Proceedings of the Japan Academy, Series A, Mathematical Sciences

Corestriction principle for non-abelian cohomology of reductive group schemes over arithmetical rings

Nguyêñ Quôć Thăńg

Full-text: Open access

Abstract

We prove some new results on Corestriction principle for non-abelian cohomology of group schemes over local and global fields or the rings of integers thereof.

Article information

Source
Proc. Japan Acad. Ser. A Math. Sci., Volume 82, Number 9 (2006), 147-151.

Dates
First available in Project Euclid: 4 December 2006

Permanent link to this document
https://projecteuclid.org/euclid.pja/1165244962

Digital Object Identifier
doi:10.3792/pjaa.82.147

Mathematical Reviews number (MathSciNet)
MR2293500

Zentralblatt MATH identifier
1143.11014

Subjects
Primary: 11E72: Galois cohomology of linear algebraic groups [See also 20G10] 14F20: Étale and other Grothendieck topologies and (co)homologies 14L15: Group schemes
Secondary: 14G20: Local ground fields 14G25: Global ground fields 18G50: Nonabelian homological algebra 20G10: Cohomology theory

Keywords
Corestriction principle norm principle group schemes arithmetical rings

Citation

Thăńg, Nguyêñ Quôć. Corestriction principle for non-abelian cohomology of reductive group schemes over arithmetical rings. Proc. Japan Acad. Ser. A Math. Sci. 82 (2006), no. 9, 147--151. doi:10.3792/pjaa.82.147. https://projecteuclid.org/euclid.pja/1165244962


Export citation

References

  • M. V. Borovoi, The algebraic fundamental group and abelian Galois cohomology of reductive algebraic groups, Max-Planck Inst., MPI/89–90, Bonn, 1990. (Preprint).
  • M. V. Borovoi, Abelian Galois Cohomology of Reductive Groups, Memoirs of Amer. Math. Soc. 162, 1998.
  • M. Borovoi, Non-abelian hypercohomology of a group with coefficients in a crossed module, and Galois cohomology. I. A. S. (Preprint).
  • L. Breen, Bitorseurs et cohomologie non abélienne, in The Grothendieck Festschrift, Vol. I, 401–476, Progr. Math., 86, Birkhäuser, Boston, Boston, MA, 1990.
  • P. Deligne, Variétés de Shimura: interprétation modulaire, et techniques de construction de modèles canoniques, in Automorphic forms, representations and $L$-functions (Proc. Sympos. Pure Math., Oregon State Univ., Corvallis, Ore., 1977), Part 2, 247–289, Proc. Sympos. Pure Math., XXXIII, Amer. Math. Soc., Providence, R.I., 1979.
  • J. -C. Douai, 2-Cohomologie galoisienne des groupes semi-simples, Thèse, Université des Sciences et Tech. de Lille 1, 1976.
  • P. Gille, La $R$-équivalence sur les groupes algébriques réductifs définis sur un corps global, Inst. Hautes Études Sci. Publ. Math. No. 86 (1997), 199–235.
  • G. Harder, Halbeinfache Gruppenschemata über Dedekindringen, Invent. Math. 4 (1967), 165–191.
  • K. Kato, S. Saito and, Global class field theory of arithmetic schemes, in Applications of algebraic $K$-theory to algebraic geometry and number theory, Part I, II (Boulder, Colo., 1983), 255–331, Contemp. Math., 55, Amer. Math. Soc., Providence, RI, 1986.
  • M. Kneser, Lectures on Galois cohomology of classical groups, Tata Inst. Fund. Res., Bombay, 1969.
  • Y. A. Nisnevich, Espaces homogènes principaux rationnellement triviaux et arithmétique des schémas en groupes réductifs sur les anneaux de Dedekind, C. R. Acad. Sci. Paris Sér. I Math. 299 (1984), no. 1, 5–8.
  • T. Ono, On the relative theory of Tamagawa numbers, Ann. of Math. (2) 82 (1965), 88–111.
  • E. Peyre, Galois cohomology in degree three and homogeneous varieties, $K$-Theory 15 (1998), no. 2, 99–145.
  • M. Demazure et A. Grothendieck, Schémas en groupes. Tom. 1–3, Lectures Notes in Math., vols. 151–153, Springer - Verlag, Berlin, 1970.
  • M. Artin et A. Grothendieck, Théorie des topos et cohomologie étale des schémas. Tome 3, Lecture Notes in Math., 305, Springer, Berlin, 1973.
  • N. Q. Th\v ańg, Corestriction principle in nonabelian Galois cohomology, Proc. Japan Acad. Ser. A Math. Sci. 74 (1998), no. 4, 63–67.
  • N. Q. Th\v ańg, On corestriction principle in non abelian Galois cohomology over local and global fields, J. Math. Kyoto Univ. 42 (2002), no. 2, 287–304.
  • N. Q. Th\v ańg, Weak corestriction principle for non-abelian Galois cohomology, Homology Homotopy Appl. 5 (2003), no. 1, 219–249. (Electronic).
  • F. Xu, Corestriction map for spinor norms. (Preprint). \beginthebibliography99
  • M. V. Borovoi, The algebraic fundamental group and abelian Galois cohomology of reductive algebraic groups. Preprint Max-Plank Inst., MPI/89–90, Bonn, 1990.
  • M. V. Borovoi, Abelian Galois Cohomology of Reductive Groups. Memoirs of Amer. Math. Soc. v. 162, 1998.
  • M. Borovoi, Non-abelian hypercohomology of a group with coefficients in a crossed module, and Galois cohomology. I. A. S. Preprint, 1991–1992.
  • L. Breen, Bitorseurs et cohomologie non-abélienne; in: Grothendieck Festschrift, v. 1, 401–476, Boston - Birkhäuser, 1990.
  • P. Deligne, Variétés de Shimura: Interprétation modulaire et techniques de construction de modèles canoniques; in: Proc. Sym. Pure Math. A. M. S. v. 33 (1979), Part 2, 247–289.
  • J. -C. Douai, 2-Cohomologie galoisienne des groupes semi-simples. Thèse, Université des Sciences et Tech. de Lille 1, 1976.
  • P. Gille, La R-équivalence sur les groupes réductifs définis sur un corps de nombres. Pub. Math. I. H. E. S., v. 86 (1997), 199–235.
  • G. Harder, Halbeinfache Gruppenschemata über Dedekindringen. Invent. Math., Bd. 4 (1967), 165–191.
  • K. Kato and S. Saito, Global class field theory of arithmetic schemes; in: Applications of algebraic $K$-theory to algebraic geometry and number theory, Contemp. Math., 55, Part II, Amer. Math. Soc., Providence, RI, 1986, 255–331.
  • M. Kneser, Lectures on Galois cohomology of classical groups. Tata Inst. Fund. Res., 1969.
  • Y. Nisnevich, Espaces homogènes principaux rationellement triviaux et arithmétique des schémas en groupes réductifs sur les anneaux de Dedekind. C. R. Acad. Sci. Paris, Sér. I Math. t. 299 (1984), no. 1, 5–8.
  • T. Ono, On relative Tamagawa numbers. Ann. Math. 82 (1965), 88–111.
  • M. Demazure et A. Grothendieck, Schémas en groupes. Tom. 1–3, Lectures Notes in Math., vols. 151–153, Springer - Verlag, 1970.
  • M. Artin et A. Grothendieck, Théorie des topos et cohomologie étale des schémas. Lecture Notes in Math. v. 305, Springer - Verlag, 1973.
  • N. Q. Th\v ańg, Corestriction Principle in non-abelian Galois Cohomology. Proceedings of the Japan Academy, v. 74 (1998), 63–67.
  • N. Q. Th\v ańg, On corestriction Principle in non-abelian Galois cohomology over local and global fields. J. Math. Kyoto Univ. v. 42 (2002), 287–304.
  • N. Q. Th\v ańg, Weak Corestriction Principle in non-abelian Galois cohomology. Homology. Homotopy and Applications, v. 5 (2003), 219–249. (Electronic).
  • F. Xu, Corestriction map for spinor norms. (Preprint).