ABHYANKAR 82nd BIRTHDAY CONFERENCE
SPEAKER: Franz-Viktor Kuhlmann (University of Saskatchewan)
TITLE: Elimination of wild ramification in immediate valued function
fields - continuing Kaplansky's work
ABSTRACT: 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".