Abelian Frobenius kernels and modules over number rings
type: Item
instance of: scholarly article
title: Abelian Frobenius kernels and modules over number rings
zbMATH Open document ID: 0912.20003
DOI: 10.1016/S0022-4049(96)00144-2
author: Ron Brown
author: David K. Harrison
published in: Journal of Pure and Applied Algebra
publication date: 1999-03-22T00:00:00Z
review text: The paper provides an interesting example for playing off groups against modules (as, e.g., Abelian groups versus \(\mathbb{Z}\)-modules). Consider finite \(C_m\)-modules \(A\), where \(C_m\) is the cyclic group \(\langle e^{2\pi i/m}\rangle\). The action of \(C_m\) on \(A\) can be extended to a module action of \(R_m=\mathbb{Z}[e^{2\pi i/m},\tfrac 1m]\) on \(A\) iff \(C_m\) acts without fixed points. The non-zero finitely generated torsion modules \(A\) over \(R_m\), \(m\in\mathbb{N}\), are objects of a natural category which, by \(A\rightsquigarrow A\rtimes C_m\), is equivalent to the category \(\mathfrak F\) of finite groups which are either Abelian or metabelian Frobenius groups. The proof requires a careful study of the morphisms in the respective categories. \(\mathfrak F\) is a subcategory of the full subcategory \(\widetilde{\mathfrak F}\) of the category of groups whose objects are (i) groups \(G\) admitting an Abelian kernel \(N\) (i.e., \(N\vartriangleleft G\), \(G/N\) is locally finite, \(C_G(n)=N\) for \(1\neq n\in N\), \(N^m=N\) for all orders \(m\) of torsion elements in \(G/N\)), and (ii) subgroups of such \(G\), which intersect \(N\) trivially. Again, \(\widetilde{\mathfrak F}\) has a module theoretic analogue. For a group \(G\) denote by \(\mathbb{Z}_G\) the localization of \(G\) with respect to the orders of the torsion elements in \(G\), and by \(R_G\) the quotient \(\mathbb{Z}_GG/{\mathfrak a}_G\) with the ideal \({\mathfrak a}_G\) generated by the sums \(\sum_{h\in H}h\) for the finite subgroups \(H\) of \(G\). A truncated module is a pair \((M,G)\) with \(G\) locally finite, \(R_G\neq 0\) and a left \(R_G\)-module \(M\). They are the objects of a natural category which is equivalent to \(\widetilde{\mathfrak F}\). The proof rests on the three facts, (a) \(G\) is a generalized Frobenius complement iff \(R_G\neq 0\), (b) \(R_G\neq 0\Rightarrow G\) is finite or countable, (c) if \(N\) is an Abelian kernel of \(G\), then there exists a complement …
Mathematics Subject Classification ID: 20C07
Mathematics Subject Classification ID: 20J15
Mathematics Subject Classification ID: 11R18
Mathematics Subject Classification ID: 20D60
zbMATH DE Number: 1121691
zbMATH Keywords: categories of finite groups
zbMATH Keywords: actions
zbMATH Keywords: finitely generated torsion modules
zbMATH Keywords: metabelian Frobenius groups
zbMATH Keywords: truncated modules
zbMATH Keywords: generalized Frobenius complements
zbMATH Keywords: group rings
zbMATH Keywords: algebraic number fields
reviewed by: Jürgen Ritter
MaRDI profile type: Publication
cites work: Q5094817
cites work: Q5515450
cites work: Q3240881
cites work: Some infinite Frobenius groups
cites work: Q5094815
cites work: Q3882586
cites work: The Lattice Point Covering Theorem for Rectangles
cites work: Q5569508
cites work: Q3697214
cites work: Q5331572
cites work: On Frobenius groups
cites work: Q3400910
full work available at URL: s0022-4049(96)00144-2
OpenAlex ID: W2075737159
Recommended article: Modular Abelian Varieties Over Number Fields
Recommended article: Kernels of polarizations of abelian varieties over finite fields
Recommended article: Q4133728
Recommended article: Q4394550
Recommended article: Q5638394
Recommended article: Extensions between simple modules for Frobenius kernels
Recommended article: Abstract Grothendieck rings
Recommended article: Modules over prime Krull rings
Recommended article: Relative Galois module structure of rings of integers of absolutely abelian number fields
Recommended article: Modules over Dedekind-like rings
description: scientific article; zbMATH DE number 1121691