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On the structure of immediate valued function fields

Le : 05/07/2013 14h00
Par : Franz-Viktor Kuhlmann (université de Sakkatoon, Canada)
Lieu : I 001
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Résumé : Elimination of wild ramification is an important part of local uniformization. In the particular case of an immediate valued function field F|K of transcendence degree 1 it means to find an element x in F such that F lies in the henselization of K(x). (An extension of valued fields is called immediate if value group and residue field do not change.) This task is trivial if the characteristic of the residue field is 0, and highly nontrivial in the positive characteristic case. In order to prove a theorem of "Henselian Rationality" over suitable valued ground fields K, we need quite a bit of technical preparation, which can be seen as a continuation of Kaplansky's important paper "Maximal fields with valuations I", Duke Math. Journal 9 (1942), 303-321. In the first step of the proof, we find the element x in the henselization F^h of F. In this step it is essential to know the degree of the extension K(x)^h|K(f(x))^h for polynomials f. In the second step of the proof, once x is found in F^h, we try to replace it by an element y in F. Here, we need to know the degree of the extension K(x)^h|K(y)^h for arbitrary elements in K(x)^h. This step can be seen as a special case of "dehenselization", corresponding to the notion of "decompletion" that appears in Temkin's work. Dehenselization in general states that if there is a finite extension F' of F within its henselization F^h such that F'|K admits local uniformization, then so does F|K. Whether this is true is an important open problem. The above special case and Temkin's "decompletion" seem to indicate that the problem could be solved to the affirmative. One tool in the proofs is to use ramification theory to reduce the problem to considering Galois extensions of degree p, by extending the ground field. Then in the end, henselian rationality has to be pulled down through such ("tame") extensions. This too can be solved within the described framework where criteria are given for the degree of the extension K(x)^h|K(f(x))^h to be 1. Concretely, it can be shown that the trace of a well chosen henselian generator "upstairs" is a henselian generator "downstairs".