范数系的上同调与伪Z_p扩张
Cohomology of normic systems and fake Z_p extensions
AI总结：
本文建立一般框架研究数域沿Z_p-扩张的伽罗瓦模Tate上同调，证明商范畴中五项正合序列的存在性，并应用于二面体类型伪Z_p-扩张的类数增长研究。
摘要：
我们建立了一个一般框架，用于研究数域沿$\mathbb{Z}_p$-扩张的伽罗瓦模的Tate上同调群。在对伽罗瓦模施加适当假设的条件下，我们在某个商范畴中确立了一个五项正合序列的存在性，该商范畴的对象同时是正向系和逆向系，并满足某种相容性条件。该正合序列尤其使得我们能够控制单位群沿$\mathbb{Z}_p$-扩张的Tate上同调群的行为。作为一个应用，我们研究了沿我们称之为“二面体类型的伪$\mathbb{Z}_p$-扩张”的类数增长。这项研究依赖于之前的一项工作，在该工作中我们以单位群的上同调群为基础建立了二面体扩张的类数公式。
英文摘要：
We set up a general framework to study Tate cohomology groups of Galois modules along $\mathbb{Z}_p$-extensions of number fields. Under suitable assumptions on the Galois modules, we establish the existence of a five-term exact sequence in a certain quotient category whose objects are simultaneously direct and inverse systems, subject to some compatibility. The exact sequence allows one, in particular, to control the behaviour of the Tate cohomology groups of the units along $\mathbb{Z}_p$-extensions. As an application, we study the growth of class numbers along what we call "fake $\mathbb{Z}_p$-extensions of dihedral type". This study relies on a previous work, where we established a class number formula for dihedral extensions in terms of the cohomology groups of the units.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
