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

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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.

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References

  1. Barquero P. (2004). Local–global norm principle for algebraic groups. Commun. Algebra 32: 829–838

    Article  MATH  MathSciNet  Google Scholar 

  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

    MATH  MathSciNet  Google Scholar 

  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)

  4. Borovoi, M.V.: Abelian Galois cohomology of reductive groups. Memoirs Amer. Math. Soc. 162 (1998)

  5. Bourbaki N. (1968). Groupes et algèbres de Lie, Chap. IV–VI. Hermann, Paris

    Google Scholar 

  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

    MATH  MathSciNet  Google Scholar 

  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)

  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. 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)

  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

    MATH  MathSciNet  Google Scholar 

  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

    MATH  MathSciNet  Google Scholar 

  12. Douai, J.-C.: 2-Cohomologie galoisienne des groupes semi-simples. Thèse, Université des Sciences et Tech. de Lille 1 (1976)

  13. Gille P. (1997). La R-équivalence sur les groupes réductifs définis sur un corps de nombres. Pub. Math. I. H. E. S. 86: 199–235

    MATH  MathSciNet  Google Scholar 

  14. Giraud J. (1971). Cohomologie non-abélienne. Grundlehren der Wiss. Math. Springer, Berlin

    Google Scholar 

  15. Harder G. (1967). Halbeinfache Gruppenschemata über Dedekindringen. Invent. Math. 4: 165–191

    Article  MATH  MathSciNet  Google Scholar 

  16. Harder G. (1975). Über die Galoiskohomologie der halbeinfacher Matrizengruppen, III. J. reine und angew. Math. 274/275: 125–138

    MathSciNet  Google Scholar 

  17. Kneser M. (1965). Galois-Kohomologie halbeinfacher algebraischer Gruppen über p-adischen Körpern, II. Math. Z. 89: 250–272

    Article  MathSciNet  Google Scholar 

  18. Kneser, M.: Lectures on Galois cohomology of classical groups. Tata Inst. Fund. Res. (1969)

  19. Kottwitz R. (1986). Stable trace formula: elliptic singular terms. Math. Annalen 275: 365–399

    Article  MATH  MathSciNet  Google Scholar 

  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. Merkurjev A.S. (1996). A norm principle for algebraic groups. St. Petersburg Math. J. 7: 243–264

    MathSciNet  Google Scholar 

  22. Milne J.S. (1980). Étale Cohomology. Princeton University Press, Princeton

    MATH  Google Scholar 

  23. Milne, J.: Arithmetic duality theorems. Prog. Math. (1980); see the online version as of 2006 at www.jmilne.org

  24. Ono T. (1965). On relative Tamagawa numbers. Ann. Math. 82: 88–111

    Article  Google Scholar 

  25. Platonov V. and Rapinchuk A. (1994). Algebraic Groups and Number Theory. Academic Press, New York

    MATH  Google Scholar 

  26. Rosset S. and Tate J. (1983). A reciprocity law for K 2-traces. Comm. Math. Helv. 58: 38–47

    Article  MATH  MathSciNet  Google Scholar 

  27. Serre J.-P. (1965). Cohomologie Galoisienne. Lecture Notes in Mathematics, vol. 5, Troisième édition. Springer, Berlin

    Google Scholar 

  28. Shatz S.S. (1972). Profinite Groups: Arithmetic and Geometry. Annals of Math. Stud., vol. 72. Princeton University Press, Princeton

    Google Scholar 

  29. Thăńg N.Q. (2000). Number of connected components of groups of real points of adjoint groups. Commun. Algebra 28: 1097–1110

    MATH  Google Scholar 

  30. Thăńg N.Q. (1998). Corestriction principle in non-abelian Galois cohomology. Proc. Japan Acad. 74: 63–67

    Article  MATH  Google Scholar 

  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

    MathSciNet  MATH  Google Scholar 

  32. Thăńg N.Q. (2003). Weak corestriction principle for non-abelian Galois cohomology. Homol. Homotopy Appl. 5: 219–249

    MATH  Google Scholar 

  33. Tits J. (1966). Classification of algebraic semisimple groups. Proc. Symp. Pure Math. A. M. S. 9: 33–62

    MathSciNet  Google Scholar 

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  1. Institute of Mathematics, 18 - Hoang Quoc Viet, Hanoi, Vietnam

    Nguyêñ Quôć Thăńg

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  1. Nguyêñ Quôć Thăńg
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Correspondence to Nguyêñ Quôć Thăńg.

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Dedicated to the memory of M. Kneser.

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Thăńg, N.Q. On Galois cohomology of semisimple groups over local and global fields of positive characteristic. Math. Z. 259, 457–467 (2008). https://doi.org/10.1007/s00209-007-0198-0

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