Characteristic ideals and Iwasawa theory
type: Item
instance of: scholarly article
title: Characteristic ideals and Iwasawa theory
zbMATH Open document ID: 1361.11065
author: Andrea Bandini
author: Ignazio Longhi
author: Francesc Bars
published in: The New York Journal of Mathematics
publication date: 2014-10-21T00:00:00Z
full work available at URL: https://arxiv.org/abs/1310.0680
full work available at URL: 20-38.html
reviewed by: Thong Nguyen Quang Do
Mathematics Subject Classification ID: 11R23
Mathematics Subject Classification ID: 13F25
zbMATH DE Number: 6359494
zbMATH Keywords: Krull domains
zbMATH Keywords: characteristic ideals
zbMATH Keywords: function fields
zbMATH Keywords: divisor classes
MaRDI profile type: Publication
arXiv ID: 1310.0680
review text: Let \(A\) be a Noetherian Krull domain and \(M\) a finitely generated torsion \(A\)-module. The structure theorem for such modules asserts that \(M\) is pseudo-isomorphic to an elementary module of the form \(\bigoplus_{1\leq i\leq n} A/\mathcal{P}_i^{e_i}\), where the \(\mathcal{P}_i\)'s are height 1 prime ideals of \(A\). This allows to define an important invariant of \(M\), the characteristic ideal \(\mathrm{Ch}_A(M) :=\prod_{1\leq i\leq n} \mathcal{P}_i^{e_i}\), which plays a major role in the (commutative) Iwasawa theory for global fields, where \(A\) is the complete group algebra \(\mathbb{Z}_p\Gamma\) of an abelian \(p\)-adic Lie group \(\Gamma\). A classical example is \(\Gamma \cong \mathbb{Z}_p^d\) when studying multiple \(\mathbb{Z}_p\)-extensions of global fields, in which case \(A\) is isomorphic (non canonically) to the power series ring \(\mathbb{Z}_pt_1,\dots,t_d\). But over a function field \(F\) of characteristic \(p\), extensions \(\mathcal{F}/F\) with \(\mathrm{Gal}(\mathcal{F}/F) \cong \mathbb{Z}_p^\infty\) occur naturally, in which case the Noetherian setting is no longer adapted and the above definition of the characteristic ideal no longer applies. One way out is to consider a filtration of \(\mathbb{Z}_p^d\)-extensions for \(\mathcal{F}\), define the characteristic ideals at the \(\mathbb{Z}_p^d\)-levels and then pass to the inverse limit.NEWLINENEWLINEThe authors' main technical theorem reads: ``Let \(\{\Lambda_d\}_{d\geq 0}\) be an inverse system of Noetherian Krull domains such that \(\Lambda_d \cong \Lambda_{d+1} /\mathcal{P}_{d+1}\) and \(\Lambda_{d+1} \cong \varprojlim \Lambda_{d+1} /\mathcal{P}_{d+1}^n\) for any \(d \geq 0\) (where \(\mathcal{P}_{d+1}\) is a principal prime ideal of \(\Lambda_{d+1}\) of height \(1\)). Let \(\Lambda := \varprojlim \Lambda_d\) and consider a finitely generated \(\Lambda\)-module \(M = \varprojlim M_d\) (where each \(M_d\) is a \(\Lambda_d\)-module). Assume for any \(d \geq 1\) the two following…
description: scientific article; zbMATH DE number 6359494