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 $\mathbb{ Q}$.
Categories: math.GR math.GN