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Abstract For important cases of algebraic extensions of valued fields, we develop presentations of the associated Kähler differentials of the extensions of their valuation rings. We compute their annihilators as well as the associated differents. We then apply the results to Galois defect extensions of prime degree. Defects can appear in finite extensions of valued fields of positive residue characteristic and are serious obstructions to several problems in positive characteristic. A classification of defects (dependent vs. independent) has been introduced by the second and the third author. It has been shown that perfectoid fields and deeply ramified fields only admit extensions with independent defect. We give several characterizations of independent defect, using ramification ideals, Kähler differentials, and traces of the maximal ideals of valuation rings. All of our results are for arbitrary valuations; in particular, we have no restrictions on their ranks or value groups.
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Erratic birational behavior of mappings in positive characteristic
Abstract Birational properties of generically finite morphisms of algebraic varieties can be understood locally by a valuation of the function field ofX. In finite extensions of algebraic local rings in characteristic zero algebraic function fields which are dominated by a valuation, there are nice monomial forms of the mapping after blowing up enough, which reflect classical invariants of the valuation. Further, these forms are stable upon suitable further blowing up. In positive characteristic algebraic function fields, it is not always possible to find a monomial form after blowing up along a valuation, even in dimension two. In dimension two and positive characteristic, after enough blowing up, there are stable forms of the mapping which hold upon suitable sequences of blowing up. We give examples showing that even within these stable forms, the forms can vary dramatically (erratically) upon further blowing up. We construct these examples in defect Artin–Schreier extensions which can have any prescribed distance.
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- Award ID(s):
- 2054394
- PAR ID:
- 10505351
- Publisher / Repository:
- Wiley-VCH GmbH
- Date Published:
- Journal Name:
- Mathematische Nachrichten
- Volume:
- 296
- Issue:
- 11
- ISSN:
- 0025-584X
- Page Range / eLocation ID:
- 5123 to 5156
- Format(s):
- Medium: X
- Sponsoring Org:
- National Science Foundation
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- Free Publicly Accessible Full Text
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Accepted Manuscript1.0
- Journal Article:
- https://doi.org/10.1002/mana.202200003
