Book contents
- Frontmatter
- Contents
- Preface
- Part I Ramification of Henselian Discrete Valuation Fields
- Part II Cyclic Extensions
- Part III Conductor and Refinements
- 6 Swan Conductor
- 7 Conductor and Differential Forms
- Part IV Geometric Applications
- Part V Upper Ramification Subgroups
- Part VI Graded Quotients and Characteristic Forms
- Solutions to Exercises
- References
- Index
6 - Swan Conductor
from Part III - Conductor and Refinements
Published online by Cambridge University Press: 29 May 2026
Book contents
- Frontmatter
- Contents
- Preface
- Part I Ramification of Henselian Discrete Valuation Fields
- Part II Cyclic Extensions
- Part III Conductor and Refinements
- 6 Swan Conductor
- 7 Conductor and Differential Forms
- Part IV Geometric Applications
- Part V Upper Ramification Subgroups
- Part VI Graded Quotients and Characteristic Forms
- Solutions to Exercises
- References
- Index
Summary
The Hasse–Arf theorem is at the heart of the classical theory of ramification. In this chapter we state it as a relation between the lower ramification subgroup and the cohomological filtration for cyclic extension. The equality holds if and only if the extension is log monogenic. The proof is by induction on the degree. The first case where the degree is p is essentially proved in Chapter 3. The proof of the induction step is based on the property that the trace mapping of differential forms with log poles is non-zero if and only if the extension is log monogenic.
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- Ramification Groups of Local Fieldswith Geometric Applications, pp. 160 - 181Publisher: Cambridge University PressPrint publication year: 2026
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