equivalence and rationality problem for semisimple adjoint classical algebraic groups
type: Item
instance of: scholarly article
title: \(R\)-equivalence and rationality problem for semisimple adjoint classical algebraic groups
zbMATH Open document ID: 0884.20029
DOI: 10.1007/BF02698837
published in: Publications Mathématiques
publication date: 1998-02-03T00:00:00Z
full work available at URL: http://www.numdam.org/item
full work available at URL: https://eudml.org/doc/104115
review text: Given an algebraic variety \(X\) defined over a non-closed field \(k\), one can try to study its rational points introducing some equivalence relations on this set. This approach was fruitfully used by \textit{Yu. I. Manin} [Cubic forms, North-Holland (1986; Zbl 0582.14010)] for cubic hypersurfaces. Two points are said to be \(R\)-equivalent if one can join them by a chain of \(k\)-rational curves. The set of classes \(X(k)/R\) is an important birational invariant of \(X\). The reviewed paper treats the case of a linear algebraic group \(G\) which was first studied by \textit{J.-L. Colliot-Thélène} and \textit{J.-J. Sansuc} [Ann. Sci. Éc. Norm. Supér., IV. Sér. 10, 175-229 (1977; Zbl 0356.14007)]. A result by \textit{V. E. Voskresenskij} [Algebraic Tori, Nauka, Moscow (1977; Zbl 0499.14013)] showed that for \(G=SL(1,A)\), \(A\) a central simple \(k\)-algebra, the above invariant \(G(k)/R\) coincides with the reduced Whitehead group \(SK_1(A)\), the invariant used by \textit{V. P. Platonov} [Proc. Int. Congr. Math., Helsinki 1978, vol. 1, 311-317 (1980; Zbl 0425.16018)] for the proof of non-rationality of \(G\). Platonov's examples of non-rational semisimple groups refer to the class of simply connected groups. Earlier arithmetic considerations allowed Serre to construct non-rational groups which are, by construction, neither simply connected, nor adjoint. It was a long-standing problem whether there exist non-rational adjoint groups. The reviewed paper answers this question. Namely, there are given examples of adjoint simple \(k\)-groups \(G\) of type \(D_n\), \(n\geq 3\), such that \(G(E)/R\neq 1\) for some extension \(E\) of \(k\) which implies that \(G\) is not \(k\)-rational. The ground field \(k\) can be taken to be an arbitrary number field for odd \(n\), and the field of rational functions in one variable over a number field for even \(n\). The proof uses techniques from the cohomology of quadratic forms (\textit{J. Arason} [J. Algebra 36, 448-491 (1975; Zbl …
Mathematics Subject Classification ID: 20G15
Mathematics Subject Classification ID: 14G05
Mathematics Subject Classification ID: 14C15
Mathematics Subject Classification ID: 20G30
zbMATH DE Number: 1057024
zbMATH Keywords: adjoint semisimple algebraic groups
zbMATH Keywords: \(R\)-equivalences
zbMATH Keywords: rational varieties
zbMATH Keywords: rational points
zbMATH Keywords: birational invariants
zbMATH Keywords: non-rational adjoint groups
author: Alexander Merkurjev
reviewed by: Boris Kunyavskiĭ
MaRDI profile type: Publication
cites work: Tensor Products of Quaternion Algebras
cites work: Hermitian forms. II
cites work: Cohomologische Invarianten quadratischer Formen
cites work: Hyperbolic involutions
cites work: Rationality properties of linear algebraic groups. II
cites work: On algebraic group varieties
cites work: Sur les multiplicateurs des similitudes
cites work: Clifford algebras for algebras with involution of Type D
cites work: Q4210545
cites work: Q4313530
cites work: Q4006651
cites work: Q5675788
cites work: Square class groups and Witt rings of central simple algebras
cites work: Q4402061
cites work: SIMPLE ALGEBRAS AND QUADRATIC FORMS
cites work: Generic element in \(SK_ 1\) for simple algebras
cites work: Index reduction formulas for twisted flag varieties. I
cites work: $ K$-COHOMOLOGY OF SEVERI-BRAUER VARIETIES AND THE NORM RESIDUE HOMOMORPHISM
cites work: The multipliers of similitudes and the Brauer group of homogeneous varieties.
cites work: Q3874404
cites work: Q3946274
cites work: Groupe de Brauer et arithmétique des groupes algébriques linéaires sur un corps de nombres.
cites work: Q3708857
cites work: The Index of a Brauer Class on a Brauer-Severi Variety
cites work: The generalized even Clifford algebra
cites work: Indecomposable algebras of prime exponent.
cites work: A Cassels-Pfister theorem for involutions on central simple algebras
cites work: Formes quadratiques, groupes orthogonaux et algèbres de Clifford
cites work: TOROIDAL FANO VARIETIES AND ROOT SYSTEMS
cites work: Q3845400
cites work: Q5728890
OpenAlex ID: W2085624349
Recommended article: Q4216759
Recommended article: Q4509518
Recommended article: Q4468784
Recommended article: $R$-equivalence in adjoint classical groups over fields of virtual cohomological dimension $2$
Recommended article: The Semisimplicity Problem for p-Adic Group Algebras
Recommended article: Q3340974
Recommended article: Rationality of almost simple algebraic groups
Recommended article: Adjoint groups over ℚ_{𝕡}(𝕏) and R-equivalence - revisited
Recommended article: Rational equivalence on adjoint groups of type 𝐷_{𝑛} over fields of virtual cohomological dimension 2
Recommended article: The \(R\)-equivalence on reductive algebraic groups defined over a global field
description: scientific article; zbMATH DE number 1057024