几乎数学，Kähler微分与深层分岐域
Almost Mathematics, Kähler differentials and deeply ramified fields
AI总结：
本文探讨了值域扩展中的分岐结构及相对Kähler微分理论，通过Gabber和Ramero的特征证明了深层分岐域的性质，并介绍了几乎数学的基本理论及弱étale扩张的证明。
摘要：
本文讨论了值域扩展中的分岐结构及相对Kähler微分理论。我们首先回顾了与Franz-Viktor Kuhlmann和Anna Rzepka近期合作工作中关于Artin-Schreier和Kummer扩展中值环扩展的相对Kähler微分理论。随后展示该理论如何用于证明Gabber和Ramero对深层分岐域的特征化。第4节介绍了几乎数学的基本理论，适合广泛读者理解。第5节给出了Gabber和Ramero关于值域扩展到其分离闭包的弱étale扩张的简单且自包含的证明。最后，我们考虑了定义为Coates和Greenberg的深层分岐域等价条件，并证明这些条件与Gabber-Ramero对局部域的条件相同。
英文摘要：
This article discusses ramification and the structure of relative Kähler differentials of extensions of valued fields. We begin by surveying the theory developed in recent work with Franz-Viktor Kuhlmann and Anna Rzepka constructing the relative Kähler differentials of extensions of valuation rings in Artin-Schreier and Kummer extensions. We then show how this theory is applied to give a simple proof of Gabber and Ramero's characterization of deeply ramified fields. Section 4 develops the basics of almost mathematics, and should be accessible to a broad audience. Section 5 gives a simple and self contained proof of Gabber and Ramero's characterization of when the extension of a rank 1 valuation of a field to its separable closure is weakly étale. In the final section, we consider the equivalent conditions characterizing deeply ramified fields, as they are defined by Coates and Greenberg, and show that they are the same as the conditions of Gabber Ramero for local fields.
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