Let G be a connected linear algebraic group over a geometric field k of cohomological dimension 2 of one of the types which were considered by Colliot-Thélène, Gille and Parimala. Basing on their results, we compute the group of classes of R-equivalence G(k /R, the defect of weak approximation A Σ(G), the first Galois cohomology H1 (k, G), and the Tate-Shafarevich kernel III1 (k, G) (for suitable k) in terms of the algebraic fundamental group π1 (G). We prove that the groups G(k)/R and A Σ(G) and the set III1 (k, G) are stably k-birational invariants of G.
- Birational invariants
- Linear algebraic group
- Tate-Shafarevick kernel
- Two-dimensional geometric field
- Weak approximation