The defect
The defect
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缺陷
DOI：
10.4337/9781786433725.00014
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发表时间：
2010
期刊：
影响因子：
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通讯作者：
F. Kuhlmann
中科院分区：
文献类型：
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作者：
F. Kuhlmann
本文介绍了估价理论中的“缺陷”现象，又称“分支缺陷”。我们描述了它在正特征的深度开放问题中所起的作用：局部均匀化（奇点解的局部形式）、值场的模型理论、值函数场的结构理论。我们给出了几个具有非平凡缺陷的代数扩展的例子。我们指出了为什么Artin-Schreier缺陷扩展起中心作用，并描述了一种对它们进行分类的方法。此外，我们还概述了有助于驯服或避免缺陷的各种结果，特别是“稳定性”定理和关于“亨塞利合理性”的定理，并说明了它们是如何应用的。最后，我们列出了一些尚未解决的问题。
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.