Unramified cohomology of classifying varieties for exceptional simply connected groups
Author:
Skip Garibaldi
Journal:
Trans. Amer. Math. Soc. 358 (2006), 359-371
MSC (2000):
Primary 11E76; Secondary 17B25, 20G10
DOI:
https://doi.org/10.1090/S0002-9947-05-03676-7
Published electronically:
March 31, 2005
MathSciNet review:
2171237
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Abstract | References | Similar Articles | Additional Information
Abstract: Let $BG$ be a classifying variety for an exceptional simple simply connected algebraic group $G$. We compute the degree 3 unramified Galois cohomology of $BG$ with values in $(\mathbb {Q}/\mathbb {Z})’(2)$ over an arbitrary field $F$. Combined with a paper by Merkurjev, this completes the computation of these cohomology groups for $G$ semisimple simply connected over all fields. These computations provide another family of examples of simple simply connected groups $G$ such that $BG$ is not stably rational.
- J.-L. Colliot-Thélène, Birational invariants, purity and the Gersten conjecture, $K$-theory and algebraic geometry: connections with quadratic forms and division algebras (Santa Barbara, CA, 1992) Proc. Sympos. Pure Math., vol. 58, Amer. Math. Soc., Providence, RI, 1995, pp. 1–64. MR 1327280
- E. B. Dynkin, Semisimple subalgebras of semisimple Lie algebras, Mat. Sbornik N.S. 30(72) (1952), 349–462 (3 plates) (Russian). MR 0047629
- R. Skip Garibaldi, Isotropic trialitarian algebraic groups, J. Algebra 210 (1998), no. 2, 385–418. MR 1662339, DOI https://doi.org/10.1006/jabr.1998.7584
- Ryan Skip Garibaldi, The Rost invariant has trivial kernel for quasi-split groups of low rank, Comment. Math. Helv. 76 (2001), no. 4, 684–711. MR 1881703, DOI https://doi.org/10.1007/s00014-001-8325-8
- Philippe Gille, Invariants cohomologiques de Rost en caractéristique positive, $K$-Theory 21 (2000), no. 1, 57–100 (French, with English and French summaries). MR 1802626, DOI https://doi.org/10.1023/A%3A1007839108933
- Philippe Gille, Algèbres simples centrales de degré 5 et $E_8$, Canad. Math. Bull. 45 (2002), no. 3, 388–398 (French, with English summary). MR 1937674, DOI https://doi.org/10.4153/CMB-2002-041-3
- N. Jacobson, Triality and Lie algebras of type ${\rm D}_{4}$, Rend. Circ. Mat. Palermo (2) 13 (1964), 129–153. MR 181705, DOI https://doi.org/10.1007/BF02849523
- Max-Albert Knus, Alexander Merkurjev, Markus Rost, and Jean-Pierre Tignol, The book of involutions, American Mathematical Society Colloquium Publications, vol. 44, American Mathematical Society, Providence, RI, 1998. With a preface in French by J. Tits. MR 1632779
- Ina Kersten and Ulf Rehmann, Generic splitting of reductive groups, Tohoku Math. J. (2) 46 (1994), no. 1, 35–70. MR 1256727, DOI https://doi.org/10.2748/tmj/1178225801
- A. S. Merkurjev, Maximal indexes of Tits algebras, Doc. Math. 1 (1996), No. 12, 229–243. MR 1405670
- Alexander Merkurjev, Unramified cohomology of classifying varieties for classical simply connected groups, Ann. Sci. École Norm. Sup. (4) 35 (2002), no. 3, 445–476 (English, with English and French summaries). MR 1914006, DOI https://doi.org/10.1016/S0012-9593%2802%2901097-2
- Alexander Merkurjev, Rost invariants of simply connected algebraic groups, Cohomological invariants in Galois cohomology, Univ. Lecture Ser., vol. 28, Amer. Math. Soc., Providence, RI, 2003, pp. 101–158. With a section by Skip Garibaldi. MR 1999385, DOI https://doi.org/10.1007/s00222-003-0292-9
- A. S. Merkurjev, R. Parimala, and J.-P. Tignol, Invariants of quasitrivial tori and the Rost invariant, Algebra i Analiz 14 (2002), no. 5, 110–151; English transl., St. Petersburg Math. J. 14 (2003), no. 5, 791–821. MR 1970336
- Holger P. Petersson and Michel L. Racine, An elementary approach to the Serre-Rost invariant of Albert algebras, Indag. Math. (N.S.) 7 (1996), no. 3, 343–365. MR 1621373, DOI https://doi.org/10.1016/0019-3577%2896%2983725-6
- Tonny Albert Springer, Jordan algebras and algebraic groups, Springer-Verlag, New York-Heidelberg, 1973. Ergebnisse der Mathematik und ihrer Grenzgebiete, Band 75. MR 0379618
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Additional Information
Skip Garibaldi
Affiliation:
Department of Mathematics & Computer Science, Emory University, Atlanta, Georgia 30322
MR Author ID:
622970
ORCID:
0000-0001-8924-5933
Email:
skip@member.ams.org
Received by editor(s):
August 15, 2003
Received by editor(s) in revised form:
March 21, 2004
Published electronically:
March 31, 2005
Article copyright:
© Copyright 2005
American Mathematical Society
