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The defect

In: Commutative Algebra

Author

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  • Franz-Viktor Kuhlmann

    (University of Saskatchewan, Department of Mathematics and Statistics)

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.

Suggested Citation

  • Franz-Viktor Kuhlmann, 2011. "The defect," Springer Books, in: Marco Fontana & Salah-Eddine Kabbaj & Bruce Olberding & Irena Swanson (ed.), Commutative Algebra, pages 277-318, Springer.
    • Handle: RePEc:spr:sprchp:978-1-4419-6990-3_11
    DOI: 10.1007/978-1-4419-6990-3_11
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