9 results on '"Franz-Viktor Kuhlmann"'
Search Results
2. Eliminating Tame Ramification: generalizations of Abhyankar's Lemma
- Author
-
Arpan Dutta and Franz-Viktor Kuhlmann
- Subjects
Lemma (mathematics) ,Pure mathematics ,General Mathematics ,Mathematics::Number Theory ,010102 general mathematics ,Mathematics - Commutative Algebra ,Commutative Algebra (math.AC) ,01 natural sciences ,Mathematics::Algebraic Geometry ,0103 physical sciences ,FOS: Mathematics ,010307 mathematical physics ,12J20, 13A18, 12J25 ,0101 mathematics ,Mathematics - Abstract
A basic version of Abhyankar's Lemma states that for two finite extensions $L$ and $F$ of a local field $K$, if $L|K$ is tamely ramified and if the ramification index of $L|K$ divides the ramification index of $F|K$, then the compositum $L.F$ is an unramified extension of $F$. In this paper, we generalize the result to valued fields with value groups of rational rank 1, and show that the latter condition is necessary. Replacing the condition on the ramification indices by the condition that the value group of $L$ be contained in that of $F$, we generalize the result further in order to give a necessary and sufficient condition for the elimination of tame ramification of an arbitrary extension $F|K$ by a suitable algebraic extension of the base field $K$. In addition, we derive more precise ramification theoretical statements and give several examples.
- Published
- 2019
- Full Text
- View/download PDF
3. NOTES ON EXTREMAL AND TAME VALUED FIELDS
- Author
-
Sylvy Anscombe and Franz-Viktor Kuhlmann
- Subjects
G110 ,Logic ,010102 general mathematics ,Mathematics - Logic ,Commutative Algebra (math.AC) ,Mathematics - Commutative Algebra ,01 natural sciences ,Philosophy ,Primary 12J20, Secondary 12J10 ,0103 physical sciences ,FOS: Mathematics ,010307 mathematical physics ,0101 mathematics ,Logic (math.LO) - Abstract
We extend the characterization of extremal valued fields given in [2] to the missing case of valued fields of mixed characteristic with perfect residue field. This leads to a complete characterization of the tame valued fields that are extremal. The key to the proof is a model theoretic result about tame valued fields in mixed characteristic. Further, we prove that in an extremal valued field of finitep-degree, the images of all additive polynomials have the optimal approximation property. This fact can be used to improve the axiom system that is suggested in [8] for the elementary theory of Laurent series fields over finite fields. Finally we give examples that demonstrate the problems we are facing when we try to characterize the extremal valued fields with imperfect residue fields. To this end, we describe several ways of constructing extremal valued fields; in particular, we show that in every ℵ1saturated valued field the valuation is a composition of extremal valuations of rank 1.
- Published
- 2016
4. The model theory of separably tame valued fields
- Author
-
Koushik Pal and Franz-Viktor Kuhlmann
- Subjects
Model theory ,Pure mathematics ,Algebra and Number Theory ,Group (mathematics) ,010102 general mathematics ,Closure (topology) ,Normal extension ,Field (mathematics) ,Mathematics - Logic ,Commutative Algebra (math.AC) ,Mathematics - Commutative Algebra ,01 natural sciences ,Residue field ,Completeness (order theory) ,Primary 03C10, 12J10, Secondary 03C60, 12J20 ,0103 physical sciences ,FOS: Mathematics ,010307 mathematical physics ,0101 mathematics ,Logic (math.LO) ,Mathematics ,Valuation (algebra) - Abstract
A henselian valued field $K$ is called separably tame if its separable-algebraic closure $K^{\operatorname{sep}}$ is a tame extension, that is, the ramification field of the normal extension $K^{\operatorname{sep}}|K$ is separable-algebraically closed. Every separable-algebraically maximal Kaplansky field is a separably tame field, but not conversely. In this paper, we prove Ax-Kochen-Ershov Principles for separably tame fields. This leads to model completeness and completeness results relative to the value group and residue field. As the maximal immediate extensions of separably tame fields are in general not unique, the proofs have to use much deeper valuation theoretical results than those for other classes of valued fields which have already been shown to satisfy Ax-Kochen-Ershov Principles. Our approach also yields alternate proofs of known results for separably closed valued fields., 30 pages. arXiv admin note: substantial text overlap with arXiv:1304.0194
- Published
- 2016
5. Counting the number of distinct distances of elements in valued field extensions
- Author
-
Franz-Viktor Kuhlmann and Anna Blaszczok
- Subjects
Model theory ,Pure mathematics ,Algebra and Number Theory ,010102 general mathematics ,Resolution of singularities ,Field (mathematics) ,Function (mathematics) ,Base (topology) ,Commutative Algebra (math.AC) ,Mathematics - Commutative Algebra ,01 natural sciences ,Measure (mathematics) ,Field extension ,0103 physical sciences ,FOS: Mathematics ,010307 mathematical physics ,0101 mathematics ,Element (category theory) ,12J10, 12J25 ,Mathematics - Abstract
The defect of valued field extensions is a major obstacle in open problems in resolution of singularities and in the model theory of valued fields, whenever positive characteristic is involved. We continue the detailed study of defect extensions through the tool of distances, which measure how well an element in an immediate extension can be approximated by elements from the base field. We show that in several situations the number of essentially distinct distances in fixed extensions, or even just over a fixed base field, is finite, and we compute upper bounds. We apply this to the special case of valued functions fields over perfect base fields. In particular, this provides important information used in forthcoming research on the ramification theory of two-dimensional valued function fields.
- Published
- 2017
6. The algebra and model theory of tame valued fields
- Author
-
Franz-Viktor Kuhlmann
- Subjects
Algebraic function field ,Applied Mathematics ,General Mathematics ,010102 general mathematics ,Normal extension ,Field (mathematics) ,Commutative Algebra (math.AC) ,Mathematics - Commutative Algebra ,01 natural sciences ,Algebraic closure ,Algebra ,Residue field ,Algebraic theory ,0103 physical sciences ,FOS: Mathematics ,12J10, 12J15 ,010307 mathematical physics ,0101 mathematics ,Algebraically closed field ,Valuation (algebra) ,Mathematics - Abstract
A henselian valued field K is called a tame field if its algebraic closure K ~ ${\tilde{K}}$ is a tame extension, that is, the ramification field of the normal extension K ~ | K ${\tilde{K}|K}$ is algebraically closed. Every algebraically maximal Kaplansky field is a tame field, but not conversely. We develop the algebraic theory of tame fields and then prove Ax–Kochen–Ershov Principles for tame fields. This leads to model completeness and completeness results relative to value group and residue field. As the maximal immediate extensions of tame fields will in general not be unique, the proofs have to use much deeper valuation theoretical results than those for other classes of valued fields which have already been shown to satisfy Ax–Kochen–Ershov Principles. The results of this paper have been applied to gain insight in the Zariski space of places of an algebraic function field, and in the model theory of large fields.
- Published
- 2014
7. The relative approximation degree in valued function fields
- Author
-
Franz-Viktor Kuhlmann and Izabela Vlahu
- Subjects
Degree (graph theory) ,Mathematics::Commutative Algebra ,General Mathematics ,Modulo ,010102 general mathematics ,Function (mathematics) ,Transcendence degree ,Commutative Algebra (math.AC) ,Mathematics - Commutative Algebra ,01 natural sciences ,Uniformization (probability theory) ,Algebra ,Field extension ,Primary 12J10, Secondary 12J20 ,0103 physical sciences ,FOS: Mathematics ,010307 mathematical physics ,0101 mathematics ,Element (category theory) ,Function field ,Mathematics - Abstract
We continue the work of Kaplansky on immediate valued field extensions and determine special properties of elements in such extensions. In particular, we are interested in the question when an immediate valued function field of transcendence degree 1 is henselian rational (i.e., generated, modulo henselization, by one element). If so, then wild ramification can be eliminated in this valued function field. The results presented in this paper are crucial for the first author's proof of henselian rationality over tame fields, which in turn is used in his work on local uniformization.
- Published
- 2013
8. Elimination of Ramification I: The Generalized Stability Theorem
- Author
-
Franz-Viktor Kuhlmann
- Subjects
Model theory ,Pure mathematics ,Applied Mathematics ,General Mathematics ,Ramification (botany) ,010102 general mathematics ,Field (mathematics) ,Extension (predicate logic) ,Function (mathematics) ,Mathematics - Commutative Algebra ,Commutative Algebra (math.AC) ,01 natural sciences ,0103 physical sciences ,Calculus ,FOS: Mathematics ,Primitive element theorem ,010307 mathematical physics ,Transcendental number ,0101 mathematics ,Uniformization (set theory) ,12J10 (Primary), 13A18, 12L12 (Secondary), 14B05 ,Mathematics - Abstract
We prove a general version of the "Stability Theorem": if $K$ is a valued field such that the ramification theoretical defect is trivial for all of its finite extensions, and if $F|K$ is a finitely generated (transcendental) extension of valued fields for which equality holds in the Abhyankar inequality, then the defect is also trivial for all finite extensions of $F$. This theorem is applied to eliminate ramification in such valued function fields. It has applications to local uniformization and to the model theory of valued fields in positive characteristic., Comment: 31 pages
- Published
- 2010
- Full Text
- View/download PDF
9. Places of algebraic function fields in arbitrary characteristic
- Author
-
Franz-Viktor Kuhlmann
- Subjects
Algebraic function field ,Discrete mathematics ,Zariski tangent space ,Zariski topology ,Mathematics(all) ,Spectrum of a ring ,Places of algebraic function fields ,General Mathematics ,010102 general mathematics ,12J10 ,Algebraic variety ,Field (mathematics) ,Mathematics - Commutative Algebra ,Commutative Algebra (math.AC) ,01 natural sciences ,Generic point ,Large fields ,Zariski space ,0103 physical sciences ,FOS: Mathematics ,010307 mathematical physics ,0101 mathematics ,Irreducible component ,Mathematics - Abstract
We consider the Zariski space of all places of an algebraic function field $F|K$ of arbitrary characteristic and investigate its structure by means of its patch topology. We show that certain sets of places with nice properties (e.g., prime divisors, places of maximal rank, zero-dimensional discrete places) lie dense in this topology. Further, we give several equivalent characterizations of fields that are large, in the sense of F. Pop's Annals paper {\it Embedding problems over large fields}. We also study the question whether a field $K$ is existentially closed in an extension field $L$ if $L$ admits a $K$-rational place. In the appendix, we prove the fact that the Zariski space with the Zariski topology is quasi-compact and that it is a spectral space., Comment: 27 pages
- Published
- 2010
- Full Text
- View/download PDF
Catalog
Discovery Service for Jio Institute Digital Library
For full access to our library's resources, please sign in.