Advertisement

Mathematische Zeitschrift

, Volume 259, Issue 2, pp 457–467 | Cite as

On Galois cohomology of semisimple groups over local and global fields of positive characteristic

  • Nguyêñ Quôć ThăńgEmail author
Article

Abstract

We prove some analogs of results due to Kneser in the case of characteristic 0 about the surjectivity of coboundary map for Galois cohomology of semisimple groups over local and global fields of characteristic p > 0 and we give also some applications to Corestriction principle and a question of surjectivity of a coboundary map.

Mathematics Subject Classification (2000)

Primary 11E72 Secondary 18G50 20G10 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Barquero P. (2004). Local–global norm principle for algebraic groups. Commun. Algebra 32: 829–838 zbMATHCrossRefMathSciNetGoogle Scholar
  2. 2.
    Borel A. and Harder G. (1978). Existence of disete cocompact subgroups of reductive groups over local fields. J. reine und angew. Math. 298: 53–64 zbMATHMathSciNetGoogle Scholar
  3. 3.
    Borovoi, M.V.: The algebraic fundamental group and abelian Galois cohomology of reductive algebraic groups. Preprint Max-Plank Inst., MPI/89-90, Bonn (1990)Google Scholar
  4. 4.
    Borovoi, M.V.: Abelian Galois cohomology of reductive groups. Memoirs Amer. Math. Soc. 162 (1998)Google Scholar
  5. 5.
    Bourbaki N. (1968). Groupes et algèbres de Lie, Chap. IV–VI. Hermann, Paris Google Scholar
  6. 6.
    Bruhat F. and Tits J. (1987). Groupes réductifs sur un corps local, Chap. III: Compléments et applications à la cohomologie galoisienne. J. Fac. Sci. Univ. Tokyo 34: 671–688 zbMATHMathSciNetGoogle Scholar
  7. 7.
    Colliot-Thélène, J.-L. (with the collaboration of J.-J. Sansuc): The rationality problem for fields of invariants under linear algebraic groups (with special regards to Brauer groups). IX Escuela Latinoamericana de Matematicas, Santiago de Chile (1988)Google Scholar
  8. 8.
    Deligne P. (1973). Cohomologie à support propre, Exp. In: SGA4. Artin, M. et al. (eds) Théorie des topos et cohomologie étale des schémas Lecture Notes in Mathematics, vol 305., pp 252–480. Springer, Heidelberg Google Scholar
  9. 9.
    Deligne, P.: Variétés de Shimura: Interprétation modulaire et techniques de construction de modèles canoniques. In: Proc. Sym. Pure Math. A. M. S., vol. 33, Part 2, pp. 247–289 (1979)Google Scholar
  10. 10.
    Douai J.-C. (1975). Cohomologie des schémas en groupes semi-simples définis sur les corps globaux. C. R. Acad. Sci. Paris Sér. A–B 281: 1077–1080 zbMATHMathSciNetGoogle Scholar
  11. 11.
    Douai J.-C. (1977). Cohomologie des schémas en groupes semi-simples sur les anneaux de Dedekind et sur les courbes lisses, complètes, irréductibles. C. R. Acad. Sci. Paris Sér. A 285: 325–328 zbMATHMathSciNetGoogle Scholar
  12. 12.
    Douai, J.-C.: 2-Cohomologie galoisienne des groupes semi-simples. Thèse, Université des Sciences et Tech. de Lille 1 (1976)Google Scholar
  13. 13.
    Gille P. (1997). La R-équivalence sur les groupes réductifs définis sur un corps de nombres. Pub. Math. I. H. E. S. 86: 199–235zbMATHMathSciNetGoogle Scholar
  14. 14.
    Giraud J. (1971). Cohomologie non-abélienne. Grundlehren der Wiss. Math. Springer, BerlinGoogle Scholar
  15. 15.
    Harder G. (1967). Halbeinfache Gruppenschemata über Dedekindringen. Invent. Math. 4: 165–191zbMATHCrossRefMathSciNetGoogle Scholar
  16. 16.
    Harder G. (1975). Über die Galoiskohomologie der halbeinfacher Matrizengruppen, III. J. reine und angew. Math. 274/275: 125–138 MathSciNetGoogle Scholar
  17. 17.
    Kneser M. (1965). Galois-Kohomologie halbeinfacher algebraischer Gruppen über p-adischen Körpern, II. Math. Z. 89: 250–272 CrossRefMathSciNetGoogle Scholar
  18. 18.
    Kneser, M.: Lectures on Galois cohomology of classical groups. Tata Inst. Fund. Res. (1969)Google Scholar
  19. 19.
    Kottwitz R. (1986). Stable trace formula: elliptic singular terms. Math. Annalen 275: 365–399 zbMATHCrossRefMathSciNetGoogle Scholar
  20. 20.
    Merkurjev, A.S.: Simple algebras and quadratic forms. Izv. Akad. Nauk SSSR Ser. Mat. 55, 218–224 (1991); translation in Math. USSR-Izv. 38, 215–221 (1992)Google Scholar
  21. 21.
    Merkurjev A.S. (1996). A norm principle for algebraic groups. St. Petersburg Math. J. 7: 243–264 MathSciNetGoogle Scholar
  22. 22.
    Milne J.S. (1980). Étale Cohomology. Princeton University Press, Princeton zbMATHGoogle Scholar
  23. 23.
    Milne, J.: Arithmetic duality theorems. Prog. Math. (1980); see the online version as of 2006 at www.jmilne.orgGoogle Scholar
  24. 24.
    Ono T. (1965). On relative Tamagawa numbers. Ann. Math. 82: 88–111 CrossRefGoogle Scholar
  25. 25.
    Platonov V. and Rapinchuk A. (1994). Algebraic Groups and Number Theory. Academic Press, New York zbMATHGoogle Scholar
  26. 26.
    Rosset S. and Tate J. (1983). A reciprocity law for K 2-traces. Comm. Math. Helv. 58: 38–47 zbMATHCrossRefMathSciNetGoogle Scholar
  27. 27.
    Serre J.-P. (1965). Cohomologie Galoisienne. Lecture Notes in Mathematics, vol. 5, Troisième édition. Springer, Berlin Google Scholar
  28. 28.
    Shatz S.S. (1972). Profinite Groups: Arithmetic and Geometry. Annals of Math. Stud., vol. 72. Princeton University Press, Princeton Google Scholar
  29. 29.
    Thăńg N.Q. (2000). Number of connected components of groups of real points of adjoint groups. Commun. Algebra 28: 1097–1110 zbMATHGoogle Scholar
  30. 30.
    Thăńg N.Q. (1998). Corestriction principle in non-abelian Galois cohomology. Proc. Japan Acad. 74: 63–67 CrossRefzbMATHGoogle Scholar
  31. 31.
    Thăńg N.Q. (2002). On corestriction principle in non-abelian Galois cohomology over local and global fields. J. Math. Kyoto Univ. 42: 287–304 MathSciNetzbMATHGoogle Scholar
  32. 32.
    Thăńg N.Q. (2003). Weak corestriction principle for non-abelian Galois cohomology. Homol. Homotopy Appl. 5: 219–249 zbMATHGoogle Scholar
  33. 33.
    Tits J. (1966). Classification of algebraic semisimple groups. Proc. Symp. Pure Math. A. M. S. 9: 33–62 MathSciNetGoogle Scholar

Copyright information

© Springer-Verlag 2007

Authors and Affiliations

  1. 1.Institute of MathematicsHanoiVietnam

Personalised recommendations