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

No.04: \fbox{Review}

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. $\displaystyle \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

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

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

\begin{displaymath}
% 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}\end{displaymath}

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

$\displaystyle \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.



2008-06-10