## Journal of Mathematics of Kyoto University

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

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

#### 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