On \(\omega\)-filtered vector spaces and their application to Abelian p- groups. I
The socle G[p] of a p-group G is an \(\omega\)-filtered vector space with the filtration \((p^ nG)[p]\), \(n=0,1,2,... \). With applications to p- groups in mind the authors investigate \(\omega\)-filtered vector spaces over at most countable fields using some of the more sophisticated tools of set theory. It is not possible to explain here the numerous categorical concepts involved and the set theoretical concepts utilized. For the novice the carefully written paper forms an excellent introduction to certain set theoretical methods and filtered vector spaces. For the initiated we mention that an invariant \(\Gamma\) (X) of the \(\omega\)-filtered vector space X of uncountable regular dimension \(\kappa\) (mostly \(\kappa =\kappa_ 1)\) plays a central role. It is an element of the Boolean algebra of all subsets of \(\kappa\) modulo the ideal of thin subsets of \(\kappa\). It is obtained by writing X as the union of a smooth ascending chain of subspaces \(X_{\alpha}\) \((\alpha <\kappa)\) of X which are ''small'' i.e. \(\dim (X_{\alpha})<\kappa\). If E is the set of all \(\alpha\) such that \(X_{\alpha}\) is not a direct summand of every (filtered) subspace of X in which \(X_{\alpha}\) has codimension less than \(\kappa\) then \(\Gamma (X)=E\) modulo thin sets. The significance of the invariant \(\Gamma\) (X) may be seen from two results: \(\Gamma (X)=0\) if and only if X is a projective filtered vector space; \(\Gamma\) (X)\(\neq 1\) if and only if every countable subspace of X is contained in a closed countable subspace and X is Hausdorff in the topology induced by its filtration. However, the invariant \(\Gamma\) in no way pins down the space: there are a large number of non-isomorphic spaces all with the same \(\Gamma\) and the same basic subspace. Among the applications is the following: A p-group is weakly \(\omega_ 1\)- separable if it is separable and every countable subset is contained in a countable closed pure subgroup. Theorem: Let G be a separable p-group of cardinality \(\kappa_ 1\) containing a pure weakly \(\omega_ 1\)- separable subgroup which is not \(\Sigma\)-cyclic, and let B be an elementary p-group of dimension \(\kappa_ 1\). Then there exist \(2^{\kappa_ 1}\) mutually non-isomorphic \(\omega\)-elongations of G by B. There are results on \(p^{\omega +1}\)-injective p-groups and on \(p^{\omega +1}\)-projective groups. E.g. an \(\omega\)-filtered Z(p)- vector space X is the socle of a \(p^{\omega +1}\)-projective p-group if and only if \(X=Y\oplus P\) where P is projective and fin dim(P)\(=fin \dim (X)\).
scientific article; zbMATH DE number 3914587