The Role of Completions in Modern Valuation Theory
by
Franz-Viktor Kuhlmann
Universität Heidelberg

In the development of valuation theory, completions have played an important role. To construct the extensions of a valuation from a field to a finite extension field, completions have been used essentially. Hensel's Lemma has first been proved for discrete complete valuations. But the history of Hensel's Lemma reflects very well the insufficiency of completions. Hensel's Lemma has also been proved for maximally valued fields of arbitrary rank, but for rank > 1 (i.e., in the case of non-archimedean ordered value groups), a complete field is not necessarily maximally valued or henselian. (A valued field is called henselian if it satisfies Hensel's Lemma). While the rank 1 valuations have remained the appropriate objects in number theory, topics like the valuation theory of formally real fields have initiated the study of valuations of arbitrary rank. In particular, the study of the model theoretical properties of valued fields has to deal with valuations of the most general types. It was also noticed that the property of being henselian is elementary in the model theoretical sense, which is not the case for the properties ``complete'' and ``maximally valued''. Thus, in modern valuation theory the latter properties are replaced by ``henselian'' wherever possible, in order to obtain a more general setting and to be able to apply the model theoretical results on valued fields. Also, the extension theory for valuations can be perfectly established without making use of completions.

However, completions have regained a quite unexpected attention since they can serve to analyze in more detail a problem that occurs for valuations with positive residue characteristic. It is known that extensions of the valuation to a finite field extension satisfy the fundamental inequality, which connects the degree of the field extension with the ramification indeces and inertia degrees of all possible extensions of the valuation. The defect (or ramification deficiency) measures how far this inequality is from being an equality. There is no defect if the residue characteristic of the valuation is 0. But otherwise, there may occur a defect, and this is a central problem in the recent study of valuations with positive residue characteristic. It has turned out that completions play a key role in the classification of the possible types of defects. If a defect vanishes by passing to completions, then it is of a relatively harmless type and can be handled quite easily. For the other case, the aim is to derive certain normal forms which admit new insights in the still unsolved problems of valuations with positive residue characteristic.

Date received: April 12, 1996

Copyright © 1996 by the author(s). The author(s) of this document and the organizers of the conference have granted their consent to include this abstract in Atlas Conferences Inc. Document # caae-50.