Speaker: Franz-Viktor Kuhlmann, University of Szczecin, Poland

Title: Higher ramification theory

Dedicated to Professor Sudesh Khanduja on the occasion of her 75th birthday

Abstract: Ramification theory describes the properties of algebraic extensions of valued fields. When we apply it to the separable algebraic closure of a valued field (K,v), we obtain:
- the absolute decomposition field, also called the henselization of (K,v), from where the extension of v to every algebraic extension of K is uniquely determined,
- the absolute inertia field, which has the same value group as K, but whose residue field is the separable-algebraic closure of the one of K, and
- the absolute ramification field, which has the same residue field as the absolute inertia field, but whose value group is the divisible closure of that of K under all primes different from the characteristic of the residue field of K.
If this characteristic is 0, then the absolute ramification field is algebraically closed, but if the residue characteristic is a prime p>0, then it may not be algebraically closed and the Galois group of its extension to the separable algebraic closure is a p-group. This is where the wild ramification appears, including the defect, which is a serious obstacle in several longstanding open problems on valued fields with positive residue characteristic, such as:
local uniformization, the local form of resolution of singularities,
decidability of the Laurent series field over the field with p elements.
Therefore, it is very important to study the algebraic extensions of the absolute ramification field, and this is the subject of higher ramification theory.
While one origin of ramification theory is number theory, where the scope up until recently was restricted to discretely valued fields, or at best valued fields with archimedean ordered value groups (rank one), another origin is the theory of ordered fields, which can have associated valuations with arbitrarily large value groups. I learned about higher ramification theory from the articles [1] and [2] of Paulo Ribenboim. He presents his computation of so-called ramification ideals (which I will define in the talk), but had problems dealing with the case of value groups of higher rank. However, I found that his construction could easily be generalized. I will describe the problem and how it can be overcome, in particular as number theorists attempting to generalize their take on higher ramification theory currently seem to struggle with the same problem. Further, I will describe applications of ramification ideals and their relation to the phenomenon of defect.
The talk is based on the preprint [3].

[1] Paulo Ribenboim: Higher ramification groups for rank one valuations, 
Math. Ann. 173 (1967), 253-259

[2] Paulo Ribenboim: Corrections to: `"Higher ramification groups for rank 
one valuations", Math. Ann. 185 (1970), 22-24

[3] Franz-Viktor Kuhlmann: Topics in higher ramification theory, I: ramification
ideals, submitted; available at https://fvkuhlmann.de/Fvkprepr.html