Hostname: page-component-586b7cd67f-g8jcs Total loading time: 0 Render date: 2024-11-24T13:09:10.219Z Has data issue: false hasContentIssue false
  • English
  • Français

The different and differentials of local fields with imperfect residue fields

Published online by Cambridge University Press:  20 January 2009

Bart de Smit
Bart de Smit
Affiliation:
Vakgroep Wiskunde, Econometrisch Instituut, Erasmus Universiteit Rotterdam, Postbus 1738, 3000 Dr Rotterdam, The Netherlands E-mail address: [email protected]
Rights & Permissions [Opens in a new window]

Abstract

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 with respect to a discrete valuation and let L be a finite Galois extension of K. If the residue field extension is separable then the different of L/K can be expressed in terms of the ramification groups by a well-known formula of Hilbert. We will identify the necessary correction term in the general case, and we give inequalities for ramification groups of subextensions L′/K in terms of those of L/K. A question of Krasner in this context is settled with a counterexample. These ramification phenomena can be related to the structure of the module of differentials of the extension of valuation rings. For the case that [L: K] = p2, where p is the residue characteristic, this module is shown to determine the correction term in Hilbert's formula.

Type
Research Article
Information
Proceedings of the Edinburgh Mathematical Society , Volume 40 , Issue 2 , June 1997 , pp. 353 - 365
Copyright
Copyright © Edinburgh Mathematical Society 1997

References

REFERENCES

1. Deligne, P., Appendix in Représentations des groupes réductifs sur un corps local (by Bernstein, Deligne, Kazhdan and Vignéras, Travaux en cours, Hermann, Paris 1984).Google Scholar
2. De Smit, B., Ramification groups of local fields with imperfect residue class fields, J. Number Theory 44 (1993), 229–236.Google Scholar
3. Grothendieck, A. (with J. Dieudonné), Éléments de géométrie algébrique IV No. 1 (Publ. Math. I.H.E.S. 20, 1964).Google Scholar
4. Hyodo, O., Wild ramification in the imperfect residue field case, in Galois representations and arithmetic algebraic geometry, Ihara, Y. (ed.), (North-Holland, Adv. Stud. Pure Math. 12, 1987).Google Scholar
5. Jacobson, N., Lectures in abstract algebra, Vol. III (Von Nostrand, Princeton, New Jersey, 1964).Google Scholar
6. Kato, K., A generalization of local class field theory by using K-groups II, J. Fac. Sci. Univ. Tokyo Sect. IA Math. 27 (1980), 603–683.Google Scholar
7. Krasner, M., Local differents of algebraic and finite extensions of valued fields, J. Number Theory 28 (1988), 17–61.Google Scholar
8. Kunz, E., Kähler differentials (Vieweg, Braunschweig, 1986).CrossRefGoogle Scholar
9. Matsumura, H., Commutative ring theory (Cambridge University Press, Cambridge, 1986).Google Scholar
10. Moriya, M., Theorie der Derivationen und Körperdifferenten, Math. J. Okayama Univ. 2 (1953), 111–148.Google Scholar
11. Serre, J.-P., Corps locaux (Hermann, Paris, 1962); English translation: Local fields (Graduate Texts in Math. 67, Springer, New York, 1979).Google Scholar
12. Zariski, O. and Samuel, P., Commutative algebra. Vol. I, II (Graduate Texts in Math. 28, 29, Springer-Verlag, New York, 1975.)Google Scholar