On TFU extensions in LCA groups

TL;DR

This paper investigates a special class of extensions in the category of locally compact abelian groups, called TFU extensions, and provides conditions for their splitting, advancing the understanding of their structure.

Contribution

It introduces the concept of TFU extensions in LCA groups and establishes criteria for when these extensions split, contributing new theoretical insights.

Findings

Characterization of TFU extensions in LCA groups

Conditions under which TFU extensions split

Introduction of new results on the structure of TFU extensions

Abstract

Let $\ell$ be the category of all locally compact abelian (LCA) groups. Let $G\in\ell$ and $H\subseteq G$. The first Ulm subgroup of $G$ is denoted by $G^{(1)}$ and the closure of $H$ by $\overline{H}$. A proper short exact sequence $0\to A\stackrel{\phi}{\to} B\stackrel{\psi}{\to} C\to 0$ in $\ell$ is said to be a $TFU$ extension if $0\to \overline{A^{(1)}}\stackrel{\overline{\phi}}{\to} \overline{B^{(1)}}\stackrel{\overline{\psi}}{\to} \overline{C^{(1)}}\to 0$ is a proper short exact sequence where $\overline{\phi}=\phi\mid_{\overline{A^{(1)}}}$ and $\overline{\psi}=\psi\mid_{\overline{B^{(1)}}}$. We introduce some results on $TFU$ extensions. Also, we establish conditions under which the $TFU$ extensions split.