Slenderness, completions, and duality for primary Abelian groups
All groups are abelian here. Let \(A\) be a group, and \(E=\text{End }A\) its endomorphism ring. For any group \(G\), set \(G^*=\text{Hom}(G,A)\) and note that \(G^*\) is naturally a left \(E\)-module. On the other hand, if \(M\) is a left \(E\)-module, then \(M^*=\text{Hom}_E(M,A)\) is an abelian group. There is the natural evaluation map \(\sigma_G\colon G\to G^{**}\): \(\sigma_G(x)(f)=f(x)\) and \(G\) is called \(A\)-reflexive if \(\sigma_G\) is an isomorphism. A left \(E\)-module \(M\) is called \(E\)-slender if every \(E\)-homomorphism \(\prod_{j\in\omega}E_j\to M\), where \(E_j\cong E\), maps some submodule \(\prod_{j\geq m}E_j\) to 0. If \(G=\bigoplus_{j\in\omega}A_j\), where \(A_j\cong A\), then \(G^*=\prod_{j\in\omega}E_j\), and \(A\) is \(E\)-slender if and only if \(G\) is \(A\)-reflexive. This indicates that there is an interaction between reflexivity and slenderness. There is also a connection with topological completion. Let \(G\) be a group and \(\mathcal T\) a neighborhood base at 0 for a linear topology on \(G\). If \(A\) is a group with discrete topology such that \(G/X\) is \(A\)-reflexive for all \(X\in\mathcal T\), and further every homomorphism \(G\to A\) is continuous, then algebraically \(G^{**}\cong\widetilde G\), where \(\widetilde G\) denotes the completion of the topological group \((G,\mathcal T)\). These are the themes of the paper under discussion. Major results are the following. Theorem 1.6. Suppose that \(A\) is an unbounded reduced \(p\)-group and \(E=\text{End }A\). Then \(A\) is \(E\)-slender if and only if \(p^\omega A\) is a slender \(E\)-module and \(A/p^\omega A\) is not torsion-complete. A corollary answers affirmatively a conjecture of the reviewer: An unbounded separable \(p\)-group \(A\) is \(\text{End }A\)-slender if and only if \(A\) is not torsion-complete. Recall that a \(p\)-group \(G\) is \(p^{\omega+n}\)-projective if it contains a \(p^n\)-bounded subgroup \(P\) such that \(G/P\) is a direct sum of cyclic groups. In particular, the \(p^\omega\)-projective groups are exactly the direct sums of cyclic \(p\)-groups. The \(\mathcal T_n\)-topology on a group \(G\) is the linear topology on \(G\) that has as a neighborhood base at 0 the set of subgroups \(X\) such that \(G/X\) is \(p^{\omega+n}\)-projective. The \(p^{\omega}\)-topology is better known as \(\oplus_c\)-topology. Theorem 2.3. The completion \(\widetilde G\) in the \(p^{\omega+n}\)-projective topology of a \(p\)-group \(G\) of non-measurable cardinality is algebraically isomorphic to the double dual with respect to a suitable group \(A\). Theorem 3.5. Let \(N\) be an arbitrary \(p\)-group. Then there is a \(p\)-group \(X\) such that \(\widetilde X/X\cong N\) where \(\widetilde X\) is the completion of \(X\) in the \(\oplus_c\)-topology. This result was motivated by the question of ``completability'', here specifically the question whether the topology of the completion \(\widetilde X\) is again the \(\oplus_c\)-topology when \(X\) carries this topology. The answer is no if the quotient \(\widetilde X/X\) is not divisible, and the author demonstrates that this can fail in a spectacular way. Corollary 3.6. If \(N\) is a \(p\)-group of non-measurable cardinality and \(A=\bigoplus_nZ_{p^n}\), then there is a group \(X\) such that \(X^{**}/\sigma_X(X)\cong N\). There are other interesting results and constructions that require too much notation to state in a review.
scientific article; zbMATH DE number 1014036
scientific article