Equivalence of inverse limit and power series definition of p-adic integers

Question

I am currently working on Neukirch's "Algebraic Number Theory", however I find that from time to time he leaves out important information. This time I sadly can't figure out what the actual argument is, he uses when saying that the two definitions for p-adic integers (the power series version and the inverse limit version) are equivalent. I am referring specifically to Proposition 1.3 in Chapter II.

He starts by defining the $p$-adics as power series in $p$:

$$\mathbb{Z}_p = \left\{\sum_{k=0}^{\infty}a_kp^k \,\middle\vert\, a_k \in \{0,1,\dotsc,p-1\}\right\}.$$

Then he proceeds to show a Proposition which states that every residue class $\bar{x} \bmod{p^n}$ can be uniquely written as partial sum $$ \sum_{k=0}^n a_kp^k \equiv \bar{x} \bmod{p^n} $$ (Proposition 1.2). This statement and the proof are quite clear to me so far.

However now the concept of inverse limits gets introduced and we consider $\varprojlim\mathbb{Z}/p^n$ where we have the canonical projections as maps between our objects which are exactly those quotients.

Here is where I am running into some problems. Because this proposition 1.3 I mentioned at the beginning states that we have a bijection $\mathbb{Z}_p \rightarrow \varprojlim\mathbb{Z}/p^n$.

This statement makes sense to me, however Neukirch claims that this immediately follows from the Proposition 1.2. Could someone now please describe to me step by step what the actual arguments are which we use when proving 1.3 through 1.2? I can see that from 1.2 it immediately follows that:

$$\mathbb{Z}/p^n \cong \left\{\sum_{k=0}^{n} a_kp^k \,\middle\vert\, a_k\in\{0,1,\dotsc,p-1\}\right\} $$ (isomorphic as sets, i.e., there is a bijection between those).

For me this by far does not suffice to show what we are claiming. A step I thought of that might be helpful is

$$\prod_{n\in\mathbb{N}}\mathbb{Z}/p^n \cong \prod_{n\in\mathbb{N}}\left\{\sum_{k=0}^{n} a_kp^k \,\middle\vert\, a_k\in\{0,1,\dotsc,p-1\}\right\}.$$

However we only have $\varprojlim\mathbb{Z}/p^n \subset \prod_{n\in\mathbb{N}}\mathbb{Z}/p^n$ and it is not clear to me why the image of the restricted map yields exactly $\mathbb{Z}_p$. Logically it makes sense to me as in the inverse limit we have exactly the sequences which are connected by the canonical projections and in $\mathbb{Z}_p$ we have exactly the (formal) series which taken by literal definition are sequences of partial sums (so we also have this connection by just reducing a partial sum to one with fewer summands). I cannot find the precise mathematical arguments which prove Proposition 1.3 by just using Proposition 1.2.

Of course any help is appreciated and if you know how one can show this bijection elementarily with a different method, I will be thankful as well! I know this is just a minor fact and I'm getting hung up on details, however I really want to understand the topic thoroughly.

Answer 1

$\varprojlim \Bbb{Z}/(p^n)$ is just the set of "sequences" $(a_n)_{n\ge 1}$ with $a_n\in \Bbb{Z}/(p^n)$ and $a_{n+1}\equiv a_n \bmod p^n$.

It is a ring with the pointwise addition and multiplication (ie. $a_n+b_n$ is done in $\Bbb{Z}/(p^n)$)

With $f_n(a_n) $ the unique representative in $0\ldots p^n-1$ you get that $(a_n)$ corresponds to the $p$-adic series $f_1(a_1)+\sum_{n\ge 2} c_n p^{n-1}$ where $c_n=\frac{f_n(a_n)-f_{n-1}(a_{n-1})}{p^{n-1}}\in 0\ldots p-1$.

A $p$-adic series $\sum_{n\ge 0} d_n p^n$ gives the sequence $a_n = \sum_{m=0}^{n-1} d_m p^m \bmod p^n$ so that $(a_n)_{n\ge 1} \in \varprojlim \Bbb{Z}/(p^n)$.

Answer 2

To add a bit to @reuns' answer: the "power series" idea of $\mathbb Z_p$ has several problems... though the general idea of it is an interesting extension/analogy of power series for holomorphic functions, and other things.

Already the addition of such "power series" is very awkard, and multiplication is worse... And, for example, how to prove that this addition and multiplication satisfy associativity, distributivity, etc.? Oof.

In fact, I'd claim that the idea of "Hensel's lemma" is a ("constructive") portrayal of $\mathbb Z_p$ as a (projective) limit of $\mathbb Z/p^n$'s.

The description @reuns notes of that limit is, to be more precise, a construction of the limit, as a sub-object of the product. With or without a construction as proof of existence, if we grant existence of the proj lim of these rings, then we obtain a (commutative) ring. No doubt. No weird multiplications... although the explicit description of the multiplication and addition is slightly indirect and arrow-theoretic.

In current math culture, there is the intermediate description of $\mathbb Z_p$ as a completion with respect to the p-adic metric. I've come to realize that the main selling point of this description is that is does mesh with the notion of "metric space", which is part of the standard curriculum. :)

To recap, in my opinion, by this point, the Hensel's lemma setting for p-adic integers (etc.) really is an instantiation of the idea of projective limit, before its time.