A noncommutative Weierstrass preparation theorem and applications to Iwasawa theory

Details A noncommutative Weierstrass preparation theorem and applications to Iwasawa theory A noncommutative Weierstrass preparation theorem and applications to Iwasawa theory

2002-04-12

arxiv.org/abs/math/0204358v1

Mathematics

Number Theory

Scientific paper

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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