关于伪零模的商
On the Quotient of a Pseudo-null Module
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中文总结 AI 辅助
本文研究非交换Iwasawa理论中伪零模的商模性质，给出伪零模除以正则中心元素后仍为伪零的充要条件，并应用于对偶精细Selmer群。
中文摘要 AI 辅助
受非交换Iwasawa理论中产生的问题的启发，设$R$为（不一定交换的）环，$T \in R$为正则中心元素，$M$为伪零$R$-模。我们研究$M/TM$作为$R/TR$-模为伪零的充分必要条件。首先，当$R$为交换环时，我们给出基于相伴素理想的充分必要条件。然后，我们将此应用于$R$为Krull整环的情形，并得到$M/TM$的特征理想与$T$-挠子模$M[T]$之间的精确关系。当$R$为非交换环时，我们给出基于$\operatorname{Ext}$群的充分必要条件，并将其与交换环情形下的判据进行比较。最后，我们给出一个在非交换Iwasawa理论中的应用。粗略地说，如果'大的'对偶精细Selmer群是伪零的，那么'大多数'特化的对偶精细Selmer群也是伪零的。
英文摘要
Motivated by questions arising in noncommutative Iwasawa theory, let $R$ be a (not necessarily commutative) ring, $T \in R$ a regular central element, and $M$ a pseudo-null $R$-module. We investigate necessary and sufficient conditions under which the quotient $M/TM$ is pseudo-null as an $R/TR$-module. We first give necessary and sufficient conditions in terms of associated prime ideals when $R$ is commutative. Then we apply this to the case when $R$ is a Krull domain and obtain a precise relationship between the characteristic ideals of $M/TM$ and the $T$-torsion submodule $M[T]$. We give necessary and sufficient conditions in terms of $\operatorname{Ext}$ groups when $R$ is a noncommutative ring and then compare them with the criterion when $R$ is commutative. Lastly, we give an application to noncommutative Iwasawa theory. Roughly speaking, if `big' dual fine Selmer group is pseudo-null, then `most' specialized dual fine Selmer group is pseudo-null.
发表机构
- Morningside Center of Mathematics, Chinese Academy of Science（中国科学院数学与系统科学研究院）
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