The defect

Franz-Viktor Kuhlmann ()
Additional contact information
Franz-Viktor Kuhlmann: University of Saskatchewan, Department of Mathematics and Statistics

A chapter in Commutative Algebra, 2011, pp 277-318 from Springer

Abstract: Abstract This work was partially supported by a Canadian NSERC grant and by a sabbatical grant from the University of Saskatchewan. I thank Bernard Teissier, Olivier Piltant, Michael Temkin and Hagen Knaf for helpful discussions and support. I thank the two referees for their careful reading and useful comments. We give an introduction to the valuation theoretical phenomenon of “defect”, also known as “ramification deficiency”. We describe the role it plays in deep open problems in positive characteristic: local uniformization (the local form of resolution of singularities), the model theory of valued fields, the structure theory of valued function fields. We give several examples of algebraic extensions with non-trivial defect. We indicate why Artin–Schreier defect extensions play a central role and describe a way to classify them. Further, we give an overview of various results about the defect that help to tame or avoid it, in particular “stability” theorems and theorems on “henselian rationality”, and show how they are applied. Finally, we include a list of open problems.

Keywords: Valuation Ring; Defect Extension; Algebraic Extension; Transcendence Degree; Separable Extension (search for similar items in EconPapers)
Date: 2011
References: Add references at CitEc
Citations:

There are no downloads for this item, see the EconPapers FAQ for hints about obtaining it.

Related works:
This item may be available elsewhere in EconPapers: Search for items with the same title.

Export reference: BibTeX RIS (EndNote, ProCite, RefMan) HTML/Text

Persistent link: https://EconPapers.repec.org/RePEc:spr:sprchp:978-1-4419-6990-3_11

Ordering information: This item can be ordered from
http://www.springer.com/9781441969903

DOI: 10.1007/978-1-4419-6990-3_11

Access Statistics for this chapter

More chapters in Springer Books from Springer
Bibliographic data for series maintained by Sonal Shukla () and Springer Nature Abstracting and Indexing ().