Topologically pure extensions (Q2738756)

The basic notions of the paper are the following: Let \(\mathcal{C}\) be the category of all LCA groups. A morphism in \(\mathcal{C}\) is said to be \textit{proper} if it is open on its image. An exact sequence \(G_{1} @>\alpha_1>>G_2 @>\alpha_2>> \cdots @>\alpha_{n-1}>>G_n\) in \(\mathcal{C}\) is called \textit{proper} if all morphisms \(\alpha_i\) are proper. A proper short exact sequence \(0\rightarrow H \rightarrow G\rightarrow K \rightarrow 0\) in the category \(\mathcal{C}\) is said to be \textit{topologically pure} provided the induced sequence \(0\rightarrow \overline{nH}\rightarrow\overline{nG}\rightarrow\overline{nK}\rightarrow 0\) is proper exact for all positive integers \(n\). A closed subgroup \(H\) of an LCA group \(G\) is said to be \textit{topologically pure} if \(\overline{nH} = H\cap\overline{nG}\) for any positive integer \(n\). An LCA group \(G\) \textit{splits} if its torsion part \(tG\) is a topological direct summand. The paper contains the following results: 1. A new proof of a theorem of Hartman and Hulanicki [\textit{S. Hartman} and \textit{A. Hulanicki}, Fundam. Math. 45, 71-77 (1957; Zbl 0083.25501)] is given saying that if \(G\) is an LCA group which is either compactly generated or has no small subgroups then a closed subgroup \(H\) of \(G\) is pure if and only if \((\widehat{G},H)\) is pure in \(\widehat{G}\). 2. It is proved (Theorem 3.1) that if \(H\) is a closed subgroup of an LCA group \(G\) then \(H\) is a topological direct summand if and only if it has the form \((\mathbb{R}/\mathbb{Z})^{m}\oplus A\), where \(m\) is a cardinal and \(A\) is a direct product of finite cyclic groups. 3. A discrete bounded topologically pure subgroup of an LCA group is constructed which is not a topological direct summand (Example 3.5). 4. A topologically pure sequence \(0\rightarrow B\rightarrow G\rightarrow K \rightarrow 0\) with bounded group \(B\) is constructed which does not split (Example 3.6). 5. A characterization of some splitting LCA groups (Theorem 3.7) is obtained: Let \(G\) be an LCA group so that \(tG\) is the topological direct sum of a compact group and a discrete group. Then \(G\) splits if and only if it possesses an ascending sequence \(A_1\subset A_2\subset\cdots\subset A_n\subset\cdots\) of open subgroups such that: 1. \(\bigcup_{ < \omega}A_n =G\); 2. \(tA_n\) is bounded for all \(n\); 3. \(t({G}/{A_n})={(tG+A_n)}/{A_n}\) for all \(n\leq\omega\) (here \(A_{\omega}=G\)). A proper short exact sequence \(E: 0\rightarrow H\rightarrow G\rightarrow K\rightarrow 0\) in \(\mathcal{C}\) is said to be *-pure provided, for each \(n\), the induced sequence \(0\rightarrow H[n]\rightarrow G[n]\rightarrow K[n]\rightarrow 0\) is properly exact. The author studies LCA groups \(G\) for which each topologically pure exact sequence (each *-pure exact sequence) \(0\rightarrow G\rightarrow H\rightarrow X\rightarrow 0\) splits.NEWLINENEWLINEFor the entire collection see [Zbl 0960.00043].