TY - JOUR
T1 - Arithmetical birational invariants of linear algebraic groups over two-dimensional geometric fields
AU - Borovoi, Mikhail
AU - Kunyavskiǐ, Boris
AU - Gille, Philippe
N1 - Funding Information:
Keywords: Two-dimensional geometric field; Linear algebraic group; Birational invariants; R-equivalence; Weak approximation; Tate–Shafarevich kernel ✩ This research was supported by the Israel Science Foundation founded by the Israel Academy of Sciences and Humanities—Center of Excellence Program and by EU RTN HPRN-CT-2002-00287. * Corresponding author. E-mail addresses: [email protected] (M. Borovoi), [email protected] (B. Kunyavski˘ı), [email protected] (P. Gille). 1 The author was partially supported by the Hermann Minkowski Center for Geometry. 2 The author was partially supported by the Ministry of Absorption (Israel), the Minerva Foundation through the Emmy Noether Research Institute of Mathematics, and INTAS 00-566. 3 The author of the Appendix.
PY - 2004/6/1
Y1 - 2004/6/1
N2 - 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.
AB - 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.
KW - Birational invariants
KW - Linear algebraic group
KW - R-equivalence
KW - Tate-Shafarevick kernel
KW - Two-dimensional geometric field
KW - Weak approximation
UR - http://www.scopus.com/inward/record.url?scp=2542505408&partnerID=8YFLogxK
U2 - 10.1016/j.jalgebra.2003.10.024
DO - 10.1016/j.jalgebra.2003.10.024
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AN - SCOPUS:2542505408
SN - 0021-8693
VL - 276
SP - 292
EP - 339
JO - Journal of Algebra
JF - Journal of Algebra
IS - 1
ER -