Shelah's singular compactness theorem.
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.
scientific article; zbMATH DE number 5302051