A non-commutative Weierstrass preparation theorem and applications to Iwasawa theory. (English) Zbl 1051.11056

Venjakob, Otmar; Vogel, Denis

J. Reine Angew. Math. 559, 153-191 (2003).

The author aims to illustrate some features of noncommutative Iwasawa theory [see e.g. J. Coates, P. Schneider and R. Sujatha, J. Inst. Math. Jussieu 2, No. 1, 73–108 (2003; Zbl 1061.11060)] in the possibly easiest noncommutative example, namely an infinite Galois extension \(k_{\infty}/k\) of number fields whose Galois group is a semi-direct product \(G = H \rtimes \Gamma,\;H \simeq \Gamma \simeq {\mathbb Z}_{p},\) with a non trivial action of \(\Gamma\) on \(H.\) For instance, take \(k = {\mathbb Q}(\mu_{p})\) and \(k_{\infty} = k ( \mu_{p^{\infty}},\, p^{p^{-\infty}} ).\)
The author first analyzes some general algebraic properties of the Iwasawa algebra \(\wedge (G),\) using a version of the Weierstrass preparation theorem for certain skew power series with coefficients in a noncommutative local ring. One remarkable phenomenon is the abundance of faithful torsion \(\wedge (G)\)-modules, i.e. non trivial torsion modules with trivial global annihilator. What is even more striking is that those modules which occur most naturally in arithmetic, namely the modules which are finitely generated over \(\wedge (H),\) are either faithful or pseudo-null. This is the case of the usual unramified Iwasawa module \(X_{nr}\) attached to \(k_{\infty} = {\mathbb Q} ( \mu_{p^{\infty}}, p^{p^{-\infty}}),\) for which the author further proves that its pseudo-nullity implies the vanishing of the \(p\)-class group of \(k_{\infty}.\) In another paper (with Y. Hachimori), it will also be shown that the Pontryagin duals of certain Selmer groups are “completely faithful modulo pseudo-null modules”, i.e. all their subquotient objects are faithful in the relevant quotient category.

Reviewer: Thong Nguyen Quang Do (Besançon)

MSC:

11R23

Iwasawa theory

Keywords:

pseudo-null; completely faithful

Citations:

Zbl 1061.11060

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