The defect

  • Franz-Viktor Kuhlmann


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.


Valuation Ring Defect Extension Algebraic Extension Transcendence Degree Separable Extension 
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© Springer Science+Business Media, LLC 2011

Authors and Affiliations

  1. 1.Department of Mathematics and StatisticsUniversity of SaskatchewanSaskatoonCanada

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