Flat and stably flat modules
type: Item
instance of: scholarly article
title: Flat and stably flat modules
zbMATH Open document ID: 0955.16036
author: E. Alcock
published in: Journal of Algebra
publication date: 2000-03-20T00:00:00Z
review text: The context of this paper is complete cohomology and modules are of type \((FP)_\infty\). Let \(\text{LH}{\mathbf F}\) denote the class of hierarchically decomposable groups and \(\text{H}_1{\mathbf F}\) the subclass of groups acting on a finite dimensional contractible CW-complex, in such a way that the setwise stabiliser of each cell is equal to the pointwise stabiliser and is finite. Let \(G\) be an \(\text{H}_1{\mathbf F}\)-group and \(R\) a strongly \(G\)-graded \(k\)-algebra such that the base ring \(R_1\) is coherent of finite global dimension. If \(N\) is an \(R\)-module, then it is proved that \(N\) is stably flat if and only if \(N\) has finite projective dimension. This result is a generalization of a result of \textit{D. J. Benson} [J. Algebra 193, No. 1, 260-287 (1997; Zbl 0886.20002)] but avoids complicated diagram chasing. The second result of importance states that for a finite group \(G\) and a strongly \(G\)-graded \(k\)-algebra \(R\) such that \(R_1\) has finite global dimension, if \(N\) is a flat \(R\)-module which is projective as an \(R_1\)-module, then \(N\) is projective.
reviewed by: Frieda Theron
Mathematics Subject Classification ID: 16W50
Mathematics Subject Classification ID: 16D40
Mathematics Subject Classification ID: 16E10
Mathematics Subject Classification ID: 16E40
zbMATH DE Number: 1384479
zbMATH Keywords: stably flat modules
zbMATH Keywords: complete cohomology
zbMATH Keywords: strongly graded algebras
zbMATH Keywords: global dimension
zbMATH Keywords: projective dimension
zbMATH Keywords: projective modules
MaRDI profile type: Publication
full work available at URL: jabr.1999.7936
OpenAlex ID: W2035074309
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DOI: 10.1006/JABR.1999.7936
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description: scientific article; zbMATH DE number 1384479