# Understanding the limit in $\mathbb{Z_p}$

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?

• Jan 21, 2017 at 10:19
• Ok.. so it's called inverse limit. It will take some time to get this concept. I am on it. Jan 21, 2017 at 10:24
• It is an inverse limit. Another example is $R[[t]] = \lim\limits_{\longleftarrow_n} R[t]/(t^n)$. You can look at math.stackexchange.com/questions/38517 Jan 21, 2017 at 10:24
• Also related: math.stackexchange.com/questions/156763 Jan 21, 2017 at 10:34

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, $$y_2p+r_2, \enspace 0\le r_2. As $$\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.

• I strongly think it'd be better to just talk about $\mathbb Z$ modulo $p^n$ and not choose representatives, nor try to do the fake-power-series expansion... Mar 9, 2020 at 2:13

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.