Boundedness and periodicity of modules over QF rings
type: Item
instance of: scholarly article
title: Boundedness and periodicity of modules over QF rings
zbMATH Open document ID: 0594.16004
DOI: 10.1016/0021-8693(86)90204-8
author: Rainer Schulz
published in: Journal of Algebra
publication date: 1986-01-01T00:00:00Z
review text: Let KG be the group algebra of a finite group with coefficients in a field K. For a finitely generated KG-module M, let \(...\to P_ 1\to^{d_ 1}P_ 0\to^{d_ 0}M\to 0\) be a minimal projective resolution. M is called bounded if the lengths of the kernels ker \(d_ i\) have a common upper bound, and M is called periodic if some ker \(d_ i\) is isomorphic to M. In case K is algebraic over its prime field then every bounded KG-module is periodic. In this paper an arbitrary QF ring R is considered instead of a group algebra and a condition is given which forces a bounded module \(M_ R\) to be periodic. 1. Let \(M_ R\) be a bounded module over a QF ring R. Let the ring \(Ext_ R^*(M;M)\) be right noetherian, and let the right \(Ext^*_ R(M;M)\)-module \(Ext^*_ R(M;N)\) be noetherian for every simple module \(N_ R\). Then M is periodic. For a QF algebra R the assumptions on M in 1. are sufficient and necessary. In particular we have: 2. For a bounded module \(M_ R\) over a QF algebra R, the following conditions are equivalent: (i) M is periodic; (ii) The ring \(Ext^*_ R(M;M)\) is right noetherian, and the right \(Ext^*_ R(M;M)\)-module \(Ext^*_ R(M;N)\) is noetherian for every simple module \(N_ R\); (iii) For every finitely generated module \(X_ R\), the right \(Ext^*_ R(M;M)\)-module \(Ext^*_ R(M;X)\) is noetherian.
Mathematics Subject Classification ID: 16S34
Mathematics Subject Classification ID: 16L60
Mathematics Subject Classification ID: 16D40
Mathematics Subject Classification ID: 16D80
zbMATH DE Number: 3955135
zbMATH Keywords: group algebra
zbMATH Keywords: finitely generated KG-module
zbMATH Keywords: minimal projective resolution
zbMATH Keywords: bounded KG-module
zbMATH Keywords: QF ring
zbMATH Keywords: QF algebra
zbMATH Keywords: periodic
MaRDI profile type: Publication
full work available at URL: 0021-8693(86)90204-8
OpenAlex ID: W2045403733
cites work: Periodicity in groups
cites work: Q3848234
cites work: The varieties and the cohomology ring of a module
cites work: Certain representation algebras
cites work: Modular representations of C2×C2
cites work: Homological Algebra on a Complete Intersection, with an Application to Group Representations
cites work: The Cohomology Ring of a Finite Group
cites work: Indecomposable representations
cites work: Q5610428
cites work: Some generalizations of finite projective dimension
cites work: Q5343374
cites work: The Classification of Algebras by Dominant Dimension
cites work: Dominant dimension of semi-primary rings.
cites work: On algebras with complete homology
cites work: Graded and filtered rings and modules
cites work: Q3050539
cites work: A Characterization of QF-3 Algebras
cites work: A Generalization of Quasi-Frobenius Rings
description: scientific article; zbMATH DE number 3955135