Arithmetical birational invariants of linear algebraic groups over two-dimensional geometric fields

Mikhail Borovoi; Boris Kunyavskiǐ; Philippe Gille

doi:10.1016/j.jalgebra.2003.10.024

Arithmetical birational invariants of linear algebraic groups over two-dimensional geometric fields

Mikhail Borovoi, Boris Kunyavskiǐ, Philippe Gille

Department of Mathematics

Research output: Contribution to journal › Article › peer-review

25 Scopus citations

Abstract

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.

Original language	English
Pages (from-to)	292-339
Number of pages	48
Journal	Journal of Algebra
Volume	276
Issue number	1
DOIs	https://doi.org/10.1016/j.jalgebra.2003.10.024
State	Published - 1 Jun 2004

Bibliographical note

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.

Funding

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.

Funders	Funder number
Hermann Minkowski Center for Geometry
INTAS	00-566
European Commission	RTN HPRN-CT-2002-00287
Minerva Foundation
Ministry of Aliyah and Immigrant Absorption
Israel Academy of Sciences and Humanities
Israel Science Foundation

Keywords

Birational invariants
Linear algebraic group
R-equivalence
Tate-Shafarevick kernel
Two-dimensional geometric field
Weak approximation

Access to Document

10.1016/j.jalgebra.2003.10.024

Cite this

@article{7c37fe5833154c52a1caf83a5b8c18ec,

title = "Arithmetical birational invariants of linear algebraic groups over two-dimensional geometric fields",

abstract = "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-Th{\'e}l{\`e}ne, 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.",

keywords = "Birational invariants, Linear algebraic group, R-equivalence, Tate-Shafarevick kernel, Two-dimensional geometric field, Weak approximation",

author = "Mikhail Borovoi and Boris Kunyavskiǐ and Philippe Gille",

note = "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.",

year = "2004",

month = jun,

day = "1",

doi = "10.1016/j.jalgebra.2003.10.024",

language = "אנגלית",

volume = "276",

pages = "292--339",

journal = "Journal of Algebra",

issn = "0021-8693",

publisher = "Academic Press Inc.",

number = "1",

}

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

M3 - ???researchoutput.researchoutputtypes.contributiontojournal.article???

AN - SCOPUS:2542505408

SN - 0021-8693

VL - 276

SP - 292

EP - 339

JO - Journal of Algebra

JF - Journal of Algebra

IS - 1

ER -

Arithmetical birational invariants of linear algebraic groups over two-dimensional geometric fields

Abstract

Bibliographical note

Funding

Keywords

Access to Document

Other files and links

Fingerprint

Cite this