Structure Theorems for locally compact modules over localizations of the integers

TL;DR

The paper establishes structure theorems for locally compact modules over localizations of integers, characterizing their form, duals, and classifying vector spaces over rationals.

Contribution

It introduces new structure theorems for LC modules over localized integers, including classifications and duality results, extending known theories for LCA groups.

Findings

Existence of a unique prime set associated with modules

Characterization of modules as inverse limits of specific families

Full classification of locally compact vector spaces over 

Abstract

Given a multiplicatively closed subset $S$ of the integers, there exist Structure Theorems for $LC$ modules over the localization $\mathbb{Z}S^{-1}$ that are "similar" to those of $LCA$ groups. The most notable one is the 1st Theorem: Given such a module $M$, there exists a unique set of prime numbers $\Sigma$ (purely dependent on $S$) for which $M \cong \mathbb{R}^n \times \sideset{}{'}\prod_{q \in \Sigma} \mathbb{Q}_p^{n_p} \times N$, where $(n, (n_p)_{p \in \Sigma})$ is a sequence of nonnegative integers and $N$ contains a compact open submodule $K$ such that $K/K_0$ is a topological module over $\prod_{ q \in \mathbb{P}\setminus{\Sigma}} \mathbb{Z}_q$. Just like for $LCA$ groups, it is also possible to characterize the locally compact, compactly generated modules over $\mathbb{Z}S^{-1}$, as well as their Pontryagin Duals (which then allows to conclude that any locally compact $\mathbb{Z}S^{-1}$-module is an inverse limit of modules within a specific family). These characterizations are given in the 2nd and 3rd Structure Theorems respectively. Furthermore, as an elementary consequence of the 1st Structure Theorem, one can obtain a full classification of locally compact vector spaces over $\mathbb{Q}$.