Can anyone suggest a good thorough reference for $p$-localization and $p$-completion of the integers? I'm an algebraic topologist who's found himself washed up without any intuition.


In particular, here are the two questions I'm trying to answer right now:


Question 1: what is $\mathbb{Q}/\mathbb{Z}\_p$ tensored with itself?


I mean $\mathbb{Z}\_p$ to be the $p$-local integers. The tensor is over $\mathbb{Z}\_p$ or $\mathbb{Z}$ - the answer should be the same.


I know $\mathbb{Q}/\mathbb{Z}\_p$ is an injective $\mathbb{Z}\_p$-module, and $\mathbb{Z}\_p$ is a nice local ring, so maybe something can be said about a flat resolution of $\mathbb{Q}/\mathbb{Z}\_p$?


Question 2: What is known about the cokernel of the map $\mathbb{Z}\_p \rightarrow \mathbb{\hat{Z}}\_p$?


I mean the map that $p$-completes the $p$-local integers. I think the cokernel is a rational vector space, but of finite or infinite dimension? Does it have any nice description?



reference for p-local and p-complete integers
I'm going to switch to a more standard notation, where the localization is $\mathbb{Z}\_{(p)}$ and the completion is $\mathbb{Z}\_p$.


For your first question, the general rule that concerns us is that the tensor product of a $p$-divisible group with a $p$-torsion group is zero. The reason is that if $a$ is a $p^n$th power and $p^n b = 0$, then $a \otimes b = p^{-n}a \otimes p^n b = 0$. Since $\mathbb{Q}/\mathbb{Z}\_{(p)}$ is both $p$-divisible and $p$-torsion, its tensor square is zero.


The cokernel of completion is a $\mathbb{Q}$-vector space, because it admits a $\mathbb{Z}\_{(p)}$-action by multiplication, and it is uniquely $p$-divisible (Proof: lift any element to $\mathbb{Z}\_p$, subtract the constant term of its $p$-adic expansion, divide by $p$, and check that different lifts yield the same answer). Its dimension over $\mathbb{Q}$ is the cardinality of the continuum, since that is its cardinality as a set.


I think most number theory texts have some discussion of $p$-adic numbers, and Gouvea even wrote a whole book about them.


