Shelah's singular compactness theorem.
type: Item
instance of: scholarly article
title: Shelah's singular compactness theorem.
zbMATH Open document ID: 1147.16001
DOI: 10.5565/PUBLMAT_52108_01
author: Paul C. Eklof
published in: Publicacions Matemàtiques
publication date: 2008-07-21T00:00:00Z
full work available at URL: https://eudml.org/doc/42260
review text: Shelah's Singular Compactness Theorem is a very useful tool for dealing with singular cardinals in inductive proofs. The current paper presents a general version (1.4) for modules, with indication of its use in deconstruction of modules. The paper does not contain any new results. Instead, it is a self-contained, short exposition of Shelah's Singular Compactness Theorem with a complete proof with minimal prerequisites: only basic knowledge of cardinals, ordinals and modules is required. In the author's words, Shelah's Singular Compactness Theorem states roughly: If \(\lambda\) is a singular cardinal and \(M\) is a \(\lambda\)-generated module such that enough submodules generated by less than \(\kappa\) elements are ``free'' for sufficiently many regular \(\kappa<\lambda\), then \(M\) is ``free''. Here ``free'' is an application-dependent special type of the modules constrained only by some axioms. For example, ``free'' can mean free in the usual sense, or a ``free'' module can mean a module having a filtration with factors in a given class of modules. The last example leads to important deconstructions of modules (filtering modules with specific factors). These applications are only indicated with emphasis on the point where the theorem is applied.
reviewed by: Gábor Braun
Mathematics Subject Classification ID: 16B70
Mathematics Subject Classification ID: 03E75
Mathematics Subject Classification ID: 16D70
Mathematics Subject Classification ID: 20K35
zbMATH DE Number: 5302051
zbMATH Keywords: singular compactness
zbMATH Keywords: almost free modules
zbMATH Keywords: deconstructions of modules
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 5302051