A noncommutative Weierstrass preparation theorem and applications to Iwasawa theory

Abstract: 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 X 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
F_p is a unique factorization domain in the sense of Chatters. |