高维局部类域论
Higher dimensional local class field theory
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中文总结 AI 辅助
本文介绍高维局部类域论，给出$\u2113$-部分的新证明并提出猜想，以理解$p$-部分，并阐明经典与高维理论的类比。
中文摘要 AI 辅助
这篇主要是综述性的文章介绍了高维局部类域论，更确切地说是局部域上光滑射影概形的类域论。我们首先简要回顾经典局部类域论的一些主要结果。然后我们解释高维情形下的主要定义和陈述。我们给出了$\u2113$-部分的一个新的且简单的证明，该证明最初归功于Jannsen--Saito和Forré。受此证明启发，我们提出了一个猜想，该猜想将蕴含剩余的$p$-系数开放情形，并增进了我们对$p$-部分的理解。本文的一个特别目标是概述经典理论与高维理论之间的类比，以及单位滤过和分歧（即$p$-部分和$\u2113$-部分）如何在高维情形下被解释。
英文摘要
In this largely expository article we give an introduction to higher dimensional local class field theory, more precisely class field theory for smooth projective schemes over local fields. We begin with a quick summary of some of the main results of classical local class field theory. Then we explain the main definitions and statements in the higher dimensional setting. We give a new and simple proof of the $\ell$-part, which is originally due to Jannsen--Saito and Forré. Inspired by this proof, we propose a conjecture which would imply the remaining open cases with $p$-coefficients and which improves our understanding of the $p$-part. A particular goal of the exposition is to outline analogies between the classical and the higher dimensional theory and on how the unit filtration and ramification, i.e. the $p$- and the $\ell$-part, can be interpreted in the higher dimensional setting.
