Groupe de Picard et groupe de Brauer des compactifications lisses d’espaces homogènes
Authors:
Jean-Louis Colliot-Thélène and Boris È. Kunyavskiĭ
Journal:
J. Algebraic Geom. 15 (2006), 733-752
DOI:
https://doi.org/10.1090/S1056-3911-06-00427-9
Published electronically:
June 12, 2006
MathSciNet review:
2237268
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Abstract: Let $k$ be a field of characteristic zero, $G$ a connected linear algebraic group over $k$ and $H$ a connected closed $k$-subgroup of $G$. Let $X$ be a smooth $k$-compactification of $Y=G/H$. We prove that the Galois lattice given by the geometric Picard group of $X$ is flasque. The result was known in the case $H=1$. We compute this Galois lattice up to addition of a permutation module. When $G$ is semisimple and simply connected, the result shows that the Brauer group of $X$ is determined by the maximal toric quotient of $H$.
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[Bo1]Bo1 M. Borovoi, Abelianization of the second nonabelian Galois cohomology, Duke Math. J. 72 (1993), 217–239.
[Bo2]Bo2 M. Borovoi, The Brauer–Manin obstructions for homogeneous spaces with connected or abelian stabilizer, J. reine angew. Math. (Crelle) 473 (1996), 181–194.
[B/K1]BK1 M. Borovoi et B. Kunyavskiĭ, Formulas for the unramified Brauer group of a principal homogeneous space of a linear algebraic group, J. Algebra 225 (2000), 804–821.
[B/K2]BK2 M. Borovoi et B. Kunyavskiĭ, Arithmetical birational invariants of linear algebraic groups over two-dimensional geometric fields (with an appendix by P. Gille), J. Al- gebra 276 (2004), 292–339.
[CT]CT J.-L. Colliot-Thélène, Résolutions flasques des groupes réductifs connexes, C. R. Acad. Sc. Paris Sér. I 339 (2004), 331–334.
[CT/Gi/Pa]CTGiPa J.-L. Colliot-Thélène, P. Gille et R. Parimala, Arithmetic of linear algebraic groups over 2-dimensional geometric fields, Duke Math. J. 121 (2004), 285–341.
[CT/K]CTK J.-L. Colliot-Thélène et B. Kunyavskiĭ, Groupe de Brauer non ramifié des espaces principaux homogènes des groupes linéaires, J. Ramanujan Math. Soc. 13 (1998), 37–49.
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[CT/San3]CTSan3 J.-L. Colliot-Thélène et J.-J. Sansuc, Principal homogeneous spaces under flasque tori: applications, J. Algebra 106 (1987), 148–205.
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[D]D A. Ducros, Dimension cohomologique et points rationnels sur les courbes, J. Algebra 203 (1998), 349–354.
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Additional Information
Jean-Louis Colliot-Thélène
Affiliation:
C.N.R.S., UMR 8628, Mathématiques, Bâtiment 425, Université de Paris-Sud, F-91405 Orsay, France
MR Author ID:
50705
Email:
colliot@math.u-psud.fr
Boris È. Kunyavskiĭ
Affiliation:
Bar-Ilan University, Department of Mathematics, 52900 Ramat Gan, Israel
Email:
kunyav@macs.biu.ac.il
Received by editor(s):
March 26, 2005
Received by editor(s) in revised form:
April 21, 2005, and June 23, 2005
Published electronically:
June 12, 2006
