$\mathbb{Z}_p$ , $\mathbb{Q}_p$ , and the ring of Witt vectors

No.04:

There are several ways to define $\mathbb{Z}_p$ :

1. $\mathbb{Z}_p$ is the completion of $\mathbb{Z}$ with respect to the $p$ -adic metric $d_p$ .
2. $\mathbb{Z}_p$ is the projective limit $\varprojlim_k (\mathbb{Z}/p^k \mathbb{Z})$ .

3. $$\mathbb{Z}_p=\{ [0.a_0 a_1 a_2 ...]_p ; a_i \in \{0,1,2,3,\dots,p-1\}\}$$

DEFINITION 04.1   We equip

$$\varprojlim_k (\mathbb{Z}/p^k \mathbb{Z}) =\{ (b_j)_{j=1}^\infty; b_j \in \mathbb{Z}/p^j \mathbb{Z}, (b_{j_1}$$    modulo $$p^{j_2})=b_{j_2}$$    whenever $$j_1 >j_2\}$$

with the following ``$p$ -addic norm''.

$$% latex2html id marker 579\vert(b_j)\vert _p = \begin{cas... ...ext{ and }b_{k+1}\neq 0 \\ 0 & \text{ if } (b_j)=0 \end{cases}$$

Then we define ``$p$ -addic metric'' $d_p$ on $\varprojlim_k \mathbb{Z}/p^k \mathbb{Z}$ in an obvious way.

LEMMA 04.2   A natural map $\iota:(\mathbb{Z},d_p) \to( \varprojlim_k (\mathbb{Z}/p^k \mathbb{Z}), d_p)$ defined by

$$\iota: \mathbb{Z}\ni n \mapsto (n,n,n\dots) \in \varprojlim_k (\mathbb{Z}/p^k \mathbb{Z})$$

is an isometry.

EXERCISE 04.1   Prove the lemma above.