A conductor formula for completed group algebras.
type: Item
instance of: scholarly article
title: A conductor formula for completed group algebras.
zbMATH Open document ID: 1306.16010
author: Andreas Nickel
published in: Documenta Mathematica
publication date: 2014-10-20T00:00:00Z
full work available at URL: https://arxiv.org/abs/1202.1091
full work available at URL: 19.html
reviewed by: Alessandro Cobbe
Mathematics Subject Classification ID: 16H10
Mathematics Subject Classification ID: 16H20
Mathematics Subject Classification ID: 11R23
zbMATH DE Number: 6358804
zbMATH Keywords: central conductor
zbMATH Keywords: completed group algebras
zbMATH Keywords: extensions of lattices
zbMATH Keywords: Fitting invariants
zbMATH Keywords: maximal orders
MaRDI profile type: Publication
arXiv ID: 1202.1091
review text: Let \(\mathfrak o\) be the ring of integers of a finite extension \(K\) of \(\mathbb Q_p\), let \(\mathfrak o[G]\) be the group ring of a finite group \(G\) over \(\mathfrak o\); then \(\mathcal F(\mathfrak o[G])=\{x\in\zeta(\mathfrak o[G]): x\Gamma\subseteq\mathfrak o[G]\}\) is called the central conductor, where \(\Gamma\subseteq K[G]\) is a maximal \(\mathfrak o\)-order containing \(\mathfrak o[G]\) and \(\zeta(\Lambda)\) denotes the center of a ring \(\Lambda\). \textit{H. Jacobinski} [Mich. Math. J. 13, 471-475 (1966; Zbl 0143.05702)] gave a complete description of \(\mathcal F(\mathfrak o[G])\) in terms of characters of \(G\) and he showed that \(\mathcal F(\mathfrak o[G])\) annihilates some Ext groups.NEWLINENEWLINE The aim of the paper under review is to extend Jacobinski's results to completed group algebras \(\mathfrak oG\) when \(G\) is a \(p\)-adic Lie group of dimension \(1\) (\(p\) is an odd prime number). In fact such a group \(G\) can be written as \(G=H\rtimes\Gamma\) with \(H\) finite and \(\Gamma\cong\mathbb Z_p\). The main result for \(\mathcal F(\mathfrak oG)\) takes the form of an inclusion, which is shown to be an equality under certain additional hypotheses or after localizations at prime ideals not containing \(p\). However it is shown by an example that equality does not hold in general. In the last section the author derives some consequences involving Ext groups, in analogy to the group ring case, though it is shown that in this setting it is not true that the central conductor annihilates the considered Ext groups. Some applications to non-commutative Fitting invariants are also described.
description: scientific article; zbMATH DE number 6358804