非对易Weierstrass准备定理及其对Iwasawa理论的应用
A noncommutative Weierstrass preparation theorem and applications to Iwasawa theory
摘要：
在本文中，以及即将与Y. Hachimori研究了数域k的无限Galois扩张K上的Iwasawa模，K的伽罗瓦群G=G（K/k）同构于两个p-adic数的半直积.首先分析了相应的Iwasawa代数的一些一般代数性质，然后将这些结果应用于K上p-Hilbert类域的伽罗瓦群.作为主要工具，我们证明了一类斜幂级数环的Weierstrass预备定理.在我们的工作中，一个引人注目的结果是发现了大量的忠实的扭模，即非平凡的扭模，其整体零化子理想为零。最后证明了系数在p元有限域上的完备群代数是Chatters意义下的唯一分解整环。
英文摘要：
In this paper and a forthcoming joint one with Y. Hachimori we study Iwasawa modules over an infinite Galois extension K of a number field k whose Galois group G=G(K/k) is isomorphic to the semidirect product of two copies of the p-adic numbers. After first analyzing some general algebraic properties of the corresponding Iwasawa algebra, we apply these results to the Galois group of the p-Hilbert class field over K. As a main tool we prove a Weierstrass preparation theorem for certain skew power series rings. One striking result in our work is the discovery of the abundance of faithful torsion modules, i.e. non-trivial torsion modules whose global annihilator ideal is zero. Finally we show that the completed group algebra with coefficients in the finite field of p elements is a unique factorization domain in the sense of Chatters.
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