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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]
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Abstract

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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

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