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Grothendieck's theorem on non-abelian $\,\, H^{2}$
and local-global principles


Authors: Yuval Z. Flicker, Claus Scheiderer and R. Sujatha
Journal: J. Amer. Math. Soc. 11 (1998), 731-750
MSC (1991): Primary 14L30, 11R34, 12G05
DOI: https://doi.org/10.1090/S0894-0347-98-00271-9
MathSciNet review: 1608617
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Abstract: A theorem of Grothendieck asserts that over a perfect field $k$ of cohomological dimension one, all non-abelian $H^{2}$-cohomology sets of algebraic groups are trivial. The purpose of this paper is to establish a formally real generalization of this theorem. The generalization - to the context of perfect fields of virtual cohomological dimension one - takes the form of a local-global principle for the $H^{2}$-sets with respect to the orderings of the field. This principle asserts in particular that an element in $H^{2}$ is neutral precisely when it is neutral in the real closure with respect to every ordering in a dense subset of the real spectrum of $k$. Our techniques provide a new proof of Grothendieck's original theorem. An application to homogeneous spaces over $k$ is also given.


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  • [BP1] E. Bayer-Fluckiger, R. Parimala, Galois cohomology of the classical groups over fields of cohomological dimension $\le 2$. Invent. math. 122, 195-229 (1995). MR 96i:11042
  • [BP2] E. Bayer-Fluckiger, R. Parimala, Classical groups and Hasse principles. Ann. Math., to appear.
  • [BS] A. Borel, J.-P. Serre, Théorèmes de finitude en cohomologie galoisienne. Comment. Math. Helv. 39, 111-164 (1964). MR 31:5870
  • [B1] M. V. Borovoi, Abelianization of the second nonabelian Galois cohomology. Duke Math. J. 72, 217-239 (1993). MR 94j:11042
  • [B2] M. V. Borovoi, Abelian Galois cohomology of reductive groups. Memoirs AMS 626, (1998). CMP 96:16
  • [Br] L. Breen, Tannakian categories. In: Motives. U. Jannsen, S. Kleiman, J.-P. Serre (eds.), Proc. Symp. Pure Math. 55, Part I, Providence, R.I., 1994, pp. 337-376. MR 95b:18009
  • [CT] J.-L. Colliot-Thélène, Groupes linéaires sur les corps de fonctions de courbes réelles. J. reine angew. Math., 139-167 (1996). MR 97c:20072
  • [DM] P. Deligne, J. Milne, Appendix to: Tannakian categories. Lect. Notes Math. 900, 220-226 (1982).
  • [DG] M. Demazure, P. Gabriel, Groupes Algébriques, North-Holland Publ. Co., Amsterdam, 1970. MR 46:1800
  • [D] J.-C. Douai, Cohomologie galoisienne des groupes semi-simples définis sur les corps globaux. C. R. Acad. Sc.Paris 281, 1077-1080 (1975). MR 53:637
  • [FJ] G. Frey, M. Jarden, Approximation theory and the rank of abelian varieties over large algebraic fields. Proc. London Math. Soc. (3) 28, 112-128 (1974). MR 49:2765
  • [G] J. Giraud, Cohomologie non abélienne. Grundlehren der mathematischen Wissenschaften 179, Springer, Berlin, 1971. MR 49:8992
  • [H] D. Haran, Closed subgroups of $G(\mathbb{Q})$ with involutions. J. Algebra 129, 393-411 (1990). MR 91f:12006
  • [M] S. Mac Lane, Homology. Grundlehren der mathematischen Wissenschaften 114, Springer, Berlin, 1963. MR 28:122
  • [R] M. Rosenlicht, Some basic theorems on algebraic groups. Am. J. Math. 78, 401-443 (1956). MR 18:514a
  • [Scha] W. Scharlau, Quadratic and Hermitian Forms. Grundlehren der mathematischen Wissenschaften 270, Springer, Berlin, 1985. MR 86k:11022
  • [Sch] C. Scheiderer, Hasse principles and approximation theorems for homogeneous spaces over fields of virtual cohomological dimension one. Invent. math. 125, 307-365 (1996). MR 97f:14019
  • [S1] J.-P. Serre, Groupes algébriques et corps de classes. Hermann, Paris, 1959. MR 21:1973
  • [S2] J.-P. Serre, Cohomologie Galoisienne. Cinquième édition. Lect. Notes Math. 5, Springer, Berlin, 1994. MR 96b:12010
  • [Sp] T. A. Springer, Nonabelian $H^{2}$ in Galois cohomology. In: Algebraic Groups and Discontinuous Subgroups, ed. A. Borel, G. D. Mostow, Proc. Symp. Pure Math. IX, Providence, R.I., 1966, pp. 164-182. MR 35:195

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Additional Information

Yuval Z. Flicker
Affiliation: Department of Mathematics, The Ohio State University, 231 W. 18th Ave., Columbus, Ohio 43210-1174
Email: flicker@math.ohio-state.edu

Claus Scheiderer
Affiliation: Fakultät für Mathematik, Universität Regensburg, 93040 Regensburg, Germany
Email: claus.scheiderer@mathematik.uni-regensburg.de

R. Sujatha
Affiliation: School of Mathematics, Tata Institute of Fundamental Research, Homi Bhabha Road, Colaba, Bombay 400005, India
Email: sujatha@math.tifr.res.in

DOI: https://doi.org/10.1090/S0894-0347-98-00271-9
Received by editor(s): September 2, 1997
Received by editor(s) in revised form: March 16, 1998
Article copyright: © Copyright 1998 American Mathematical Society

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