The defect

TL;DR

This paper introduces the concept of defect in valuation theory, explores its significance in solving key problems in positive characteristic, and reviews methods to understand and manage it.

Contribution

It provides a comprehensive overview of defect phenomena, including examples, classifications, and stability results, highlighting their importance in algebraic and model-theoretic contexts.

Findings

Examples of algebraic extensions with non-trivial defect

Classification approach for Artin-Schreier defect extensions

Summary of stability theorems and their applications

Abstract

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.