Structure Theorems for locally compact modules over localizations of the integers

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 $Σ$ (purely dependent on $S$) for which $M \cong \mathbb{R}^n \times \sideset{}{'}\prod_{q \in Σ} \mathbb{Q}_p^{n_p} \times N$, where $(n, (n_p)_{p \in Σ})$ is a sequence of nonnegative integers and $N$ contains a compact open submodule $K$ such that $K/K_0$ is a topological module over $\prod_{ q \in \mathbb{P}\setminusΣ} \mathbb{Z}_q$. 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 $\mathbb{Q}$.