Journal of Mathematics of Kyoto University

Weak approximation, Brauer and $R$-equivalence in algebraic groups over arithmetical fields

Nguyêñ Quôć Thǎńg

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Abstract

We prove some new relations between weak approximation and some rational equivalence relations (Brauer and R-equivalence) in algebraic groups over arithmetical fields. By using weak approximation and local-global approach, we compute completely the group of Brauer equivalence classes of connected linear algebraic groups over number fields, and also completely compute the group of R-equivalence classes of connected linear algebraic groups $G$, which either are defined over a totally imaginary number field, or contains no anisotropic almost simple factors of exceptional type ${}^{3,6}D_{4}$, nor $E_{6}$. We discuss some consequences derived from these, e.g., by giving some new criteria for weak approximation in algebraic groups over number fields, by indicating a new way to give examples of non stably rational algebraic groups over local fields and application to norm principle. Some related questions and relations with groups of Brauer and R-equivalence classes over arbitrary fields of characteristic 0 are also discussed.

Article information

Source
J. Math. Kyoto Univ., Volume 40, Number 2 (2000), 247-291.

Dates
First available in Project Euclid: 17 August 2009

Permanent link to this document
https://projecteuclid.org/euclid.kjm/1250517714

Digital Object Identifier
doi:10.1215/kjm/1250517714

Mathematical Reviews number (MathSciNet)
MR1787872

Zentralblatt MATH identifier
1014.20019

Subjects
Primary: 14F22: Brauer groups of schemes [See also 12G05, 16K50]
Secondary: 14G20: Local ground fields 14G27: Other nonalgebraically closed ground fields 18G50: Nonabelian homological algebra

Citation

Thǎńg, Nguyêñ Quôć. Weak approximation, Brauer and $R$-equivalence in algebraic groups over arithmetical fields. J. Math. Kyoto Univ. 40 (2000), no. 2, 247--291. doi:10.1215/kjm/1250517714. https://projecteuclid.org/euclid.kjm/1250517714


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