TITLE: The isomorphism class of $\mathrm{Ext}^1_\mathbb{Z}(\mathbb{R}/\mathbb{Z},\mathbb{Z})$
QUESTION [18 upvotes]: In a recent conversation with a colleague, the following question arose:


What is the isomorphism class of $\mathrm{Ext}^1_\mathbb{Z}(\mathbb{R}/\mathbb{Z},\mathbb{Z})$?  That is to say, what is $\mathrm{Ext}^1_\mathbb{Z}(\mathbb{R}/\mathbb{Z},\mathbb{Z})$ as an Abelian group?


I did not have any immediate response other than “That’s interesting; I’ll have to think about it.”  Afterwards, I spent some time recalling some standard information about $\mathbb{R}/\mathbb{Z}$ and the Ext functor, and have arrived at the following:
$$\mathrm{Ext}^1_\mathbb{Z}(\mathbb{R}/\mathbb{Z},\mathbb{Z})\simeq {\left(\prod_{x\in\mathbb{R}}{\mathrm{Ext}^1_\mathbb{Z}{\left(\mathbb{Q}_x,\mathbb{Z}\right)}}\right)}\times{\left(\prod_{p\in\mathbb{P}}{\mathrm{Ext}^1_\mathbb{Z}{\left(\mathbb{Z}{\left(p^\infty\right)},\mathbb{Z}\right)}}\right)}\ .$$
Here, $\mathbb{Q}_x=\mathbb{Q}$ for all $x\in\mathbb{R}$, $\mathbb{P}$ denotes the set of prime numbers, and for a prime $p$, $\mathbb{Z}{\left(p^\infty\right)}$ is the Prüfer $p$-group.  This decomposition reduces the original problem to the following two questions:


What is the isomorphism class of $\mathrm{Ext}^1_\mathbb{Z}(\mathbb{Q},\mathbb{Z})$?
Given a prime number $p$, what is the isomorphism class of $\mathrm{Ext}^1_\mathbb{Z}(\mathbb{Z}{\left(p^\infty\right)},\mathbb{Z})$?


I was curious to know if anyone happens to know the answer to any of questions (1), (2), or (3), or can point me to such in the literature.

REPLY [7 votes]: For context: the group $LA=\text{Ext}(\mathbb{Z}/p^\infty,A)$ is called the derived $p$-completion of $A$.  It has a natural map to the ordinary completion $CA=\lim_n A/p^nA$ which is often an isomorphism.  In particular it is an isomorphism if $A$ is a free abelian group, or if it is finitely generated, or if there exists $n$ such that $\text{ann}(p^n,A)=\text{ann}(p^{n+1},A)$.  In the cases where $CA\neq LA$ it typically works out that $LA$ has better behaviour and is more relevant for applications, especially in algebraic topology and homological algebra.  One reference is the book "Homotopy Limits, Completion and Localization" by Bousfield and Kan.