Understanding the limit in $\mathbb{Z_p}$

Question

Let $A_n = \mathbb{Z}/p^n \mathbb{Z}$ . We define the ring of p-adic integers $\mathbb{Z_p}$ as $$\mathbb{Z_p} := \lim_{\longleftarrow} (A_n , \phi_n)$$ where $\phi_n : A_n \rightarrow A_{n-1}$ is a homomorphism.

It's called projective limit of the system $(A_n , \phi_n)$. Any element of $\mathbb{Z_p}$ is a sequence $x = (... , x_n, ...,x_1)$ with $x_n \in A_n$ and $\phi_n(x_n) = x_{n-1}$ for $n >1$.

I do not understand what kind of limit is defined for $\mathbb{Z_p}$. Can someone please explain to me this notation?

Answer 1

This is a particular (and concrete) example or Inverse Limit of Rings (Bourbaki Algebra Ch 1 § 10 nb. 1). Take an inverse system of rings that is the data of

• A preordered indexing set $(I,\leq)$ right directed i.e. $$ (\forall \alpha,\beta\in I)(\exists \gamma\in I)(\alpha\leq \gamma\mbox{ and }\beta\leq \gamma) $$
• Families $(R_\alpha)_{\alpha\in I}$ $(f_{\alpha,\beta})_{\alpha,\beta\in I\atop \alpha\leq \beta}$ where $R_\alpha$ are rings (in your case $A_n$) and $f_{\alpha,\beta}\in \operatorname{Hom}_\text{ring}(R_\beta,R_\alpha)$

The limit is the set of families (in your case sequences) $(x_\alpha)_{\alpha\in I}$ such that, for all $\alpha\leq \beta$ one has $f_{\alpha,\beta}(x_\beta)=x_\alpha$.
Note that, in your case, $\phi_n$ is not arbitrary. It is the canonical morphism.

Answer 2

I'll examine in some detail what it means from an example.

An element in $\mathbf Z_p$ is formally a sequence $(x_1,x_2,\dots,x_n,\dots)$, where each $x_n\in\mathbf Z/p^n\mathbf Z$, satisfying the condition that the canonical image of $x_n\in \mathbf Z/p^n\mathbf Z$ in $\mathbf Z/p^{n-1}\mathbf Z$ is $x_{n-1}$.

First consider $x_1\in\mathbf Z/p\mathbf Z$: this congruence class is represented by an integer $a_0\in\{0,1,\dots,p-1\}$.

Next $x_2\in\mathbf Z/p^2\mathbf Z$: it is represented by an integer in $\{0,\dots,p-1,p,\dots,p^2-1\}$, which we can write, by Euclidean division, $y_2p+r_2, \enspace 0\le r_2<p-1$. As $\phi_2(x_2)=x_1$, this implies $r_2=a_0$, and any two values of $y_2p$ are congruent modulo $p^2$, hence any two values of $y_2$ are congruent modulo $p$. Hence $y_2$ has a unique representative $a_1$ in $\{0,\dots, p-1\}$, and $x_2$ is represented by the integer: $$a_0+a_1p,\quad 0\le a_0,a_1\le p-1.$$

More generally, an easy induction shows that is $x_n\in\mathbf Z/p^n\mathbf Z$ represented by the integer $$\sum_{i=0}^{n-1}a_ip^i,\quad 0\le a_0,a_1,\dots, a_{i-1}\le p-1,$$ and the $p$-adic integer $x$ is represented the infinite series: $$x=\sum_{i=0}^{\infty}a_ip^i,\quad \forall i,\;0\le a_i\le p-1.$$ This is somewhat like the infinite decimal expansion of a real number, but in the other direction. Of course, it is not convergent for the usual topology induced by $\mathbf R$, but for the metric deduced from the $p$-adic valuation.