Hostname: page-component-848d4c4894-cjp7w Total loading time: 0 Render date: 2024-06-14T16:32:53.082Z Has data issue: false hasContentIssue false
  • English
  • Français

A Note on Galois Cohomology Groups of Algebraic Tori

Published online by Cambridge University Press:  22 January 2016

Kazuo Amano
Kazuo Amano*
Affiliation:
Mathematical Institute, Nagoya University
Rights & Permissions [Opens in a new window]

Extract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

Let k be a complete field of characteristic 0 whose topology is defined by a discrete valuation and let T be an algebraic torus of dimension d defined over k. As is well known, T has a splitting field K which is a finite Galois extension of k with Galois group . For a ring R, denote by TR the subgroup of R-rational points of T. Then TK and T0K, DK being a valuation ring of K, become -modules in the usual manner.

Type
Research Article
Information
Nagoya Mathematical Journal , Volume 34 , March 1969 , pp. 121 - 127
Copyright
Copyright © Editorial Board of Nagoya Mathematical Journal 1969

References

[1] Artin, E. and Tate, J., Class field theory. Harvard (1961).Google Scholar
[2] Borel, A., Groupes linéaires algébriques. Ann. of Math. 64 (1956), 20–82.Google Scholar
[3] Chevalley, Séminaire C., Classification des groupes de Lie algébriques. Ecole Norm. Sup. Paris (1958).Google Scholar
[4] Harder, G., Über die Galoiskohomologie halbeinfachen Matrizengruppen. I. Math. Z. 90 (1965), 404–428; II. Math. Z. 92 (1966).CrossRefGoogle Scholar
[5] Kawada, Y., Algebraic number theory. Publ. Kyoritu Japan (1957).Google Scholar
[6] Kneser, M., Galois-Kohomologie halbeinfacher algebraischer Gruppen über p-adischen Körpern. I. Math. Z. 88 (1965), 40–47; II. Math. Z. 89 (1965), 250–272.CrossRefGoogle Scholar
[7] Lang, S., Algebraic groups over finite fields. Amer. J. Math. 78 (1956), 555–563.Google Scholar
[8] Nakayama, T., Cohomology of class field theory and tensor product modules. I. Ann. of Math. 65 (1957), 255–267.Google Scholar
[9] Ono, T., On some arithmetic properties of linear algebraic groups. Ann. of Math. 70 (1959), 266–290.Google Scholar
[10] Ono, T., Arithmetic of algebraic tori. Ann. of Math. 74 (1961), 101–139.CrossRefGoogle Scholar
[11] Rosen, M., Two theorem on Galois cohomology. Proc. Amer. Math. Soc. 17 (1966), 1183–1185.CrossRefGoogle Scholar
[12] Serre, J-P., Corps locaux. Paris Hermann (1962).Google Scholar
[13] Serre, J-P., Cohomologie galoisienne. Springer Verlag Berlin (1964).Google Scholar
[14] Springer, T.A., Galois cohomology of linear algebraic groups. Proc. Sympos. Pure Math. Vol. 9 Amer. Math. Soc. (1966) 149–158.Google Scholar
[15] Yokoi, H., A note on the Galois cohomology group of the ring of integers in an algebraic number field. Proc. Japan Akad. 40 (1964), 245–246.Google Scholar